The mesoscale eddies Generation

.4).

. The growing eddies (around the Rd scale) extract their energy from the MAPE and act for the flattening of the isopycnals. A measure of the energy transfer realized by the baroclinic instability is the Vertical Buoyancy Flux (V BF ) : w ′ b ′ with w the vertical velocity component, b the buoyancy, the overbar denotes a time average and the primes denote fluctuation to the time average.

2. The barotropic and centrifugal/inertial instabilities are mean horizontal shear instabilities and they develop from unstable horizontally-sheared mean currents. The growing eddies extract their energy from the MKE, a measure of which is the Horizontal Reynolds Stress (HRS) :

 7. The measure of the energy transfer realized by the centrifugal/inertial instability is given by HRS and the one of the energy transfer realized by the symmetric instabilitity is the Vertical Reynolds Stress (V RS) : et al., 2013). 9. The processing of the satellite altimetry measure by AVISO (Archiving, Validation and Interpretation of Satellite Oceanographic) provides a 25-years long (1993-currently ∼ 2018) SSH time series at the global scale at the spatial resolution of O(100) km (Figure 1.2).

.

Question I : What drives the submesoscale variability at the transition between the Northern and Southern branches of the Agulhas Current ?

, to confirm the barotropic instability has their generation mechanism and to identify the factors controlling the spatial The second study carried out in this work, focuses on the mesoscale EKE budget in the Agulhas Current system and in particular on the following questions :

Question II : What are the processes driving the mesoscale eddy generation, dissipation and energy transfers with smaller vertical structures in the Northern and Southern Agulhas Currents ?

This question is addressed in the Chapter 3 : "Study II : Mesoscale Kinetic Eddy Energy budget and transfers between vertical modes in a Western Boundary Current". The mesoscale eddy energy reservoir is characterized from the eddy energy (EKE and EAP E) vertical paritioning and the mesoscale EKE budget is characterized and interpreted, in terms of physical processes, for the Agulhas Current region using a numerical simulation. This study uses the vertical modes and focuses on the KE transfers between

ACKNOWLEDGEMENT

Je tiens à remercier chaleureusement en tout premier lieu Pierrick Penven, Jonathan Gula et Claire Ménesguen qui ont dirigé mon travail de recherche pendant ces trois années. Merci de m'avoir donné l'opportunité de travailler sur ce sujet, de m'avoir donné la liberté d'explorer mes idées et de m'avoir aidé à les mener à bien. Les nombreuses discussions scientifiques, vos conseils, votre rigueur et votre exigence ont particulièrement été formateurs et m'ont beaucoup appris. Merci pour m'avoir poussé et montré le champs des possibles. Merci également pour votre disponibilité, votre écoute et votre bienveillance et confiance constantes qui m'ont été très précieuses pendant ces trois années. Merci d'avoir partagé avec moi votre curiosité scientifique enthousiasmante qui était au coeur de mon désir de faire cette thèse et qui est au coeur de mon désir de faire de la recherche. Un merci tout particulier pour votre soutien lors de ma péripétie en deux roues sous la pluie. Je tiens également à remercier chaleureusement les rapporteurs (Julien Jouanno et Patrick Marchesiello) et les examinateurs (Xavier Capet, Yves Morel et Marjolaine Krug) de ma thèse pour avoir accepté d'évaluer mon travail, pour vos commentaires constructifs et pour vos appréciations. Merci à Xavier Carton d'avoir accepté de présider mon jury de thèse et pour ton suivi bienveillant depuis mon Master. Merci également aux membres de mon comité de suivi individuel de thèse (Xavier Carton, Xavier Capet et Yves Morel) pour les discussions annuelles enrichissantes. Merci à Clément Vic et Noé Lahaye pour les quelques discussions (non moins importantes !) sur les modes verticaux. Merci à Mathieu Lecorre pour le template du manuscrit de thèse (encore plus important !). Merci à René Schubert pour avoir partagé son travail sur le Courant des Aiguilles. Merci à Tristan pour avoir solutionné mes problèmes informatiques, à chaque fois incongru, avec bonne humeur. Et pour finir, un énorme merci à toutes les personnes du LOPS que j'ai pu croiser pendant ces trois années (qui j'en suis sûre se reconnaîtront) et qui les ont embelli.

INTRODUCTION

In this section, I present the dissertation topic. I first review the study context, present the region of study as well as its interest for the dissertation topic. I finally define the resulting problematics addressed in this dissertation.

Oceanic motions

The oceans support motions over a large ranges of spatial and temporal scales. The large-scale mean circulation in the upper-ocean corresponds to wind-driven gyres. They take place in each oceanic basin and correspond to an anticyclonic (subtropical) or a cyclonic (subpolar) gyre depending on the wind patterns which vary latitudinally. The gyres are asymmetrical, due to an intensified current on the basins western boundary closing their circulations. A scheme of the South Indian subtropical gyre is shown on Figure 1.1. The understanding of the oceanic dynamics has largely evolved with the advances in satellite altimetry in the late 90s [START_REF] Le Traon | An improved mapping method of multisatellite altimeter data[END_REF]. Altimetry provided for the first time a global-scale picture of the surface mesoscale oceanic dynamics [START_REF] Ducet | Global high-resolution mapping of ocean circulation from TOPEX/Poseidon and ERS-1 and-2[END_REF]. It evidenced the ubiquity of mesoscale eddies at the ocean surface (Figure 1.2), which are the counterpart of the atmospheric storms, and allowed to characterize their dynamics at scales down to O(100) km [START_REF] Chelton | Global observations of large oceanic eddies[END_REF](Chelton et al., , 2011b)).

The understanding of the oceanic dynamics at smaller scales has evolved since the 2000s with the increasing horizontal resolution of numerical models and of remote sensing [START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF]. The submesoscale is an intermediate scale range between the mesoscale and the turbulent boundary layers scale and supports a variety of motions : eddies, fronts, filaments, tides and internal-gravity waves [START_REF] Müller | Routes to dissipation in the ocean: The 2D/3D turbulence conundrum[END_REF]. The continuum between the large-scale mean circulation, the mesoscale eddies and the submesoscale currents is shown from a high-resolution numerical simulation of the Agulhas Current region [START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF] 1.2: Snapshot (28-Aug-1996) of the global-scale eddy SSH field [cm] from AVISO constructed from the merged T/P and ERS-1 data following a procedure described in Chelton et al. (2011b). Note the ubiquity of the mesoscale eddies. Figure from Chelton et al. (2011b). f computed from a high-resolution numerical simulation (dx ∼ 2 km) of the Agulhas Current region. Note that in most regions, the vorticity is intense over a wide scales range : from mesoscale meanders and eddies to submesoscale eddies within mesoscale eddies, submesoscale fronts, filaments and eddies in the "submesoscale soup". In few regions, the vorticity is weaker and is dominated by mesoscale eddies and submesoscale fronts and filaments. Figure from [START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF] Introduction

The oceanic Kinetic Energy budget

The energy of oceanic motions partitions into two energy types : the Kinetic Energy (KE) and the Available Potential Energy (APE) [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF][START_REF] Vallis | Atmospheric and oceanic fluid dynamics[END_REF]. The first one depends on the velocity field (KE = 0.5(u 2 + v 2 )) and represents the fluid motion. The second one depends on the density field (AP E = g 1 ρ 0 ρz) and represents the energy contained in the fluid stratification which can potentially be converted into KE. Both reservoirs can subsequently be divided into a mean (time average) and an eddy (fluctuations relative to the time average) sub-reservoirs. The KE reservoir is the most directly related to the oceanic motions and the oceanic energy budget is usually investigated via this energy reservoir [START_REF] Ferrari | The distribution of eddy kinetic and potential energies in the global ocean[END_REF].

Sources of energy

The general circulation in the upper ocean mainly corresponds to the wind-driven oceanic gyres. They corresponds to basin-scale geostrophically-balanced mean currents. At mid-latitudes, the easterlies (30 • S -30 • N) and westerlies (30 • N -60 • N) induce a converging meridional Ekman drift turning into a downward Ekman pumping at the gyre center. The Ekman pumping is compensated, in term of Potential Vorticity, by the latitudinal variation of the Coriolis parameter which explains the presence of an anticyclonic subtropical gyre [START_REF] Stommel | The westward intensification of wind-driven ocean currents[END_REF][START_REF] Munk | On the wind-driven ocean circulation[END_REF]. A view of the wind-diven subtropical gyre in the South Indian Ocean is shown in Figure 1.1. This wind-driven circulation closes on the basin western boundary by the so-called western boundary current -the Agulhas Current in the South Indian Ocean (Figure 1.1). In terms of energy, the wind-induced circulation corresponds to the mean KE (MKE) reservoir and the downward Ekman pumping works against the stratification transfering energy to the mean APE (MAPE) reservoir (MAPE estimated of the order of O(1000) MKE by [START_REF] Gill | Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies[END_REF] from a two-layer quasi-geostrophic model). It must be noted that the wind-driven circulation also consists in subpolar gyres, the polar counterpart of the mid-latitude subtropical gyres.

Introduction

This intense mesoscale variability has historically been related to the baroclinic instability 1 [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. [START_REF] Gill | Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies[END_REF]; [START_REF] Smith | The geography of linear baroclinic instability in Earth's oceans[END_REF] showed from linear stability analysis of a simulated and a climatological mean density that the general circulation and the areas where the EKE is the largest are baroclinically unstable. The maximum growth rates for both stability analysis correspond to scales proportional to, but smaller than, Rd. The spatial coincidence between the mean currents and the mesoscale eddies (Figure 1.4), the APE reservoir being the largest mean energy reservoir and the mean density being baroclinically unstable all supports the conclusion that the baroclinic instability is the main mesoscale eddies generation mechanism. The wind transfers energy to the oceanic MAPE which is then converted into EKE.

The list of the processes involved in the generation of mesoscale eddies and the factors controlling their development has been expanded by recent studies. The topographic constraint has been evidenced as a factor shaping the stability and mesoscale variability of western boundary currents. These currents flows closely from the shelf, resulting in an intense horizontal shear on their inshore sides which sets the favourable conditions for horizontal-shear instabilities 2 . The Gulf Stream mesoscale variability has been related to a mixed baroclinic-barotropic instability process triggering downstream of a location where the shelf curvature locally increases (Charleston Bump) and at the current separation from the shelf (Cape Outlook) (Gula et al., 2015a). The Agulhas Current mesoscale variability has been related to the barotropic instability triggering at a location where the shelf locally widens (Natal Bight) (Lutjeharms et al., 2003a;[START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF], where the shelf curvature locally increases (Eastern Agulhas Bank Bight) (Lutjeharms et al., 2003a) and to a flow detachment process at the current separation from the shelf (Agulhas Bank southern tip) [START_REF] Penven | Generation of cyclonic eddies by the Agulhas Current in the Lee of the Agulhas Bank[END_REF]Lutjeharms et al., 2003a). In addition of these classic instability processes, mesoscale eddies can be generated by the momentum conservation required for the retroflecting currents [START_REF] Pichevin | Why are there Agulhas rings[END_REF]. The intense mesoscale variability of the Agulhas Current Retroflection is related to the intermittent shedding of mesoscale anticyclones from the retroflecting current [START_REF] Lutjeharms | Three decades of research on the greater Agulhas Current[END_REF].

Balanced dynamics

The mesoscale eddies are oceanic features of spatial scales around the 1 st Rossby deformation radius (Rd), or larger, and to temporal scales from weeks to months (subinertial frequency band). Rd is inversely proportional to the latitudes and corresponds to scales of O (30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40) km in the Agulhas current system [START_REF] Chelton | Geographical variability of the first baroclinic Rossby radius of deformation[END_REF]. The mesoscale flows are vertically-sheared flows in geostrophic balance3 . Their dynamics is dominated by the rotation4 (Ro << 1) and by the stratification 5 (Ri >> 1) effects and is described by the quasigeostrophic theory [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF].

Mesoscale eddies vertical structure

A convenient way to describe the mesoscale motion vertical structure is the decomposition into vertical modes. In a continously stratified medium with a flat bottom, the vertical variability (vertical modes) is separable from the horizontal and time variabilities (modal coefficients). The vertical modes (φ n ) originate from the quasigeostrophic theory and correspond to a set of orthogonal eigenfunctions solution of a Sturm-Liouville problem depending on the background buoyancy frequency (N 2 ) [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF]. The traditional vertical modes of the horizontal velocity components (flat-bottom resulting in no vertical velocity) are constituted of a barotropic mode n = 0, representing the depth average, and of nth baroclinic modes, having n zero-crossing in the vertical direction and representing vertical structures intensified at different depths. The vertical modes eigenvalues correspond to the inverse Rossby deformation radius [START_REF] Chelton | Geographical variability of the first baroclinic Rossby radius of deformation[END_REF]. The 1 st baroclinic vertical mode φ 1 therefore represents the vertical structure of oceanic features of horizontal scales around Rd. Another definition of the vertical modes has been recently established by replacing the null vertical velocity at the bottom by null horizontal velocity [START_REF] De La Lama | The vertical structure of ocean eddies[END_REF][START_REF] Lacasce | The prevalence of Oceanic Surface Modes[END_REF]. This vertical mode base is only constituted of 'rough bottom' baroclinic modes and bottom-intensified flows are not represented by a barotropic mode, but by bottom topographic waves which are not solution of the Sturm-Liouville problem [START_REF] Charney | Oceanic analogues of large-scale atmospheric motions[END_REF][START_REF] Rhines | Edge-, bottom-, and Rossby waves in a rotating stratified fluid[END_REF]. [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF] showed from in situ data, sampled in different dynamics regimes (Gulf Stream and Agulhas Current, the Mid-Atlantic Ridge, the Atlantic subpolar gyre, the Tropics, the Brazil Basin and the North Pacific), that the mesoscale eddies mainly decompose into the barotropic and the 1 st baroclinic (n = 1) traditional vertical modes. The prevalence of these two traditional vertical modes is also predicted by theory [START_REF] Fu | Nonlinear energy and enstrophy transfers in a realistically stratified ocean[END_REF] and has been seen from quasi-geostrophic and realistic numerical simulations [START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF][START_REF] Venaille | Baroclinic turbulence in the ocean: Analysis with primitive equation and quasigeostrophic simulations[END_REF]. [START_REF] De La Lama | The vertical structure of ocean eddies[END_REF] showed from in situ data, also sampled in different dynamics regimes (Gulf Stream, Kuroshio, North Atlantic, South Atlantic, Greenlands, Antarctic Circumpolar Current, North Pacific, Equator, Arctic), that the velocity fluctuations decomposes into the 1 st baroclinic 'rough bottom' mode, being a combination of the traditional barotropic and 1 st baroclinic modes at the condition they compensate each other at the bottom, in locations of non-bottom flow. This result is supported by [START_REF] Stanley | Vertical structure of ocean mesoscale eddies with implications for parameterizations of tracer transport[END_REF], who showed from a global ocean model that the combinations of the barotropic and 1 st baroclinic traditional modes or 'rough bottom' 1 st baroclinic mode both result in accurate reconstructions of eddy velocity vertical profiles.

The mesoscale eddies contain 90% of the oceanic KE [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF] and represent therefore a major component of the general circulation KE budget [START_REF] Müller | Routes to dissipation in the ocean: The 2D/3D turbulence conundrum[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. They also exert a feedback on the mean circulation vorticity balance [START_REF] Le Corre | Barotropic vorticity balance of the North Atlantic subpolar gyre in an eddy-resolving model[END_REF] and are a significant component of the global climate (transport of heat and salt) [START_REF] Hausmann | The observed signature of mesoscale eddies in sea surface temperature and the associated heat transport[END_REF] and of the oceanic biology (transport of biogeochemical tracers and nutrients) (Chelton et al., 2011a;[START_REF] Gaube | Satellite observations of chlorophyll, phytoplankton biomass, and Ekman pumping in nonlinear mesoscale eddies[END_REF].

The geostrophic turbulence paradigm

The geostrophic turbulence theory characterizes the non-linear interactions of stably-stratified flows in near geostrophic balance, also referred as the interior quasigeostrophy [START_REF] Charney | Geostrophic turbulence[END_REF]. In a stratified medium, the non-linear interactions occur between different horizontal wavenumbers and different vertical structures [START_REF] Rhines | The dynamics of unsteady currents[END_REF][START_REF] Rhines | Geostrophic turbulence[END_REF][START_REF] Salmon | Baroclinic instability and geostrophic turbulence[END_REF].

The geostrophic turbulence theory predicts horizontal KE turbulent cascades for the barotropic and baroclinic traditional vertical modes which connect by a vertical KE transfer [START_REF] Rhines | The dynamics of unsteady currents[END_REF][START_REF] Rhines | Geostrophic turbulence[END_REF][START_REF] Salmon | Baroclinic instability and geostrophic turbulence[END_REF][START_REF] Fu | Nonlinear energy and enstrophy transfers in a realistically stratified ocean[END_REF]. The baroclinic instability injects energy around the scale of Rd. The baroclinic total energy fluxes toward Rd (Figure 1.5). At scales around Rd the baroclinic KE is transfered to the barotropic mode : this is the barotropization [START_REF] Fu | Nonlinear energy and enstrophy transfers in a realistically stratified ocean[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF] (Figure 1.5). The barotropic KE fluxes toward larger horizontal scales (Figure 1.5). This upscale (inverse) KE turbulent cascade is stopped by the bottom drag, dissipating the barotropic KE (Figure 1.5) [START_REF] Salmon | Baroclinic instability and geostrophic turbulence[END_REF][START_REF] Sen | Global energy dissipation rate of deep-ocean low-frequency flows by quadratic bottom boundary layer drag: Computations from current-meter data[END_REF]. A part of the baroclinic KE leaks toward the higher baroclinic modes. These ones are Introduction characterized by a downscale cascade (Figure 1.5).

Figure 1.5: Scheme of the total energy (APE + KE) pathways in the geostrophic turbulence paradigm. The horizontal and vertical directions respectively represent the horizontal wavenumbers and the traditional vertical modes. Note the barotropization occurring between the barotropic and the 1 st baroclinic vertical modes and the inverse cascade of the barotropic energy stopped by the bottom drag. Scheme from [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF].

The wind stress partially dissipates barotropic and baroclinic KE at meso and submesoscale [START_REF] Renault | Dampening of submesoscale currents by air-sea stress coupling in the Californian upwelling system[END_REF][START_REF] Hughes | Wind work on the geostrophic ocean circulation: An observational study of the effect of small scales in the wind stress[END_REF]. The surface geostrophic currents feedback on the wind acts as a "mechanical dampening" [START_REF] Seo | Eddy-wind interaction in the California Current System: Dynamics and impacts[END_REF][START_REF] Eden | Effects of mesoscale eddy/wind interactions on biological new production and eddy kinetic energy[END_REF], leading to a lower and to a more realistic mesoscale EKE content in western boundary currents [START_REF] Renault | Remarkable control of western boundary currents by eddy killing, a mechanical air-sea coupling process[END_REF] and especially in the Agulhas Current region [START_REF] Renault | Modulation of the Agulhas Current Retroflection and Leakage by Oceanic Current Interaction with the Atmosphere in Coupled Simulations[END_REF]. Numerical studies, using quasigeostrophic and primitive-equation models, have showed that the inverse turbulent cascade is not confined to the barotropic mode and also occur for the 1 st baroclinic mode at scales larger than Rd [START_REF] Scott | Spectral energy fluxes in geostrophic turbulence: Implications for ocean energetics[END_REF] or than 200 km [START_REF] Venaille | Baroclinic turbulence in the ocean: Analysis with primitive equation and quasigeostrophic simulations[END_REF]. The inverse turbulent cascade refers, to a broad sense, to upscale energy transfers occurring at large horizontal scales. In that sense, the inverse turbulent cascade has been extensively described from surface fields, using global and regional ocean models, as occurring at large scales, either larger than Rd [START_REF] Tulloch | Scales, growth rates, and spectral fluxes of baroclinic instability in the ocean[END_REF][START_REF] Arbic | On eddy viscosity, energy cascades, and the horizontal resolution of gridded satellite altimeter products[END_REF] or larger than O( 200) km [START_REF] Aluie | Mapping the energy cascade in the North Atlantic Ocean: The coarse-graining approach[END_REF][START_REF] Arbic | Geostrophic turbulence in the frequency-wavenumber domain: Eddy-driven low-frequency variability[END_REF] and than 300 km [START_REF] Schlösser | Diagnosing the energy cascade in a model of the North Atlantic[END_REF].

Introduction

In a nutshell, the geostrophic turbulence paradigm states that the baroclinic instability of the mean currents injects KE around Rd, from where it is transferred to the barotropic mode and then to larger scales of the barotropic mode. Numerical studies have completed this vision, by showing that the inverse energy cascade occurs at large barotropic and baroclinic scales. The turbulent energy cascades are stopped by the dissipation of barotropic KE by the bottom drag and of the surface KE by the wind stress. However, a large part of the KE dissipation occurs at very small scales (molecular scales) and how the energy is transferred from the large to these small scales is still an open question [START_REF] Müller | Routes to dissipation in the ocean: The 2D/3D turbulence conundrum[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF].

Interactions between scales

The ocean steady-state requires that the KE of the general circulation is transferred from the forcing (basin-scale) to the dissipation (molecular) scale. This downscale energy transfer must also come with a KE transfer in the vertical, in order to redistribute the energy injected at the oceans surface by the wind forcing toward the interior and abyssal oceans [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. This KE redistribution across horizontal and vertical scales is supported by the observation of KE over all the frequency and wavenumber bands and at all depths [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF].

The mesoscale eddies represent the first stage of the KE downscale redistribution, an energy transfer from basin-scale toward the Rd scales. However, the direct KE turbulent cascade is not realized by the geostrophic turbulence. How is energy transferred from the mesoscale eddies to smaller scales and so on to ultimately be dissipated at microscale ? The mesoscale eddies loses energy by transfers toward smaller scales when there is a breakdown of geostrophic balance. The mixed-layer and the bottom ocean are locations where the interior quasigeostrophic turbulence (geostrophic turbulence paradigm) does not hold [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF] and where the turbulence is characterized by an ageostrophic component.

Submesoscale dynamics

The submesoscale currents correspond to spatial scales inferior to Rd having temporal scales from hours to days to weeks (superinertial frequency band) [START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF]. They occur at an intermediate scale range (O(0.1-10) km) between the mesoscale (O(10 -100) km) and scales corresponding to the turbulent boundary layer depth (O(10-100) m), below which the flow becomes isotropic. The submesoscale currents are only partially geostrophically-balanced, depending on the currents amplitude and horizontal scales. Their dynamics is then characterized by an equivalent effect of the rotation with respect to inertia (Ro ∼ 1) Introduction and of the stratification with respect to the vertical shear (Ri ∼ 1). The quasigeostrophic theory holds in part at the submesoscale and the surface quasigeostrophic theory [START_REF] Held | Surface quasi-geostrophic dynamics[END_REF][START_REF] Lapeyre | Surface quasi-geostrophy[END_REF] can be used to describe the quasi-geostrophic version of the frontogenetic processes [START_REF] Callies | Interpreting energy and tracer spectra of upper-ocean turbulence in the submesoscale range (1-200 km)[END_REF].

Interactions between the meso and submesoscale

The interactions between the meso and submesoscale occurring in the mixed layer (O(100) m), the interior ocean (O(1500 -2000) m) and the bottom ocean are presented in the following.

The mixed-layer is a vertically quasi-homogenised density layer at the oceans surface (e.g., [START_REF] De Boyer Montégut | Mixed layer depth over the global ocean: An examination of profile data and a profile-based climatology[END_REF]) driven by the atmospheric forcings [START_REF] Mahadevan | Ocean science: Eddy effects on biogeochemistry[END_REF]. The mixed-layer hosts a seasonally-varying submesoscale turbulence which either works for or against the mixed-layer stratification and can drive a downscale (Capet et al., 2008c;[START_REF] Thomas | Reduction of the usable wind-work on the general circulation by forced symmetric instability[END_REF][START_REF] Molemaker | Balanced and unbalanced routes to dissipation in an equilibrated Eady flow[END_REF][START_REF] D'asaro | Enhanced turbulence and energy dissipation at ocean fronts[END_REF][START_REF] Thomas | Symmetric instability in the Gulf Stream[END_REF] or an upscale KE transfer (Capet et al., 2008c;[START_REF] Sasaki | Regionality and seasonality of submesoscale and mesoscale turbulence in the North Pacific Ocean[END_REF][START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF][START_REF] Dong | The seasonality of submesoscale energy production, content, and cascade[END_REF] depending on the amplitude of ageostrophic components. Fronts are sharp horizontal density gradients which can be created by the strain of mesoscale eddies [START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF]. The geostrophic motions in frontogenetic processes are described by the surface quasigeostrophic theory [START_REF] Callies | Interpreting energy and tracer spectra of upper-ocean turbulence in the submesoscale range (1-200 km)[END_REF], which is characterized by an upscale turbulent cascade (Capet et al., 2008c,a). The ageostrophic motions in frontogenetic processes have been related to a direct KE turbulent cascade using idealized (Capet et al., 2008c;[START_REF] Molemaker | Balanced and unbalanced routes to dissipation in an equilibrated Eady flow[END_REF] and realistic numerical simulations [START_REF] D'asaro | Enhanced turbulence and energy dissipation at ocean fronts[END_REF][START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF]. The mixed-layer baroclinic instability (MLI) is the counterpart of the baroclinic instability of the largescale general circulation (cf. the section 1.2.2). It triggers from thermal-wind balanced submesoscale flows and transfers energy from MAPE to EKE 6 by generating submesoscale eddies around the mixed-layer Rossby deformation radius [START_REF] Stone | On non-geostrophic baroclinic stability: Part II[END_REF]. MLI has been related to an upscale turbulent cascade in the Agulhas Current region [START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF] and in the Kuroshio region [START_REF] Sasaki | Regionality and seasonality of submesoscale and mesoscale turbulence in the North Pacific Ocean[END_REF][START_REF] Dong | The seasonality of submesoscale energy production, content, and cascade[END_REF] using regional numerical simulations. Secondary (frontal) instabilities can develop from fronts. The centrifugal/inertial and symmetric instability conditions are set by down-front winds inducing an Ekman transport of dense over light waters [START_REF] Thomas | Symmetric instability in the Gulf Stream[END_REF]. They generate submesoscale eddies by extracting KE from the mean flow shear 7 . The barotropic instability is also a secondary (frontal) instability under the condition of an unstable frontal horizontal shear. It is the counterpart of the barotropic instability of the large-scale mean currents (cf. the section 1.2.2) and generates submesoscale eddies by extracting KE from the mean flow shear 8 . The symmetrical instability has been related to the KE dissipation using an idealized numerical simulation [START_REF] Thomas | Symmetric instability in the Gulf Stream[END_REF] and observations [START_REF] Thomas | Reduction of the usable wind-work on the general circulation by forced symmetric instability[END_REF]. The role of the submesoscale eddies originating from the centrifugal/inertial and barotropic instabilities in the turbulent cascade is not evidenced yet and they can either realize a subsequent upscale or downscale KE transfer.

The interior ocean is mainly populated by internal-gravity waves. They are ubiquitous features in the O(0.1 -10) km range generated by various mechanisms (wind-induced inertial waves, barotropic tide and geostrophic currents scattering over topographic roughness, spontaneous emission from the mesoscale eddies and from the geostrophic-adjustment of atmospheric-induced horizontal density gradient) which radiate into the interior ocean. The non-linear interactions between the mesoscale eddies and the internalwaves fields can result in the eddy energy loss [START_REF] Polzin | Mesoscale eddy-internal wave coupling. Part II: Energetics and results from PolyMode[END_REF][START_REF] Shakespeare | Spontaneous surface generation and interior amplification of internal waves in a regional-scale ocean model[END_REF][START_REF] Rocha | Stimulated generation: extraction of energy from balanced flow by near-inertial waves[END_REF]. [START_REF] Cusack | Observed Eddy-Internal Wave Interactions in the Southern Ocean[END_REF] recently showed, using observations, that eddies-waves interactions result in eddy energy gain by straining and in eddy energy loss by vertical shear. The internal-gravity waves fields are also related to the geostrophic KE direct turbulent cascade by enhancing the KE dissipation at internal waves breaking sites [START_REF] Müller | Routes to dissipation in the ocean: The 2D/3D turbulence conundrum[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF].

The dynamics at the bottom ocean are characterized by flows interactions with topographic features, which range from sloping bottom, topographic roughness to seamounts. An along-slope current gains vertical vorticity on its inshore front, generated in the turbulent bottom boundary layer (thickness of O(10) m) [START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF]. The inshore front destabilizes by shear instability processes after the current separation from the sloping bottom [START_REF] Molemaker | Submesoscale instability and generation of mesoscale anticyclones near a separation of the California Undercurrent[END_REF]Gula et al., 2015b). This process extracts energy from the mean currents and from the mesoscale eddies by generating submesoscale eddies. These ones have been simulated to further dissipate [START_REF] Dewar | Centrifugal instability and mixing in the California Undercurrent[END_REF]Gula et al., 2016b), to merge and form mesoscale eddies [START_REF] Molemaker | Submesoscale instability and generation of mesoscale anticyclones near a separation of the California Undercurrent[END_REF] or to form long-lived submesoscale coherent vortices [START_REF] D'asaro | Generation of submesoscale vortices: A new mechanism[END_REF][START_REF] Bower | Direct observations of formation and propagation of subpolar eddies into the Subtropical North Atlantic[END_REF][START_REF] Bosse | Spreading of Levantine Intermediate Waters by submesoscale coherent vortices in the northwestern Mediterranean Sea as observed with gliders[END_REF][START_REF] Vic | Eddy-topography interactions and the fate of the Persian Gulf Outflow[END_REF]. The topographic interactions can also result in mesoscale eddies driving mean currents over a sloping topography. Such process has been described as the Neptune effect [START_REF] Holloway | Systematic forcing of large-scale geophysical flows by eddy-topography interaction[END_REF][START_REF] Adcock | Interactions between geostrophic eddies and the mean circulation over large-scale bottom topography[END_REF] and although the underlying energy transfers have not yet been explicited, it would correspond to a transfer from EKE to MKE. Mesoscale eddies flowing over topographic roughness and seamounts generate internal-gravity waves -socalled lee waves -by scattering, which radiate into the ocean interior [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF][START_REF] Nikurashin | Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory[END_REF]. This scattering process results in the an energy loss for mesoscale eddies and can feed a direct tur-8. The measure of the energy transfer realized by the centrifugal/inertial instability is given by HRS bulent cascade due to the internal-gravity waves subsequent breaking and energy dissipation [START_REF] Nikurashin | Routes to energy dissipation for geostrophic flows in the Southern Ocean[END_REF][START_REF] Klymak | A simple mixing scheme for models that resolve breaking internal waves[END_REF][START_REF] Garabato | Widespread intense turbulent mixing in the southern ocean[END_REF]. The flows interactions with seamounts also lead to submesoscale eddies shedding in the lee of the seamount [START_REF] Perfect | Vortex structures in the wake of an idealized seamount in rotating, stratified flow[END_REF][START_REF] Srinivasan | Submesoscale vortical wakes in the lee of topography[END_REF] which may be of importance for the direct KE turbulent cascade [START_REF] Perfect | Energetics of Seamount Wakes. Part I: Energy Exchange[END_REF]

Observations of the turbulent cascades and of the eddy sources and sinks

Satellite altimetry 9 allowed to significantly improve our understanding of the mesoscale variability and of the mesoscale turbulent cascades [START_REF] Fu | Eddy dynamics from satellite altimetry[END_REF].

Turbulent cascades

The geostrophic turbulence paradigm [START_REF] Rhines | The dynamics of unsteady currents[END_REF][START_REF] Rhines | Geostrophic turbulence[END_REF][START_REF] Salmon | Baroclinic instability and geostrophic turbulence[END_REF][START_REF] Fu | Nonlinear energy and enstrophy transfers in a realistically stratified ocean[END_REF] states that baroclinic instability of the mean currents injects KE around Rd in the form of mesoscale eddies and these scales support an inverse energy cascade. This paradigm shows an approximate agreement with satellite altimetry observations. The observed scale of the mesoscale eddies, inferred from a 16-years long altimetry dataset, ranges between 50 and 150 km (Chelton et al., 2011b). This shift between Rd and the eddy equilibration scale supports an inverse turbulent cascade from a source scale (O(Rd) as stated by the geostrophic turbulence paradigm). Spectral KE fluxes allow for characterizing the turbulent cascade direction according to the wavenumber ranges. [START_REF] Scott | Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry[END_REF]; [START_REF] Arbic | On eddy viscosity, energy cascades, and the horizontal resolution of gridded satellite altimeter products[END_REF][START_REF] Arbic | Geostrophic turbulence in the frequency-wavenumber domain: Eddy-driven low-frequency variability[END_REF] showed that the surface inverse turbulent cascade is supported down to Rd, consistently with the linear stability theory of the baroclinic instability. [START_REF] Tulloch | Scales, growth rates, and spectral fluxes of baroclinic instability in the ocean[END_REF] showed from linear theory that the fastest growing mode of the baroclinic instability injects energy at a scale smaller than Rd, which lies or not in the inverse turbulent cascade scale range depending on the latitude, and that the eddy scale is in the range of the surface inverse turbulent cascade. [START_REF] Khatri | Surface ocean enstrophy, kinetic energy fluxes, and spectra from satellite altimetry[END_REF] showed that the surface inverse turbulent cascade is supported at scales larger than O(200) km, but not down to Rd. However, the altimetry measure processing leading to the AVISO dataset has been suggested to alter the turbulent energy cascade directions and magnitudes [START_REF] Arbic | On eddy viscosity, energy cascades, and the horizontal resolution of gridded satellite altimeter products[END_REF][START_REF] Arbic | Geostrophic turbulence in the frequency-wavenumber domain: Eddy-driven low-frequency variability[END_REF] as well as scales smaller than altimetry resolution alter the wavenumber range supporting the inverse turbulent cascade [START_REF] Arbic | Nonlinear cascades of surface oceanic geostrophic kinetic energy in the frequency domain[END_REF][START_REF] Arbic | On eddy viscosity, energy cascades, and the horizontal resolution of gridded satellite altimeter products[END_REF].

The SSH wavenumber (k) spectra allows for characterising the dynamics of the turbulent cascade (inertial range) 10 . Evidences of a non universal SSH wavenumber spectra were provided since the early beginning of the satellite altimetry [START_REF] Le Traon | Comments on "Mesoscale Variability in the Atlantic Ocean from Geosat Altimetry and WOCE High-Resolution Numerical Modeling[END_REF] and were supported by several subsequent studies. SSH wavenumber spectra following a k -11/3 power law, typical of surface quasigeostrophy, were determined in high energetic area [START_REF] Le Traon | Do altimeter wavenumber spectra agree with the interior or surface quasigeostrophic theory[END_REF]. [START_REF] Xu | The effects of altimeter instrument noise on the estimation of the wavenumber spectrum of sea surface height[END_REF] emphasized strong regional variations of SSH wavenumber spectra, therefore confirming its non-universal shape. They determined power laws between k -5 (interior quasigeostrophy) and k -11/3 (surface quasigeostrophy) in the high energetic areas of the western boundary currents. The various power laws indicate the surface turbulent cascade cannot only be explained by the geostrophic turbulence and that smaller scale turbulence (surface quasigeostrophy theory) also significantly contributes [START_REF] Dufau | Mesoscale resolution capability of altimetry: Present and future[END_REF]. [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] used another approach to study the mesoscale eddies KE balance from satellite altimetry. They diagnosed the divergence of the eddy-energy fluxes to identify where the mesoscale eddies gain and lose energy, which would point toward the processes involved in the direct turbulent cascade.

Eddy sources and sinks

Their eddy-energy fluxes divergence, diagnosed from satellite altimetry data, show ubiquitous mesoscale eddy sinks at western boundaries (Figure 1.6). This result is coherent with the mesoscale eddies originating almost everywhere, propagating westward (Chelton et al., 2011b) and decaying upon western boundaries due to eddy-topographic interactions catalyzing a direct energy cascade [START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF]. This would suggest that the 1 st baroclinic mode KE transfer toward small scales and/or dissipation at the western boundaries shortcut the KE barotropization and dissipation by the bottom drag predicted by the geostrophic turbulence paradigm [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. However, this study is based on several assumptions : geostrophy, the 1 st baroclinic mode being the main contribution to the surface dynamics and accurately representing the mesoscale eddy vertical structure and mesoscale eddy dynamics being mostly linear. These assumptions are partly inherent to satellite altimetry, but they strongly constrain the eddy-energy fluxes interpretation.

The Agulhas Current system

The mesoscale variability is a significant component of the Agulhas Current system [START_REF] Lutjeharms | The Agulhas Current[END_REF] which therefore makes it an excellent object of study for the mesoscale eddies and for understanding their role in the oceanic Kinetic Energy budget.

The mean circulation of the Agulhas Current system

The Agulhas Current system is the returning branch of the wind-driven anticyclonic circulation of the South Indian Subtropical Gyre (Figure 1.1) and it is constituted of the Agulhas Current, the Retroflection and of the Agulhas Return Current [START_REF] Lutjeharms | The Agulhas Current[END_REF]

(Figure 1.7).
The Agulhas Current is the western boundary current of the South Indian Subtropical Gyre. It flows poleward along the South African coast from the southern extremity of the Mozambique Channel (∼ 32.5 • E and 26.5 S S) to the southern extremity of the African continent (∼ 20 • and 37 • S) [START_REF] Lutjeharms | The Agulhas Current[END_REF]. The Agulhas Current is fed from three sources : the Mozambique Current, the East Madagascar Current Figure 1.7: Scheme of the Agulhas Current system circulation. The Agulhas Current, the Agulhas Retroflection and the Agulhas Return Current are denoted in black and the background circulation features are denote in white (note the wind-driven upwelling along the African western coast). Note the Agulhas Current three sources : the Mozambique Current (black dashed line), the East Madagascar Current (black line) and the South West Indian Subgyre (white arrows). The anticylonic and cyclonic eddies are respectively denoted in black and white. The dotted and dashed lines respectively denote the 500 and 3000 m isobaths and the areas shallower than 3000 m are shaded. Scheme from [START_REF] Lutjeharms | The Agulhas Current[END_REF]. and the South West Indian Ocean Subgyre (Figure 1.7). The two first sources force the current at its origin in the form of mesoscale eddies [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Saetre | The circulation of the Mozambique Channel[END_REF]. The last source forces the Agulhas Current along its northern portion, over the whole water column, resulting in an increasing of the current volume flux downstream [START_REF] Lutjeharms | The Agulhas Current[END_REF]. [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF] estimated the Agulhas Current transport to be 84 ± 2 Sv, which makes it the most intense western boundary current of the Southern Introduction Hemisphere. Downstream of the southern African continent extremity, the Agulhas Current kinematics drastically differ from the other western boundary currents. The poleward flow overshoots the continental shelf tip and this is likely due to a combination of inertia [START_REF] De Ruijter | Asymptotic analysis of the Agulhas and Brazil Current systems[END_REF], β-effect [START_REF] Boudra | The wind-driven circulation of the South Atlantic-Indian Ocean-II. Experiments using a multi-layer numerical model[END_REF] and topographic torque [START_REF] Matano | A numerical study of the Agulhas retroflection: The role of bottom topography[END_REF]. The Retroflection dynamics has been suggested to be inherent to the intermittent shedding of Agulhas Rings [START_REF] Pichevin | Why are there Agulhas rings[END_REF]. The Retroflection location is relatively stable, but with extreme events of retroflection occurring south of Port Elizabeth (26 • E) [START_REF] Lutjeharms | Anomalous upstream retroflection in the Agulhas Current[END_REF]. These Upstream Retroflection events have been observed in 2000-2001 [START_REF] Quartly | SST observations of the Agulhas and East Madagascar retroflections by the TRMM Microwave Imager[END_REF][START_REF] Rouault | Estimation of sea-surface temperature around southern Africa from satellite-derived microwave observations[END_REF] and in 2008.

Once the Agulhas Current flow has retroflected, it becomes an eastward current flowing back into the South Indian Ocean at the latitude of the Subtropical Convergence : this is the Agulhas Return Current [START_REF] Lutjeharms | The agulhas return current[END_REF]) (Figure 1.7). It extends eastward up to about 70 • E, at the eastern extremity of the South West Indian Ocean Subgyre, where it connects with the South Indian Current [START_REF] Stramma | The South Indian Ocean Current[END_REF]. The current dynamics is controlled by the topography and exhibits two persistent equatorward meanders at the Agulhas Plateau (26 • E) and at the Southern Mozambique Ridge (32 • E) [START_REF] Boebel | Path and variability of the Agulhas Return Current[END_REF]. Numerical studies also suggested that it has a significant inertial component [START_REF] Lutjeharms | The Agulhas Current[END_REF]. The Agulhas Current system is an important branch of the South West Indian Ocean circulation and it plays a significant role in the local and global climate. The warm waters and fronts of the Agulhas Current system affect the regional South East African summer rainfalls [START_REF] Lutjeharms | The Agulhas Current[END_REF], especially where the current flows closely to the coast down to Port Elizabeth (26 • E) [START_REF] Jury | Influence of the Agulhas Current on summer rainfall along the southeast coast of South Africa[END_REF]. The Agulhas Current system also induces a heat transport at the basin-and at the inter basin-scales. The first one corresponds to the Agulhas Current transporting warm waters poleward [START_REF] Lutjeharms | The Agulhas Current[END_REF]. The latter one corresponds to the Agulhas leakage, associated with the warm and salty Agulhas Rings developing at the Agulhas Current Retroflection and propagating into the South East Atlantic Ocean, which makes part of the global overturning circulation [START_REF] Beal | On the role of the Agulhas system in ocean circulation and climate[END_REF][START_REF] Biastoch | Agulhas leakage dynamics affects decadal variability in Atlantic overturning circulation[END_REF][START_REF] Weijer | Response of the Atlantic Overturning Circulation to South Atlantic sources of buoyancy[END_REF].

The meso and submesoscale variability of the Agulhas Current system

A particular characteristic of the Agulhas Current system circulation, in addition to the Agulhas Retroflection, is its turbulent component. The Agulhas Current mesoscale variability is largely documented and allows to characterize the current dynamics into two branches : the Northern and Southern Agulhas Currents [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF], whereas its submesoscale variability is the subject of recent studies.

The mesoscale variability

The Agulhas Current stability is partitioned into a Northern and Southern branches connecting at Port Elizabeth (26 • E) [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]. The Northern Agulhas Current (32.5 • E -26 • E) flows closely to the narrow shelf and steep shelf slope and it is characterized by a low mesoscale variability. This one is mainly due to the downstream propagation of Natal Pulses (Figure 1.8) : solitary cyclonic eddies, with diameters of about 50-200 km and extending over the whole water column (Lutjeharms et al., 2003a;[START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF], generated at the Natal Bight (∼ 31 • E) four to five times a year [START_REF] Lutjeharms | The Agulhas Current[END_REF]) by a barotropic instability process [START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF] related to interactions between the mean Agulhas Current current with anticyclonic eddies generated in the Mozambique Channel [START_REF] Tsugawa | Generation and growth mechanism of the Natal Pulse[END_REF].

The Southern Agulhas Current (26 • E -20 • E) flows along the Agulhas Bank, a widening topographical feature resulting in the relaxing of the topographical constraint on the current, and it is characterized by a higher variability than the Northern Agulhas Current [START_REF] Lutjeharms | The Agulhas Current[END_REF]. This one manifests from several border features (meanders, eddies and plumes), of which the Shear-edge eddies [START_REF] Lutjeharms | Eddies and other boundary phenomena of the Agulhas Current[END_REF][START_REF] Goschen | Agulhas current variability and inshore structures off the Cape Province, South Africa[END_REF] are a typical example (Figure 1.8). They are quasi-permanent cyclonic eddies, with diameters of about 50-100 km and extending over the first 500 m depths (Lutjeharms et al., 2003a), generated at the eastern Agulhas Bank Bight by horizontal shear processes controlled by the topography (Lutjeharms et al., 2003b,a). The eastern Agulhas Bank Bight is also affected by a remotely generated variability. A fraction of the Natal Pulses, 1.7 per year [START_REF] Krug | Satellite observations of an annual cycle in the Agulhas Current[END_REF][START_REF] Rouault | New perspectives on Natal Pulses from satellite observations[END_REF][START_REF] Krug | Interactions between the Agulhas Current and the eastern margin of the Agulhas Bank[END_REF], get trapped at the eastern Agulhas Bank Bight and can grow in size before leaking downstream. The eastern Agulhas Bank Bight variability is as well likely connected to the one of the western Agulhas Bank Bight. The western Agulhas Bank Bight is the place of cyclonic lee eddies (Figure 1.8), with diameters of about 50-200 km extending over the first 500 m depths (Lutjeharms et al., 2003a), generated intermittently at the current separation from the Agulhas Bank (∼ 20 • ) likely by a flow detachment process [START_REF] Penven | Generation of cyclonic eddies by the Agulhas Current in the Lee of the Agulhas Bank[END_REF] related to the leakage of the Shear-edge eddies (Lutjeharms et al., 2003b).

The Agulhas Current Retroflection (15 • E -20 • E) is characterized by an extremely high mesoscale variability, visible within any dataset [START_REF] Weeks | Seasonal and spatial SST variability in the Agulhas retroflection and Agulhas return current[END_REF][START_REF] Wedepohl | Surface drift in the south-east atlantic ocean[END_REF][START_REF] Lutjeharms | A statistical analysis of the meso-scale dynamics of the Southern Ocean. Deep Sea Research Part A[END_REF]. The high variability is due to the globally meandering dynamics of the flow, which leads to the shedding of Agulhas Rings about six time a year [START_REF] De Ruijter | Indian-Atlantic interocean exchange: Dynamics, estimation and impact[END_REF]. These anticyclones, with average diameters of O( 200) km [START_REF] Ducombe | Agulhas retroflection rings in the South Atlantic Ocean: an overview[END_REF], can have unusual large scales as shown by two particular rings which respectively had a diameter of 500 km [START_REF] Arhan | On the deep water circulation of the eastern South Atlantic Ocean[END_REF] and a baroclinic vertical structure with a strong barotropic component [START_REF] Van Aken | Observations of a young Agulhas ring, Astrid, during MARE in March 2000[END_REF]. An analytical study suggested that these rings are not generated by classical instabilities, but compensate the zonal momentum flux of the retroflective flow [START_REF] Pichevin | Why are there Agulhas rings[END_REF] and are likely inherent to these dynamics.

Introduction

The Agulhas Return Current (20 • E -70 • E) is characterized by a high mesoscale variability that forms a a continuum from the Agulhas Current Retroflection decreasing eastward [START_REF] Lutjeharms | The Agulhas Current[END_REF]. It manifests from a variety of mesoscale eddies developing at stable geographical areas along the current meanders [START_REF] Gründlingh | Drift of a satellite-tracked buoy in the southern Agulhas Current and Agulhas Return Current[END_REF]. Noticeable features are cyclonic eddies, with diameters of 100-150 km, developing around the Agulhas Plateau [START_REF] Lutjeharms | Eddies at the subtropical convergence south of Africa[END_REF].

The submesoscale variability

The Agulhas Current submesoscale variability has been documented for the first time with the "wake eddies" (Lutjeharms et al., 2003a), referring to submesoscale eddies sampled by floats in the wake of Shear-edge eddies (24-24.5 • E along front portion). This type of submesoscale features are routinely observed with sea surface temperature satellite images (Figure 1.9) around Port Elizabeth (26 • E), at the transition between the Northern and the Southern Agulhas Currents. Submesoscale frontal cyclonic eddies, generated by a barotropic instability process, have recently been sampled by Seagliders (25-26.25 • E along front portion) [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF]

The Agulhas Current system upper ocean submesoscale variability has also recently been documented

Introduction in the areas away from the intense Agulhas Current circulation. The interactions between the mesoscale and the submesoscale turbulences have been characterized in the South East Atlantic Ocean, affected by the Agulhas Rings, and in the South West Indian Ocean Subgyre, affected by the mesoscale eddies shedding from the Agulhas Return Current, from a high-resolution numerical simulation [START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF].

Problematics and Approaches

Study I

The submesoscale instabilities can extract energy from the large-scale balanced mean currents toward unbalanced motions at smaller scales, from which energy can be transferred downscale and so on to the dissipation scale. They can therefore be important for the oceanic KE budget [START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF]. Few studies of the mean circulation submesoscale instabilities and their implications in the KE budget have been realized from high-resolution numerical simulations over the last decade [START_REF] Molemaker | Submesoscale instability and generation of mesoscale anticyclones near a separation of the California Undercurrent[END_REF][START_REF] Dewar | Centrifugal instability and mixing in the California Undercurrent[END_REF]Gula et al., 2015bGula et al., , 2016b)).

In the Agulhas Current, the submesoscale variability have been sampled by RAFOS floats (Lutjeharms et al., 2003a) and Seagliders [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF] at the Agulhas inshore front around Port Elizabeth (26 • E). This location has historically been described as marking the transition between the Northern (stable) and Southern (unstable) Agulhas Current branches [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]. The presence of submesoscale eddies at the transition between the stable and unstable current branches, suggests the Agulhas Current dynamics is less dichotomous than previously described. The submesoscale variability may be a transient dynamics or may add to the mesoscale variability that develop downstream of Port Elizabeth.

The first study carried out in this work, focuses on the Agulhas Current submesoscale variability and in particular on the following question : and temporal variations of the instability triggering. This Chapter corresponds to the scientific publication : [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF].

Study II

The understanding of the oceanic KE budget requires a complete view of the mesoscale eddies connections with the forcings and surrounding ocean (Ferrari andWunsch, 2009, 2010). This includes the mesoscale eddy energy generation and dissipation as well as the energy transfers with other physical features and energy reservoirs. Regional mesoscale eddies dynamics have usually been studied from the perspective of one energetic route, such as the eddies generation by instability processes (Gula et al., 2015a;[START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF], dissipation by the wind stress [START_REF] Renault | Modulation of the Agulhas Current Retroflection and Leakage by Oceanic Current Interaction with the Atmosphere in Coupled Simulations[END_REF] or one route of the energy transfers with smaller scales [START_REF] Nikurashin | Routes to energy dissipation for geostrophic flows in the Southern Ocean[END_REF][START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF]. While the studies evaluating all the different routes of the mesoscale eddy dissipation usually focuses on a specific feature which has been the subject of in situ measurements [START_REF] Zhang | Observed 3d structure, generation, and dissipation of oceanic mesoscale eddies in the south china sea[END_REF][START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF].

The Agulhas Current mesoscale variability is largely documented [START_REF] Lutjeharms | The Agulhas Current[END_REF] and characterizes the current into a low mesoscale variability branch : the Northern Agulhas Current (Natal Pulses) and into an intense mesoscale variability one : the Southern Agulhas Current (Shear-edge eddies, cyclonic lee eddies, Agulhas Rings and Eddies) [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]. The Agulhas Current is therefore a natural subject of study of the mesoscale eddy dynamics and allows to characterize the mesoscale EKE budget at a regional-scale. This region is even more interesting, because a study suggested that the western boundary currents are key locations for the mesoscale eddies dissipation [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF]. The EKE budget in the Agulhas Current region would possibly gives insights on generic mesoscale eddies dynamics.

Introduction

the different vertical structures in addition to the ones between energy reservoirs and to the KE generation and dissipation processes. This Chapter corresponds to the final version of a scientific paper draft and will be soon submitted to the Journal of Physical Oceanography.

Study III

Satellite altimetry allowed to improve our understanding of large-mesoscale motions and turbulence [START_REF] Fu | Eddy dynamics from satellite altimetry[END_REF] and it is used on a daily basis for evaluating the surface circulation and KE content of the modeled mesoscale dynamics. Recently, [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] showed, using satellite altimetry data and assuming a linear dynamics, that western boundaries basins are ubiquitous mesoscale eddy-energy sinks. This result would close the mesoscale eddy life cycle : generation by instability processes of currents, westward propagation and dissipation on the basins western boundaries, and would contribute to our understanding of the oceanic KE budget.

Altimetry data (AVISO) are an indirect measure of the geostrophically-balanced surface dynamics down to O(100) km on a global scale (Chelton et al., 2011b). This does not give a synoptic view of the mesoscale eddy dynamics, knowing the significance of the unbalanced dynamics at smaller scales for sustaining their energetic balance [START_REF] Müller | Routes to dissipation in the ocean: The 2D/3D turbulence conundrum[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF][START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF]. In addition to the lack of scales smaller than O(100) km, the altimetry measure processing by AVISO has been suggested to alter the diagnostic of the turbulent energy cascade [START_REF] Arbic | Geostrophic turbulence in the frequency-wavenumber domain: Eddy-driven low-frequency variability[END_REF][START_REF] Arbic | On eddy viscosity, energy cascades, and the horizontal resolution of gridded satellite altimeter products[END_REF]. This raises the question to what extent satellite altimetry data are representative of the mesoscale eddy energy dynamics, in particular in the context of [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF]'s result. Assessing the capability of satellite altimetry data to diagnose the mesoscale eddy-energy sources and sinks will point toward the significant elements of dynamics for the mesoscale eddy dynamics. Numerical models directly simulate the velocity field at relative high resolution (O(1) km) on a regional scale and therefore represent a natural tools for assessing the possible limitations of the altimetry data for evaluating the mesoscale eddy-energy sources and sinks. In particular in the coming satellite altimetry mission SWOT11 the use of numerical models would allow to preliminary infer the possible breakthroughs this mission could lead to [START_REF] Morrow | Global observations of fine-scale ocean surface topography with the Surface Water and Ocean Topography (SWOT) mission[END_REF]d'Ovidio et al., 2019).

The third study carried out in this work, focuses on the diagnostic of the mesoscale eddy energetic dynamics from the satellite altimetry data (AVISO) and in particular on the following question :

Question III : What is the significance and interpretation of the mesoscale eddy-energy sources and sinks diagnosed from satellite altimetry data ?

Partie , Introduction

This question is addressed in the Chapter IV : "Study III : Can the mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?". [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF]'s diagnostic of the mesoscale eddy-energy sources and sinks is reproduced using modeled and altimetry sea surface height fields and is compared to the same diagnostic using the full modeled 3D-velocity fields. The limitations of the assumptions, required by the use of altimetry data (geostrophically-balanced dynamics and the mesoscale eddy vertical structure described by the 1 st baroclinic mode) and the assumption of a linear dynamics, are then evaluated. This Chapter corresponds to a collection of results in preparation for a scientific publication which will be submitted to a journal after my doctorate.

Chapter 2

METHODS : PRESENTATION OF THE APPROACHES

In this Chapter, I present the approach used, its design and I define the diagnostics realized to address the dissertation problematics.

The Agulhas Current numerical simulations

The numerical simulations used to conduct this regional study of the Agulhas Current are presented in this section. The numerical model (CROCO) is presented first, followed by the numerical choices made for the numerical simulations.

The Coastal and Regional Ocean COmmunity model -CROCO

The Coastal and Regional Ocean COmmunity model (CROCO) [START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF] is introduced in this subsection.

The CROCO model is an oceanic modeling system based on ROMS [START_REF] Shchepetkin | The Regional Oceanic Modeling System (ROMS): A splitexplicit, free-surface, topography-following-coordinate ocean model[END_REF]. It is designed to realistically resolve ocean dynamics at coastal and regional scales. This implies to resolve fine scales and their interactions with larger scales. This is made possible by AGRIF tool [START_REF] Debreu | Two-way embedding algorithms: a review[END_REF][START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF], allowing grid nesting and their 2-ways communications, at a relatively low computational cost. CROCO is a split-explicit, free surface and topographic following coordinate model which solves the primitive equations in the Boussinesq and hydrostatic approximations (Shchepetkin andMcWilliams, 2003, 2005). Seawater density anomalies (ρ) do not significantly vary around a reference value (ρ 0 ) : | ρ |<< ρ 0 . The Boussinesq approximation disregards the density variations for all the momentum terms except for the buoyancy term (2.1), which corresponds to the water column weight and plays a crucial role in the oceanic dynamics. This approximation also results in considering the seawater as incompressible Partie , Chapter 2 -Methods : Presentation of the Approaches (2.4) and thus to filter the acoustic waves. The hydrostatic balance originates from the small aspect ratio between the vertical and the horizontal scales of oceanic motion. The vertical component of the momentum equations corresponds then to the balance between the pressure gradient and the buoyancy terms (2.3). The primitive equations are coupled to a non-linear equation of state (2.5) [START_REF] Shchepetkin | A method for computing horizontal pressure-gradient force in an oceanic model with a nonaligned vertical coordinate[END_REF] and advective-diffusive equations of the potential temperature and salinity (2.2).

∂ ∂t u

T ime rate

+ ∇ H .(uu)
Horizontal advection

+ ∂ ∂z (wu) V ertical advection + f k ∧ u Coriolis = - 1 ρ 0 ∇ H p P ressure gradient + F u F orcing + D u Dissipation (2.1)

∂ ∂t C

T ime rate

+ u.∇ H C Horizontal advection + w ∂ ∂z C V ertical advection = F C F orcing + D C Dissipation (2.2) ∂ ∂z p = -gρ (2.3) ∇ H .u + ∂ ∂z w = 0 (2.4) ρ = ρ(T, S, z) (2.5)
with u the horizontal velocity vector, w the vertical velocity component, f the Coriolis parameter, p the total pressure, g the acceleration of gravity, ρ(x, z, t) the density variations relative to the density reference ρ 0 , T the potential temperature, S the salinity, C any scalar concentration field, k the unit vector in the z direction and ∇ H the horizontal divergence operator.

In the model, equations are solved for the time averaged variables and the sub-grid turbulent processes are represented by the Reynolds stress term. To close the primitive equations, this term is parameterized by the vertical mixing and horizontal dissipation terms in the model. On the vertical, the boundary conditions at the surface and the bottom are respectively the wind stress and the bottom friction. On the horizontal, the horizontal dissipation can be explicitly defined or be implicitly induced by the horizontal advective scheme.

The numerical grid coordinate system is curvilinear on the horizontal and stretched on the vertical following the topographic contours, such that the height-coordinate transforms to : z = z(x, σ), with σ the fractional depth spanning from -1 (bottom) to 0 (surface). This coordinate system adjusts to land and topography geometries and allows a realistic representation of the domain. The stretching options at the surface and at the bottom [START_REF] Haidvogel | Numerical ocean circulation modeling[END_REF][START_REF] Shchepetkin | Correction and commentary for "ocean forecasting in terrainfollowing coordinates: Formulation and skill assessment of the regional ocean modeling system[END_REF] as well participate to the boundary layers representation. However, this vertical grid coordinate system has the disadvantage of generating spurious errors due to the pressure horizontal derivatives discretization in areas of strong slopes [START_REF] Mccalpin | A comparison of second-order and fourth-order pressure gradient algorithms in a σ-co-ordinate ocean model[END_REF]. These errors are limited by formulating the pressure horizontal derivatives from the seawater density [START_REF] Shchepetkin | A method for computing horizontal pressure-gradient force in an oceanic model with a nonaligned vertical coordinate[END_REF] and by limiting truncations errors using high order schemes for its discretization [START_REF] Mccalpin | A comparison of second-order and fourth-order pressure gradient algorithms in a σ-co-ordinate ocean model[END_REF][START_REF] Shchepetkin | A method for computing horizontal pressure-gradient force in an oceanic model with a nonaligned vertical coordinate[END_REF]. In practice, the errors limitation is reinforced by the respect of criteria for the smoothing of topography and the vertical stretching of the vertical grid [START_REF] Shchepetkin | A method for computing horizontal pressure-gradient force in an oceanic model with a nonaligned vertical coordinate[END_REF].

The different simulated variables are discretized on staggered horizontal and vertical grids. Both the Arakawa -C horizontal grid [START_REF] Arakawa | Computational design of the basic dynamical processes of the UCLA general circulation model[END_REF], favorising the accuracy of the horizontal derivatives discretization, and the momentum terms flux formulation optimize the mass conservation.

The free-surface model uses a split-explicit time stepping. The barotropic mode (barotropic momentum and free surface elevation) and the baroclinic mode (baroclinic momentum and tracers) are advanced in time following respectively fast and slow time steps. The barotropic mode is resolved, using a predictorcorrector scheme, for a finite number of time steps within two slow baroclinic time steps. The barotropic variables are time averaged following a 2-way time-averaging procedure in order to satisfy the continuity equation. The baroclinic mode follows a Leapfrog (LF) predictor scheme with a third order Adams-Molton (AM-03) interpolation, which is specific to the ROMS model (Shchepetkin andMcWilliams, 2005, 2009). The barotropic and baroclinic modes coupling happens during the baroclinic corrector stage. This time stepping scheme is a tradeoff between the time stepping increasing without loss of physical accuracy and increasing of computational cost, allows to resolve efficiently the primitive equations with a high numerical precision.

The numerical simulations set up

A set of four regional numerical simulations has been designed using the ROMS TOOLS [START_REF] Penven | Software tools for pre-and post-processing of oceanic regional simulations[END_REF]. The numerical choices relevant for their realism are presented in this subsection. These include the 2-way nesting procedure, the vertical grid, the forcings, the numerical advection scheme and the associated effective resolution.

The four nested regional simulations (dx ∼ 22.5, 7.5 2.5 and 0.75 km in Figure 2.1) have been designed to study the dynamics of the greater Agulhas Current as a part of the global circulation (dx ∼ 22.5 and 7.5 km) and along its path at the mesoscale (dx ∼ 2.5 km) and submesoscale (dx ∼ 0.75 km) resolving numerical resolutions. -1 ] in the Agulhas Current region as simulated by CROCO for the four nested grids : dx ∼ 22.5, 7.5, 2.5 and 0.75 km and time averaged over the 1994-1999 period. The dx ∼ 22.5 km grid simulates the Agulhas Current as a part of the global circulation (grey features), the dx ∼ 7.5 km grid simulates the current as a part of the South Indian Ocean subtropical gyre (white features), the dx ∼ 2.5 km grid focuses on the Agulhas Current (purple features) and the dx ∼ 0.75 km grid is centered on the current portion around Port Elizabeth (purple dot). The circulation scheme is largely inspired from [START_REF] Lutjeharms | The Agulhas Current[END_REF]; [START_REF] Talley | Descriptive Physical Oceanography: An Introduction[END_REF] and EMC : East Madagascar Current, NEMC : North East Madagascar Current and EACC : East African Coastal Current.

The dx ∼ 22.5 km grid extends over most of the South Equatorial Indian Ocean (102 • E) to the South East Atlantic Ocean (-10 • E) from the Equatorial (-3 • S) to the subpolar (-56 • S) latitudes. It simulates the Agulhas Current as a part of the global circulation connecting the Indian to the Atlantic Ocean through the Agulhas leakage [START_REF] Beal | On the role of the Agulhas system in ocean circulation and climate[END_REF][START_REF] Peeters | Vigorous exchange between the Indian and Atlantic oceans at the end of the past five glacial periods[END_REF][START_REF] Weijer | Response of the Atlantic Overturning Circulation to South Atlantic sources of buoyancy[END_REF] and connecting the Indian to the Southern Ocean through the Agulhas Return Current [START_REF] Lutjeharms | The Agulhas Current[END_REF] (grey features in Figure 2.1). The dx ∼ 7.5 km grid is centered on the African continent tip (6 • E -55 • E), enclosing Madagascar, from the Equatorial (-7 • S) to the subpolar (-46 • S) latitude. It simulates the Agulhas Current as a part of the South Indian Ocean subtropical gyre and includes the flows from the Mozambique Channel, from the East of Madagascar and from the South-West Indian Ocean sub-gyre, all of three being sources of the Agulhas Current [START_REF] Lutjeharms | The Agulhas Current[END_REF] (white features in Figure 2.1). The dx ∼ 2.5 km grid is centered on the most intense western boundary current of the South Hemisphere [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF] : the Agulhas Current. It is simulated from its source, north of Durban (34 2.1). The mesoscale eddies vertical structure is described by the barotropic and 1 st baroclinic modes [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF], which on the horizontal correspond to the external (O(100) km, not shown) and 1 st internal (Figure 2.2) Rossby deformation radii. With an effective resolution about 10dx ∼ 25 km [START_REF] Soufflet | On effective resolution in ocean models[END_REF], this grid solves the mesoscale horizontal scales and is thus adequate for studying the mesoscale turbulence in the region of the Agulhas Current.

The dx ∼ 0.75 km grid is centered on the Agulhas Current portion around Port Elizabeth (26 • E ; -34 • S) (purple dot in Figure 2.1). This specific location marks the transition between the stable Upstream Agulhas Current branch and the unstable Downstream Agulhas Current branch [START_REF] Lutjeharms | The Agulhas Current[END_REF] and allows to give a new perspective on this transitional area, resolving its dynamics at the submesoscale range [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF].

• 2-way nesting procedure

The three coarser simulations (dx ∼ 22.5, 7.5 and 2.5 km) use an online 2-way nesting procedure [START_REF] Debreu | Two-way embedding algorithms: a review[END_REF][START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF], while the dx ∼ 0.75 km has been implemented following an offline 1-way nesting procedure [START_REF] Mason | Procedures for offline grid nesting in regional ocean models[END_REF].

The operational capability AGRIF tool of the CROCO model provides the online 2-way nesting algorithm allowing spatial and time refinement, in a split-explicit free surface model, using several nesting levels. The nesting procedure allows for modeling at high resolution for a low computational cost and the 2-way procedure allows for the solution to be dynamically adjusted for time dependent problems (Figure 2.1). The classic nesting procedure consists in a full overlap between a local high resolution grid (child grids : dx ∼ 7.5, 2.5 and 0.75 km grids) embedded in a coarse resolution grid (parent grids : dx ∼ 22.5, 7.5 and 2.5 km grids). Information is transferred from the parent toward the child grid as boundary conditions (for the child grids : dx ∼ 7.5, 2.5 and 0.75 km) and information is transferred from the child toward the parent grid (update for the parent grids : dx ∼ 22.5 and 7.5 km). This feedback from the fine to the large scales requires a scale selective update operator, which allows the transfer of the maximum information to the coarse grid at the scales well resolved by this one and filter the unresolved scales [START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF]. With this procedure, the volume and tracers conservation need a special care, especially for long-term integrations. This property is reinforced by the use of a correction flux algorithm.

• Momentum and tracers numerical schemes and turbulent closure

In the CROCO model, the numerical schemes for advection are based on finite differences, which are a compromise between accuracy, encountered for high orders of discretization, and numerical cost.

For the four nested grids, the momentum advection is discretized using a third-order upwind biased advection scheme on the horizontal and using a semi-implicit, with an adjustment between the explicit and implicit formulations according to the model stability, on the vertical. On the horizontal, the numerical scheme introduces an implicit numerical dissipation. This ones act for the dissipation of the processes not resolved by the horizontal grid and can be computed as the difference between the third-order upwind numerical advective momentum scheme and a fourth-order centered scheme [START_REF] Gula | Submesoscale cold filaments in the Gulf Stream[END_REF].

For the four nested grids, the tracers advection is discretized using a split and rotated third-order upwind-biased scheme [START_REF] Marchesiello | Spurious diapycnal mixing in terrain-following coordinate models: The problem and a solution[END_REF][START_REF] Lemarié | Are there inescapable issues prohibiting the use of terrain-following coordinates in climate models[END_REF] on the horizontal and using a fourth-order compact scheme on the vertical. On the horizontal, the numerical scheme reduces the spurious diapycnal mixing inherent to terrain-following vertical coordinate models [START_REF] Marchesiello | Spurious diapycnal mixing in terrain-following coordinate models: The problem and a solution[END_REF]. At intermediate numerical resolutions (typical from regional simulations), the Veronis effect induces diapycnal mixing depending on the σ-levels orientation with the isopycnals. Splitting the diffusive part of the numerical scheme from the advective part allows to reduce the diapycnal mixing by injecting the dissipation along isopycnals. This allows to conserve the water masses properties and thus to contribute to the simulations realism.

For the four nested grids, the K-profile parameterization [START_REF] Large | Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization[END_REF] is used for the turbulent closure of the primitive equations. This 1D profile parameterizes the vertical mixing by determining the "eddy" (subgrid scale) viscosity parameter (K M ) in the surface and the bottom boundary layers and in the ocean interior. Both boundary layers thickness are estimated from an Ekman depth estimate and from a Richardson number criterion, using [START_REF] Lemarié | Are there inescapable issues prohibiting the use of terrain-following coordinates in climate models[END_REF] modification of the critical Richardson number value (Ri cr =0.15). In the bottom boundary layer, the K-profile has a parabolic shape proportionals to the boundary layer thickness following [START_REF] Durski | Vertical mixing schemes in the coastal ocean: Comparison of the level 2.5 Mellor-Yamada scheme with an enhanced version of the K profile parameterization[END_REF]'s adjustment.

• Bathymetry and vertical grid

For the four nested grids, the bathymetry is built from the General Bathymetric Chart of the Oceans 30-s resolution [START_REF] Weatherall | A new digital bathymetric model of the world's oceans[END_REF].

For the three larger grids (dx ∼ 22.5, 7.5 and 2.5 km grids), the vertical grid consists in 60 σ levels distributed following [START_REF] Haidvogel | Numerical ocean circulation modeling[END_REF]. They are stretched at the surface (θ s =5 and θ b =0) with a transition depth between flat z and σ-levels of h cline =10 m. For the nested simulation dx ∼ 0.75 km, the vertical resolution consists in 100 σ levels distributed following [START_REF] Lemarié | Are there inescapable issues prohibiting the use of terrain-following coordinates in climate models[END_REF]'s method. They are stretched at the surface (θ s =6) and at the bottom (θ b =4) with a transition depth of h cline =300 m. The grid stretching allows for a finer vertical resolution at the surface and at the bottom and then for a better representation of the upper and bottom layers processes. The main disadvantage of the terrain-following vertical coordinate system is the generation of pressure gradient errors. To limit these numerical errors, the topography is locally smoothed when the steepness (r = ∇H H ) exceeds 0.2 [START_REF] Beckmann | Numerical simulation of flow around a tall isolated seamount. Part I: Problem formulation and model accuracy[END_REF].

• Surface and boundary forcings

At the surface, the four nested simulations are forced by the [START_REF] Fairall | Bulk parameterization of air-sea fluxes for tropical ocean-global atmosphere coupled-ocean atmosphere response experiment[END_REF] bulk formula which uses the daily variables (temperature at 2 m, wind velocity components at 10 m, specific humidity, thermal and solar radiations, precipitation rate and surface stress components) of the global atmospheric re-analysis of ERA-Interim from ECMFW [START_REF] Dee | The ERA-Interim reanalysis: Configuration and performance of the data assimilation system[END_REF]. This re-analysis, generated by a data assimilation system based on the 2006 version of the Integrated Forecasting System (IFS), provides a time serie of daily atmospheric variables from 1979 to 2019 at a spatial resolution of about 80 km. The wind forcing is provided by the relative winds. The use of relative winds, including the current feedback on the wind velocities instead of absolute wind, contributes to the numerical simulations realism. The wind participates in the mesoscale eddies dissipation and [START_REF] Renault | Modulation of the Agulhas Current Retroflection and Leakage by Oceanic Current Interaction with the Atmosphere in Coupled Simulations[END_REF] showed that with relative winds the modeled mesoscale eddies energy reservoir is weaker than with absolute winds, but it is in better agreement with observations. This influence on the mesoscale eddies energy reservoir is also related to the Agulhas Current stability decreasing the number of occurrence of spurious early retroflections.

At the boundaries, the dynamical and state variables of the nested simulations (dx ∼ 7.5 and 2.5 km) are forced at each time step using the closest parent grids (dx ∼ 22.5 and 7.5 km). The simulation dx ∼ 0.75 km is forced at the boundaries using daily fields from the closest parent grid (dx ∼ 2.5 km). The largest parent grid (dx ∼ 22.5 km) is forced using the oceanic re-analysis GLORYS [START_REF] Ferry | Glorys2v1 global ocean reanalysis of the altimetric era (1992-2009) at meso scale[END_REF]. This re-analysis, a product of Mercator Ocean generated by the ORCA025 configuration of the NEMO model using data assimilation of in situ and satellite observations, provides a time series of oceanic properties for the 20 last years at the spatial resolution of 0.25 • . The forcing at the boundaries of the nested simulations involves a sponge layer, located at the grids interfaces, where an explicit horizontal dissipation and nudging conditions are applied [START_REF] Marchesiello | Open boundary conditions for long-term integration of regional oceanic models[END_REF] in order to limit the discontinuity between the grid borders and interior.

After their initializations, the coarser simulations (dx ∼ 22.5 et 7.5 km) are spun up over 3 years and the dx ∼ 2.5 km is spun up over 1 year. After their spin-up, the 3 grids are run over 20 years . The dx ∼ 0.75 km grid is spun-up over 3 months and then run for 1 year (1996).

Convergence of the Agulhas Current modeling

The parallel efforts of two oceanic model communities : CROCO [START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF] and NEMO [START_REF] Madec | NEMO ocean engine[END_REF], for simulating the oceanic dynamics at high resolution, led to the convergence of the submesoscaleresolving regional simulation of the Agulhas Current system.

The CROCO (dx ∼ 2.5 km and 0.75 km grids from [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF]) and NEMO (dx ∼ 1.5 km grid from [START_REF] Schubert | Submesoscale impacts on mesoscale Agulhas dynamics[END_REF]) simulations show a similar vertical structure of the Agulhas Current system in good agreement with the Agulhas Current Time series Experiment (ACT) [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF] in situ data (Figure 2.3). The Agulhas Current (light blue contours in Figure 2.3) extends from the shore, where it is the most intense, to about 200 km offshore. On the vertical, it has the typical V-shape of the surface-intensified currents, with speeds peaking at 150 cm s -1 , and extends to about 1500 m depth. The agreement between the Agulhas Current vertical structures is also found in its net volume flux : -77 ± 5 Sv [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF] (Figure 2.3 top left panel), -73 ± 26 Sv [START_REF] Schubert | Submesoscale impacts on mesoscale Agulhas dynamics[END_REF]) (Figure 2.3 top right panel) and -79.6 Sv [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF]) (Figure 2.3 bottom left panel). The Agulhas Undercurrent and the Subgyre recirculation, both flowing equatorward opposite to the Agulhas Current, are also captured by the ACT section. The Agulhas Undercurrent flows between 1500 and 4000 m along the continental slope at a maximum speed of 1 cm s -1 . The cyclonic circulation of the Subgyre takes place at about 200 km offshore and extends over the whole water column with flow intensified at the surface and at the bottom.

The convergence between the two oceanic models, based on different numerical methods (vertical grid, advection and time schemes, numerical dissipation, bottom boundary layer parameterization) strengthens our confidence in the realistic numerical models accuracy and robustness. This allows in particular to contrast the respective numerical artefacts of the models, for example the pressure gradient errors at strong topographic slopes for the CROCO model [START_REF] Mccalpin | A comparison of second-order and fourth-order pressure gradient algorithms in a σ-co-ordinate ocean model[END_REF], and to dissociate the physical dynamics from the one inherent to the numerics. This evolution of the oceanic modeling at high resolution need to be systematically evaluated. However, there is currently no global or regional observational baseline at the submesoscale resolution on a regular grid. The SWOT project aims to build a global time serie of sea surface height at the 15-30 km resolution and thus could help fulfill this need [START_REF] Morrow | Global observations of fine-scale ocean surface topography with the Surface Water and Ocean Topography (SWOT) mission[END_REF]d'Ovidio et al., 2019). [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF], from (top right) the dx ∼ 1.5 km NEMO simulation [START_REF] Schubert | Submesoscale impacts on mesoscale Agulhas dynamics[END_REF] and from (bottom panels) the dx ∼ 0.75 and 2.5 km CROCO simulations [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF]. The red contours denote the "equatorward" current with a contours interval of 10 cm s -1 and the black and blue contours denote the "poleward" current with a contours interval respectively of 5 and 10 cm s -1 . The cyan contour denotes the 5 cm s -1 isotach from the ACT data set. The bold black contour denotes the 0 cm s -1 isotach. The mean along-front velocity is defined for the May 2010-February 2013 period (top and bottom right panles) and over the 1996 1-year period the dx ∼ 0.75 km CROCO (bottom left panel).

Mean Eddy Kinetic Energy equation (EKE)

The Kinetic Energy equation is decomposed into a mean and an eddy part, following [START_REF] Harrison | Energy analysis of open regions of turbulent flows-Mean eddy energetics of a numerical ocean circulation experiment[END_REF], such that : EKE = KE -M KE, with KE = 1 2 (u 2 + v 2 ) the mean Kinetic Energy of the total flow, M KE = 1 2 (u 2 + v 2 ) the Kinetic Energy of the mean flow, and EKE = 1 2 (u ′2 + v ′2 ) the mean Eddy Kinetic Energy. The overbar denotes a time average (mean part) and the prime denotes time fluctuations relative to the time average (eddy part). The EKE equation is developed in this section. The KE and the M KE equations are derived to finally obtain the EKE equation.

Derivation of the EKE equation

KE equation

The KE equation is obtained by taking the inner product of the horizontal velocities (u) with the horizontal momentum equations and time averaging.

u.

∂ Here, the vertical mixing terms are expressed as :

V = ∂ ∂z (K M v ∂u ∂z
) with K M v the "eddy" (subgrid scale) diffusivity. The horizontal dissipation terms : D = (D u , D v ) correspond to the effect of the numerical dissipation. The forcing terms: S = (S u , S v ) correspond to other sources and sinks such as explicit dissipation and nudging, acting in the grid sponge layers, and the boundary conditions. The numerical dissipation is implicit in the model and is diagnosed as the difference between the third-order upwind numerical advective momentum scheme and a fourth-order centered scheme [START_REF] Gula | Submesoscale cold filaments in the Gulf Stream[END_REF].

M KE equation

The M KE equation is obtained by taking the inner product of the mean horizontal velocities (u) with the mean horizontal momentum equations.

-u. 1 ρ 0 ∇ H p + u. 1 ρ 0 ∇ H p + u.V -u.V + u.D -u.D + u.S -u.S
Which simplifies into :

u ′ . ∂ ∂t u + u ′ .(u.∇ H )u + u ′ .w ∂ ∂z u = -u ′ . 1 ρ 0 ∇ H p ′ + u ′ .V + u ′ .D + u ′ .S (2.9)
The advection and pressure gradient terms are further decomposed in the following :

• Advection u ′ .(u.∇ H )u + u ′ .w ∂ ∂z u = u ′ .(u.∇ H )u ′ + u ′ .w ∂ ∂z u ′ + u ′ .(u ′ .∇ H )u + u ′ .w ′ ∂ ∂z u = u.∇ H 1 2 u ′2 + w ∂ ∂z 1 2 u ′2 + u ′ .(u ′ .∇ H )u + u ′ .w ′ ∂ ∂z u = ∇ H .( 1 2 uu ′2 ) + ∂ ∂z ( 1 2 wu ′2 ) + u ′ .(u ′ .∇ H )u + u ′ .w ′ ∂ ∂z u (2.10)
Here, the continuity equation (2.4) has been used.

• Pressure gradient

-u ′ . 1 ρ 0 ∇ H p ′ = -∇ H .( 1 ρ 0 u ′ p ′ ) + p ′ 1 ρ 0 ∇ H .u ′ = -∇ H .( 1 ρ 0 u ′ p ′ ) -p ′ 1 ρ 0 ∂ ∂z w ′ = -∇ H .( 1 ρ 0 u ′ p ′ ) - ∂ ∂z ( 1 ρ 0 p ′ w ′ ) + w ′ 1 ρ 0 ∂ ∂z p ′ = -∇ H .( 1 ρ 0 u ′ p ′ ) - ∂ ∂z ( 1 ρ 0 p ′ w ′ ) + w ′ b ′ (2.11)
Here the continuity (2.4) and the hydrostatic (2.3) equations have been used.

Finally, replacing the advection (2.10) and the pressure gradient (2.11) terms into the EKE equation (2.9) leads to :

u ′ . ∂ ∂t u T ime rate + ∇ H .( 1 2 uu ′2 + 1 ρ 0 u ′ p ′ ) + ∂ ∂z ( 1 2 wu ′2 + 1 ρ 0 p ′ w ′ )
Divergences of the eddy-energy f luxes = (2.12)

-u ′ .(u ′ .∇)u HRS + u ′ .w ′ ∂ ∂z u V RS M KE→EKE + w ′ b ′ EP E→EKE (V BF ) + u ′ .V V ertical mixing + u ′ .D N umerical dissipation + u ′ .S F orcings
The left-hand side of the equation (2.12) represents the EKE evolution (storage and transport) and the right-hand side gathers its sources and sinks. The physical meaning of the terms is explained in the following :

EKE evolution :

-Time rate : represents the EKE storage over time and tends toward zero over long time averages.

-Divergences of the eddy-energy fluxes : represent the EKE changes done by the total flow advection and by the eddy pressure work. When integrated over the volume of the domain, this term can be interpreted as the EKE fluxes at the domain boundaries.

Transfers between the EKE, M KE and EP E reservoirs :

-M KE → EKE (HRS) term : Horizontal Reynolds Stress-mean flow non-linear interactions. The energy conversion is done by the mean horizontal shear and the HRS term can be a proxy for barotropic (Gula et al., 2015a(Gula et al., ,b, 2016a) ) and centrifugal/inertial (Gula et al., 2016b) instabilities.

-M KE → EKE (V RS) term : Vertical Reynolds Stress-mean flow non-linear interactions. The energy conversion is done by the mean vertical shear and the V RS term can be a proxy for Kelvin-Helmholtz type and symmetric instabilities [START_REF] Haine | Gravitational, symmetric, and baroclinic instability of the ocean mixed layer[END_REF][START_REF] Thomas | Symmetric instability in the Gulf Stream[END_REF].

-EP E → EKE (V BF ) term : eddy energy conversion between the potential and kinetic reservoirs. It corresponds to vertical buoyancy fluxes and can be a proxy for baroclinic instability (Capet et al., 2008b;Gula et al., 2015aGula et al., , 2016a)), frontogenesis (Capet et al., 2008b) and filamentogenesis [START_REF] Gula | Submesoscale cold filaments in the Gulf Stream[END_REF].

EKE generation and dissipation :

-Vertical mixing : EKE dissipation and/or generation by wind stress, interior vertical mixing and bottom drag (Gula et al., 2016b).

-Numerical dissipation : EKE dissipation by the horizontal viscosity of the model.

-Forcings : EKE dissipation and/or generation (due to explicit horizontal dissipation and nudging) in the grid sponge layers close to the open boundaries.

Examples of EKE budget in regional numerical simulations can be found in [START_REF] Kang | Energetics of eddy-mean flow interactions in the Gulf Stream region[END_REF]; Gula et al. (2016b); [START_REF] Yan | Energetics of eddy-mean flow interactions along the western boundary currents in the North Pacific[END_REF].

Mean modal Eddy Kinetic Energy (EKE n ) equation

The EKE equation is developed within the mathematical framework of the vertical modes (subsequently referred as EKE n ) in this section. First, the mathematical framework of the vertical modes is defined, then the EKE n demonstration is developed following [START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF] and finally how the diagnostics of the EKE n terms are performed in the regional numerical simulation is detailed.

Definition of the vertical modes

The mathematical framework of the traditional vertical modes [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF] is defined in this subsection.

The separation of variables can be used for writing the variables as an expansion of vertical modes in a continuously stratified medium with a flat-bottom. This latter assumption constrains the scope of application of this method, but several studies have shown that the flat-bottom assumption can be relaxed and that the vertical modes can be used to study baroclinic waves and tides generation over topographic features [START_REF] Kelly | Internal-tide energy over topography[END_REF][START_REF] Kelly | The cascade of tidal energy from low to high modes on a continental slope[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF][START_REF] Lahaye | Modal analysis of internal wave propagation and scattering over large-amplitude topography[END_REF]. The traditional vertical modes definition is evaluated as valid in the context of the dx ∼ 2.5 km regional simulation in Appendix A.

The modal expansion is a combination of vertical modes, normal to each others, describing the variables vertical structure and of modal amplitudes, describing the variables horizontal structure. The variables projection onto the vertical modes orthogonal basis is as follow:

[u(x, z, t), 1 ρ 0 p(x, z, t)] = ∞ n=0 [u n (x, t), 1 ρ 0 p n (x, t)]φ n (x, z) (2.13) [w(x, z, t), b(x, z, t)] = ∞ n=0 [w n (x, t), b n (x, t) N 2 (x, z)]Φ n (x, z) (2.14)
with p the dynamic pressure, b = -g ρ ρ 0 the dynamic buoyancy and N 2 the static Brunt-Väisälä frequency. φ n and Φ n are respectively the vertical modes basis of the horizontal velocity/dynamic pressure and of the vertical velocity/dynamic buoyancy while (u n , p n , w n , b n ) are the modal amplitudes of the horizontal velocity, the dynamic pressure, the vertical velocity and of the dynamic buoyancy. The vertical modes basis (φ n and Φ n ) originates from the Quasi-Geostrophic (subsequently referred as QG) framework. In this respect, the vertical structure of the two set of variables : (u, p ′ ) and (w, b ′ ) are described by two different vertical modes basis related to each other by :

φ n = ∂ ∂z Φ n .
Partie , Chapter 2 -Methods : Presentation of the Approaches

The modal expansion of w (2.14) is injected into the QG equation for the potential vorticity linearized around a background state of rest [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF]. Doing so, the resulting QG equation is equivalent to the one describing the vertical velocity of a freely propagating linear internal waves [START_REF] Mysak | Waves in the Ocean[END_REF][START_REF] Cushman-Roisin | Introduction to geophysical fluid dynamics: physical and numerical aspects[END_REF][START_REF] Kundu | Fluid Mechanics, 5th Version[END_REF]. That is why, this traditional vertical modes mathematical framework is usually used for studies on internal waves and tides [START_REF] Kelly | Internal-tide energy over topography[END_REF][START_REF] Kelly | The cascade of tidal energy from low to high modes on a continental slope[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF][START_REF] Lahaye | Modal analysis of internal wave propagation and scattering over large-amplitude topography[END_REF]. The resulting QG equation corresponds to a Sturm-Liouville type equation (2.15). The vertical modes (Φ n ) basis is formed of the orthogonal eigenfunctions solution of this Sturm-Liouville equation with a linearized free-surface ((w) z=η = ∂ ∂t η) and a flat bottom ((w) z=-H = 0).

∂ 2 ∂z 2 Φ n + N 2 c 2 n Φ n = 0 (2.15) (Φ n ) z=η = ( c 2 n g ∂ ∂z Φ n ) z=η (2.16) (Φ n ) z=-H = 0 (2.17)
with c n the eigenspeeds. The vertical modes (φ n ) base is determined using the relation between the two vertical modes basis (φ n = ∂ ∂z Φ n ) and the boundary condition of the flat-bottom : (Φ n ) z=-H = 0. The system of equations (2.18) becomes for the Φ n basis :

∂ ∂z ( 1
N 2 ∂ ∂z φ n ) + 1 c 2 n φ n = 0 (2.18) ( ∂ ∂z φ n ) z=η = ( -N 2 g φ n ) z=η (2.19) ( ∂ ∂z φ n ) z=-H = 0 (2.20)
The orthogonality conditions of the vertical modes basis are the following [START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF] :

η -H φ m φ n dz = δ mn h (2.21) and g c 2 n (Φ m Φ n ) z=η + η -H N 2 c 2 n Φ m Φ n dz = δ mn h (2.
w n (x, y, t) = 1 hc 2 n ( η -H N 2 w(x, y, z, t)Φ n (z) dz + g(wΦ n ) z=η ) (2.24) b n (x, y, t) = 1 hc 2 n ( η -H b(x, y, z, t)Φ n (z) dz + g( b N 2 Φ n ) z=η ) (2.25)

Derivation of the EKE n equation

The EKE n equation is derived in this subsection similarly to EKE, as follows : (2.27)

EKE n = KE n -M KE n , with KE n = η -H 1 2 (u 2 n + v 2 n ) dz
These two expressions of the density are reconciled by approximating the static density by the mean total density ( ρ(z) + ρ(x, z) ∼ ρ 0 + ρ(x, z)) and the dynamic density by the time fluctuations of the density ( ρ ′ (x, z, t) ∼ ρ ′ (x, z, t)). These approximations are not valid for permanent currents because the mean total density includes the static density and the mean dynamic density corresponding to the permanent circulation. However, our study focuses on eddies for which the time fluctuating dynamic density is fully captured by the time fluctuations of the density.

Modal momentum equations

The projection of the momentum equations onto a vertical mode n is done by multiplying the horizontal momentum equations and the continuity equation (2.4) by the vertical mode φ n , the hydrostatic equation (2.3) by the vertical mode Φ n and depth integrating ( η(x,t) -H(x) . dz).

• Modal horizontal momentum equations

η -H ( ∂ ∂t u)φ n dz + η -H (u.∇ H u)φ n dz + η -H (w ∂ ∂z u)φ n dz + η -H (f k ∧ u)φ n dz = (2.28) - η -H (∇ H p)φ n dz + η -H Vφ n dz + η -H Dφ n dz + η -H Sφ n dz • Modal continuity equation 0 = η -H (∇ H .u)φ n dz + η -H ( ∂ ∂z w)φ n dz 0 = η -H ∇ H .(uφ n ) dz - η -H u.∇ H φ n dz + η -H ∂ ∂z (wφ n ) dz - η -H w ∂ ∂z φ n dz 0 = ∇ H . η -H uφ n dz + ∇ H η.(uφ n ) z=η -∇ H H.(uφ n ) z=-H - ∞ m=0 u m .( η -H φ m ∇ H φ n dz) + (wφ n ) z=η -(wφ n ) z=-H + 1 c 2 n η -H wN 2 Φ n dz 0 = ∇ H .(hu n ) + ∇ H η.(uφ n ) z=η -∇ H H.(uφ n ) z=-H - ∞ m=0 u m .( η -H φ m ∇ H φ n dz) + g c 2 n (wΦ n ) z=η + ∇ H H.(uφ n ) z=-H + hw n - g c 2 n (wΦ n ) z=η ∇ H .(hu n ) = ∞ m=0 u m .( η -H φ m ∇ H φ n dz) -hw n -∇ H η.(uφ n ) z=η (2.29)
Here, the Leibniz's rule :

∇ H η(x) -H(x) f (x, z) dz = η(x) -H(x) ∇ H f (x, z) dz+∇ H η(x)f (x, z = η)-∇ H H(x)f (x, z = -H
) and the modal expansion of the variable u (2.13) have been used. The vertical integration η -H . dz has been simplified to η -H . dz because the vertical modes are determined from the mean Brunt-Väisälä frequency (N 2 ). The relation between the two vertical modes base given by the continuity equation (2.3.1), the Sturm-Liouville equation for the vertical modes Φ n (2.15), the kinematic boundary conditions at the surface (2.16) and at the bottom (2.17) and the modal amplitude of the variable w n (2.24) have been used.

• Modal hydrostatic equation

η -H ( ∂ ∂z p ′ )Φ n dz = η -H b ′ Φ n dz η -H ∂ ∂z (p ′ Φ n ) dz - η -H p ′ ∂ ∂z Φ n dz = hc 2 n b ′ n -g( b ′ N 2 Φ n ) z=η (p ′ Φ n ) z=η -(p ′ Φ n ) z=-H - η -H p ′ φ n dz = hc 2 n b ′ n -g( b ′ N 2 Φ n ) z=η hp ′ n = -hc 2 n b ′ n + g( b ′ N 2 Φ n ) z=η (2.30)
Here, the free-surface dynamic boundary condition (p ′ z=η = 0), the kinematic boundary condition at the bottom (2.17) and the modal amplitude of the variable b ′ n (2.25) have been used.

KE n equation

The KE n equation is obtained by taking the inner product of the horizontal velocities for a vertical mode n (u n ) with the horizontal momentum equations for a vertical mode n (2.28) and time averaging.

u n .(h( ∂ ∂t u) n ) T ime rate + u n . η -H (u.∇ H u)φ n dz Horizontal advection + u n . η -H w ∂ ∂z uφ n dz V ertical advection = (2.31) -u n . η -H (∇ H p)φ n dz P ressure gradient + u n .(hV n ) V ertical mixing + u n .(hD n ) N umerical dissipation + u n .(hS n ) F orcing

M KE n equation

The M KE n equation is obtained by taking the inner product of the mean horizontal velocities for a vertical mode n (u n ) with the mean horizontal momentum equations for a vertical mode n.

u n .(h( ∂ ∂t u) n ) T ime rate + u n . η -H (u.∇ H u)φ n dz Horizontal advection + u n . η -H w ∂ ∂z uφ n dz V ertical advection + u n .(f k ∧ hu n ) Coriolis = (2.32) -u n . η -H (∇ H p)φ n dz P ressure gradient + u n .(hV n ) V ertical mixing + u n (h.D n ) N umerical dissipation + u n (h.S n ) F orcing
Partie , Chapter 2 -Methods : Presentation of the Approaches

EKE n equation

The EKE n equation is obtained by subtracting M KE n (2.32) to KE n (2.31).

u n .(h( ∂ ∂t u) n ) -u n .(h( ∂ ∂t u) n ) + u n . η -H (u.∇ H u)φ n dz -u n . η -H (u.∇ H u)φ n dz + u n . η -H (w ∂ ∂z u)φ n dz -u n . η -H (w ∂ ∂z u)φ n dz -u n .(f k ∧ hu n ) = (2.33) -( u n . η -H (∇ H p)φ n dz -u n . η -H (∇ H p)φ n dz ) + u n .(hV n ) -u n .(hV n ) + u n .(hD n ) -u n (h.D n ) + u n .(hS n ) -u n (h.S n )
Which simplifies into :

u ′ n .(h( ∂ ∂t u) n ) T ime rate + u ′ n . η -H (u.∇ H u)φ n dz Horizontal advection + u ′ n . η -H (w ∂ ∂z u)φ n dz V ertical advection -u n .(f k ∧ hu n ) Coriolis = (2.34) -u ′ n . η -H (∇ H p)φ n dz P ressure gradient + u ′ n .(hV n ) V ertical mixing + u ′ n .(hD n ) N umerical dissipation + u ′ n .(hS n ) F orcing
The horizontal and vertical advections and the pressure gradient modal terms are further decomposed in the following :

• Horizontal advection u ′ n . η -H (u.∇ H u)φ n dz = u ′ n . η -H ( ∞ m=0 u m φ m ).∇ H ( ∞ l=0 u l φ l )φ n dz = ∞ m=0 ∞ l=0 u ′ n .(u m .∇ H u l ) η -H φ n φ m φ l dz Horizontal advection 1 (2.35) + ∞ m=0 ∞ l=0 u ′ n .(u l u m ). η -H φ n φ m ∇ H φ l dz
Horizontal advection 2 52

Mean modal Eddy Kinetic Energy (EKE n ) equation

Here, the modal expansion of the variable u (2.13) has been used twice. The vertical integration η -H . dz has been simplified to η -H . dz, because the vertical modes are determined from the mean Brunt-Väisälä frequency (N 2 ).

The two contributions of the horizontal advection (2.35) are further decomposed as follows :

• Horizontal advection 1 ∞ m=0 ∞ l=0 u ′ n .(u m .∇ H u l ) η -H φ n φ m φ l dz = u ′ n .(u n .∇ H )u n η -H φ n φ n φ n dz + ∞ m =n ∞ l =n u ′ n .(u m .∇ H )u l η -H φ n φ m φ l dz = u ′ n .(u ′ n .∇ H )u n η -H φ n φ n φ n dz (2.36) + ∇ H .((u n + u ′ n ) 1 2 ||u ′ n || 2 η -H φ n φ n φ n dz) - 1 2 ||u ′ n || 2 ∇ H .(u n η -H φ n φ n φ n dz) + ∞ m =n ∞ l =n u ′ n .(u ′ m .∇ H )u ′ l η -H φ n φ m φ l dz + ∞ m =n ∞ l =n [u ′ n .(u m .∇ H )u ′ l + u ′ n .(u ′ m .∇ H )u l ] η -H φ n φ m φ l dz • Horizontal advection 2 ∞ m=0 ∞ l=0 u ′ n .u l u m . η -H φ n φ m ∇ H φ l dz = ∞ m=0 ∞ l=0 u ′ n .u ′ l u ′ m . η -H φ n φ m ∇ H φ l dz (2.37) + ∞ m=0 ∞ l=0 u ′ n .(u ′ l u m + u l u ′ m ). η -H φ n φ m ∇ H φ l dz • Vertical advection u ′ n . η -H (w ∂ ∂z u)φ n dz = u ′ n . η -H ( ∞ m=0 w m Φ m ) ∂ ∂z ( ∞ l=0 u l φ l )φ n dz = ∞ m=0 ∞ l=0 u ′ n .(u l w m ) η -H φ n Φ m ∂ ∂z φ l dz = - ∞ m=0 ∞ l=0 u ′ n .(u l w m ) η -H N 2 c 2 n φ n Φ m Φ l dz = -u ′ n .(u n w ′ n ) η -H N 2 c 2 n φ n Φ n Φ n dz (2.38) -w n ||u ′ n || 2 η -H N 2 c 2 n φ n Φ n Φ n dz - m =n l =n u ′ n .(u l w m ) η -H N 2 c 2 n φ n Φ m Φ l dz
Here, the modal expansion of the variables u (2.13) and w (2.14) have been used. The relation between the two vertical modes basis, given by the continuity equation (2.29), and the Sturm-Liouville equation for the vertical modes Φ n (2.15) have been used. As previously, the vertical integration η -H . dz has been simplified to η -H . dz, because the vertical modes are determined from the mean Brunt-Väisälä frequency (N 2 in 2.18).

• Pressure gradient

-u ′ n . η -H (∇ H p)φ n dz = -u ′ n . η -H (∇ H p ′ )φ n dz = -u ′ n . η -H ∇ H (p ′ φ n ) dz + u ′ n . η -H p ′ ∇ H φ n dz = -u ′ n .( ∇ H η -H p ′ φ n dz -∇ H η(p ′ φ n ) z=η -∇ H H(p ′ φ n ) z=-H ) + ∞ m=0 (u ′ n p ′ m ). η -H φ m ∇ H φ n dz = -u ′ n .∇ H (hp ′ n ) -u ′ n .∇ H η(p ′ φ n ) z=η -u ′ n .∇ H H(p ′ φ n ) z=-H + ∞ m=0 (u ′ n p ′ m ). η -H ∇ H (φ m φ n ) dz - ∞ m=0 (u ′ n p ′ m ). η -H φ n ∇ H φ m dz = -u ′ n .∇ H (hp ′ n ) -u ′ n .∇ H η(p ′ φ n ) z=η -u ′ n .∇ H H(p ′ φ n ) z=-H + ∞ m=0 (u ′ n p ′ m ).∇ H η -H φ m φ n dz + ∞ m=0 (u ′ n p ′ m ).∇ H η(φ m φ n ) z=η + ∞ m=0 (u ′ n p ′ m ).∇ H H(φ m φ n ) z=-H - ∞ m=0 (u ′ n p ′ m ). η -H φ n ∇ H φ m dz = -u ′ n .∇ H (hp ′ n ) + u ′ n .(p ′ n ∇ H h) - ∞ m=0 (u ′ n p ′ m ). η -H φ n ∇ H φ m dz = -u ′ n .(h∇ H p ′ n ) - ∞ m=0 (u ′ n p ′ m ). η -H φ n ∇ H φ m dz = -∇ H .(hu ′ n p ′ n ) + p ′ n ∇ H .(hu ′ n ) - ∞ m=0 (u ′ n p ′ m ). η -H φ n ∇ H φ m dz (2.39)
Here, the Leibniz's rule, the modal expansion of the variable p ′ (2.13) and the orthogonality condition of the vertical modes φ n (2.21) have been used. The vertical integration η -H . dz has been simplified to η -H . dz because the vertical modes are determined from the mean Brunt-Väisälä frequency (N 2 ). The contributions of the pressure gradient (2.39) can be further developed by replacing the second term with the continuity equation for a vertical mode n (2.29) :

(2.39) = -∇ H .(hu ′ n p ′ n ) + ∞ m=0 (u ′ m p ′ n ). η -H φ m ∇ H φ n dz - ∞ m=0 (u ′ n p ′ m ). η -H φ n ∇ H φ m dz -hp ′ n w ′ n -(∇ H ηp ′ n .(u ′ φ n ) z=η )
Partie , Chapter 2 -Methods : Presentation of the Approaches

The final form of the pressure gradient contributions (2.39) is given after replacing the fourth term with the hydrostatic equation for a vertical mode n (2.30) :

(2.39) = -∇ H .(hu ′ n p ′ n ) + ∞ m=0 (u ′ m p ′ n ). η -H φ m ∇ H φ n dz - ∞ m=0 (u ′ n p ′ m ). η -H φ n ∇ H φ m dz + c 2 n hb ′ n w ′ n -gw ′ n ( b ′ N 2 Φ n ) z=η -(∇ H ηp ′ n .(u ′ φ n ) z=η )
In the EKE n equation final expression, the term (

∇ H ηp ′ n .(u ′ φ n ) z=η ) is distributed in the Forcing (u ′ n .(hS n ))
. This term has a mathematical (2.29) and a numerical significance, but it only represents 8 % of the smallest contribution (the numerical dissipation : u ′ n .(hD n )) to the EKE n budget. Then it does not represent a significant physical contribution to the EKE n budget.

The final expression of the EKE n equation is obtained by replacing the different contributions of the horizontal (2.36 and 2.37) and vertical advection (2.38) and of the pressure gradient terms (2.39) into the former expression of the EKE n equation (2.34). The terms of the final expression of the EKE n equation are organised into the 4 following categories :

A EKEn Evolution = B
T ransf ers between the EKEn, M KEn and EP En reservoirs

+ C

T ransf ers between vertical modes

+ D

EKEn generation and dissipation

(2.40) leading to the following final expression of the EKE n equation : 56

Mean modal Eddy Kinetic Energy (EKE

n ) equation u ′ n .(h( ∂ ∂t u) n ) T ime rate (A.1)
(2.41)

+ ∇ H .(hu ′ n p ′ n ) Eddy-pressure work (A.2) + ∇ H .[(u n + u ′ n ) 1 2 ||u ′ n || 2 η -H φ n φ n φ n dz] -(w n + w ′ n )||u ′ n || 2 η -H N 2 c 2 n φ n Φ n Φ n dz Advection of EKEn (A.3)
Divergences of the eddy-energy f luxes (A.2+A.1)

= -u ′ n .(u ′ n .∇ H )u n η -H φ n φ n φ n dz M KEn→EKEn (HRSn) (B.1) + u ′ n .(u n w ′ n ) η -H N 2 c 2 n φ n Φ n Φ n dz M KEn→EKEn (V RSn) (B.2) + c 2 n hb ′ n w ′ n -gw ′ n ( b ′ N 2 Φ n ) z=η EP En→EKEn (V BFn) (B.3) - ∞ m =n ∞ l =n [ u ′ n .(u ′ m .∇ H )u ′ l Eddy-eddy interactions + u ′ n .(u m .∇ H )u ′ l + u ′ n .(u ′ m .∇ H )u l Eddy-mean interactions ] η -H φ n φ m φ l dz Horizontal triadic interactions (C.1.1) + ∞ m =n ∞ l =n u ′ n .(u l w m ) η -H N 2 c 2 n φ n Φ m Φ l dz V ertical triadic interactions (C.1.2) - ∞ m=0 ∞ l=0 u ′ n .u ′ l u ′ m . η -H φ n φ m ∇ H φ l dz Advective eddy-eddy interactions on slopes (C.2) - ∞ m=0 ∞ l=0 u ′ n .(u ′ l u m + u l u ′ m ). η -H φ n φ m ∇ H φ l dz
Advective eddy-mean interactions on slopes (C.3)

+ ∞ m=0 (u ′ m p ′ n ). η -H φ m ∇ H φ n dz - ∞ m=0 (u ′ n p ′ m ). η -H
φ n ∇ H φ m dz P ressure-gradient eddy-eddy interactions on slopes (C.4)

+ p ′ n ∇ H η.(u ′ φ n ) z=η Contribution of the f ree-surf ace <<1 (C.5) + 1 2 ||u ′ n || 2 ∇ H .(u n η -H φ n φ n φ n dz)
Complement to the advection of EKEn (C.6)

+ u ′ n .(hV n ) V ertical mixing (D.1) + u ′ n .(hD n ) N umerical dissipation (D.2) + u ′ n .(hS n ) F orcings (D.3) + u n .(f k ∧ hu n ) Coriolis (D.4)
Similarly to the EKE equation (2.12), the equation (2.41) l.h.s. represents the EKE n evolution (category A) and the r.h.s. gathers the local EKE n sources and sinks (category B, C and D). The contributions of the different terms to the EKE n budget (equation 2.41) and the related physical processes are further detailed in the following : -Divergences of the eddy-energy fluxes (A.2 + A.3): EKE n transport which can be interpreted as the EKE n fluxes at the domain boundaries when integrated over a closed volume. It has two contributions : the eddy-pressure work, which represents the energy transport done by the linear components of the dynamics, and the advection of EKE n by the total 3D flow, which represents the energy transport done by the non-linear component of the dynamics. In the frame of linear theories of internal waves [START_REF] Kelly | Internal-tide energy over topography[END_REF][START_REF] Kelly | The cascade of tidal energy from low to high modes on a continental slope[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF], Rossby waves [START_REF] Masuda | Group velocity and energy transport by Rossby waves[END_REF] and of open-ocean dynamics [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] the eddy-pressure work is the only contribution to the EKE n transport.

(B) Transfers between the EKE n , M KE n and EP E n reservoirs :

-M KE n → EKE n (HRS n ) term (B.1): Horizontal Reynolds Stress-mean flow non-linear interactions within a vertical mode n. It is the counterpart of the Horizontal Reynolds Stress (HRS), which is used as an indicator of horizontal shear instabilities such as the barotropic (Gula et al., 2015a,b) and the centrifugal/inertial (Gula et al., 2016b) instabilities.

-

M KE n → EKE n (V RS n ) term (B.
2): Vertical Reynolds stress-mean flow non-linear interactions within a vertical mode n. It is the counterpart of the Vertical Reynolds Stress (V RS), which is used as an indicator of vertical-shear related instabilities such as Kelvin-Helmholtz or symmetric instability [START_REF] Thomas | Symmetric instability in the Gulf Stream[END_REF].

-

EP E n → EKE n (V BF n ) term (B.
3): Vertical Buoyancy Fluxes reflecting eddy energy conversion between the potential and kinetic reservoirs within a vertical mode n. It is the counterpart of the V BF term, which is used as an indicator of the baroclinic instability at meso and submesoscale [START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF].

(C) Transfers between vertical modes :

-Horizontal triadic interactions (C.1.1) : Eddy-eddy and eddy-mean non-linear interactions between vertical modes. The intermodal coupling is represented by the triadic coefficient : η -H φ n φ m φ l dz and corresponds to the following vertical modes combinations : (n = m = l), (n=l = m), (n = m = l) and (n = m = l). These triadic interactions are the counterpart of the horizontal component of the advective operator which transfers KE across scales and realizes the turbulent energy cascades [START_REF] Salmon | Baroclinic instability and geostrophic turbulence[END_REF][START_REF] Vallis | Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation[END_REF]. The eddy-eddy interactions represent the turbulent energy cascades and the eddy-mean interactions represent unforced energy transfers across scales and reservoirs.

-Vertical triadic interactions (C.1.2) : Eddy-total flow non-linear interactions between vertical modes. The intermodal coupling is represented by the triadic coefficient : -Advective eddy-eddy flow interactions on slopes (C.2) : Eddy-eddy non-linear interactions between vertical modes associated with stratification-topography horizontal gradients. The intermodal coupling is represented by : η -H φ n φ m ∇ H φ l dz and corresponds to the vertical modes combinations : (n = m = l), (n=l = m), (n = m = l) and (n = m = l). This term points toward eddy-eddy and eddy-waves non-linear interactions forced by the stratification-topography horizontal gradient. The (n = m = l) case does not represent an energy transfer between reservoirs or vertical structures and likely corresponds to EKE n advection. We will show that this case does not represent a main contribution to the EKE 0-1 budget (Figure 4.8 in Chapter 3).

η -H N 2 c 2 n φ n Φ m Φ l dz
-Advective eddy-mean interactions on slopes (C.3) : Eddy-mean non-linear interactions between vertical modes associated with stratification-topography horizontal gradients. The intermodal coupling is represented by : η -H φ n φ m ∇ H φ l dz and corresponds to the vertical modes combinations : (n = m = l), (n=l = m), (n = m = l) and (n = m = l). This term points toward eddy-mean non-linear interactions forced by the stratification-topography horizontal gradients. These interactions are mainly interpreted as eddies driving mean currents over sloping topography [START_REF] Holloway | Systematic forcing of large-scale geophysical flows by eddy-topography interaction[END_REF][START_REF] Adcock | Interactions between geostrophic eddies and the mean circulation over large-scale bottom topography[END_REF] if the vertical modes combination (n = m = l) has an equal contribution as the other ones to the energy transfer. Indeed, bottom-intensified flows decompose into a barotropic mode, allowing a nonzero bottom velocity, and into baroclinic modes, to cancel or to shape velocity in the upper layers, as tested by [START_REF] Stanley | Vertical structure of ocean mesoscale eddies with implications for parameterizations of tracer transport[END_REF].

-Pressure-gradient eddy-eddy interactions on slopes (C.4) : Eddy-eddy linear interactions between vertical modes associated with stratification-topography horizontal gradients. The intermodal coupling is represented by : η -H φ m ∇ H φ n dz and η -H φ n ∇ H φ m dz and corresponds to the vertical modes combination : (n = m). This term points toward eddy-eddy and eddy-waves linear interactions forced by the stratification-topography horizontal gradients. These interactions correspond to eddies inducing Topographic Rossby Waves on a sloping bottom and to eddies scattering over seamounts, topographic features, accidents and slopes leading to the generation of internal-gravity waves -such as lee waves - [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF][START_REF] Nikurashin | Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory[END_REF]. This latter process is similar to the tidal scattering leading to internal tides generation [START_REF] Kelly | Internal-tide energy over topography[END_REF][START_REF] Kelly | The cascade of tidal energy from low to high modes on a continental slope[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF].

-Contribution of the free-surface « 1 (C.5) : Eddy-eddy interactions between vertical modes at the free-surface. This term has a mathematical origin (Eq. 2.29), but we will show that it has a negligible effect on our EKE 0-1 budget (Figure 4.8 in Chapter 3).

-Complement to the advection of EKE n (C.6) : Term derived from the horizontal advection contribution, which allows to write the horizontal advection of EKE n as a divergence of eddy-energy flux (Eq. 4.25). This contribution does not cancel out in the framework of the vertical modes, contrary to the EKE budget where the continuity equation is used to cancel out a similar contribution [START_REF] Harrison | Energy analysis of open regions of turbulent flows-Mean eddy energetics of a numerical ocean circulation experiment[END_REF].

(D) EKE n generation and dissipation :

-Vertical mixing (D.1) : EKE n dissipation and/or generation by wind stress, interior vertical mixing and bottom drag. The wind stress dissipates surface energy regardless of the oceanic structures [START_REF] Hughes | Wind work on the geostrophic ocean circulation: An observational study of the effect of small scales in the wind stress[END_REF][START_REF] Renault | Dampening of submesoscale currents by air-sea stress coupling in the Californian upwelling system[END_REF] and the bottom drag dissipates barotropic [START_REF] Fu | Nonlinear energy and enstrophy transfers in a realistically stratified ocean[END_REF][START_REF] Sen | Global energy dissipation rate of deep-ocean low-frequency flows by quadratic bottom boundary layer drag: Computations from current-meter data[END_REF] and non-null baroclinic EKE.

-Numerical dissipation (D.2) : EKE n dissipation by the horizontal implicit viscosity of the model.

-Forcings (D.3) : EKE n dissipation and generation (due to explicit horizontal dissipation and nudging) in the grid sponge layers close to the open boundaries.

-Coriolis term (D.4) : EKE n dissipation and/or generation by the Coriolis term. This contribution does not cancel out in the framework of the vertical modes, contrary to the EKE budget [START_REF] Harrison | Energy analysis of open regions of turbulent flows-Mean eddy energetics of a numerical ocean circulation experiment[END_REF].

EKE n budget in the dx ∼ 2.5 km numerical simulation

The diagnostics of the EKE n equation (2.41) terms in a regional numerical simulation is presented in this subsection.

The dynamical (u, w, η) and state (T , S) variables and the online momentum terms (Time rate, Advection, Coriolis, Pressure-gradient, Vertical mixing, Numerical dissipation and Forcing gathering the horizontal mixing and the nudging in the sponge layers) are recorded as daily time averages over the simulation period (1994 -1995). Although this choice has the disadvantage of filtering out the dynamics having a shorter period than the day (internal waves and some submesoscale processes), it is a good compromise between the simulation of the mesoscale eddy dynamics at relatively high resolution (dx ∼ 2.5 km) and the storage of the numerical outputs.

Firstly, the vertical modes are computed. The barotropic mode and the first 9 baroclinic modes are defined from the Brunt-Väisälä frequency (N 2 ), diagnosed offline from the T and S numerical outputs, and time averaged over the 1994-1999 period (Figure 2.4). The accuracy of the modal decomposition of the modeled EKE using the 10 first vertical modes is evaluated in Appendix A. Secondly, the EKE n equation (2.41) terms are diagnosed. To do so, the terms of the KE n (2.31) and M KE n (2.32) equations 60 -1000 For the KE n terms, the online momentum diagnostics and the horizontal velocity vector (u) are projected onto the vertical modes, separately, then their inner product is computed. The KE n terms are time averaged by repeating this procedure for each numerical outputs. The budget closure is evaluated by the root mean square of the residual, defined as the difference between the Time rate (Figure 2.5 top left panel) and the sum of the rest of the terms (Figure 2.5 top right, middle and bottom panels), normalized by the root mean square of the Time rate and converted into percentages [%]. The KE n budget closes with a residual of 1.1 10 -4 % (Figure 2.5) and the accuracy of the modeled KE n terms modal decomposition onto the 10 first vertical modes is evaluated in Appendix B. It has to be noted that due to the discretization of the u and f variables on the Arakawa-C grid, the Coriolis contribution is not null pointwise, but its contribution does not play a significant role in the EKE n budget.

Mean modal Eddy Kinetic Energy (EKE

0 0 x10 3 [m] 1 x10 3 [m] 2 x10 3 [m] 3 x10 3 [m] 4 x10 3 [m] -4 -2 0 2 4 -4000 -3000 -2000 -1000 0 5 x10 3 [m] -4 -2 0 2 4 6 x10 3 [m] -4 -2 0 2 4 7 x10 3 [m] -4 -2 0 2 4 8 x10 3 [m] -4 -2 0 2 4 9 x10 3 [m]
For the M KE n terms, the inner product is taken between the mean online momentum diagnostics projected onto the vertical modes and the mean horizontal velocity vector projected onto vertical modes (u n ). The M KE n budget closes with a residual of 1.2 10 -4 % (Figure 2.6) and the accuracy of the modeled M KE n terms modal decomposition onto the 10 first vertical modes is evaluated in Appendix B.

The EKE n terms (2.41) are obtained from the substraction between the KE n (2.31) and the M KE n (2.32) terms. The EKE n budget also closes with a residual of 9.2 10 -5 % (Figure 2.7) and the accuracy of the modeled EKE n terms modal decomposition onto the 10 first vertical modes is evaluated in Appendix B.

The procedure for getting the EKE n terms (2.34) is straightforward for the Time rate, Vertical mixing, Numerical dissipation, Coriolis and the Forcing (Horizontal mixing and Nudging in the sponge layers), but not for the horizontal advection (2.36, 2.37), the vertical advection (2.38) and the pressure gradient (2.11) terms, because all three need to be decomposed into several contributions. As a first step, each contribution is explicitly diagnosed offline from the numerical outputs. The u and w variables are recorded as numerical outputs, but the seawater density (ρ) has to be diagnosed offline. Density is computed, in the exact same way as in the model, from the T and S numerical outputs. The buoyancy (b) and pressure (p) are deduced from this offline seawater density. The spatial derivatives are discretized as first order finite differences. At first order, the advection and pressure gradient terms diagnosed explicitly offline, plotted as the sum of their respective contributions (Figure 2.8), show comparable patterns and magnitudes with the advection and pressure gradient terms diagnosed from the online momentum diagnostics (Figure 2.7). This first order agreement allows us to be confident in the analytical development of their respective contributions (2.36, 2.37), 2.38 and 2.39). However, the numerical artefacts generated by the derivatives discretization and by the time sampling differences between the online momentum diagnostics and the offline explicit computations, using the numerical outputs are not exactly zero (residuals of 16 % and 5 % between the online and offline diagnostics of the advection and pressure gradient terms respectively). As a second step, the decomposition of the advection and pressure gradient contributions is performed using the offline explicit diagnostics combined with online momentum diagnostics. In order to limit the computation of spatial derivatives, the HRS n and V RS n terms (B. 3) is defined as the residual between these offline terms and the online advection diagnostic. In the same way, the V BF n term (B.3), the pressure-gradient eddy-eddy interactions on slopes (C.4) and the contribution of the free-surface (C.5) are explicitly computed offline and the eddy-pressure work (A.2) is defined as the residual between these offline terms and the online pressure gradient diagnostic. This method, for diagnosing the different contributions of the advection and of the pressure gradient to the EKE n budget (2.41), allows to close the EKE n budget (Figure 2.9). A more rigorous method could be to implement the online diagnostic of the advection and pressure gradient contributions in the CROCO model. To do so, the vertical modes, derived from the mean modeled stratification, could be given as a model input in the same way as the vertical grid, the initial and restart files and as the forcings. -3 ] for the 1 st baroclinic mode (n = 1) and computed over the 1994-1999 period. The overbar denotes a time average over the simulation period (1994)(1995)(1996)(1997)(1998)(1999) and the prime denotes fluctuations relative to the time average. The top panels show the advection and pressure gradient terms diagnosed from the online momentum diagnostics (2.9), the middle panels show the sum of the advection (2.36 and 2.37) and pressure gradient (2.39) contributions explicitly diagnosed offline and the bottom panels show the residual between the online (top panels) and offline (middle panels) diagnostics. The black contours denote the 1000 and 3000 m isobaths. The residuals (16 % and 5 % for the advection and pressure gradient terms, respectively) originate from the time sampling and from the spatial derivative discretization which both differ from the online and the offline diagnostics. 2.41), for the 1 st baroclinic mode (n = 1) and computed over the 1994-1999 period. The overbar denotes a time average (over the 1994-1999 period) and the prime denotes fluctuations relative to the time average. All the terms are diagnosed from the online momentum diagnostics, except the advection and the pressure gradient terms. The advection and pressure gradient terms are the sum of their respective contributions which are explicitly diagnosed offline, except one contribution which is defined as a residual using the online advection and pressure gradient diagnostics. The black contours denote the 1000 and 3000 m isobaths.

′ n . η -H (u.∇ H u + w ∂ ∂z u)φ n dz) and pressure gradient (u ′ n . η -H (∇ H p)φ n dz) terms [m 3 s
Chapter 3

STUDY I : GENERATION OF FRONTAL SUBMESOSCALE EDDIES IN THE AGULHAS CURRENT

In this Chapter, I present the study addressing the first question of this dissertation :

Question I : What drives the submesoscale variability at the transition between the Northern and Southern branches of the Agulhas Current ?

This Chapter corresponds to the scientific publication of [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF].

Key points

• Submesoscale frontal eddies are generated due to barotropic instability in the Agulhas Current

• The background strain provides favorable conditions for instability generation

• A new regime of variability is highlighted in addition to the two modes described by [START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]

Abstract

This study addresses the dynamics of the Agulhas inshore front in the submesoscale range upstream of 26°E. Submesoscale frontal eddies are observed in the vicinity of Port Elizabeth (26°E) from satellite images and in observations collected from under-water gliders. Using a submesoscale-resolving numerical model (dx ∼ 0.75 km) we are able to simulate similar submesoscale eddies. Barotropic instability is confirmed as the generation mechanism by a 1D linear stability analysis and an eddy kinetic energy budget. Kinetic energy is transferred from the mean flow to the eddies through the mean horizontal shear, which is a signature of barotropic instability. When the Agulhas Current is in a non-meandering state, submesoscale eddy generation is a recurrent process which locally drives the front's variability. Along the front, the spatial variability of barotropic instability is shaped by the background strain. A large strain aligned with the frontal axis intensifies the frontal shear upstream of 28°E while a weakening of the strain allows for barotropic instability to be triggered downstream. Although an intermittent process, the barotropic instability shows a dominant period of variability comparable with the variability of the Agulhas Current and Undercurrent.

Introduction

The Agulhas Current is the most intense western boundary current of the South Hemisphere with a mean transport of ∼ 80 Sv [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF]. It flows poleward along the South African coastline between 32°E and 19°E [START_REF] Lutjeharms | The Agulhas Current[END_REF] and forms part of the South Indian Ocean subtropical gyre (Figure 3.1). The Agulhas Current finds its origin in the Mozambique Channel and from south east Madagascar. Between about 19-20°E, the Agulhas Current retroflects and flows eastward back into the south Indian Ocean as the Agulhas Return Current. Large anticyclones, named Agulhas Rings, which are shed at the Agulhas Retroflection transport warm and salty waters into the South Atlantic Ocean [START_REF] Lutjeharms | Shedding of an Agulhas ring observed at sea[END_REF]. This heat and salt flux, named the Agulhas leakage, is thought to be a key component of the upper branch of the global overturning circulation [START_REF] Weijer | Response of the Atlantic Overturning Circulation to South Atlantic sources of buoyancy[END_REF][START_REF] Peeters | Vigorous exchange between the Indian and Atlantic oceans at the end of the past five glacial periods[END_REF][START_REF] Beal | On the role of the Agulhas system in ocean circulation and climate[END_REF]. Upstream of Port Elizabeth (26°E), the jet is narrow and flows close to a steep and straight topography. The Agulhas Current in this region is occasionally disturbed by the passage of Natal Pulses (Figure 3.1), solitary meanders with diameters of 50-200 km developing near the Natal Bight (∼ 31°E, Figure 3.1) 4-5 times a year and traveling downstream [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Schouten | Upstream control of Agulhas Ring shedding[END_REF]. Due to dissipation and/or merging processes, only a fraction of Natal Pulses (around 1.7 per year) reaches the eastern margin of the Agulhas Bank (∼ 24°E, Figure 3.1) [START_REF] Rouault | New perspectives on Natal Pulses from satellite observations[END_REF][START_REF] Krug | Satellite observations of an annual cycle in the Agulhas Current[END_REF][START_REF] Krug | Interactions between the Agulhas Current and the eastern margin of the Agulhas Bank[END_REF]. Downstream of Port Elizabeth (26°E), the current becomes increasingly unstable. In addition to Natal Pulses, frontal eddies [START_REF] Schumann | Observations of Agulhas Current frontal features south of Africa, October 1983[END_REF], also called shear-edge eddies [START_REF] Lutjeharms | Eddies and other boundary phenomena of the Agulhas Current[END_REF], with a diameter of 50-100 km, develop along the eastern margin of the Agulhas Bank (Figure 3.1). Further downstream, cyclonic eddies are generated from the detachment of the Agulhas Current at the tip of the Agulhas Bank [START_REF] Penven | Generation of cyclonic eddies by the Agulhas Current in the Lee of the Agulhas Bank[END_REF]. All these meanders and eddies are associated with features such as filaments and plumes which can intrude onto the continental shelf [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Krug | Interactions between the Agulhas Current and the eastern margin of the Agulhas Bank[END_REF]. The Agulhas retroflection (20-19°E, Figure 3.1) has a high level of eddy variability due to the spawning of Agulhas Rings [START_REF] Lutjeharms | Shedding of an Agulhas ring observed at sea[END_REF].

The first baroclinic Rossby radius of deformation (Rd) is O(20-40) km in the region [START_REF] Chelton | Geographical variability of the first baroclinic Rossby radius of deformation[END_REF]. As the features described above are typically larger than Rd [START_REF] Lutjeharms | The Agulhas Current[END_REF]; [START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF] historical descriptions of the Agulhas Current dynamics accounts for the mesoscale range. However, the Shelf Agulhas Glider Experiment (SAGE) [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF] provides recent observations of submesoscale instabilities (L < Rd) in the region. In April 2015, two Seagliders sampled submesoscale frontal cyclonic eddies, with diameters ∼ 15 km, offshore Port Elizabeth (Figure 3.2a). Features similar to the SAGE's eddies are regularly visible on maps of Sea Surface Temperature (SST). Figure 3.2b shows submesoscale meanders along a portion of the cyclonic inshore front (28.5-26°E), just upstream of Port Elizabeth.

Observations of these frontal submesoscale eddies give a new insight on the historical description of the Agulhas Current dynamics [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]. They show features at scales smaller that the previously documented at a location where the current was reported "stable". The investigation of these submesoscale features should give a more comprehensive understanding of the Agulhas Current frontal dynamics. As a consequence of, developing at the Agulhas inshore front, these submesoscale eddies can influence the shelf dynamics and variability. Plumes, filaments and eddy shedding affect the shelf by driving horizontal water masses transport [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Krug | Interactions between the Agulhas Current and the eastern margin of the Agulhas Bank[END_REF] and so might do the frontal submesoscale eddies. In addition, the ageostrophic components of the submesoscale frontal flow lead to large vertical velocities and enhanced vertical exchanges, which can impact biogeochemistry and the primary production of the upper ocean [START_REF] Mahadevan | Ocean science: Eddy effects on biogeochemistry[END_REF]. Similar examples of submesoscale frontal cyclonic eddies are reported for two other western boundary currents. In the Gulf Stream, eddies with diameters of 10-15 km were observed on satellite SST images and modeled using a submesoscale-resolving simulation (Gula et al., 2015b). In the Eastern Australian Current, eddies with diameters of 10 km were observed with coastal HF radars and satellites [START_REF] Schaeffer | Characterizing frontal eddies along the East Australian Current from HF radar observations[END_REF]. In addition, frontal submesoscale eddies may also be of interest for the ocean energy budget as they can trigger an energy route from the large-scale geostrophic circulation toward smaller scales [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF].

In this study, we address the dynamics of the Agulhas inshore front around Port Elizabeth (26°E) using a submesoscale-resolving numerical model. We aim to characterize the frontal submesoscale eddy generation mechanism, its spatial and time variability and the factors involved in their triggering. In section 2, we present the numerical simulations. In section 3, we compare the modeled mean dynamics of the Agulhas Current with in situ and satellite observations. In section 4, we investigate the generation mechanism of the submesoscale eddies in the model. Finally, in section 5 we discuss the spatial and time variability of the submesoscale eddy generation mechanism.

Simulation setup

The realistic simulations are run with the CROCO (Coastal and Regional Ocean COmmunity model) model [START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF], which is based on ROMS [START_REF] Shchepetkin | The Regional Oceanic Modeling System (ROMS): A splitexplicit, free-surface, topography-following-coordinate ocean model[END_REF]. This model solves the hydrostatic primitive equations for the momentum and the state variables and uses topographic following vertical coordinates. To set up a submesoscale eddy-resolving simulation, we use a nesting approach. Four nests with respective horizontal resolutions of dx ∼ 22.5, 7.5 2.5 and 0.75 km are embedded (Figure 3.3). The highest-resolution grid (dx ∼ 0.75 km) has a sufficient effective horizontal resolution [START_REF] Soufflet | On effective resolution in ocean models[END_REF] to solve the submesoscale range (< O(30)km) in the Agulhas region. The three first nests (dx ∼ 22, 7.5 and 2.5 km) are set up with the same nesting procedure, forcings and vertical grid. The nesting procedure is a two-way online nesting based on AGRIF [START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF]. At each time step each "child" grid is forced by its "parent" grid and feedbacks from the "child" grids (finer scales) to larger grids (larger scales) are taken into account. The GLORYS oceanic reanalysis [START_REF] Ferry | Glorys2v1 global ocean reanalysis of the altimetric era (1992-2009) at meso scale[END_REF] provides the lateral boundary conditions for the largest grid (dx ∼ 22 km). The surface forcings are provided by a bulk formulation [START_REF] Fairall | Bulk parameterization of air-sea fluxes for tropical ocean-global atmosphere coupled-ocean atmosphere response experiment[END_REF] using the daily ERA-ECMWF's reanalysis [START_REF] Dee | The ERA-Interim reanalysis: Configuration and performance of the data assimilation system[END_REF]. Relative winds are preferred to absolute winds due to their implications in a more realistic dissipation of mesoscale eddies in the Agulhas Current [START_REF] Renault | Modulation of the Agulhas Current Retroflection and Leakage by Oceanic Current Interaction with the Atmosphere in Coupled Simulations[END_REF]. The bathymetry is built from the General Bathymetric Chart of the Oceans 30-sec resolution (Gebco 2014). The topography is locally smoothed when steepness exceeds the threshold r max = 0.2, with r max = ∇h h the slope parameter defined by [START_REF] Beckmann | Numerical simulation of flow around a tall isolated seamount. Part I: Problem formulation and model accuracy[END_REF]. The vertical grids are designed following the [START_REF] Haidvogel | Numerical ocean circulation modeling[END_REF] method. Parameters driving the levels refinement at the surface and the bottom are respectively θ s = 5, θ b = 0. The transition depth between flat-z levels and terrain following sigma level is set to h cline = 10 m.

The highest-resolution grid (dx ∼ 0.75 km) set up differs from the three first grids on several points. The nesting procedure is an offline one-way nesting as described by Gula et al. (2015a). In this case, the parent forces the child grid without feedback from the finer toward the larger scales. The surface forcing and the bathymetry are derived from the same datasets as for parent grids. The boundary conditions are provided by the nearest parent grid (dx ∼ 2.5 km) at a daily rate. The vertical resolution is increased relative to the horizontal resolution with 100 vertical levels. They are distributed following [START_REF] Lemarié | Are there inescapable issues prohibiting the use of terrain-following coordinates in climate models[END_REF] method with parameters θ s = 6, θ b = 4 and h cline = 300 m.

The horizontal advection of momentum is discretized using a 3 rd order upstream biased numerical scheme and horizontal advection of tracers is discretized using a split and rotated 3 rd order upstreambiased numerical scheme [START_REF] Marchesiello | Spurious diapycnal mixing in terrain-following coordinate models: The problem and a solution[END_REF]. The later scheme reduces spurious diapycnal mixing. On the vertical, the momentum advection scheme is semi-implicit, with an adjustment between explicit and implicit formulation according to the model stability.

A K-profile parameterization (KPP) [START_REF] Large | Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization[END_REF] parameterizes the vertical mixing of tracers and momentum. The adjusted KPP by [START_REF] Durski | Vertical mixing schemes in the coastal ocean: Comparison of the level 2.5 Mellor-Yamada scheme with an enhanced version of the K profile parameterization[END_REF] is used for the bottom boundary layer. Both parameterizations take into account the [START_REF] Lemarié | Are there inescapable issues prohibiting the use of terrain-following coordinates in climate models[END_REF]'s modification of the Richardson number criterion with the critical value (Ri cr = 0.15). The bottom viscous stress has a quadratic form.

The parent grid (dx ∼ 22.5 km has 450 x 258 x 60 points) covers most of the South Equatorial Indian Ocean and a part of the Atlantic Ocean. The first child grid (dx ∼ 7.5 km), with 578 x 542 x 60 points, encompasses the African continent from the Equator to the Antarctic Circumpolar Current. It includes the Mozambique Channel, where large anticyclones are generated [START_REF] Halo | Eddy properties in the Mozambique Channel: A comparison between observations and two numerical ocean circulation models[END_REF] and related to Natal Pulses development in the Agulhas Current [START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF]. Both nests are spun-up from their initial state for 3 years. The third child grid (dx ∼ 2.5 km), with 674 x 578 x 60 points, encompasses the Agulhas Current from upstream the Natal Bight (∼ 31°E) to the Agulhas retroflection region (∼ 19-20°E). This simulation is spun up for a year and run from 1993 to 2014. The highest-resolution grid (dx ∼ 0.75 km has 1352 x 652 x 100 points) is centered on the SAGE area (Port Elizabeth, 26 °E) and resolves more precisely the submesoscale dynamics of the current and their potential interactions with the topography. This grid is run from 01/09/1995 to 31/03/1997. This period is centered on the year 1996, during which no upstream Agulhas retroflection is observed or simulated.

Partie , Chapter 3 -Study I : Generation of Frontal Submesoscale Eddies in the Agulhas Current

Observed and modeled Agulhas dynamics

In this section, we evaluate the mean Agulhas Current dynamics. First, we compare the mean modeled surface Agulhas Current to satellite data and second, we compare the vertical and the surface structures of the mean modeled Agulhas inshore front, in the vicinity of our area of interest (Port Elizabeth, 26°E) with in situ data.

The mean Agulhas Current: surface description

The mean Eddy Kinetic Energy (EKE) is defined as EKE = 0.5(u ′2 + v ′2 ) with u and v the horizontal velocity components. The overbar denotes a time mean and the prime denotes fluctuations relative to this mean. We compare the surface EKE from the dx ∼ 2.5 km nest with the one derived from satellite altimetry observations of absolute dynamic topography gridded at 1/4°provided by AVISO (Figure 3.4). Both observed and modeled EKE are defined from geostrophic velocity anomalies diagnosed from a 10year long subset (1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003) of sea surface height anomaly.

The modeled and observed surface EKE patterns match the spatially "bi-modal" dynamics of the Agulhas Current. Upstream of Port Elizabeth (26°E), EKE is low (O(0.5-2) 10 3 s -2 ) and downstream it is more intense (O >(3.5) 10 3 cm 2 s -2 ). The Agulhas retroflection region, where Agulhas Rings and Agulhas Eddies are generated, has the largest values of EKE. The Agulhas Return Current EKE has moderate values (O(2-3) 10 3 cm 2 s -2 ) reflecting the two standing meanders found at the Agulhas Plateau (26°E) and at 32°E. The observed and the modeled EKE magnitudes differ upstream of 23°E. The difference is the largest at the Agulhas Bank eastern margin (25°E-23°E). The model shows an intense EKE, with values of the same order as at the Retroflection area, whereas AVISO shows a lower magnitude (O(1.5) 10 3 cm 2 s -2 ). This area is where Natal Pulses can be trapped and grow in size [START_REF] Krug | Interactions between the Agulhas Current and the eastern margin of the Agulhas Bank[END_REF] and where Shear-Edge eddies can develop [START_REF] Lutjeharms | Eddies and other boundary phenomena of the Agulhas Current[END_REF] and potentially leak downstream. The amplitude difference between the two datasets can be explained by their different spatial resolutions. AVISO spatial effective resolution is λ ∼ 100 -150 km (Chelton et al., 2011b) whereas the dx ∼ 2.5 km nest has an effective resolution of about λ ∼ 10dx ∼ 25 km [START_REF] Soufflet | On effective resolution in ocean models[END_REF]. Hence, scales in the range of 10 -70 km seem to drive the Agulhas Current variability at the Agulhas Bank eastern margin. The relationship between the relaxation of the topographic constraint on the current and the development of intense perturbations [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF] is visible in the modeled EKE. Areas of intense EKE are associated with the current path (sea surface height contours in green) separating from the topography (1000 m isobath is the most inshore black contour). The Agulhas Current mean vertical structure from the 1 year-long simulation of the highest-resolution nest (dx ∼ 0.75 km) is compared to the Agulhas Current Time-series Experiment (ACT) data (Figure 3.5). ACT is a 34 month-long time series (April 2010-February 2013) of current measurements undertaken from a mooring array across the Agulhas Current at 28.5°E [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF].

The modeled Agulhas Current has a vertical structure close to the one observed during the ACT experiment. They both have a similar width of 200 km, defined as the distance between the 0 m s -1 velocity contour (Figures 3.5a,c), in the upper ocean. In the model, the current extends slightly further offshore than in the observations and does not intrude significantly onto the continental shelf. This results in a clear delimitation between the shelf waters and the Agulhas Current. The modeled and observed currents have a similar vertical structure, with a v-shape typically associated with surface intensified flows. The mean Agulhas Current velocities reach values larger than -1.5 m s -1 at the surface and the current core, defined by the -0.5 m s -1 contour, extends over a width of about 100 km at the surface and is 800 m deep. These similarities in the mean Agulhas structure are reflected in the net volume flux (T box in [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF]) : -77 ± 5 Sv, for the ACT data and -79.6 Sv for the highest-resolution nest.

The modeled and observed Agulhas inshore fronts have high values of cyclonic relative vorticity which is uniform from the surface to about 500 m depth (Figures 3.5b and d). The modeled front is more intense with relative vorticity values up to O(3f) compared to the observations of O(0.5f). This magnitude difference could suggests different mean current structures. The modeled current does not spread onto the continental shelf and this results in a sharper front at the shelf break. However, the coarser horizontal resolution of the ACT data, which cannot resolve the sharp current gradients at the inshore front of the Agulhas Current, must also contribute to this difference.

The agreement between the highest-resolution nest and the ACT data also includes the Agulhas Undercurrent mean vertical structure. Below the Agulhas Current flows a weaker equatorward current : the Agulhas Undercurrent (Figures 3. 5a andc). Both modeled and observed deep currents show a vein of weak north-easterly flow between 1500 and 3000 m over a width of 50 km. This undercurrent embraces the topography and is associated with mean velocities of about 0.03 m s -1 and peaking up to 0.08 m s -1 where it is the closest to the topography.

Mean surface structure of the Agulhas cyclonic front at 25.75 • E: comparison with SAGE data

The Agulhas Current mean surface structure, along the SAGE section (25.75°E), from the four model nests (dx ∼ 22.5, 7.5, 2.5 and 0.75 km) are compared to glider data from SAGE (Figure 3.6). With the SAGE dataset, the cross-front section is computed by averaging the data in the along-front direction and over the one month-long record. The model sections are computed following the same steps for a period equivalent to the SAGE one. In the model, the one-month period is chosen in order to get similar dynamical conditions as during the SAGE experiment regardless of the dates. This one-month period corresponds to a period without mesoscale meanders, i.e a nonmeandering state, the 1000 m isobath marks the mean inshore front position (not shown here) and submesoscale eddies are present. Contrary to the SAGE period, no northeastward current on the shelf is present during the one-month period in the model. The origin of this feature on the shelf is unclear [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF] and it is shown in the next sections that it is not a limiting factor for our study.

The highest-resolution nest (dx ∼ 0.75 km) best fits with SAGE observations. The modeled and observed along-front velocities have close cross-front patterns and magnitude offshore of the shelf (about -1 m s -1 at the inshore front location). The modeled vorticity shows a pattern typical of sheared flows, with a peak of O(2f) at the front location, whereas the observed vorticity reaches a threshold at the front Partie , Chapter 3 -Study I : Generation of Frontal Submesoscale Eddies in the Agulhas Current (O(1f)) and increases offshore. The limited number of observations collected during SAGE could explain why the modeled and observed relative vorticity cross-front profiles differ. The gliders did not go further offshore to catch the maximum magnitude of the current and hence cannot show a decrease after reaching a relative vorticity maximum. All the modeled cross-front sections, relative to different numerical resolutions, show different patterns. This suggests that the highest-resolution nest did not converge yet. While the dx ∼ 22.5 and 7.5 km nests have flat cross-front patterns, the dx ∼ 2.5 and 0.75 km nests show gradients of mean along-front velocity and mean vorticity peaking at the mean Agulhas inshore front position (1000 m isobath). The lack of convergence means the highest-resolution nest could be numerical resolution-dependent. This could also explain the different relative vorticity cross-front patterns between the model and SAGE. However, in the next sections it is shown that in spite of the lack of convergence, the highest-resolution nest and SAGE captured the same frontal dynamics.

In this section, we show that the model captures the mesoscale variability of the Agulhas Current (Figure 3.4). Perturbations are the most energetic along the downstream branch of the current: at the Agulhas Bank, at the Retroflection and along the Agulhas Return Current. In the vicinity of Port Elizabeth (26°E), the model shows, in good agreement with in situ data, that the mean Agulhas inshore front has an intense cyclonic relative vorticity (O(> f )) that is uniform from the surface to about 500 m depth (Figure 3.5). Despite the respective limitations of the model and SAGE data -the model might not have converged yet and the glider data are sampled over a limited area -the modeled mean-state is similar to the SAGE one (Figure 3.6). We also validated the modeled water masses at the ACT array against the CARS climatology (K.R. [START_REF] Ridgway | Ocean interpolation by four-dimensional weighted least squares-application to the waters around australasia[END_REF] (not shown here).

Partie , Chapter 3 -Study I : Generation of Frontal Submesoscale Eddies in the Agulhas Current

Submesocale eddies along the Agulhas cyclonic front

In this section, we describe the submesoscale frontal eddies in the model, we investigate barotropic instability as the generation mechanism proposed by [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF], and we evaluate its importance in driving the Agulhas inshore front variability.

Description of the submesoscale cyclonic eddies in the SAGE area

Submesoscale eddies (diameters of 10-20 km) develop along the Agulhas inshore front upstream of Port Elizabeth (26°E) in the highest-resolution nest (Figure 3.7). Several eddies develop simultaneously between 28°E and 26°E, over a portion of the front where the current flows closely to the topography, and forms a vortex street. In the model, submesoscale eddies are generated on a quasi-regular basis. They have a limited life-time (∼ 3-4 days). Once developed, they propagate downstream, in the opposite direction of Kelvin waves, over 300-400 km before dissipating. Along the Agulhas Bank (westward of Port Elizabeth, 26°E), the topographic constraint on the front relaxes. As a result, the submesoscale eddies can grow to diameters of 40-50 km, beyond the submesoscale range.

The size and location of the modeled submesoscale eddies are consistent with the submesoscale eddies captured during SAGE [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF] and observed on SST satellite images (Figure 3.2).

The submesoscale eddies are intensified at the surface and associated with a cold temperature anomaly (Figure 3.8). With values of (O(5-10)f), the relative vorticity is the largest in the eddy cores and within the upper 25 m (Figure 3.8 top panel). Their core temperatures range from 15 to 19°C and at the edges it ranges from 20 to 23°C (Figure 3.8 bottom panel). The cold temperature anomaly is related to the cyclonic motion of the eddies which induces an upward doming of the isotherms. Water mass horizontal transfer during the process of the front destabilization can also reinforce the eddy temperature anomaly. In fact, the Agulhas front, characterized by warm water, develops cross-frontal meanders which extend onto the shelf and trap local cold water when they grow into finite amplitude and form the cyclonic eddies.

Barotropic instability as the generation mechanism

The Agulhas inshore front is barotropically unstable before the submesoscale eddy generation. The submesoscale eddies develop along a current portion where the mean velocity shear is intense (Figures 3.5 typical velocity profile U (y), selected from the highest-resolution nest below the surface (5 m) upstream of the eddy generation site (28°E) and at a time just before eddies develop (Figure 3.9, top panel), satisfies the instability condition.

Perturbations originating from the barotropically unstable velocity profile are coherent with the submesoscale eddies. A linear stability analysis is performed with the modeled velocity profile shown in Figure 3.9. The Boussinesq eigenvalue problem described by [START_REF] Ménesguen | Ageostrophic instability in a rotating stratified interior jet[END_REF] is modified to a 1D horizontal problem. The linear stability problem is solved for a typical velocity state U (y) and normal modes for the perturbations {u(y), v(y), p(y)} exp i(my) exp σt , with y the cross-front direction, u and v the along and cross-front velocity components, p the pressure, m the non-dimensional x-wavenumber, t the time and σ the perturbation growth rate. For this 1D configuration, the adimensionalized Boussinesq equations for momentum are :

Ro[σv + imU v] -u + ∂ y p -Ek∆ ′ v = 0 Ro[σu + ∂ y U v -imU u] + v + imp -Ek∆ ′ u = 0 ∂ y v + imu = 0 (3.1) with ∆ ′ = ∂ 2 x 2 -m 2 .
The initial velocity profile U (y) corresponds to the function : U (y) = U 0 tanh( y d ) + U 1 fitted to the velocity profile from the highest-resolution nest (Figure 3.9, top panel), with U 0 = -1.09 m s -1 the typical velocity magnitude, d = 830 m the shear width and U 1 = -0.32 m s -1 a constant. For the system of equations 3.1, the 1D flow is governed by two non-dimensional numbers : the Rossby and the Ekman numbers. The Rossby number Ro = U 0 f d , with f the Coriolis parameter, is the ratio of inertial to rotation terms and is as large as Ro = 16. The Ekman number Ek = ν f d 2 , with ν the viscosity, is the ratio of diffusion to rotation terms and is kept very small (10 -6 ) to correspond to inviscid flow.

The fastest growing modes have wavelengths of 10-13 km (Figure 3.9). The 1D linear stability analysis points out perturbations in the same scale range as the submesoscale eddies in the highest-resolution nest (10-20 km) and as the ones captured by SAGE and observed on the SST satellite images (∼ 15 km).

The parameter d corresponds to the shear width and it is close to the numerical resolution: dx ∼ 0.75 km. This is consistent with the possible non-convergence of the highest-resolution nest, as hinted by Figure 3.6 and this means that the mean velocity shear amplitude could be limited by the model viscosity. However, this is also counterbalanced by the modeled shear being of the same order of magnitude as the shear measured during SAGE (Figure 3.6). Figure 3.9: The cross-front velocity profile (U (y) [m s -1 ] in plain black line) and its fit (dashed black line) used as initial conditions in the 1D linear stability analysis are plotted (top). It is picked from the model at 5 m depth, at 28°E, just upstream of the submesoscale eddies generation, for a snapshot just before the submesoscale eddies develop. The relative vorticity gradient,

d 2 U dy 2 [m -1 s -1 ]
, computed for the initial velocity profile (plain grey line) and its fit (dashed grey line) are also plotted. This shows that the cross-front velocity profile satisfies the barotropic instability necessary condition. The perturbation growth rate [s -1 ], resulting from the 1D linear stability analysis, is plotted as a function of wavelength [km] (bottom). This 1D linear stability analysis shows perturbations at the same scale (15 km) than the submesoscale eddies seen in the model (10-20 km) and in the SAGE dataset (15 km). This preliminary analysis, led for a specific time of submesoscale eddy generation, supports [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF]'s suggestion that barotropic instability is the main generation mechanism.

The eddy energy source, identified for the same episode of submesoscale eddy generation, confirms the prevalence of barotropic instability. The EKE equation is (Gula et al., 2016b):

1 2 ∂u ′2 i ∂t ∂EKE i ∂t + ∂ ∂x i ( 1 2 u j u ′2 i + 1 2 u ′ j u ′2 i + 1 ρ 0 u ′ j p ′ ) Boundary transport = -u ′ j u ′ i ∂u i ∂x j M KE→EKE + w ′ b ′ EP E→EKE + V ′ i u ′ i V ertical mixing + D ′ i u ′ i Horizontal dif f usion (3.2)
Partie , Chapter 3 -Study I : Generation of Frontal Submesoscale Eddies in the Agulhas Current

with M KE → EKE = HRS -u ′ i u ′ i ∂u i ∂x i - V RS u ′ i w ′ ∂u i ∂z and EP E → EKE = V BF
w ′ b ′ The Cartesian tensor notation with summation convention is used with i = 1,2 and j=1,2,3. u i are the horizontal velocity components, u 3 =w is the vertical velocity, p is the pressure anomaly, b = -gρ ρ 0 is the buoyancy anomaly, V ′ i is the vertical mixing, with

V ′ i = ∂ ∂z K M v ( ∂u ′ i ∂z )
and D ′ i is the horizontal diffusion term in the horizontal momentum equations, computed as the difference between the third-order upwind numerical advective momentum scheme and the fourth-order centered scheme [START_REF] Gula | Submesoscale cold filaments in the Gulf Stream[END_REF]. The overbar denotes a time mean and the prime denotes fluctuations relative to this mean.

In Equation 3 The three possible EKE sources (HRS, V RS and V BF ) are averaged over a one-week period which corresponds to the full period of the submesoscale eddy generation event previously investigated (Figure 3.10). Along the front portion, where the submesoscale eddies develop (28-26°E), HRS is positive and has the highest magnitude whereas V RS and V BF are lower and mainly negative. HRS is here the dominant energy source term for the perturbations (M KE → EKE). This confirms the role of barotropic instability for the generation of submesoscale eddies along the Agulhas Current inshore front. The instability process is thus very similar to the development of barotropic instabilities in the submesoscale range described for the Gulf Stream (Gula et al., 2015b).

The vertical structure of the energy conversion term HRS matches the surface-intensified structure of the submesoscale eddies (Figure 3.10). Indeed, along the vortex street (29.5-25°E) the energy conversion term is intensified from the surface to a depth of about 50 m.

In this subsection, we focus on a specific event of submesoscale eddy generation (Figures 3.7 and 3.8). In the next subsection, we show that the generation of submesoscale eddies by barotropic instability is recurrent in this region and we discuss its predominance in driving the local variability of the Agulhas inshore front.

Barotropic instability as a locally preponderant mechanism

The triggering of the barotropic instability along the Agulhas inshore front is compared between the model and the SAGE data. With the one-month long glider dataset, a partial HRS term can be computed along a cross-front section at 25.75°E [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF]. This partial term accounts for the cross-front derivative subterms :

-u ′ v ′ ∂ y u -v ′ v ′ ∂ y v,
the overbar denotes a monthly time average, which corresponds to the full SAGE period, and the prime denote fluctuations relative to this time mean. This partial HRS term is positive and reaches a maximum at the Agulhas front. This result should be considered with caution since it involves several approximations on spatial and time averages and their findings might be limited by their use of a partial HRS. The modeled partial HRS term, computed over a equivalent period to the SAGE one (presented in the subsection 3.2.2, Figure 3.6), is compared to the SAGE one along the SAGE section (25.75°E) (Figure 3.11). The modeled (blue lines) and the observed (red line) HRS have similar patterns. They reach a maximum energy conversion rate inshore the 1000 m isobath, corresponding to the mean front location, with close magnitudes : 6 10 -6 and 1.08 10 -5 m 2 s -3 respectively. These maximum values are also in agreement with those found in the Gulf Stream (1 10 -5 m 2 s -3 ) by Gula et al. (2015b). Hence, the approximations on spatial and time averages required by the SAGE data for the HRS computation do not create a strong bias in the energy conversion term. Similarly, the presence of a northeastward current on the shelf during the SAGE experiment (Figure 3.6), which is absent in the model, does not create discrepancies relevant for the Agulhas inshore frontal dynamics.

The energy transfer M KE → EKE realized by the HRS term is mainly attributable to the mean current lateral shear : -u ′ v ′ ∂ y u. In the model, the partial and the full versions of the HRS term:

-u ′ u ′ ∂ x u -v ′ u ′ ∂ y u,
with u the vector of the horizontal velocity components, are compared. Both terms show the same patterns with only small magnitude differences. Since the upstream Agulhas Current is a coastal jet constrained by a steep and straight topography, its frontal structure is dominated by the crossshelf structure. Hence, [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF]'s estimation of the energy conversion at the Agulhas inshore front using a partial HRS term is valid at the first order. In a configuration such as the upstream Agulhas Current, the estimation of the HRS term using only the cross-front subterms is a workable approximation.

Submesoscale eddy generation by barotropic instability is a recurrent process when the Agulhas Current is not in a meandering state. The one-year long numerical simulation provides the ability to check the impact of time-averaging on the cross-front HRS pattern. The one-month average is defined in order to fit the SAGE's conditions: non-meandering state, the mean front position along the 1000 isobath and submesoscale eddies developing. The five-month average is instead defined as the consecutive months for which the current is not in a meandering state. Both time averages show similar patterns with slightly larger values for the five-month average. Submesoscale eddy generation by barotropic instability is a recurrent process along the Agulhas inshore front between 28°E and 26°E. The HRS term averaged over a five-month period (Figure 3.12) is very similar to the HRS term averaged over the one-week period corresponding to the submesoscale eddy generation event of Figure 3.10. For both time periods, HRS has a very similar pattern with positive values (O(2.10 -3 ) m 3 s -3 ) all along the front. Other energy conversion terms V RS and V BF have lower magnitudes (about O(0.25 10 -3 ) m 3 s -3 ) and have respectively negative and an alternation of negative and positive patterns along the front (not shown here). In addition to being a recurrent process, the generation of submesoscale eddies is the major process driving the local dynamics and variability of the Agulhas inshore front in a non-meandering state in the vicinity of Port Elizabeth (26°E). In the model, Natal Pulses inhibited the development of submesoscale eddies, but these mesoscale meanders can still host submesoscale activity following the mechanisms described by Gula et al. (2016a) In this section, we confirm barotropic instability as the submesoscale eddy generation mechanism. The submesoscale eddies are recurrent features which locally drive the Agulhas frontal dynamics and variability. Comparisons between the SAGE data and the model show that they captured the same frontal dynamics despite their respective limitations. 92

Spatial and Temporal variability of the instability

Spatial and Temporal variability of the instability

In this section we aim to better characterize the barotropic instability spatial and time variability.

Spatial variability of the instability

Where does barotropic instability occur along the Agulhas front ?

To investigate where the barotropic instability is triggered along the front, we determine the contribution of HRS to the spatial evolution of EKE. If HRS is the main source of EKE (Figures 3.10 Westward of Port Elizabeth (26°E), a combination of barotropic and baroclinic processes are responsible for the EKE generation, but they combine with other processes acting as sinks of EKE. At first order, the evolution of EKE still follows the pattern of HRS. HRS decreases to zero from 26°E to 25°E and, as a result, EKE is flatter between 25.5°E and 24.5°E. Downstream of 24.5°E, HRS increases strongly again and V BF becomes positive and increases as well. Both terms reach a maximum at 23.5°E. Accordingly, EKE increases strongly between 24.5°E and 23.5°E. However, EKE does not increase as strongly as suggested by the HRS and V BF terms (EKE < x

0

HRS+V BF u dx) as other processes acting as EKE sinks are also at play. Sinks include effects of the bottom friction, wind stress, interior vertical mixing, horizontal diffusion, and of the nonlinear terms (Eq. 3.2) advecting EKE out of the integration domain. Along this second front portion the flow still gains EKE, but some is also strongly dissipated. This indicates a dynamical shift of the current towards a more turbulent nature. dx [m 3 s -2 ] (a). The along-front distributions result from a cross-front average over the width (of about 100 km) of the black dashed box centered on the inshore front shown in Figure 3.12. Cross-front maximum of mean relative vorticity, ζ f , along the Agulhas front at the surface, 25, 50 and 75 m depth (b). In both panels, the reading direction for the along-front distributions corresponds to the current westward direction. The significant increase of HRS and EKE between 28°E and 26°E reflects the submesoscale eddies generation. Upstream of this front portion, the relative vorticity is large suggesting an unstable. This dynamical shift of the Agulhas current at 26°E is related to a topographical shift. The eastern margin of the Agulhas Bank, which begins westward of 26°E, is associated with an abrupt topography change including a sudden widening of the shelf break and a curving of the topography (Figure 3.1). When reaching the eastern Agulhas Bank, the current is known to increasingly meander in the mesoscale range [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF][START_REF] Krug | Interactions between the Agulhas Current and the eastern margin of the Agulhas Bank[END_REF]. This is reflected in the evolution of the vortex street. The modeled submesoscale eddies grow in size, potentially interact with each other and dissipate. Other perturbations, such as Shear-Edge eddies, are known to grow and occasionally leak down-stream (Lutjeharms et al., 2003b,a;[START_REF] Lutjeharms | The Agulhas Current[END_REF]. To sum up, when flowing close to the topography (upstream of 26°E) the barotropic instability leads to submesoscale eddy generation. When the Agulhas Current is in a non-meandering state they are the only EKE source. When the topographic constraint relaxes on the current (downstream of 26°E), the current develops meanders in the mesoscale range. The EKE sources are shared between barotropic and baroclinic processes and sinks also come into play.

The triggering of the barotropic instability between 28°E and 26°E is associated with intense relative vorticity upstream. Barotropic instability extracts energy from the mean horizontal shear. In the context of the upstream Agulhas Current, the main contribution to the horizontal shear is the lateral shear: ∂ y u (Figure 3.11). This term is also the main contributor to the current's relative vorticity. Cross-front maxima of relative vorticity, for different depths between the surface and 75 m, are plotted along the Agulhas front on Figure 3.13b. The relative vorticity is more intense at the surface than at depth. It suggests that the front is more unstable at the very surface. This is coherent with HRS being more intense between the surface and 50 m depth (Figure 3.10) and the submesoscale eddies being surface-intensified (Figures 3.8a and b).

The along-front relative vorticity provides an indication of where barotropic instability can be triggered and lead to submesoscale eddy generation. Upstream of the location of the barotropic instability triggering (28°E), the relative vorticity is the largest, with values of about O(6f ) at the surface, indicative of a potentially unstable front. Downstream of 28°E, the relative vorticity decreases in two steps. Where the submesoscale eddies develop (between 28°E and 26°E) the relative vorticity starts to decrease. Submesoscale eddies induce meanders in the front and in doing so they spread the mean frontal relative vorticity. The spread is the strongest at the surface where the submesoscale eddies are the most intense. Westward of 26°E, the relative vorticity decreases, with values O(< f ), and becomes vertically homogeneous. This decrease is due to the formation of Agulhas frontal meanders of larger scales, downstream of Port Elizabeth (26°E) where the topography constraint on the current is relaxed.

What controls the triggering of the barotropic instability around 28 • E ?

The barotropic instability originates from an unstable shear, itself set by the background strain. The mean strain, S f , is defined as :

  ∂ x u -∂ y v ∂ x v + ∂ y u  
Partie , Chapter 3 -Study I : Generation of Frontal Submesoscale Eddies in the Agulhas Current with u and v the velocity components in the x and y directions, f the Coriolis parameter and the overbar denoting a five-month time average when no Natal Pulse is present. The background strain reflects the large scale straining and stretching acting on the flow. It can actively intensify the velocity and the horizontal gradients of tracers if its axis is aligned with the frontal axis. Both axes are aligned if their angle is inferior to π 4 [START_REF] Hoskins | The mathematical theory of frontogenesis[END_REF] and in that case the background strain is frontogenetic. Conversely, if the strain and the frontal axis are not aligned the strain spreads the front and is frontolytic. The Agulhas inshore front is associated with an intense background strain at 25 m depth (Figure 3.14). Upstream of 28°E, the strain is the strongest and is well aligned with the front, whereas downstream of 26°E, it is weaker and the strain and frontal axis alignment is lost. The along-front distribution of the mean relative vorticity (Figure 3.13b) originates from the combined effect of the strain magnitude and of its alignment with the frontal axis. The portion where the front is the most sheared (upstream of 28°E) is consistent with the background strain being the most efficient at sharpening and intensifying the Agulhas front. The barotropic instability is triggered where the background strain weakens. Where the front destabilizes, downstream of 28°E, the alignment of the strain with the front decreases, meaning the strain is less efficient at driving the front intensification. A frontogenetic background strain has the conjoint effect of intensifying the front and preventing mean horizontal shear-related instabilities to be triggered. In doing so it "stabilizes" a potentially unstable 2D relative vorticity band [START_REF] Dritschel | The stability of a two-dimensional vorticity filament under uniform strain[END_REF]. Similarly, a frontogenetic tendency can prevent the triggering of shear instabilities on the upstream face of a Gulf Stream frontal eddy (Gula et al., 2016a), as well as the development of submesoscale cross-front disturbances on the upstream face of jet meandering troughs [START_REF] Mcwilliams | The Gulf Stream North Wall: Ageostrophic Circulation and Frontogenesis[END_REF]. By stretching and straining the front, the background strain exerts a stress in the along-front direction which inhibits any cross-front momentum and energy transfer. To sum up, the barotropic instability develops just downstream of a front portion where the relative vorticity is the most intense. This large relative vorticity originates from a frontogenetic background strain. The instability triggering is concomitant with the background strain weakening. When the front is released from the background strain constraint there is no external stress restraining the development of cross-frontal features and the front destabilizes.

A frontogenetic background strain usually results from the combination of the background mesoscale advection field and of the topographic constraint on the flow. Several studies connect front destabilizations with a weakening of the background strain [START_REF] Molemaker | Submesoscale instability and generation of mesoscale anticyclones near a separation of the California Undercurrent[END_REF]Gula et al., 2015b). In these studies, a straight and steep topography is identified to be at the origin of an intense background strain which is reinforced by the effect of the topographic drag on the current shear. An abrupt separation of the current from the topography leads to the strain weakening and thus to the triggering of barotropic instabilities just downstream the separation.

The Agulhas Current case differs slightly. The instability is triggered along a front portion where the current flows closely to the topography. In fact, the topographic constraint on the flow relaxes further downstream (at Port Elizabeth 26°E). The triggering of the barotropic instability between 28°E and 26°E is thus not directly related to a topographic change. This mismatch between the front dynamics and the topographic constraint is also visible from the vertical structure of the strain (not shown here). Similarly to the relative vorticity (Figure 3.11), the strain is the strongest at the surface even though along this portion of the front, the front is not in contact with the topography at depths shallower than 150-200 m (not shown here). Therefore the background mesoscale advection field would be the main contributor to the intense background strain at the surface. The contribution of the topography in the front intensification and destabilization between 28°E and 26°E in the submesoscale range is less evident.

Time variability of the instability

While submesoscale eddy generation by barotropic instability is a recurrent process (Figures 3.11 and 3.12), which drives the Agulhas front variability in the submesoscale range in the vicinity of Port Elizabeth (26°E), it is an intermittent process with a strong time variability.

Partie , Chapter 3 -Study I : Generation of Frontal Submesoscale Eddies in the Agulhas Current

The Agulhas front variability at relatively high frequencies is investigated with a five-month long time series of the energy conversion term HRS (Figure 3.12). The time series is built with the HRS term averaged over the vertical and over the frontal area between 28°E and 26°E from July to November when no Natal Pulse is present. Hence, this time series takes place during the austral winter, when [START_REF] Krug | Satellite observations of an annual cycle in the Agulhas Current[END_REF]; [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF] observed the current to be less intense. .15: Time series of energy conversion rate HRS over a 5 month-long period when no Natal Pulse is present. The time series results from a volume average : on the vertical, HRS is averaged over the whole water column and on the horizontal over the 100 km wide black dashed box centered on the Agulhas front between 29°E and 26°E shown in Figure 3.12. The thin dashed and bold plain lines represent the raw and the smoothed (over a 10-point window) time series. The grey-shaded areas correspond to submesoscale eddies generation events identified from a movie a surface relative vorticity. Submesoscale eddies are generated when HRS is peaking, but the reverse is not true. This shows that a large HRS is not a sufficient condition to trigger instabilities and that submesoscale eddies generation result from a combination of several factors. Submesoscale eddy generation shows a periodicity of about 10 days, on average, from the HRS time series. The HRS time series shows a constantly positive fluctuating signal (Figure 3.15). Not all the HRS peaks correspond to submesoscale eddy generation and ten events of actual submesoscale eddy generation are identified from a surface relative vorticity movie. They are marked in shades of grey on the HRS time series. These events are spaced by about 10 days on average. This periodicity also comes out from a five-month long time series of the surface relative vorticity upstream of the instability triggering location 98 3.7. Spatial and Temporal variability of the instability (28°E) (not shown here). The triggering of the barotropic instability is then connected to the fluctuations, with a 10 days period, of the Agulhas inshore front structure.

The submesoscale eddy generation periodicity is relatively close to the Agulhas Current and to the Agulhas Undercurrent's ones. [START_REF] Bryden | Structure and transport of the Agulhas Current and its temporal variability[END_REF] and [START_REF] Beal | A time series of Agulhas Undercurrent transport[END_REF] respectively looked at the Agulhas Current and Agulhas Undercurrent time variability using 12 and a 17 month-long in situ records of current velocity. They show that both currents have the largest variability for the Natal Pulse period (60 days). At lower periods the Agulhas Current shows a large variability for periods between 8-27 days and the Agulhas Undercurrent for periods of 25 days and between 8-20 days. The variability of the submesoscale eddy generation is associated with large scale variations of the Agulhas Current and Undercurrent rather than with local variations of the Agulhas inshore front.

The few listed possible mechanisms involved in the time variability of the submesoscale eddy generation are a remote and a local forcings : the Coastal Trapped Waves or Rossby Topographic Waves and the wind stress respectively. [START_REF] Schumann | Coastal-trapped waves off the coast of south africa: generation, propagation and current structures[END_REF] showed, from in situ measurements of sea level along the Southern African coastline, that Coastal Trapped Waves affect the South African continental shelf dynamics and possibly interact with the Agulhas Current upstream of Port Elizabeth. Even though no study investigated the Rossby Topographic Waves along the South African continental slope, the location where the submesoscale eddies develop (28-26°E) is favourable to their development in theory. Both waves can affect the Agulhas inshore front dynamics and then be involved or interact with the submesoscale eddies.

Rather than one process setting the conditions for the instability to be triggered, this can be the result of a combination of different processes all acting to destabilize the front. Local wind is another possible factor impacting the frontal dynamics. [START_REF] Thomas | Intensification of ocean fronts by down-front winds[END_REF] showed that down-front winds induce an Ekman transport of dense waters in the top of light waters resulting in the frontogenetic process enhancement. [START_REF] Schaeffer | Characterizing frontal eddies along the East Australian Current from HF radar observations[END_REF] observed the Eastern Australian Current mainly meandered and destabilized under up-front winds conditions leading to the formation of submesoscale frontal cyclonic eddies. However, in this study local wind forcing is not favourable before and during the submesoscale eddy generation.

In this section we address the spatial and time variability of the barotropic instability triggering. The barotropic instability triggering in the submesoscale range and leading to the generation of frontal submesoscale eddies is the only source of variability for the Agulhas inshore front in the absence of Natal Pulses. The instability originates from an intense relative vorticity at the surface which itself is driven by the frontogenetic background strain. However, the relative contributions of the topography and of the background mesoscale advection field on driving the background strain are not completely disentangled and need further investigations. In the same way, further studies are needed to determine in which proportion remote and local mechanisms contribute to the 10 days variability of the instability triggering. 100

Conclusion

Conclusion

Our study addresses the Agulhas Current frontal dynamics in the submesoscale range. We investigated and characterized the generation mechanism of frontal submesoscale eddies along a non-meandering Agulhas Current using a high-resolution (dx ∼ 0.75 km) numerical simulation. The simulation reproduces frontal cyclonic submesoscale eddies in the vicinity of Port Elizabeth (26°E) with a diameter range of 10-20 km in agreement with satellite and in situ observations. A 1D linear stability analysis, of the cross-front shear upstream of the instability triggering location (28°E) shows that it is barotropically unstable and it develops perturbations with wavelengths of 10-13 km matching the scale of the submesoscale eddies. Barotropic instability is confirmed as the simulated submesoscale eddy generation mechanism by an energy transfer analysis. The barotropic instability is a horizontal shear instability which draws its energy from the mean kinetic energy field when the mean horizontal shear is unstable (HRS = M KE → EKE). Comparison between the modeled barotropic instabilities and the ones observed during SAGE shows that the lateral shear is the main contribution to the energy conversion term HRS.

Our energy transfer analysis over longer time periods (one and five-month long) indicates that the submesoscale frontal eddy generation by barotropic instability is a recurrent process which drives most of the variability around Port Elizabeth (26°E).

This result gives new insights on the Agulhas Current along-stream dynamics. When considering a broader scale range, from the mesoscale to the submesoscale, the current dynamics are less dichotomous than previously described [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF].

This study shows that in addition to the Natal Pulses (mesoscale solitary meanders), frontal submesoscale eddies also locally drive the "stable" branch variability around Port Elizabeth (26°E). Rather than a dynamical shift, Port Elizabeth is the location of a smooth transition, between a "stable" and an "unstable" branch in the mesoscale range, via the triggering of instabilities in the submesoscale range. Hence, the so far known sources of variability for the upstream "stable" branch are the Natal Pulses and the frontal submesoscale eddies which both originate from barotropic instability [START_REF] Biastoch | The role of mesoscale eddies in the source regions of the Agulhas Current[END_REF][START_REF] Tsugawa | Generation and growth mechanism of the Natal Pulse[END_REF][START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF].

The spatial variability of the barotropic instability triggering originates from the background strain. Be-tween 28°E and 26°E, barotropic instability is triggered and leads to the generation of frontal submesoscale eddies. Upstream of the instability triggering location (28°E), the background strain is frontogenetic and the front is intensively sheared. The frontogenetic strain has the double effect of intensifying the shear and of preventing the development of cross-frontal perturbations. Downstream, the frontogenetic influence of the strain weakens allowing for barotropic instability to be triggered along the front.

The origin of the frontogenetic background strain slightly differs between the Agulhas Current and the Gulf Stream. The bottom friction does not play a leading role in generating the current's horizontal shear and the topographic constraint role is not as evident as in the Gulf Stream (Gula et al., 2015b).

The triggering of barotropic instability has a periodicity of about 10 days on average. Possible mechanisms driving this time variability are Coastal Trapped Waves or Rossby Topographic Waves and wind stress. Their relative contributions need further investigation.

The submesoscale frontal eddies could drive exchanges of material and momentum between shelf waters and the Agulhas Current, with potential consequences on coastal dynamics, marine life and large scale circulation. Additional observations and modeling studies are required to address these aspects.

Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a Western Boundary Current

Abstract

Western boundary currents are hotspots of mesoscale variability and eddy-topography interactions, which canalize energy transfers toward small scales, and eventually down to dissipation. Here, we assess the role of topographic interactions in mesoscale eddies energy loss in the Agulhas Current region from a regional numerical simulation. We derive an Eddy Kinetic Energy (EKE) budget in the frame of the vertical modes. It includes energy transfers between energy reservoirs and vertical modes, including transfers canalized by topography. The EKE vertical partitioning is dominated by mesoscale eddies (barotropic and 1 st baroclinic modes) in the path of intense mean currents. Eddy-topography interactions result in a major mesoscale eddy energy sink in the Agulhas Current region, along the three following paths, which have a comparable importance : energy transfers toward bottom intensified time-mean currents, generation of eddies of smaller vertical scales, and bottom friction. The generation of eddies of smaller vertical scales takes different forms in the Northern Agulhas Current, where it corresponds to non-linear transfers to smaller vertical eddies on the slope, and in the Southern Agulhas Current, where it is dominated by a (linear) generation of internal-gravity waves over topographic accidents. Away from the shelf, mesoscale eddies gain energy by an inverse vertical turbulent cascade. However, the Agulhas Current region remains a net source of mesoscale eddy energy due to the strong generation of eddies, modulated by the topography, especially in the Southern Agulhas Current. This challenges the vision of Western Boundaries as ubiquitous net energy sinks of mesoscale eddies.

Introduction

Mesoscale eddies, with length scales on the order of the Rossby radius of deformation (Rd) or larger (Chelton et al., 2011b), represent 90 % of the Kinetic Energy (KE) reservoir and are a major component of the global oceanic energy budget [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. Understanding how the mesoscale eddies are sustained is a fundamental issue for the phenomenological and quantitative understanding of the global oceanic energy budget.

Wind drives the upper-ocean basin-scales mean circulation. Energy is then injected around the Rossby deformation radius Rd (O(30-100) km [START_REF] Chelton | Geographical variability of the first baroclinic Rossby radius of deformation[END_REF]), as mesoscale eddies, by instability processes of the mean currents. Oceanic dynamics at the mesoscale (L ∼ Rd) are characterized by the dominant effects of the rotation and stratification and are described by the quasi-geostrophic balance [START_REF] Charney | Geostrophic turbulence[END_REF]. In this dynamical regime, non-linear interactions result in an inverse turbulent cascade [START_REF] Rhines | The dynamics of unsteady currents[END_REF][START_REF] Rhines | Geostrophic turbulence[END_REF], i.e., an energy transfer toward larger horizontal scales, and in an energy barotropization [START_REF] Fu | Nonlinear energy and enstrophy transfers in a realistically stratified ocean[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF], i.e., an energy transfer from the 1 st baroclinic (upper layersintensified structure) to the barotropic (vertically homogeneous structures) vertical modes [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF]). An inverse turbulent cascade for the baroclinic energy has been inferred from the surface fields measured by satellite altimetry [START_REF] Scott | Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry[END_REF][START_REF] Scott | Spectral energy fluxes in geostrophic turbulence: Implications for ocean energetics[END_REF]. This inverse turbulent cascade occurring at the surface has been extensively investigated using satellite altimetry [START_REF] Tulloch | Scales, growth rates, and spectral fluxes of baroclinic instability in the ocean[END_REF][START_REF] Khatri | Surface ocean enstrophy, kinetic energy fluxes, and spectra from satellite altimetry[END_REF] and models based on quasigeostrophic [START_REF] Scott | Spectral energy fluxes in geostrophic turbulence: Implications for ocean energetics[END_REF] and primitive equations [START_REF] Schlösser | Diagnosing the energy cascade in a model of the North Atlantic[END_REF][START_REF] Aluie | Mapping the energy cascade in the North Atlantic Ocean: The coarse-graining approach[END_REF].

Energy injected around Rd by the currents instabilities takes therefore the form of barotropic and baroclinic eddies. The barotropic component of this energy reservoir is partially dissipated by the bottom drag [START_REF] Salmon | Baroclinic instability and geostrophic turbulence[END_REF][START_REF] Sen | Global energy dissipation rate of deep-ocean low-frequency flows by quadratic bottom boundary layer drag: Computations from current-meter data[END_REF] and both barotropic and baroclinic components are partially dissipated by the wind stress [START_REF] Eden | Effects of mesoscale eddy/wind interactions on biological new production and eddy kinetic energy[END_REF][START_REF] Seo | Eddy-wind interaction in the California Current System: Dynamics and impacts[END_REF][START_REF] Renault | Modulation of the Agulhas Current Retroflection and Leakage by Oceanic Current Interaction with the Atmosphere in Coupled Simulations[END_REF]. However, a large part of the energy reservoir dissipates at very small (micro) scales. The known energy paths from "balanced" motions (i.e., geostrophically balanced) at large-scales toward "unbalanced" motions at smaller scales involve ageostrophic turbulence and interactions with internal-gravity waves [START_REF] Müller | Routes to dissipation in the ocean: The 2D/3D turbulence conundrum[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. Mesoscale eddies can lose energy to smaller scales by : scattering over rough bottom topography leading to internal lee waves generation (Nikurashin andFerrari, 2010, 2011), generation of internal waves along western boundaries [START_REF] Clément | Generation of internal waves by eddies impinging on the western boundary of the North Atlantic[END_REF], straining induced by the eddies-internal waves interactions [START_REF] Rocha | Stimulated generation: extraction of energy from balanced flow by near-inertial waves[END_REF], hydraulic jumps related-dynamics generated by eddy-boundary waves interactions [START_REF] Dewar | Topographic inviscid dissipation of balanced flow[END_REF][START_REF] Gula | Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow-water model. Part 1. Passive lower layer[END_REF], submesoscale mean horizontal shear instabilities triggered by topographic-interactions [START_REF] Dewar | Centrifugal instability and mixing in the California Undercurrent[END_REF]Gula et al., 2016b), and by the frontogenesis enhanced by the mesoscale eddy field background strain (Capet et al., 2008c;[START_REF] Molemaker | Balanced and unbalanced routes to dissipation in an equilibrated Eady flow[END_REF][START_REF] D'asaro | Enhanced turbulence and energy dissipation at ocean fronts[END_REF][START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF] and possibly developing secondary frontal instabilities [START_REF] Thomas | Symmetric instability in the Gulf Stream[END_REF]. They can as well lose energy to the time-mean circulation by the so-called "Neptune effect" [START_REF] Holloway | Systematic forcing of large-scale geophysical flows by eddy-topography interaction[END_REF][START_REF] Adcock | Interactions between geostrophic eddies and the mean circulation over large-scale bottom topography[END_REF].

Pointing toward the locations where mesoscale eddies predominantly loose energy provides a new perspective for investigating the mesoscale eddies energy loss. [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF]'s suggested Western boundaries as ubiquitous sinks of mesoscale eddy energy and pointed that eddy-topography interactions could be responsible for the mesoscale eddy energy scattering toward smaller scales and eventually down to dissipation. [START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF] provides a reference observational study case for the phenomenological understanding of the eddy-topography interactions involved in mesoscale eddies decay on western boundaries, in the absence of a western boundary current. However, the phenomenological and quantitative descriptions of these processes do not make yet a complete picture of the direct turbulent cascade, i.e., the energy transfer toward smaller scales [START_REF] Ferrari | The distribution of eddy kinetic and potential energies in the global ocean[END_REF].

In this study, we evaluate the energetics of the mesoscale eddies by deriving a Kinetic Energy (KE) budget for the different vertical structures described by the traditional vertical modes [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF]. This method allows to characterize energy transfers between the different vertical modes and highlights the vertical component of the turbulent cascades, including the scattering, i.e., when energy is transferred to higher baroclinic modes. Energy scattering has often been studied in the context of internal tides and lee waves generation [START_REF] Kelly | Internal-tide energy over topography[END_REF][START_REF] Kelly | The cascade of tidal energy from low to high modes on a continental slope[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF][START_REF] Lahaye | Modal analysis of internal wave propagation and scattering over large-amplitude topography[END_REF], but more seldomly in the context of mesoscale eddies. This framework gives a complementary vision of the turbulent energy cascade to the one traditionally evaluated across horizontal scales by classic spectral analysis [START_REF] Scott | Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry[END_REF][START_REF] Schlösser | Diagnosing the energy cascade in a model of the North Atlantic[END_REF][START_REF] Tulloch | Scales, growth rates, and spectral fluxes of baroclinic instability in the ocean[END_REF][START_REF] Arbic | On eddy viscosity, energy cascades, and the horizontal resolution of gridded satellite altimeter products[END_REF][START_REF] Arbic | Geostrophic turbulence in the frequency-wavenumber domain: Eddy-driven low-frequency variability[END_REF][START_REF] Khatri | Surface ocean enstrophy, kinetic energy fluxes, and spectra from satellite altimetry[END_REF] and by coarse-graining methods [START_REF] Aluie | Mapping the energy cascade in the North Atlantic Ocean: The coarse-graining approach[END_REF][START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF].

Our mesoscale EKE budget is based on a regional numerical simulation of the Agulhas Current region, built upon a primitive equations model (CROCO) [START_REF] Shchepetkin | The Regional Oceanic Modeling System (ROMS): A splitexplicit, free-surface, topography-following-coordinate ocean model[END_REF][START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF]. The Agulhas Current is the most intense western boundary current of the Southern hemisphere [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF]. It flows poleward along the South African coast, from the Mozambique Channel (26.5

• S) to the Southern African tip (37 • S) where it separates from the continental shelf [START_REF] Lutjeharms | The Agulhas Current[END_REF],

Introduction

retroflects eastward and becomes the Agulhas Return Current when re-entering in the South Indian Ocean [START_REF] Lutjeharms | The agulhas return current[END_REF]. Based on its mesoscale variability, the Agulhas Current can be separated into a Northern and Southern Agulhas Current branches, connecting at Port Elizabeth (33.5 • S) [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]. The mesoscale variability of the Northern Agulhas Current is low and mainly due to the intermittent passage of solitary meanders (Natal Pulses) (Lutjeharms et al., 2003a) whereas mesoscale eddies such as Shear-edge eddies [START_REF] Lutjeharms | Eddies and other boundary phenomena of the Agulhas Current[END_REF](Lutjeharms et al., , 2003a) ) are common features of the Southern Agulhas Current. From the Agulhas Retroflection, large anticyclonic eddies (Agulhas Rings) are generated and propagate into the South East Atlantic Ocean [START_REF] Lutjeharms | The Agulhas Current[END_REF]. A variety of mesoscale eddies are generated in the Agulhas Return Current permanent meanders [START_REF] Gründlingh | Drift of a satellite-tracked buoy in the southern Agulhas Current and Agulhas Return Current[END_REF][START_REF] Lutjeharms | Eddies at the subtropical convergence south of Africa[END_REF]. The Agulhas Current is therefore a suitable region for studying the mesoscale eddy dynamics, due to its intense local mesoscale variability and because it is suspected to catalize the energy loss of remotely formed mesoscale eddies [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF].

In the present study, we aim to characterize the mesoscale EKE budget in the Agulhas Current region, focusing on the Northern Agulhas Current (NAC) and the Southern Agulhas Current (SAC). We particularly address the following questions : (1) What is the vertical structure of eddy energy in the Agulhas Current region ? (2) What is the mesoscale EKE budget in the NAC and SAC ? (3) What are the processes driving the mesoscale eddies generation, dissipation and energy transfers toward smaller vertical structures in the NAC and SAC ? The study is organised as follows : the regional numerical simulation set up is presented and the mesoscale variability characteristics in the simulation are evaluated in section 2.a. The vertical modes are defined and put into the context of the simulations eddy energy in section 2.b. The derivation and computation of the modal EKE budget are presented in section 2.c. The mesoscale eddy energy reservoir is characterized from the eddy vertical structures in section 4.4 and the mesoscale eddy energy budget is evaluated and interpreted, in terms of physical processes, in section 4.5. The results are summarized and discussed in section 4.6.

Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a Western Boundary Current

Methods

The regional numerical simulations

We present in this section the regional numerical simulations and the modeled mesoscale eddy dynamics statistical evaluation against remote satellite and in situ data.

The simulation setup

Methods

The numerical model used in this study is the Coastal and Regional COmmunity model (CROCO). It is a free surface model, based on ROMS [START_REF] Shchepetkin | The Regional Oceanic Modeling System (ROMS): A splitexplicit, free-surface, topography-following-coordinate ocean model[END_REF], which solves the primitive equations in the Boussinesq and hydrostatic approximations using a terrain following coordinate system [START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF]. We use a nesting approach with successive horizontal grid refinements from a parent grid resolution of dx ∼ 22.5 km, covering most of the South Indian Ocean, to successive child grids resolutions of dx ∼ 7.5 km, 2.5 km and 0.75 km (Figure 4.1). The nesting procedure is online 2-way based on AGRIF [START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF] for the dx ∼ 22.5 km, 7.5 km and 2.5 km grids and offline 1-way [START_REF] Mason | Procedures for offline grid nesting in regional ocean models[END_REF] for the dx ∼ 0.75 km grid. The surface forcings of the four grids are provided by a bulk-formulation [START_REF] Fairall | Bulk parameterization of air-sea fluxes for tropical ocean-global atmosphere coupled-ocean atmosphere response experiment[END_REF] using the ERA-ECMWF's reanalysis [START_REF] Dee | The ERA-Interim reanalysis: Configuration and performance of the data assimilation system[END_REF] with relative winds [START_REF] Renault | Modulation of the Agulhas Current Retroflection and Leakage by Oceanic Current Interaction with the Atmosphere in Coupled Simulations[END_REF]. The dx ∼ 22.5 km, 7.5 km and 2.5 km grids have 60 vertical levels stretched at the surface following [START_REF] Haidvogel | Numerical ocean circulation modeling[END_REF]'s method and the dx ∼ 0.75 km grid has 100 vertical levels stretched at the surface and at the bottom following Lemarié et al. ( 2012)'s method. The dx ∼ 22.5 km, 7.5 km and 2.5 km grids are run for the 1993-2014 period after a spin-up of 3 years (dx ∼ 22.5 km and 7.5 km) and 1 year (dx ∼ 2.5 km) from their initial states and the dx ∼ 0.75 km grid is run for the year 1996 after a 3 months spin-up. The four numerical simulations and the numerical choices contributing to their realism (bathymetry, grid, parameterization of the turbulent closure, momentum and advective numerical schemes) are detailed in [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF].

In the present study, we investigate the mesoscale eddy energetics in the Agulhas Current region based on the 1994-1999 period of the dx ∼ 2.5 km grid. The other grids are used to evaluate the sensitivity of the modeled eddy energy vertical structure to the numerical resolution (section 2.b.3).

Observed and modeled mesoscale eddies

The mesoscale eddy dynamics are inferred from their signature on the eddy energy. Regardless of their generation mechanism, mesoscale eddies sign the Eddy Kinetic Energy : EKE = 1 2 || u ′ || 2 , with u the horizontal velocity vector, the overbar denotes a time average and the prime denotes fluctuations relative to this average. Surface geostrophic EKE from the dx ∼ 2.5 km grid is compared to the one derived from the altimetric data gridded onto a 1/4 • regular grid by AVISO (Archiving, Validation, and Interpretation of Satellite Oceanographic Data) (Figure 4.2 top panels). Both EKE are computed from geostrophic velocity derived from sea surface height anomalies. The baroclinic component of the mesoscale eddy dynamics signs at depths the Eddy Available Potential Energy : EAP E = 1 2 b ′ ζ ′ , with b the buoyancy and ζ the vertical isopycnals displacement [START_REF] Roullet | Global interior eddy available potential energy diagnosed from Argo floats[END_REF]. EAP E at 500 m depth from the dx ∼ 2.5 km grid is compared to Roullet (2020)'s Atlas derived from ARGO data and gridded onto a 1/4 • regular grid (Figure 4.2 bottom panels). Both EKE and the modeled EAP E are defined relative to a 1-year rolling mean with a 1-month time step over the 1994-1999 period and the observed EAP E is defined relative to a time average over the whole 112

Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a Western Boundary Current

Methods

ARGO dataset (∼ 10 6 profiles over the last 15 years, Roullet ( 2020)). The eddy component defined by the 1-year time rolling mean filters the low frequency dynamics and points toward the mesoscale dynamics, which corresponds to time scales of weeks to months.

Both modeled surface geostrophic EKE and 500 m depth EAP E have patterns and magnitudes in fairly good agreement with the one derived from AVISO and ARGO data, denoting the known areas of low and high mesoscale variability in the Agulhas Current region (Figure 4.2). The eddy energy is the highest along the Agulhas Current and the Agulhas Return Current (EKE : O(0.05-0.5) m 2 s -2 and EAP E :

O(0.05-0.3) m 2 s -2
) and it is the lowest in the South-West Indian Ocean Subgyre (EKE : O(0-0.05) m 2 s -2 and EAP E : O(0-0.05) m 2 s -2 ) in the model and in the observations. The Agulhas Current mesoscale variability has historically been characterized into a Northern (stable) and Southern (unstable) branches connecting around Port Elizabeth (26 • E) [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]. The stability of the Northern Agulhas branch (upstream of 26 • E) is denoted by lower eddy energy level (EKE : O(0-0.1) m 2 s -2 and EAP E : O(0-0.05) m 2 s -2 ) than the Southern branch, which has the highest eddy energy level of the Agulhas Current region (EKE : O(0.05-0.5) m 2 s -2 and EAP E : O(0.05-0.3) m 2 s -2 ) in the model and the observations. The eddy energy level gradually increases downstream of the Southern Agulhas branch leading to a maximum at the Retroflection for the observed EKE model and both EAP Es (EKE : O(0.2-0.5) m 2 s -2 and EAP E : O(0.2-0.3) m 2 s -2 ). The intense mesoscale variability of the Agulhas Return Current is denoted by a high eddy energy level, fairly uniform along the entire length of the Agulhas Return Current (EKE : O(0.1-0.15) m 2 s -2 and EAP E : O(0.05-0.15) m 2 s -2 ), but lower than that of the Retroflection. The intense mesoscale variability along the Southern Agulhas Current, at the Agulhas Retroflection and along the Agulhas Return Current is visible at the surface (EKE in Figure 4.2 top panels) and at 500 m depth (EAP E in Figure 4.2 bottom panels), suggesting that mesoscale eddies have a vertical structure extending from the surface to at least 500 m depth in these areas.

A few discrepancies are noticeable between the modeled and observed surface EKEs and 500 m depth EAP Es. The modeled surface EKE has a higher magnitude along the entire Agulhas Current than the observed one (Figure 4.2 top panels). It shows a moderate signal (O(∼ 0.1) m 2 s -2 ) along the Northern branch, inshore the 1000 m isobath and a strong signal along the Southern branch, that are both absent in the observations. However, the modeled EKE is highly similar to the observed one if we smooth the model sea surface height using a length scale of 100 km, which mimics the processing of altimetric data by AVISO (not shown). It evidences the major role played by horizontal scales < 100 km for the Southern Agulhas Current variability, which are not measured by AVISO. The observed EAP E shows patterns of moderate amplitude (O(∼ 0.05) m 2 s -2 ) in the Subgyre that are absent from the model and a signal associated to the Agulhas Return Current (O(0.05-0.1) m 2 s -2 ) that covers a larger area than in the model (Figure 4.2 bottom panels). These EAP E differences are not located specifically in under-sampled areas (20-35 profiles) compared to the ARGO profiles density over the whole domain (10-60 profiles). The different definitions of the eddy components between the observed EAP E (relative to a time average over the whole ARGO dataset collected over the last 15 years) and the modeled EAP E (relative to a 1-year rolling mean with a 1-month time step over a 5-years period) could explain some of these differences. The modeled EAP E defined relative to a 5-years time average does not filter the interannual variability and show weaker differences with the observed EAP E at these locations (not shown). However, the overall agreement in both EAP E patterns and magnitudes, suggests that the fluctuation definition by a rolling mean is appropriate for investigating the mesoscale variability at depths in the Agulhas Current region.

Based on the different local mesoscale eddy dynamics, we define two main areas in the Agulhas Current region : the NAC and SAC (Figure 4.2 top right panel). The NAC covers the Northern Agulhas branch and the Northern part of the Subgyre and is characterized by a low mesoscale variability. The SAC covers the Southern Agulhas branch, the Agulhas Retroflection and the Southern part of the Benguela Current and is characterized by a high mesoscale variability. In the following, the mesoscale eddy dynamics of the Agulhas Current region is characterized separately in these two areas.

Vertical Modes

We define in this section the vertical modes that are used to derive a modal EKE budget in our regional numerical simulations. We first give the mathematical definition and properties of the vertical modes bases, we then derive the EKE and EAP E modal expansions, we describe and classify the vertical structures corresponding to the barotropic and the first 9 baroclinic vertical modes and we finally evaluate the modal EKE resolution for our four nested grids.

Definition of the vertical modes

In the following, we give the mathematical definition and properties of the vertical modes.

We use the traditional vertical modes [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF]. An alternative definition of vertical modes has been proposed by LaCasce (2017), using a condition of no bottom-flow, but we prefer to use the traditional 4.3. Methods vertical modes to straightforwardly account for bottom-intensified flows [START_REF] Stanley | Vertical structure of ocean mesoscale eddies with implications for parameterizations of tracer transport[END_REF]. A convenient approach to describe the vertical structure of mesoscale motions is the modal decomposition [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. In a continuously stratified ocean over a flat-bottom, the vertical variability of the dynamical and state variables can be projected onto an orthogonal basis of n vertical modes (φ n (z) or Φ n (z)) as follows :

[u(x, z, t), 1 ρ 0 p ′ (x, z, t)] = ∞ n=0 [u n (x, t), 1 ρ 0 p ′ n (x, t)]φ n (z) (4.1) [w(x, z, t), b ′ (x, z, t)] = ∞ n=0 [w n (x, t), b ′ n (x, t)N 2 (x, z)]Φ n (z) (4.2)
with p the total pressure, w the vertical velocity component, N 2 the mean Brunt-Väisälä frequency, (u n ,

p ′ n , w n , b ′ n )
the respective modal amplitudes of the horizontal velocity components, the eddy pressure, the vertical velocity component and the eddy buoyancy for a vertical mode n. Injecting the modal decomposition in the linearized quasigeostrophic equation leads to two equivalent Sturm-Liouville eigenvalue problems (2.18 and 2.15). The normal eigenfunctions solution are the φ n (z) and Φ n (z) vertical modes using linearized free-surface ((w) z=η = ∂ ∂t η) and flat bottom ((w) z=-H = 0) boundary conditions as in [START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF] :

∂ ∂z [ 1 N 2 ∂ ∂z φ n ] + 1 c 2 n φ n = 0 (4.3) ( ∂ ∂z φ n ) z=η = ( -N 2 g φ n ) z=η (4.4) ( ∂ ∂z φ n ) z=-H = 0 (4.5) ∂ 2 ∂z 2 Φ n + N 2 c 2 n Φ n = 0 (4.6) (Φ n ) z=η = ( c 2 n g ∂ ∂z Φ n ) z=η (4.7) (Φ n ) z=-H = 0 (4.8)
with g the acceleration of gravity and c 2 n the eigenvalues of the corresponding vertical modes. The two modal bases are related via the continuity equation (4.9) and they respectively satisfy an orthog-

Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a

Western Boundary Current onality condition (4.10 and 4.11) :

φ n = ∂ ∂z Φ n (4.9) and η -H φ m φ n dz = δ mn h, (4.10) g c 2 n (Φ m Φ n ) z=η + η -H N 2 c 2 n Φ m Φ n dz = δ mn h (4.11)
with δ mn the usual Kronecker symbol and h = η + H the water column depth.

The modal amplitudes of the variables are derived by combining their modal expansions (4.1 and 4.2) with the modes orthogonality conditions (4.10 and 4.11) :

[u n (x, t), 1 ρ 0 p ′ n (x, t)] = 1 h η -H [u(x, z, t), 1 ρ 0 p ′ (x, z, t)]φ n (z) dz (4.12) w n (x, t) = 1 hc 2 n [ η -H N 2 w(x, z, t)Φ n (z) dz + g(wΦ n ) z=η ] (4.13) b ′ n (x, t) = 1 hc 2 n [ η -H b ′ (x, z, t)Φ n (z) dz + g( b ′ N 2 Φ n ) z=η ] (4.14)
The vertical modes are related to horizontal scales via their eigenvalues, which are good approximations of the Rossby baroclinic deformation radii [START_REF] Chelton | Geographical variability of the first baroclinic Rossby radius of deformation[END_REF] :

Rd n≥1 = c n |f | = 1 nπ | f | η -H N (x, z) dz (4.15)
with f the Coriolis parameter.

The traditional vertical modes definition is based on a flat bottom assumption, which is relaxed in regional numerical simulations. This leads to spatially varying vertical modes (φ n (x, z) and Φ n (x, z)) and thus to non null vertical modes horizontal gradients. These latter represent an intermodal coupling forced by the stratification-topographic spatial variations, which is of particular interest for internal tides generation and scattering [START_REF] Kelly | Internal-tide energy over topography[END_REF][START_REF] Kelly | The cascade of tidal energy from low to high modes on a continental slope[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF][START_REF] Lahaye | Modal analysis of internal wave propagation and scattering over large-amplitude topography[END_REF]. The validity of the traditional vertical modes definition is confirmed in the context of the dx ∼ 2.5 km regional simulation in Appendix A. The barotropic and the first 9 baroclinic modes capture 80-100 % of the variability of the

Modal expansions of eddy energies

In the following, we derive the EKE and EAP E modal expansions (subsequently referred as EKE n and EAP E n ) and describe and classify the vertical structures corresponding to the barotropic and 9 first baroclinic vertical modes of the two modal bases : φ n and Φ n .

The EKE n and EAP E n , expressed with the free-surface boundary condition, respectively depend on the two vertical mode bases (φ n and Φ n ) as following :

η -H EKE dz = η -H 1 2 || ∞ n=0 (u ′ n φ n ) ∞ m=0 (u ′ m φ m ) || 2 dz = ∞ n=0 h 2 || u ′ n || 2 = ∞ n=0 EKE n (4.16) η -H EAP E dz = η -H 1 2 ∞ n=0 (b ′ n N 2 Φ n ) ∞ m=0 (ζ ′ m Φ m ) dz = ∞ n=0 [ 1 2 b ′ n ζ ′ n hc 2 n - ∞ m=0 g 2 b ′ n ζ ′ m (Φ n Φ m ) z=η ]
Intermodal coupling at the f ree-surf ace (Cnm)

= ∞ n=0 [EAP E n + C nm ] (4.17)
The EAP E n expression (4.17) is more complex than EKE n (4.16), because it involves a term of intermodal coupling at the free-surface (C nm ). It originates from the orthogonality condition of the Φ n bases expressed with the free-surface boundary condition (2.22).

The two vertical modes bases (φ n and Φ n ), and therefore EKE n and EAP E n , correspond to different vertical structures for a given vertical mode n (Figure 4.3). The EKE (Figure 4.3 top panels) decomposes into a barotropic mode (φ 0 ) representing a vertically homogeneous structure, a 1 st baroclinic mode (φ 1 ) representing a structure intensified in the upper layers to the main thermocline depth (1000 m for a 5000 m deep ocean) and into higher baroclinic modes (φ 2-9 ) representing structures intensified over surface layers (O(100) m) whose vertical extension becomes shallower with the baroclinicity level. The EAP E (Figure 4.3 bottom panels), decomposes into a barotropic mode (Φ 0 ) representing a structure linearly decreasing from a maximum at the free-surface toward zero at the bottom, a 1 st baroclinic mode (Φ 1 ) representing a structure intensified at intermediate depths of O(500 -2000) m, and into higher baroclinic modes (Φ 2-9 ) representing structures intensified below the surface. The (φ 0 ,Φ 1 ) categories represent interior-intensified vertical structures, the φ 1 represents upper layerintensified vertical structures, the (φ 2-9 ,Φ 0,2-9 ) categories represent surface-intensified vertical structures and the Φ 0 represents surface-intensified linearly decreasing vertical structures.

Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a Western Boundary Current

Based on these three categories, both φ 0 and Φ 1 correspond to interior-intensified vertical structures, the φ 1 corresponds to upper layers-intensified vertical structures and the φ 2-9 and Φ 2-9 correspond to surface-intensified vertical structures. Φ 0 represents a category of its own, of surface-intensified vertical structures whose energy linearly decreases with depth. This last category originates from the use of the free-surface boundary condition (2.16), but its physical meaning remains unclear especially since it accounts for almost no potential energy [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF].

Sensitivity of the EKE n to the numerical resolution

The Rossby deformation radii expression (4.15) shows that high baroclinic modes correspond to small horizontal scales and thus require high horizontal numerical resolution to be resolved. In the following, we evaluate the sensitivity of the vertical modes to the numerical resolution by comparing the EKE 4.3. Methods partitioning, between the φ 0 , φ 1 and φ 2-9 vertical modes categories, for our four nested simulations (dx ∼ 22.5, 7.5, 2.5 and 0.75 km) in three areas : NAC, Alongshore Agulhas Current and SAC (areas defined in between the three vertical modes categories : φ 0 (plain lines), φ 1 (dashed lines) and φ 2-9 (dashed-dotted lines), for the NAC, Alongshore Agulhas Current and SAC areas (located on Figure 4.2) and for the four grids : dx ∼ 22.5 (black lines), 7.5 (grey lines), 2.5 (green lines) and 0.75 km (blue lines, only for the Alongshore Agulhas Current). The overbar denotes a 1-year rolling mean with a 1-month time stepping over the 1994-1999 period. The EKE partitioning is sensitive to the numerical resolution in the NAC and Alongshore Agulhas Current areas, contrary to the SAC, and it shows a convergence for the dx ∼ 2.5 km grid in the Alongshore Agulhas Current.

in Figure 4.4). The effect of the horizontal resolution on the EKE partitioning differs between the three areas, suggesting contrasted local eddy dynamics. In the SAC area, the energy partition does not change significantly when the resolution increases (Figure 4.4 bottom left panel). In the NAC area, the barotropic mode loses energy to the 1 st baroclinic mode when the resolution increases, such that energy is almost equipartitioned between both modes at dx ∼ 2.5 km (Figure 4.4 top left panel). In the Alongshore Agulhas Current area, the barotropic mode strongly dominates at low resolution (80-90 %), but loses energy to the 1 st baroclinic (up to 30-40 %) as well as the higher baroclinic modes (up to 15 %) at higher resolution (Figure 4.4 top right panel). The partition of energy in the Alongshore Agulhas Current area does not change between the dx ∼ 2.5 km (green lines) and the dx ∼ 0.75 km grids (blue lines), showing a convergence in the representation of the 10 first vertical modes. It must be noted that the vertical resolution is also increased between dx ∼ 2.5 km and dx ∼ 0.75 km grids (from 60 to 100 vertical levels). The convergence suggests that the dx ∼ 2.5 km grid vertical resolution (60 vertical levels) is adequate to represent the vertical modes at this horizontal resolution and that this grid resolves the barotropic and the first 9 baroclinic modes, validating its choice for investigating the mesoscale eddies dynamics.

The mean modal Eddy Kinetic Energy (EKE n ) budget

We develop in this section the modal EKE (EKE n ) evolution equation corresponding to the classic EKE evolution equation, but derived in the vertical modes frame. We also present the EKE n budget computation using the dx ∼ 2.5 km grid numerical outputs.

Derivation of the EKE n equation

The modal Kinetic Energy equation is decomposed into a mean and an eddy part, in the same way than the classical EKE equation [START_REF] Harrison | Energy analysis of open regions of turbulent flows-Mean eddy energetics of a numerical ocean circulation experiment[END_REF]Gula et al., 2016b), such as :

KE n = M KE n + EKE n , with KE n = η -H 1 2 || u n || 2 dz the mean modal KE, M KE n = η -H 1 2 || u n || 2 dz the modal KE of the mean flow and EKE n = η -H 1 2 || u ′ n || 2 dz the mean modal EKE.
The CROCO model solves the primitive equations under the Boussinesq and hydrostatic approximations :

∂ ∂t u + u.∇ H u + w ∂ ∂z u + f k ∧ u = - 1 ρ 0 ∇ H p + V + D (4.18) ∂ ∂z p + gρ = 0 (4.19) ∇ H .u + ∂ ∂z w = 0 (4.20)
with ρ(x, z, t) the density variation relative to the density reference : ρ 0 = 1025 kg m -3 , V the vertical mixing expressed as :

V = ∂ ∂z (K M v ∂u ∂z ) with K M v the
h( ∂ ∂t u) n + η -H (u.∇ H u + w ∂ ∂z u)φ n dz + f k ∧ hu n = (4.21) - η -H (∇ H p)φ n dz + hV n + hD n hp ′ n + hc 2 n b ′ n -g( b ′ N 2 Φ n ) z=η = 0 (4.22) ∇ H .(hu n ) - ∞ m=0 u m . η -H φ m ∇ H φ n dz + hw n -∇ H η.(uφ n ) z=η = 0 (4.23)
The EKE n equation is obtained by doing EKE n = KE n -M KE n in the same way than for the classic EKE equation (Gula et al., 2016b). The KE n equation is derived by taking the inner product between the modal horizontal velocity vector (u n ) and the modal horizontal momentum equation (4.21) and by time averaging. The M KE n equation is derived by taking the inner product between the mean modal horizontal velocity vector (u n ) and the mean modal horizontal momentum equations (4.21 time averaged).

The EKE n equation is finally obtained by substracting M KE n to KE n , such as:

u ′ n .h( ∂ ∂t u) n T ime rate + u ′ n . η -H (u.∇ H u + w ∂ ∂z u)φ n dz Advection -u n .(f k ∧ hu n ) Coriolis (4.24) = -u ′ n . η -H (∇ H p)φ n dz P ressure gradient + u ′ n .(hV n ) V ertical mixing + u ′ n .(hD n ) N umerical dissipation
The advection and the pressure gradient terms of the EKE n equation (4.24) can be further decomposed into different contributions.

The horizontal advection term is decomposed into six contributions : the Horizontal Reynolds Stress within a vertical mode n (HRS n ), the horizontal advection of EKE n , the complementary term to the advection of EKE n , the horizontal triadic interactions and the advective eddy-eddy and eddy-mean interactions on

Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a

Western Boundary Current slopes, using the modal expansion of the horizontal velocities (4.1) :

u ′ n . η -H ( u.∇ H u )φ n dz = u ′ n .(u ′ n .∇ H )u n η -H φ n φ n φ n dz HRSn + ∇ H .((u n + u ′ n ) 1 2 ||u ′ n || 2 η -H φ n φ n φ n dz) Horizontal advection of EKEn - 1 2 ||u ′ n || 2 ∇ H .(u n η -H φ n φ n φ n dz) Complement to the horizontal advection of EKEn + ∞ m =n ∞ l =n [ u ′ n .(u ′ m .∇ H )u ′ l Eddy-eddy interactions + u ′ n .(u m .∇ H )u ′ l + u ′ n .(u ′ m .∇ H )u l Eddy-mean interactions ] η -H φ n φ m φ l dz Horizontal triadic interactions + ∞ m=0 ∞ l=0 u ′ n .u ′ l u ′ m . η -H φ n φ m ∇ H φ l dz
Advective eddy-eddy interactions on slopes

+ ∞ m=0 ∞ l=0 u ′ n .(u ′ l u m + u l u ′ m ). η -H φ n φ m ∇ H φ l dz
Advective eddy-mean interactions on slopes (4.25)

The vertical advection term is decomposed into 3 contributions : the Vertical Reynolds Stress within a vertical mode n (V RS n ), the vertical advection of EKE n and the vertical triadic interactions, using the modal expansion of the horizontal and vertical velocities (4.1 and 4.2), the relation between the two vertical mode basis (4.9) and the Sturm-Liouville equation (4.6) :

u ′ n . η -H (w ∂ ∂z u)φ n dz = -u ′ n .(u n w ′ n ) η -H N 2 c 2 n φ n Φ n Φ n dz V RSn -w n ||u ′ n || 2 η -H N 2 c 2 n φ n Φ n Φ n dz V ertical advection of EKEn - m =n l =n u ′ n .(u l w m ) η -H N 2 c 2 n φ n Φ m Φ l dz V ertical triadic interactions (4.26)
The pressure gradient term is decomposed into 4 contributions : the eddy-pressure work, the Vertical Buoyancy Fluxes within a vertical mode n (V BF n ), the eddy-eddy interactions on slopes and a small term reflecting the free-surface contribution, using the Leibniz's rule, the modal expansion of the pressure anomaly (4.1) , the orthogonality condition of the vertical modes (4.10), the modal continuity equation -

u ′ n . η -H (∇ H p)φ n dz = -∇ H .(hu ′ n p ′ n )
Eddy-pressure work

+ c 2 n hb ′ n w ′ n -gw ′ n ( b ′ N 2 Φ n ) z=η V BFn (4.27) + ∞ m=0 (u ′ m p ′ n ). η -H φ m ∇ H φ n dz - ∞ m=0 (u ′ n p ′ m ). η -H φ n ∇ H φ
+ ∇ H .[(u n + u ′ n ) 1 2 ||u ′ n || 2 η -H φ n φ n φ n dz] -(w n + w ′ n )||u ′ n || 2 η -H N 2 c 2 n φ n Φ n Φ n dz Advection of EKEn (A.3)
Divergences of the eddy-energy f luxes (A.2+A.1)

= -u ′ n .(u ′ n .∇ H )u n η -H φ n φ n φ n dz M KEn→EKEn (HRSn) (B.1) + u ′ n .(u n w ′ n ) η -H N 2 c 2 n φ n Φ n Φ n dz M KEn→EKEn (V RSn) (B.2) + c 2 n hb ′ n w ′ n -gw ′ n ( b ′ N 2 Φ n ) z=η EP En→EKEn (V BFn) (B.3) - ∞ m =n ∞ l =n [ u ′ n .(u ′ m .∇ H )u ′ l Eddy-eddy interactions + u ′ n .(u m .∇ H )u ′ l + u ′ n .(u ′ m .∇ H )u l Eddy-mean interactions ] η -H φ n φ m φ l dz Horizontal triadic interactions (C.1.1) + ∞ m =n ∞ l =n u ′ n .(u l w m ) η -H N 2 c 2 n φ n Φ m Φ l dz V ertical triadic interactions (C.1.2) - ∞ m=0 ∞ l=0 u ′ n .u ′ l u ′ m . η -H φ n φ m ∇ H φ l dz
Advective eddy-eddy interactions on slopes (C.2)

- ∞ m=0 ∞ l=0 u ′ n .(u ′ l u m + u l u ′ m ). η -H φ n φ m ∇ H φ l dz
Advective eddy-mean interactions on slopes (C.3)

+ ∞ m=0 (u ′ m p ′ n ). η -H φ m ∇ H φ n dz - ∞ m=0 (u ′ n p ′ m ). η -H
φ n ∇ H φ m dz P ressure-gradient eddy-eddy interactions on slopes (C.4)

+ p ′ n ∇ H η.(u ′ φ n ) z=η Contribution of the f ree-surf ace <<1 (C.5) + 1 2 ||u ′ n || 2 ∇ H .(u n η -H φ n φ n φ n dz)
Complement to the advection of EKEn (C.6)

+ u ′ n .(hV n ) V ertical mixing (D.1) + u ′ n .(hD n ) N umerical dissipation (D.2) + u n .(f k ∧ hu n ) Coriolis (D.3)
The different contributions to the EKE n budget (equation 4.29) and the related physical processes are further detailed in the following :

(A) EKE n evolution :

-Time rate (A.1) : Evolution of the EKE storage within the vertical mode n over time, which tends toward zero for a long time average.

-Divergences of the eddy-energy fluxes (A.2 + A.3): EKE n transport which can be interpreted as the EKE n fluxes at the domain boundaries when integrated over a closed volume. It has two contri-

Methods

butions : the eddy-pressure work, which represents the energy transport done by the linear components of the dynamics, and the advection of EKE n by the total 3D flow, which represents the energy transport done by the non-linear component of the dynamics. In the frame of linear theories of internal waves [START_REF] Kelly | Internal-tide energy over topography[END_REF][START_REF] Kelly | The cascade of tidal energy from low to high modes on a continental slope[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF], Rossby waves [START_REF] Masuda | Group velocity and energy transport by Rossby waves[END_REF] and of open-ocean dynamics [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] the eddy-pressure work is the only contribution to the EKE n transport.

(B) Transfers between the EKE n , M KE n and EP E n reservoirs :

-M KE n → EKE n (HRS n ) term (B.1): Horizontal Reynolds Stress-mean flow non-linear interactions within a vertical mode n. It is the counterpart of the Horizontal Reynolds Stress (HRS), which is used as an indicator of horizontal shear instabilities such as the barotropic (Gula et al., 2015a,b) and the centrifugal/inertial (Gula et al., 2016b) instabilities.

-

M KE n → EKE n (V RS n ) term (B.
2): Vertical Reynolds stress-mean flow non-linear interactions within a vertical mode n. It is the counterpart of the Vertical Reynolds Stress (V RS), which is used as an indicator of vertical-shear related instabilities such as Kelvin-Helmholtz or symmetric instability [START_REF] Thomas | Symmetric instability in the Gulf Stream[END_REF].

-

EP E n → EKE n (V BF n ) term (B.
3): Vertical Buoyancy Fluxes reflecting eddy energy conversion between the potential and kinetic reservoirs within a vertical mode n. It is the counterpart of the V BF term, which is used as an indicator of the baroclinic instability at meso and submesoscale [START_REF] Mcwilliams | Submesoscale currents in the ocean[END_REF].

(C) Transfers between vertical modes :

-Horizontal triadic interactions (C.1.1) : Eddy-eddy and eddy-mean non-linear interactions between vertical modes. The intermodal coupling is represented by the triadic coefficient : η -H φ n φ m φ l dz and corresponds to the following vertical modes combinations : (n = m = l), (n=l = m), (n = m = l) and (n = m = l). These triadic interactions are the counterpart of the horizontal component of the advective operator which transfers KE across scales and realizes the turbulent energy cascades [START_REF] Salmon | Baroclinic instability and geostrophic turbulence[END_REF][START_REF] Vallis | Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation[END_REF]. The eddy-eddy interactions represent the turbulent energy cascades and the eddy-mean interactions represent unforced energy transfers across scales and reservoirs.

-Vertical triadic interactions (C.1.2) : Eddy-total flow non-linear interactions between vertical modes. The intermodal coupling is represented by the triadic coefficient :

η -H N 2 c 2 n
φ n Φ m Φ l dz and corresponds to the following vertical modes combinations : (n = m = l), (n=l = m), (n = m = l) and (n = m = l). These triadic interactions point toward the contribution of the vertical advection to the turbulent energy cascades and to the unforced energy transfers across scales and reservoirs, as a complement to the horizontal triadic interactions (C.1.1).

Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a Western Boundary Current

-Advective eddy-eddy flow interactions on slopes (C.2) : Eddy-eddy non-linear interactions between vertical modes associated with stratification-topography horizontal gradients. The intermodal coupling is represented by : η -H φ n φ m ∇ H φ l dz and corresponds to the vertical modes combinations : (n = m = l), (n=l = m), (n = m = l) and (n = m = l). This term points toward eddy-eddy and eddy-waves non-linear interactions forced by the stratification-topography horizontal gradient. The (n = m = l) case does not represent an energy transfer between reservoirs or vertical structures and likely corresponds to EKE n advection. This case does not represent a main contribution to the EKE 0-1 budget (Figure 4.8).

-Advective eddy-mean interactions on slopes (C.3) : Eddy-mean non-linear interactions between vertical modes associated with stratification-topography horizontal gradients. The intermodal coupling is represented by : η -H φ n φ m ∇ H φ l dz and corresponds to the vertical modes combinations : (n = m = l), (n=l = m), (n = m = l) and (n = m = l). This term points toward eddy-mean non-linear interactions forced by the stratification-topography horizontal gradients. These interactions are mainly interpreted as eddies driving mean currents over sloping topography [START_REF] Holloway | Systematic forcing of large-scale geophysical flows by eddy-topography interaction[END_REF][START_REF] Adcock | Interactions between geostrophic eddies and the mean circulation over large-scale bottom topography[END_REF] if the vertical modes combination (n = m = l) has an equal contribution as the other ones to the energy transfer. Indeed, bottom-intensified flows decompose into a barotropic mode, allowing a nonzero bottom velocity, and into baroclinic modes, to cancel or to shape velocity in the upper layers, as tested by [START_REF] Stanley | Vertical structure of ocean mesoscale eddies with implications for parameterizations of tracer transport[END_REF].

-Pressure-gradient eddy-eddy interactions on slopes (C.4) : Eddy-eddy linear interactions between vertical modes associated with stratification-topography horizontal gradients. The intermodal coupling is represented by : η -H φ m ∇ H φ n dz and η -H φ n ∇ H φ m dz and corresponds to the vertical modes combination : (n = m). This term points toward eddy-eddy and eddy-waves linear interactions forced by the stratification-topography horizontal gradients. These interactions correspond to eddies inducing Topographic Rossby Waves on a sloping bottom and to eddies scattering over seamounts, topographic features, accidents and slopes leading to the generation of internal-gravity waves -such as lee waves - [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF][START_REF] Nikurashin | Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory[END_REF]. This latter process is similar to the tidal scattering leading to internal tides generation [START_REF] Kelly | Internal-tide energy over topography[END_REF][START_REF] Kelly | The cascade of tidal energy from low to high modes on a continental slope[END_REF][START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF].

-Contribution of the free-surface « 1 (C.5) : Eddy-eddy interactions between vertical modes at the free-surface. This term has a mathematical origin (Eq. 2.29), but it has a negligible effect on our EKE 0-1 budget (Figure 4.8).

-Complement to the advection of EKE n (C.6) : Term derived from the horizontal advection contribution, which allows to write the horizontal advection of EKE n as a divergence of eddy-energy flux (Eq. 4.25). This contribution does not cancel out in the framework of the vertical modes, contrary to the EKE budget where the continuity equation is used to cancel out a similar contribution [START_REF] Harrison | Energy analysis of open regions of turbulent flows-Mean eddy energetics of a numerical ocean circulation experiment[END_REF]. We will show that it has a small effect on our EKE 0-1 budget (Figure 4.8).

Methods

(D) EKE n generation and dissipation :

-Vertical mixing (D.1) : EKE n dissipation and/or generation by wind stress, interior vertical mixing and bottom drag. The wind stress dissipates surface energy regardless of the oceanic structures [START_REF] Hughes | Wind work on the geostrophic ocean circulation: An observational study of the effect of small scales in the wind stress[END_REF][START_REF] Renault | Dampening of submesoscale currents by air-sea stress coupling in the Californian upwelling system[END_REF] and the bottom drag dissipates barotropic [START_REF] Fu | Nonlinear energy and enstrophy transfers in a realistically stratified ocean[END_REF][START_REF] Sen | Global energy dissipation rate of deep-ocean low-frequency flows by quadratic bottom boundary layer drag: Computations from current-meter data[END_REF] and non-null baroclinic EKE.

-Numerical dissipation (D.2) : EKE n dissipation by the horizontal implicit viscosity of the model.

-Coriolis term (D.3) : EKE n dissipation and/or generation by the Coriolis term. This contribution does not cancel out in the framework of the vertical modes, contrary to the EKE budget [START_REF] Harrison | Energy analysis of open regions of turbulent flows-Mean eddy energetics of a numerical ocean circulation experiment[END_REF]. We will show that it has a negligible effect on our EKE n budget (Figure 4.8).

EKE n budget in the regional numerical simulation

The EKE n budget (Eq. 4.29) is derived using the dx ∼ 2.5 km grid numerical outputs for the 1994-1999 period. The numerical outputs are daily averages of the u, w, temperature (T ) and salinity (S) variables and of the online momentum tendency terms. The vertical mixing term is parameterized using the K-profile parameterization [START_REF] Large | Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization[END_REF] and the numerical dissipation term is defined as the difference between the third-order upwind numerical advective momentum scheme used for our simulations and a fourth-order centered scheme (Gula et al., 2016b). The barotropic and the first 9 baroclinic modes are derived from the mean Brunt-Väisälä frequency (N 2 ) diagnosed offline from the numerical outputs. The eddy component is defined relative to a 1-year rolling mean with a 1-month time step, as for Figure 4.2.

The time rate, the vertical mixing, the numerical dissipation and the Coriolis contributions to the EKE n budget (equation 4.29) are computed using their respective online momentum diagnostics. The contributions of the horizontal (4.25) and vertical advections (4.26) and of the pressure gradient (4.27) are computed using a residual method combining the advection and the pressure gradient online diagnostics with offline explicit diagnostics. The accuracy of the analytical development of the contributions of the horizontal advection term (4.25), the vertical advection term (4.26) and of the pressure gradient term (4.27) is evaluated by explicitly computing the contributions offline. The offline diagnostic has the disadvantage to generate artefacts. They come from the time sampling difference and the finite difference schemes used for discretizing the spatial derivative between the online and the offline diagnostics. Although these artefacts do not affect the first order agreement between the advection and the pressure gradient terms, computed using online diagnostics or explicitly computed offline (Figure 2.8 in Chapter 2), the numerical precision of the EKE n budget closure must be preserved to keep accuracy. The residual method consists in explicitly computing offline all the contributions of the advection and the pressure gradient terms except one, computed as a residual between the sum of the explicitly computed offline contributions and the advection and pressure gradient terms computed using online momentum diagnostics. To limit the offline computation of spatial derivatives, the HRS n and V RS n terms (B.1 and B.2), the vertical advection of EKE n (component of A.3), the horizontal and vertical triadic interactions (C.1.1 and C.1.2), the advective eddy-eddy and eddy-mean interactions on slopes (C.2 and C.3) and the complement of the advection of EKE n (C.6) are explicitly computed offline and the horizontal advection of EKE n (component of A.3) is determined as a residual. In the same way, the V BF n term (B.3), the pressure-gradient eddy-eddy interactions on slopes (C.4) and the contribution of the free-surface (C.5) are explicitly computed offline and the eddy-pressure work (A.2) is determined as a residual. The EKE n budget closure is evaluated by the ratio between the budget residual (defined as the difference between the time rate and the sum of the rest of the EKE n terms) root mean square and the time rate root mean square. It is of 1.3 10 -6 for the 1 st baroclinic mode (n = 1) and attests of the closure of the EKE 1 budget derived from the dx ∼ 2.5 km grid numerical outputs.

Results I : Characterization of the mesoscale eddy energy reservoirs

We characterize in this section the vertical structure of the mesoscale eddy energy reservoirs (EKE and EAP E) using the vertical modes. We characterize first the eddy energy vertical partitioning between the vertical modes categories (n = 0, n = 1 and n = 2 -9) and then the local eddy energy vertical structures for 8 areas of contrasted eddy energy vertical partitioning.

Eddy energy vertical partitioning

The EKE and EAP E partitioning between the vertical modes categories (n = 0, n = 1 and n = 2 -9) differ over the Agulhas Current region, with an EAP E partitioning more contrasted than that of EKE (Figure 4.5).

The EKE mainly partitions into the φ 0 and φ 1 categories (O(70-100) %) and weakly into the φ 2-9 category (< O(30) %) (Figure 4.5 top panels). This is consistent with the mesoscale eddies vertical structure being dominated by the barotropic and 1 st baroclinic modes [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF] and with mesoscale eddies being the most energetic eddy flows [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. The EKE is uniformly equipartitioned between both categories over the Agulhas Current region, except at the Retroflection where the φ 0 category dominates.

The EAP E partitions significantly into the 3 vertical modes categories (Φ 0 : (0-60) %, Φ 1 : (15-89) % and Φ 2-9 : (5-57) %) (Figure 4.5 bottom panels). The Φ 1 category contains most of the EAP E (O(70-89) %) and has the closest geographical distribution to that of EKE into φ 0 and φ 1 in areas of intense mean circulation (Agulhas Current, Agulhas Retroflection and Agulhas Return Current). In contrast, the Φ 0 and Φ 2-9 categories are the most energetic (O(40-60) %) in the Subgyre. The vertical structure of the mesoscale eddy is described by the φ 0 , φ 1 (EKE) and Φ 1 (EAP E) vertical modes, which represent interior-and upper layers-intensified vertical structures.

Local vertical structures of eddy energy

The local vertical structures of eddy energy in the Agulhas Current region are characterized for 8 areas (A-H) located in areas of contrasted EKE and EAP E vertical partitionings (Figure 4.5). These areas sample regions of documented physical processes of the Agulhas Current region [START_REF] Lutjeharms | The Agulhas Current[END_REF] : Northern Agulhas Current branch (A), Eastern Agulhas Bank Bight in the Southern Agulhas branch (B), Agulhas Retroflection (C), Agulhas Return Current (D), Benguela area (E) and the South-West Indian Ocean Subgyre (F, G and H). The EKE and EAP E vertical partitionings for the 8 areas are classified into 3 combinations : φ 0 > φ 1 and Φ 1 >> (Φ 0 , Φ 2-9 ) for A, B and C, φ 0 ∼ φ 1 and Φ 1 >> (Φ 0 , Φ 2-9 ) for D and E and φ 0 ∼ φ 1 and (Φ 0 , Φ 1 , Φ 2-9 ) in variable proportions for F, G and H (Table 4.1). The EAP E partitioning is more contrasted between the 8 areas than that of EKE (Table 4.1) and it is used to characterize the local eddies vertical structures more precisely than with the EKE partitioning alone. It especially allows to dissociate the D and E areas from the F, G and H ones (Figure 4.6). Both EKE partitionings (φ 0 > φ 1 and φ 0 ∼ φ 1 ) correspond to upper layers-intensified vertical structures with a non-zero velocity at depth and at the bottom (Figure 4.6). The relative proportions of φ 0 and

(n = 0, n = 1, n = 2 -9) spatially averaged over the (1 • x 1 • ) boxes (A-H) delimited in Figure 4.5.
The standard deviation of the spatial averages within the boxes are also indicated. The overbar denotes a 1-year rolling mean with a 1-month stepping over the 1994-1999 period.

EKE n A: Northern Agulhas Current φ 0 (56.9 ± 10.2 %) > φ 1 (28.5 ± 8.6 %) B: Eastern Agulhas Bank Bight φ 0 (60.0 ± 4.9 %) > φ 1 (36.9

± 4.3 %) C: Retroflection φ 0 (68.6 ± 1.5 %) > φ 1 (28.7 ± 1.4 %) D: Agulhas Return Current φ 0 (49.4 ± 1.4 %) ∼ φ 1 (46.0 ± 1.4 %) E: Benguela area φ 0 (51.2 ± 2.3 %) ∼ φ 1 (43.7 ± 2.5 %) F: Northern Subgyre φ 0 (46.9 ± 2.4 %) ∼ φ 1 (39.2 ± 2.7 %) G: Middle Subgyre φ 0 (48.5 ± 2.3 %) ∼ φ 1 (45.2 ± 1.9 %) H: Southern Subgyre φ 0 (54.6 ± 3.0 %) ∼ φ 1 (38.9 ± 3.2 %) EAP E n A: Φ 0 (6.1 ± 2.2 %) < < Φ 1 (68.3 ± 4.3 %) > > Φ 2-9 (25.5 ± 4.4 %) B: Φ 0 (8.9 ± 1.9 %) < < Φ 1 (77.3 ± 3.8 %) > > Φ 2-9 (13.9 ± 3.6 %) C: Φ 0 (6.0 ± 1.0 %) < < Φ 1 (80.1 ± 1.4 %) > > Φ 2-9 (13.8 ± 1.0 %) D: Φ 0 (14.1 ± 1.7 %) < < Φ 1 (68.4 ± 1.5 %) > > Φ 2-9 (17.5 ± 0.9 %) E: Φ 0 (7.3 ± 2.3 %) < < Φ 1 (75.6 ± 3.4 %) > > Φ 2-9 (17.1 ± 1.5 %) F:
Φ 0 (26.9 ± 3.1 %) ∼ Φ 1 (37.6 ± 2.7 %) ∼ Φ 2-9 (35.5 ± 2.3 %) G:

Φ 0 (24.9 ± 2.1 %) ∼ Φ 2-9 (27 ± 0.8 %) < Φ 1 (48.1 ± 2.7 %) H: Φ 0 (47.3 ± 16.0 %) > Φ 1 (31.4 ± 11.7 %) > > Φ 2-9 (16.3 ± 7.0 %) φ 1 drive the rate of decrease of energy with depth. The vertical structures corresponding to the different EAP E partitionings are described for the three area combinations in the following.

Mesoscale eddies

The A, B and C areas are located in the path of the Agulhas Current and show an EKE and an EAP E mostly in one vertical mode : φ 0 (O(60) %) and Φ 1 (O(75) %) (Table 4.1). The latter corresponds to profiles intensified at intermediate depths (A : O(0-500) m and B and C : O(500) m ) decreasing with depth toward a minimum value at the bottom (Figure 4.6 top panels). Such vertical structures denote an intense mesoscale variability in these areas, as already hinted from the surface geostrophic EKE and the 500 m depth EAP E (Figure 4.2). The NAC (A) mesoscale variability is mainly due to the passage of Natal Pulses. They are solitary cyclonic eddies, with diameters of about 50-200 km, extending over the whole water column (Lutjeharms et al., 2003a;[START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF], which are generated at the Natal Bight (∼ 31 • E). The Eastern Agulhas Bank Bight (B) high mesoscale variability manifests from several border features [START_REF] Lutjeharms | Eddies and other boundary phenomena of the Agulhas Current[END_REF](Lutjeharms et al., , 2003a)). One is the Shear-edge eddies [START_REF] Lutjeharms | Eddies and other boundary phenomena of the Agulhas Current[END_REF][START_REF] Goschen | Agulhas current variability and inshore structures off the Cape Province, South Africa[END_REF], which are quasi permanent cyclonic eddies, with diameters of about 50-100 km, extending at least over the first 500 m deep (Lutjeharms et al., 2003a). This area is also affected by the trapping of a Natal Pulses fraction [START_REF] Rouault | New perspectives on Natal Pulses from satellite observations[END_REF][START_REF] Krug | Interactions between the Agulhas Current and the eastern margin of the Agulhas Bank[END_REF]. The Agulhas Retroflection (C) is known for its extremely high mesoscale variability due to the Agulhas Rings, which are anticyclones, with diameters of O( 200) km [START_REF] Ducombe | Agulhas retroflection rings in the South Atlantic Ocean: an overview[END_REF], that may have a baroclinic vertical structure with a strong barotropic component [START_REF] Van Aken | Observations of a young Agulhas ring, Astrid, during MARE in March 2000[END_REF].

The D and E areas are respectively located in the paths of the Agulhas Return Current and the Benguela Current. They both show an EKE equipartitioned between the φ 0 and φ 1 categories and an EAP E mostly into Φ 1 (O(70) %) (Table 4.1). The EKE partitioning corresponds to surface-intensified profiles with a lower energy level at depths than in the A, B and C areas while the EAP E partitionning is the same than in the A, B and C areas (Figure 4.6 middle panels). The Agulhas Return Current (D) is the place of permanent meanders, controlled by topography at 26. • E (Agulhas Plateau) and at 32.6 • E [START_REF] Boebel | Path and variability of the Agulhas Return Current[END_REF]. Its mesoscale variability is due to a variety of mesoscale eddies developing from the meanders [START_REF] Gründlingh | Drift of a satellite-tracked buoy in the southern Agulhas Current and Agulhas Return Current[END_REF]. The EKE vertical structure denotes a moderate mesoscale variability, compared to the SAC and to the Retroflection (as shown in Figure 4.2). The Benguela area (E) mesoscale variability results from locally and remotely generated mesoscale eddies [START_REF] Lutjeharms | The Agulhas Current[END_REF]. The area is affected by the Agulhas Rings drifting northeastward and by cyclonic mesoscale eddies developing along the coastal upwelling fronts (Cape Town ∼ 33 • S northern and southern cells) at topographically-controlled locations [START_REF] Lutjeharms | The Agulhas Current[END_REF]. Typical features are cyclonic mesoscale eddies with diameters about 60 km in areas deeper than 2500 m depth and about 20 km in area between 500-2500 m [START_REF] Rubio | Mesoscale eddy activity in the southern Benguela upwelling system from satellite altimetry and model data[END_REF].

Small vertical scale eddies

The F and G areas are located in the Northern and Middle Subgyre away from the Agulhas Current mean circulation. They both have an EKE equipartitioned between the φ 0 and φ 1 categories and an EAP E which respectively partitions as : Φ 0 ∼ Φ 1 ∼ Φ 2-9 and Φ 1 > (Φ 0 ∼ Φ 2-9 ). These EAP E profiles correspond to profiles intensified over the first 100 m and at intermediate depths (F : O(1000) m and G : O(500) m) and an energy decrease with depth toward a minimum value at the bottom (Figure 4.6 bottom left and middle panels). These EAP E vertical structures denote a higher variability at small vertical scales confined to the surface than at the mesoscale. The Subgyre (F and G) low mesoscale variability is due to the absence of local Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a Western Boundary Current generation mechanism and likely corresponds to the propagation of mesoscale eddies generated in the Agulhas Return Current or in the Subtropical South Indian Gyre. The small vertical scales confined to the surface can be a signature of the mixed-layer related variability, which significance has been recently highlighted in the Subgyre region [START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF].

The H area is located in the Southern Subgyre and shows a singular EAP E partitioning : Φ 0 > Φ 1 >> Φ 2-9 . This one corresponds to a profile intensified over the first 100 m, decreasing with depth toward a very weak value at the bottom without a distinct secondary intensification at intermediate depths (Figure 4.6 bottom right panel). This is a unique area where both EKE and EAP E are mainly partitioned into their barotropic modes, although they represent different vertical structures (Figure 4.4). This area is located into a closed circulation contour (0.75 m sea surface height contour in Figure 4.5) and may denote a region of recirculation.

The EKE and EAP E partitionings also denote together a variability at small vertical scales (φ 2-9 : O(30) % and Φ 2-9 : O(40) %) at two locations along the Agulhas Current : 26 • E and 23 • E (Figure 4.5 top right panel). Around 26 • E, frontal submesoscale cyclonic eddies have been observed and simulated [START_REF] Krug | Submesoscale cyclones in the Agulhas Current[END_REF][START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF].

The eddy energies vertical partitioning (EKE n and EAP E n ) allows to characterize the different eddy structures present in the Agulhas Current region. In the NAC, eddies have contrasted vertical structures between areas along the shelf in the path of the Agulhas Current (A) and offshore in the Northern Subgyre (F). These latter are respectively dominated by intense mesoscale eddies (φ 0 > φ 1 and Φ 1 ) and smaller vertical scale eddies (Φ 2-9 ). In the SAC, eddies have more homogeneous vertical structures. This region is dominated by intense mesoscale eddies (φ 0 > φ 1 and Φ 1 ) in the path of the Agulhas Current (B and C) and by moderate mesoscale eddies (φ 0 ∼ φ 1 and Φ 1 ) in the path of the Benguela Current (E). The contrasted eddy vertical structures in the different regions are the result of different eddy dynamics, which we investigate using our modal EKE budget in the following section. Vector fields denote the EKE 0-1 fluxes (note that there is a factor 10 between the vector fields of the bottom right panel and of the others). The NAC and SAC boxes colors denote the sign of the spatially averaged divergences (red/blue colors : positive/negative) (Figure 4.8). The black isolines denote the 1000 and 3000 m isobaths. The NAC and SAC are respectively moderate (0.17 10 -5 m 3 s -3 ) and high (2.06 10 -5 m 3 s -3 ) mesoscale eddy generation regions. The mesoscale eddy transport by the linear (eddy-pressure work, A.2) and non-linear (advection of EKE 0-1 by the total flow, A.3) dynamics follows the Agulhas Current circulation and denotes inter-oceans exchanges in the SAC area (toward the South-East Atlantic Ocean and the Southern Ocean for the non-linear transport).

Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a Western Boundary Current

Results II : Mesoscale Eddy Kinetic Energy (EKE 0-1 ) budget in the Northern (NAC) and Southern Agulhas

Current (SAC) corresponds to a background signal (Figure 4.7 bottom right panel). The eddy energy transported by the linear and non-linear components of the dynamics follow different evolutions : the linear dynamics (A.2) transports eddy energy that is partially dissipated along the shelf and the non-linear dynamics (A.3) transports eddy energy downstream, outside of the NAC.

In the SAC (Southern box in Figure 4.7), both linear and non-linear incoming EKE 0-1 fluxes follow the Agulhas Current circulation (Eastern face) and the outgoing fluxes are directed toward the South-East Atlantic Ocean (Western and Northern faces) and toward the Southern Ocean (Southern face, at the exception of the linear EKE 0-1 fluxes resulting in an incoming flux). This eddy energy circulation is coherent with the advection of Agulhas Rings and cyclonic lee eddies in the South-East Atlantic Ocean [START_REF] Penven | Generation of cyclonic eddies by the Agulhas Current in the Lee of the Agulhas Bank[END_REF][START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Rubio | Mesoscale eddy activity in the southern Benguela upwelling system from satellite altimetry and model data[END_REF]. Both EKE 0-1 fluxes divergences have positive contributions with the same order of magnitude (eddy-pressure work : 0.97 10 -5 m 3 s -3 and EKE 0-1 advection by the total flow : 1.09 10 -5 m 3 s -3 ), much larger than in the NAC and resulting in an intense EKE 0-1 source. The positive contribution of the eddy-pressure work (A.2) is spread over the SAC, whereas the contribution of the advection of EKE 0-1 by the total flow (A.3) in a source at the southern tip of the Agulhas Bank (19 • -23 • E and 37.15 • -39 • S) where the current separates from the shelf. Both linear and non-linear dynamics transport the locally generated mesoscale eddies outside of the SAC.

Leading-order EKE 0-1 sources and sinks (terms B, C and D)

The leading-order balance between the EKE 0-1 sources and sinks is determined from the spatially averaged EKE 0-1 terms of the r.h.s of EKE n equation (B, C and D terms in Eq. 4.29) over the NAC and SAC (Figure 4.8).

In the NAC and SAC areas combined, the main energy sources for mesoscale eddies are instability processes (B.1 + B.3 : 74 % of the total EKE 0-1 source) and the non-linear unforced interactions (C.1 : 25 % of the total EKE 0-1 source). The main energy sinks are the vertical mixing (D.1 : 41 % of the total EKE 0-1 source) and the interactions canalized by topography (C.2 + C.3 + C.4 : 59 % of the total EKE 0-1 source) (Figure 4.8 top panel). This indicates that mesoscale eddies are mainly generated by instabilities extracting energy from the mean currents and lose energy, in similar proportions, due to the bottom friction-and wind-induced dissipation and due to transfers between energy reservoirs and vertical scales in the presence of topography. The combination of the contribution of the free-surface (C.5), the complement to the advection of EKE n (C.6), the numerical dissipation (D.2) and Coriolis (D.3) represents a low contribution (3 %) of the EKE 0-1 budget compared to the other terms (Figure 4.8 top panel). It confirms that terms C.5 and C.6 are not related to explicit physical processes and that the numerical

(A. 1) (A. 2 + A. 3) (B. 1) (B. 2) (B. 3) (C. 1.1 + C. 1.2) (C. 2) (C. 3) (C. 4) (C. 5) (C. 6) (D. 1) (D. 2) (D. 3) 1 10 4 m 3 s 3
Northern Agulhas Current (NAC)

2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 ), but inhibit it at the same time [START_REF] Dritschel | The stability of a two-dimensional vorticity filament under uniform strain[END_REF]Gula et al., 2015b;[START_REF] Molemaker | Submesoscale instability and generation of mesoscale anticyclones near a separation of the California Undercurrent[END_REF].

x10 5 [m 3 s 3 ] (A. 1) (A. 2 + A. 3) (B. 1) (B. 2) (B. 3) (C. 1.1 + C. 1.2) (C. 2) (C. 3) (C.
The EKE 0-1 source in the Natal Bight mainly corresponds to the HRS 0-1 (B.1) contribution, consistently with the generation of Natal Pulses by a barotropic instability process [START_REF] Van Der Vaart | Stability of western boundary currents with an application to pulselike behavior of the Agulhas Current[END_REF][START_REF] Tsugawa | Generation and growth mechanism of the Natal Pulse[END_REF][START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF], and more weakly to eddy-mean triadic interactions (eddy-mean C.1.1) (Figures 4.9 and 4.10). This EKE 0-1 source is compensated in part by the negative contribution of the advective eddy-mean interactions on slopes (C.3) (Figures 4.9 and 4.10). This term equally transfers energy from EKE to M KE at the mesoscale and from EKE at mesoscale toward M KE at smaller vertical scale (respectively light and dark blues in Figure 4.9). It therefore supports the mesoscale eddies driving mean currents over sloping bottom [START_REF] Holloway | Systematic forcing of large-scale geophysical flows by eddy-topography interaction[END_REF][START_REF] Adcock | Interactions between geostrophic eddies and the mean circulation over large-scale bottom topography[END_REF] and is mainly interpreted as a transfer between energy reservoirs at this location. Indeed, bottom-intensified flows decompose into a barotropic mode, allowing a nonzero bottom velocity, and into baroclinic modes, to cancel or to shape velocity in the upper layers, as tested by [START_REF] Stanley | Vertical structure of ocean mesoscale eddies with implications for parameterizations of tracer transport[END_REF]. In the Natal Bight, the barotropic instability (B.1) and the non-linear eddy-mean interactions across vertical scales (C.1.1) transfers M KE to EKE at mesoscale and the shelf slope partially canalizes a transfer from EKE at mesoscale to M KE (C.3).

The low EKE 0-1 source in the Western Subgyre is mainly constituted of the positive contributions of the triadic interactions (C.1.1) (Figures 4.9 and 4.10). In this offshore area, the effect of the mean current weakens (eddy-mean C.1.1) and the mesoscale eddies dynamics is dominated by non-linear interactions between eddy vertical structures (eddy-eddy C.1.1), which reinforce the mesoscale eddies by realizing an inverse vertical turbulent cascade.

The Southern Agulhas Current (SAC)

The EKE 0-1 source at the Eastern Agulhas Bank Bight and off the Agulhas Bank tip includes the positive contributions of HRS 0-1 (B.1) and V BF 0-1 (B.3) (Figures 4.9 and 4.9). This indicates that mesoscale In the Eastern Agulhas Bank Bight, the advective eddy-mean interactions on slopes reinforces the mesoscale eddies (C.3 > 0). It is mainly interpreted as an energy transfer between vertical structures, because energy is mainly transferred from the M KE at small vertical scales to EKE at mesoscale. The mesoscale eddy dynamics in the Eastern Agulhas Bank Bight is partly consistent with the Shear-edge eddies generation documented by a horizontal shear process controlled by the topography (Lutjeharms et al., 2003b,a). The SAC areas away from the mesoscale eddy generation areas (Eastern Agulhas Bank Bight and Agulhas Bank tip), are characterized by the positive contribution of the eddy-eddy triadic interactions (eddy-eddy C.1.1) (Figure 4.9). This indicates that the mesoscale eddies are reinforced by the non-linear interactions between vertical structures, realizing an inverse vertical turbulent cascade, while they propagate (Figure 4.7).

The SAC area, is characterized by the negative contributions of the vertical mixing (D.1) and of the pressure-gradient eddy-eddy interactions on slopes (C.4) (Figures 4.8 and 4.9). The vertical mixing (D.1) contribution is not confined to shallow areas (81 % of D.1 contribution is in areas of depths > 1000 m) (Figure 4.9). The vertical mixing is therefore attributable to the wind, because the wind dissipates surface energy at all scales [START_REF] Renault | Dampening of submesoscale currents by air-sea stress coupling in the Californian upwelling system[END_REF] and the vertical mixing is also a significant sink for smaller vertical structures in this area (EKE 2-9 , not shown). The vertical mixing is also attributable to the bottom friction, because the barotropic mode accounts for O(70) % of the EKE in the SAC area (Figures 4.4 and 4.5). The pressure-gradient eddy-eddy interactions on slopes (C.4) contribution is localised around topographic accidents and seamounts (Figure 4.9). It represents the EKE transfer toward small vertical structures due to the mesoscale eddy scattering over topographic roughness, generating internal-gravity waves -lee waves - [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF][START_REF] Nikurashin | Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory[END_REF]. The internal-gravity waves represent a significant direct route of the mesoscale eddies dissipation, by breaking and locally dissipating their energy [START_REF] Nikurashin | Global energy conversion rate from geostrophic flows into internal lee waves in the deep ocean[END_REF][START_REF] Scott | Global rate and spectral characteristics of internal gravity wave generation by geostrophic flow over topography[END_REF][START_REF] Garabato | Widespread intense turbulent mixing in the southern ocean[END_REF][START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF]. The sequence of events realizing the direct route of mesoscale eddies dissipation may therefore take place here as well.

The eddy-mean triadic interactions (eddy-mean C.1.1) and the advective eddy-mean interactions on slopes (C.3) show intense signals around topographic accidents and seamounts (Figure 4.9). Each term compensates between the seamounts faces and there is compensation between the two terms on each faces. This double compensation suggests that these signals around topographic roughness do not significantly contribute to the net EKE 0-1 budget. The latter compensation likely comes from the separation of the horizontal velocity gradient into a 2D-horizontal (x) (eddy-mean triadic interactions, C.1.1) and 3D (x, z) (advective eddy-mean interactions on slopes, C.3) components, which are not relevant for these particular areas.

vertical scales in the NAC and SAC are evaluated from the leading-order EKE 0-1 sources and sinks (Figures 4.9,4.10 and 4.11). The intensity of the mesoscale eddy generation by instability processes of the Agulhas Current gradually increases downstream : Natal Bight, Eastern Agulhas Bank Bight and Agulhas Bank tip (Figure 4.9). This agrees with historical characterization [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF] and our description, from observed and modeled EKE (Figure 4.2), of the low and high mesoscale variability level of the Northern and Southern Agulhas Current branches, respectively.

In the NAC, mesoscale eddies mainly lose energy along the shelf (Alongshore Agulhas Current) at the exception of the Natal Bight, where they are locally generated, and they weakly gain energy offshore (Northern Subgyre). This makes the NAC a moderate mesoscale eddies generation location (Figure 4.8). The low mesoscale variability of the Northern Agulhas branch has historically been attributed to the straight and steep shelf slope exerting a strong topographic constraint on the current [START_REF] Lutjeharms | The Agulhas Current[END_REF]; [START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]. The main mesoscale variability of this branch is driven by the Natal Pulse, solitary meanders locally generated at the Natal Bight [START_REF] Schouten | Upstream control of Agulhas Ring shedding[END_REF][START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Tsugawa | Generation and growth mechanism of the Natal Pulse[END_REF] where the shelf curvature widens. Our results show that at the Natal Bight, mesoscale eddies are generated by barotropic instability (B.1 in Figures 4.9 and 4.10), consistently with the Natal Pulse generation process [START_REF] Van Der Vaart | Stability of western boundary currents with an application to pulselike behavior of the Agulhas Current[END_REF][START_REF] Tsugawa | Generation and growth mechanism of the Natal Pulse[END_REF][START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF]. We have also shown that along the straight and steep shelf slope (Alongshore Agulhas Current), the topographic constraint modulates energy transfers from mesoscale eddies to the mean currents (B.1 + eddy-mean C.1.1 : 50 % of the EKE 0-1 sink), mesoscale eddies energy dissipation by bottom friction (D.1 : 24 % of the EKE 0-1 sink) and the eddy-topography interactions canalizes energy transfers from mesoscale eddies to smaller vertical scales (C.2 : 18 % of the EKE 0-1 sink) (Figures 4.9 and 4.10). Although, the small vertical scales numerical dissipation (D.2 2-9 , not shown) is locally high, we cannot conclude on a direct energy route realized by the C.2 contribution, because the mean current (eddy-mean C.1.1 2-9 + C.3 2-9 , not shown) represent the main energy source for small vertical scales in this area.

In the SAC, mesoscale eddies are intensively generated at the Eastern Agulhas Bank Bight and off the Agulhas Bank tip, which makes it an intense mesoscale eddies generation location (Figure 4.8). The high mesoscale variability of the Southern Agulhas branch has historically been attributed to the increase of the shelf curvature, relaxing the topographic constraint on the current [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF]. Our results show that mesoscale eddies are generated where the current separates locally (Eastern Agulhas Bank Bight) and permanently (Agulhas Bank tip) from the shelf by mixed baroclinic-barotropic instabilities (B. 1 + B.3 in Figures 4.9 and 4.11). This instability process is also documented to drive the Gulf Stream mesoscale variability [START_REF] Cronin | Eddy-mean flow interaction in the Gulf Stream at 68 W. Part II: Eddy forcing on the time-mean flow[END_REF]Gula et al., 2015a). We have also shown that for the SAC region, mesoscale eddies mainly lose energy due to eddy-topography interactions (C.3 + C.4 : 57 % of the EKE 0-1 sink) followed by bottom and surface frictional processes (D.1 : 42 % of the EKE 0-1 sink) (Figures 4.8 bottom right panel). These interactions include the energy loss due to mesoscale eddies inducing mean deep currents mainly over the Agulhas Bank tip sloping bottom (C.3) and due to mesoscale eddies generating internal-gravity waves by scattering over seamounts and topographic accidents (C.4). These latter process represents the main energy source for small vertical scales (C.4 2-9 , not shown) whereas bottom friction-and wind-induced dissipation and numerical dissipation are their main energy sink (D.1 2-9 + D.2 2-9 , not shown). Both EKE 0-1 (Figures 4.8 bottom right panel) and EKE 2-9 budgets (not shown) are therefore consistent with the internal-gravity waves being a significant direct route of mesoscale eddies dissipation by breaking, as characterized for the Southern Ocean [START_REF] Garabato | Widespread intense turbulent mixing in the southern ocean[END_REF][START_REF] Nikurashin | Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory[END_REF][START_REF] Scott | Global rate and spectral characteristics of internal gravity wave generation by geostrophic flow over topography[END_REF] and for isolated eddies [START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF]. Even though we did not quantify the local and non-local energy dissipation, respectively due to lee waves and propagating internal waves breaking, this is likely the main direct route of mesoscale eddy dissipation in the SAC region.

In the absence of topographic constraint and away from intense mean currents (Northern Subgyre in NAC and offshore areas in SAC), mesoscale eddies are reinforced by an inverse vertical turbulent cascade (Figures 4.10 and 4.11). In the Northern Subgyre (NAC), the inverse cascade is partially fed by energy at submesoscale. Indeed, the variability in this region is dominated by small vertical scales surface-intensified eddies (Figures 4.5 and 4.6), which are generated by baroclinic instability (B.3 n=2-9 , not shown) and whose energy is partially transferred to mesoscale by the triadic interactions (C.1 n=2-9 < 0 not shown and C.1 n=0-1 > 0 in Figure 4.9). This sequence of events is consistent with the submesoscale inverse turbulent cascade characterized in this region by [START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF] with a different approach : mixed-layer baroclinic instability generates submesoscale eddies which subsequently realize an inverse turbulent cascade.

More generally, our modal EKE budget allows to focus on topographic interactions and to evaluate their significance for the mesoscale eddies energy loss in a Western Boundary Current region. The EKE 0-1 budget of the NAC and SAC regions combined is dominated by that of the SAC and shows that mesoscale eddies mainly lose energy due to eddy-topography interactions (C.2 + C.3 + C.4 : 59 % of the EKE 0-1 sink) followed by bottom friction-and wind-induced dissipation (D.1 : 41 % of the EKE 0-1 sink) (Figure 4.8). Eddy-topography interactions canalize in similar proportions, an energy transfer from mesoscale eddies to the mean currents (C.3 : 27 % of the EKE 0-1 sink) and to smaller vertical scales, likely realizing a direct route to dissipation (C.2 and C.4 : 6 % and 26 % of the EKE 0-1 sink). Our results are therefore a complement to the phenomenological understanding of mesoscale eddy dynamics upon western boundaries. In a western boundary region without a western boundary current, [START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF] evidenced the mesoscale eddies decay by topographic interactions, canalizing a direct energy route toward small scales and dissipation. Our study suggests that in the presence of a western boundary current, mesoscale eddies lose as much energy to the mean current than to smaller vertical scales.

Additionally, our modal EKE budget allows to evaluate the mesoscale eddy-energy net sources and sinks and to test the vision of the mesoscale eddies decay upon Western Boundaries for the Agulhas Current region [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF]Chelton et al., 2011b;[START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF]. The EKE 0-1 flux divergence shows that the Agulhas Current region results, in average, in a mesoscale eddies generation region (Figures 4.7 and 4.8). This result is inconsistent with [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] which suggests Western Boundary regions as ubiquitous mesoscale eddy-energy sinks and the Agulhas Current region should be the largest sink. Our results support a different vision of the mesoscale eddy dynamics in Western Boundary Current regions. The local mesoscale eddies generation can overcome the local dissipation of mesoscale eddies generated remotely, almost everywhere basin-wide, which propagate westward (Chelton et al., 2011b). The discrepancies between our study and [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] come from the different methods used to diagnose the mesoscale eddy-energy sources and sinks. Using a regional numerical model, we will investigate in a future study the ability of both methods to account for the mesoscale eddy-energy sources and sinks.

Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

Abstract

Western Boundary regions were suggested as graveyard for remotely-generated mesoscale eddy using altimetry data. The mesoscale eddy-energy sources and sinks are inferred by the divergence of the eddyenergy fluxes (eddy-pressure work and advection of eddy-energy) and their diagnostic using altimetry data alone requires several approximations. Using a realistic oceanic numerical simulation (dx ∼ 2.5 km), we diagnose the eddy-pressure work in the South Indian Ocean western boundary area, the Agulhas Current region, and compare the results to diagnostics based on altimetry data. For geostrophic velocities, the eddy pressure-work results in a residual meridional eddy-energy flux. At the mesoscale range, it is a negligible contribution to the eddy-energy sources and sinks compared to the contribution of the ageostrophic dynamics. For the 1 st baroclinic mode, the eddy-pressure work characterizes the Agulhas Current region as an eddy-energy sink and for the two vertical modes describing the entire mesoscale eddy reservoir vertical structure (barotropic and 1 st baroclinic mode), it characterizes this region as an eddy-energy source. Ours results suggest that the mesoscale eddy-energy sources and sinks inferred from the eddy-pressure work based on altimetry data alone are biased. For the linear divergence (eddy-pressure work), the mesoscale net eddy-energy source of the Agulhas Current is largely underestimated (96 %). This divergence component dominates in the Interior region, where an energy sink of remotely-generated mesoscale eddies is found. Our results suggest that the local generation of mesoscale eddy overcomes the local dissipation of the remotely-generated ones in the Agulhas Current region. This calls for further investigations to quantify the locally-and remotely-generated mesoscale eddies contributions to the dynamics in other Western Boundary Current regions.

Introduction

Our understanding of the oceanic dynamics has been revolutionized by the advances in satellite altimetry in the late 90s [START_REF] Le Traon | An improved mapping method of multisatellite altimeter data[END_REF]. The processing of the altimetry measure by AVISO (Archiving, Validation and Interpretation of Satellite Oceanographic) provided the first global-scale picture of the surface oceanic dynamics [START_REF] Ducet | Global high-resolution mapping of ocean circulation from TOPEX/Poseidon and ERS-1 and-2[END_REF]. These data evidenced the ubiquity of mesoscale eddies at the ocean surface and allowed to characterize their dynamics at scales down to O(100) km [START_REF] Chelton | Global observations of large oceanic eddies[END_REF](Chelton et al., , 2011b)).

The scale of the mesoscale eddies is given by the 1 st Rossby deformation radius (Rd), about O( 40) km at mid-latitudes [START_REF] Chelton | Geographical variability of the first baroclinic Rossby radius of deformation[END_REF], or larger (Chelton et al., 2011b). At these scales, motions are in geostrophic and hydrostatic balances, this is the so-called balanced dynamics. Mesoscale eddies have surface-intensified vertical structures which are described by the barotropic and 1 st baroclinic modes [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF][START_REF] Venaille | Baroclinic turbulence in the ocean: Analysis with primitive equation and quasigeostrophic simulations[END_REF]Tedesco et al., submitted). Because the latter is surface-intensified, the altimetry data are usually interpreted as primarily reflecting this mode [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF].

Mesoscale eddies are a major component of the global ocean energy budget [START_REF] Müller | Routes to dissipation in the ocean: The 2D/3D turbulence conundrum[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. Wind drives the upper-ocean basin-scales mean circulation. Energy is then injected around Rd in the form of mesoscale eddies by instability processes of the mean currents. The approximate agreement between linear theory and the mesoscale eddies scale observed by satellite altimetry supports the view of the mesoscale eddies generated by baroclinic instability [START_REF] Gill | Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies[END_REF][START_REF] Smith | The geography of linear baroclinic instability in Earth's oceans[END_REF][START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. In particular, [START_REF] Gill | Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies[END_REF] showed that the ocean is baroclinically unstable almost everywhere. This supports the observation by satellite altimetry of mesoscale eddies originating almost everywhere (Chelton et al., 2011b). Different dynamics and instability processes take place in western boundary currents, such as the Gulf Stream (Gula et al., 2015a) and the Agulhas Current [START_REF] Pichevin | Why are there Agulhas rings[END_REF][START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Elipot | Characteristics, Energetics, and Origins of Agulhas Current Meanders and Their Limited Influence on Ring Shedding[END_REF], completing the view of the mesoscale eddies generation.

The picture of the global ocean energy budget is completed by the geostrophic turbulence theory [START_REF] Rhines | The dynamics of unsteady currents[END_REF][START_REF] Rhines | Geostrophic turbulence[END_REF][START_REF] Salmon | Baroclinic instability and geostrophic turbulence[END_REF][START_REF] Fu | Nonlinear energy and enstrophy transfers in a realistically stratified ocean[END_REF]. It characterizes the non-linear interactions between mesoscale eddies and predicts that around Rd, the 1 st baroclinic mode energy is transferred to the barotropic mode from where it is transferred to larger horizontal scales by an inverse turbulent cascade. The barotropic energy is then dissipated by friction at the bottom. Altimetry evidenced an inverse turbulent cascade at the surface [START_REF] Scott | Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry[END_REF]. It is an inverse turbulent cascade of baroclinic Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

energy and therefore completes the geostrophic turbulence theory. The horizontal scale at which this inverse turbulent cascade starts at the surface is variable. [START_REF] Scott | Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry[END_REF]; [START_REF] Arbic | On eddy viscosity, energy cascades, and the horizontal resolution of gridded satellite altimeter products[END_REF][START_REF] Arbic | Geostrophic turbulence in the frequency-wavenumber domain: Eddy-driven low-frequency variability[END_REF] determined the inverse cascade extended from affects large scales down to Rd, while [START_REF] Khatri | Surface ocean enstrophy, kinetic energy fluxes, and spectra from satellite altimetry[END_REF] determined that it was effective only at scales larger than O(200) km. [START_REF] Tulloch | Scales, growth rates, and spectral fluxes of baroclinic instability in the ocean[END_REF] showed, from a combination of linear theory, satellite altimetry observations and a primitive-equation model, that baroclinic instability injects energy at a scale smaller than Rd, which lies in the inverse turbulent cascade scale range at mid-latitude, and that the eddy equilibrated scale is in the range of the surface inverse turbulent cascade.

Mesoscale eddy-energy dissipation is an important aspect of the mesoscale eddy dynamics, which has been seldomly investigated using satellite altimetry. In the context of the mean modal Eddy Kinetic Energy budget (EKE n ), the eddy-energy sources and sinks for a vertical mode n are evaluated by the modal eddy-energy fluxes divergence (Tedesco et al., submitted). This term corresponds to the sum of the local EKE n contributions and according to its sign it indicates a local EKE n generation (> 0) or a local EKE n dissipation (< 0). The divergence is constituted of two components accounting for the linear and non-linear part of the dynamics. Diagnosing the linear divergence term from altimetry data, [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] suggested that western boundaries are ubiquitous mesoscale eddy-energy sinks, with the South Indian Ocean Western Boundary representing the largest one. This result would close the mesoscale eddies life cycle in the following way : mesoscale eddies are generated by instability of the mean currents, they transfer their energy toward larger horizontal and vertical scales while propagating westward and they decay upon western boundaries due to eddy-topographic interactions. This result is therefore of importance for the global ocean energy budget. [START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF] completed the phenomenological understanding of the mesoscale eddies decay on western boundaries by evidencing a direct energy cascade realized by eddy-topographic interactions.

However, [START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF] study focused on a western boundary region excluding western boundary currents and [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] study is based on satellite altimetry data. These data are a measure of the large-mesoscale balanced motions mostly of the 1 st baroclinic mode, using the usual interpretation of the surface primarily reflecting this surface-intensified mode [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF]. [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] diagnostic therefore points toward the contribution of a specific dynamics to the mesoscale eddy-energy sources and sinks. This calls for further investigations to evaluate the validity of these various dynamical assumptions and investigate how ell the total eddy-energy sources and sinks can be characterized with such computation. [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] estimation of the South Indian Ocean western boundary as the largest mesoscale eddyenergy sink designates the Agulhas Current as a suitable study region to evaluate the possible limitations of satellite altimetry based computations. In this study, we evaluate the significance of the altimetry-based energy budgets for the mesoscale eddy sources and sinks against a complete energy budget using regional numerical simulation in the Agulhas Current region. We particularly address the following questions :

(1) Is the 1 st baroclinic mode linear divergence (eddy-pressure work) diagnosed from altimetry data is representative of the mesoscale eddy-energy sources and sinks ? (2) Which approximation, among the ones implied by the use of satellite altimetry data, limit the evaluation of the mesoscale eddy-energy sources and sinks ?

The study is organised as follows : the regional numerical simulation is first presented, the vertical modes are defined and the eddy-energy fluxes divergence linear components, inferred from the total-velocity and the sea surface height fields, are then defined and interpreted in the frame of the EKE n budget in section 2. The eddy-energy sources and sinks for the 1 st baroclinic mode diagnosed for both expressions of the linear divergence are compared in section 3. The three assumptions required for diagnosing the mesoscale eddy-energy sources and sinks using the sea surface height field : geostrophically-balanced dynamics, the vertical structure of the mesoscale eddies inferred from the 1 st baroclinic mode and the linear dynamics, are evaluated in section 4. The results are summarized and discussed in section 5.

Methods

∼ 7.5 km grid and exerts a feedback on this one at each time step. The surface forcing is provided by a bulk-formulation [START_REF] Fairall | Bulk parameterization of air-sea fluxes for tropical ocean-global atmosphere coupled-ocean atmosphere response experiment[END_REF] using the ERA-ECMWF's reanalysis [START_REF] Dee | The ERA-Interim reanalysis: Configuration and performance of the data assimilation system[END_REF] with relative winds [START_REF] Renault | Modulation of the Agulhas Current Retroflection and Leakage by Oceanic Current Interaction with the Atmosphere in Coupled Simulations[END_REF] and the bathymetry is built from the the General Bathymetric Chart of the Oceans 30-s resolution [START_REF] Weatherall | A new digital bathymetric model of the world's oceans[END_REF]. The numerical simulation is run for the 1993-2014 period after a 1-year spin up from its initial state. The four nested grids and the numerical choices contributing to their realisms (bathymetry, grid, parameterization of the turbulent closure, momentum and advective numerical schemes) are presented in more details in [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF].

The dx ∼ 2.5 km grid simulates the Agulhas Current from its source, north of the Natal Bight (∼ 31 • E), to its Retroflection, south of the Agulhas Bank (∼ 19 • E -20 • E), where it becomes the Agulhas Return Current and flows eastward (Figure 5.1). With an effective resolution of about 10 dx ∼ 25 km [START_REF] Soufflet | On effective resolution in ocean models[END_REF], the grid solves the 1 st internal Rossby deformation radius (Rd : O(30-40) km Figure 5.1) and is therefore adequate to investigate the 1 st baroclinic mode dynamics. In addition, the mesoscale variability characteristics in the dx ∼ 2.5 km numerical simulation is shown to be in good agreement with satellite altimetry and in situ data in Tedesco et al. (2019, submitted).

The mean modal Eddy Kinetic Energy (EKE n ) budget

We present in the following the EKE n budget and focus in particular on the eddy-energy fluxes divergence. We first define the traditional vertical modes [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF]. We then define the modal eddy-energy fluxes divergence and interpret it in the frame of the EKE n budget. Finally, we define the modal eddy-energy fluxes divergence linear component (eddy-pressure work) in the frame of satellite altimetry data [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF].

Vertical modes

A convenient approach to describe the vertical structure of mesoscale motions is a vertical modes decomposition [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF]. The vertical modes (φ n ) of the horizontal velocity component (u) and of the eddy pressure (p ′ ) are the eigenfunctions solution of the following Sturm-Liouville problem, Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

with the linearized free-surface (5.2) and flat-bottom boundary conditions (5.3) :

∂ ∂z ( 1 N 2 ∂ ∂z φ n ) + 1 c 2 n φ n = 0 (5.1) ( ∂ ∂z φ n ) z=η = ( -N 2 g φ n ) z=η (5.2) ( ∂ ∂z φ n ) z=-H = 0 (5.3)
with N 2 the buoyancy frequency, g the acceleration of gravity and c 2 n the vertical modes corresponding eigenvalues. The overbar denotes a time average and the prime denotes fluctuations relative to this average. The vertical modes are normal to each other and satisfy the orthogonality condition :

η -H φ m φ n dz = δ mn h (5.4)
with δ mn the usual Kronecker symbol and h = η + H the water column depth.

The vertical modes are related to horizontal scales via their eigenvalues, which are good approximations of the Rossby baroclinic deformation radii :

Rd n≥1 = cn |f | = 1 nπ|f | η -H N (x, z
) dz, with f the Coriolis parameter [START_REF] Chelton | Geographical variability of the first baroclinic Rossby radius of deformation[END_REF].

We diagnosed the vertical modes from the buoyancy frequency computed from the modeled temperature and salinity outputs and averaged over the 1995-2004 period. The u and p ′ variables are projected onto the vertical modes (u n ,p ′ n ) with the following method :

[u n (x, t), p ′ n (x, t)] = 1 h η -H [u(x, z, t), p ′ (x, z, t)]φ n (x, z) dz (5.5)
In the present study, we use the barotropic and first 9 baroclinic modes which capture 80-95 % of the modeled dynamics by the dx ∼ 2.5 km grid (Tedesco et al., submitted).

The eddy-energy fluxes divergence in the mean modal Eddy Kinetic Energy (EKE n ) budget

• The eddy-energy fluxes divergence exact expression

The modal EKE n budget, corresponds to the classic EKE equation [START_REF] Harrison | Energy analysis of open regions of turbulent flows-Mean eddy energetics of a numerical ocean circulation experiment[END_REF],

Methods

projected on vertical modes (Tedesco et al., submitted). The EKE n equation : • The eddy-pressure work term altimetry-based expression [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] derived an alternative expression of the eddy-pressure work to diagnose it from satellite altimetry. Using the hydrostatic balance (Eq. 5.7) and assuming that the surface primarily reflects the 1 st baroclinic mode [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF], leading to Equation (5.8), results in the following approximation (Eq. 5.9) between the pressure anomaly modal coefficient for the 1 st baroclinic mode (p ′ 1 (x, t)) and the sea surface height (η(x, t)) :

ρ 0 u ′ n .(h( ∂ ∂t u) n ) T ime rate + ∇ H .(u ′ n p ′ n η -H φ 2 n dz) Eddy-pressure work + ∇ H .(u n ρ 0 2 ||u ′ n || 2 η -H φ 3 n dz)
∂ ∂z p(x, z, t) = -gρ(x, z, t) (5.7) p ′ (x, z = 0, t) = gρ 0 η ′ (x, t) ∼ p ′ 1 (x, t)φ 1 (x, z = 0) (5.8) p ′ 1 (x, t) ∼ ρ 0 gη ′ (x, t) φ 1 (x, z = 0) (5.9)
with ρ(x, z, t) the density variations relative to the density reference ρ 0 . The expression for the 1 st baroclinic mode 3D horizontal velocity field depending on the 2D sea surface height field (Eq. 5.11) is obtained by replacing (Eq. 5.9) into the modal geostrophic balance (Eq. 5.10) : .11) with f = f 0 + βy expressed for the β-plan approximation.

u ′ 1 (x, t)φ 1 (x, z) = k ∧ 1 ρ 0 f ∇ H (p ′ 1 (x, t)φ 1 (x, z)) (5.10) u ′ 1 (x, t)φ 1 (x, z) ∼ k ∧ g f ∇ H (η ′ (x, t) φ 1 (x, z) φ 1 (x, z = 0) ) ( 5
The expression of the eddy-pressure work diagnosable from satellite altimetry (Eq. 5.15) is finally obtained by replacing the total-velocity field in the exact expression of the eddy-pressure work (Eq. 5.6) by the geostrophic-velocities defined using sea surface height field (Eq. 5.11) and assuming that the horizontal variations of topography and sea surface height (terms from Leibniz rule in Eq. 5.12) are negligible relative to the eddy-energy fluxes ones :

∇ H .(u ′ 1 p ′ 1 η -H φ 2 1 dz) Eddy-pressure work f rom velocity f ield = ∇ H . η -H u ′ 1 p ′ 1 φ 2 1 dz = η -H ∇ H .(u ′ 1 p ′ 1 φ 2 1 ) dz + ∇ H .ηu ′ 1 p ′ 1 |φ 2 1 | z=η + ∇ H .Hu ′ 1 p ′ 1 |φ 2 1 | z=-H terms f rom Leibniz rule (5.12) ∼ - β 2ρ 0 f 2 η -H ∂ x (p ′2 1 φ 2 1 ) dz (5.13) ∼ - βρ 0 g 2 2f 2 η -H ∂ ∂ x ( φ 2 1 (x, y, z) φ 2 1 (x, y, z = 0) η ′2 ) dz (5.14) ∼ - βρ 0 g 2 2f 2 ∂ ∂ x ( η -H φ 2 1 (x, y, z) dz φ 2 1 (x, y, z = 0) η ′2 )
Eddy-pressure work f rom η f ield (5.15) 5.4. Result I : 1 st baroclinic mode eddy-energy sources and sinks diagnosed from the exact and altimetry-based eddy-pressure work terms 1 st baroclinic eddy-energy sink integrated over the South Indian Ocean western boundary to be -0.046 10 12 W. This is consistent with our estimations of the energy sink diagnosed from the altimetry (-0.0017 10 12 W) and simulated η fields (-0.00081 10 12 W), knowing that our integration region along the western boundary (south of 28 • S, Figure 5.2) is less extended than the one in [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] (south of 10 • S).

The simulated velocity and η eddy-pressure work terms result in a 1 st baroclinic mode eddy-energy sink in the Western Boundary region (velocity field : -1.19 10 -2 W m -2 and η field : -5.01 10 -3 W m -2 ) and in an eddy-energy source in the Interior region (velocity field : 6.93 10 -3 W m -2 and η field : 1.32 10 -4 W m -2 ), but of different magnitudes (Figures 5.2b and 5.2d).

The comparison between the velocity and η eddy-pressure work terms is not direct and their largescale signals shows fairly different local patterns, even after smoothing (Figures 5.2a,5.2b and 5.2d). This questions the altimetry-based diagnostics (geostrophically-balanced linear dynamics of the 1 st baroclinic mode) as the leading-order contribution to the 1 st baroclinic eddy-energy sources and sinks. The differences between the eddy-pressure work diagnosed from the total-velocity and η are attributable to the assumptions made in deriving the equations, because these differences are robust to the data (numerical outputs vs. altimetry data).

Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

Result II : Contribution of the altimetry-based eddy-pressure work term to the mesoscale eddy-energy sources and sinks

In this section, we interpret the eddy-pressure work divergence diagnosed from altimetry data (Figure 5.2a) by evaluating the biases induced by each approximations separately. The approximations implied by the use of altimetry data : geostrophically-balanced dynamics (subsection 4.1) and the mesoscale eddy vertical structure described by the 1 st baroclinic mode alone (subsection 4.2) are first evaluated. The approximation of a linear dynamics (subsection 4.3) is also evaluated.

Is geostrophically-balanced dynamics the main contribution to the 1 st baroclinic mode eddy-pressure work term ?

We evaluate the bias induced by the geostrophically-balanced dynamics on the 1 st baroclinic eddy-energy sources and sinks by comparing the eddy-pressure work terms diagnosed from the simulated total-(Eq. 5.6) and geostrophic-velocity fields (Eq. 5.17

Result II : Contribution of the altimetry-based eddy-pressure work term to the mesoscale eddy-energy sources and sinks

The eddy-pressure work terms diagnosed from the total-(Figure 5.3a) and geostrophic-velocity fields (Figure 5.3b) show strong discrepancies. The latter has similar patterns to the simulated-η eddy-pressure work (Figure 5.2b), but with uniformly lower magnitudes. This suggests that assuming that the surface primarily reflects the 1 st baroclinic mode [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF], leading to (Eq. 5.8), results in an intensification of the geostrophic-velocity eddy-pressure work. The geostrophic-velocity eddy-pressure work results in a weak 1 st baroclinic mode eddy-energy sink in the Western Boundary region (-5.14 10 -5 W m -2 ), compared to the total-velocity one (-1.19 10 -2 W m -2 ). While in the Interior region, it results in a low net contribution (-4.06 10 -4 W m -2 ) opposite to the total-velocity one (6.93 10 -3 W m -2 ). Therefore, this term does not represent the leading-order eddy-energy sources and sinks, suggesting that the geostrophically-balanced part of the dynamics is not the main contribution to the eddy-energy divergence.

The exact eddy-pressure work term (Eq. 5.6) expresses as follows for the 1 st baroclinic mode when replacing the total-by the geostrophic-velocity fields (Eq. 5.10) :

∇ H .(u ′ 1 p ′ 1 η -H φ 2 1 dz) ∼ η -H ∇ H .(u ′ 1,geos p ′ 1 φ 2 1 ) dz ∼ η -H (u ′ 1,geos φ 1 ).∇ H (p ′ 1 φ 1 ) dz P ressure-gradient term (A) + η -H (p ′ 1 φ 1 )∇ H .(u ′ 1,geos φ 1 ) dz Divergence term (B) (5.16) ∼ - β 2ρ 0 f 2 η -H ∂ ∂ x (p ′2 1 φ 2 1 ) dz Divergence term (B)
(5.17)

∼ - β f v ′ 1,geos p ′ 1 η -H φ 2 1 dz
Divergence term (B)

(5.18)

The geostrophic-velocity eddy-pressure work decomposes into two components (A and B in 5.16). For geostrophic velocities, the pressure-gradient term (A) is null whereas the divergence term (B) is null for the f -plan approximation (f = f 0 , a constant) and results in a residual depending on the β parameter in the β-plan approximation (f = f 0 + βy latitudinally varies). The eddy-pressure work is not null for the geostrophic balance only in the β-plan approximation and corresponds then to a term of meridional EKE 1 -flux driven by the β parameter (Eq. 5.18). This contribution of the geostrophically-balanced dynamics to the 1 st baroclinic mode eddy-energy sources and sinks is low compared to the ageostrophic dynamics ones. For ageostrophic velocities, term A is not null and term B has a larger contribution than for the geostrophic velocities, as suggested by a scaling of the geostrophic (Eq. 5.18) and ageostrophic B terms ratio : The scaling of the ratio of the geostrophic and ageostrophic B terms of the eddy-pressure work confirms that the mesoscale eddy-energy sources and sinks are dominated by the ageostrophic dynamics. This scaling can also be expressed as the ratio of the motion length scale (L) to the Rhines scale (Rh). This latter denotes the shift between non-linear eddy dynamics (turbulent energy cascade) and linear waves dynamics (radiating energy) [START_REF] Rhines | Waves and turbulence on a beta-plane[END_REF]. According to the Rhines scale, the mesoscale range (L < Rh) is characterized by the non-linear eddy dynamics and our scaling shows that this one is dominated by its ageostrophic part.

-β f v ′ 1,geos p ′ 1 η -H φ 2 1 dz ∇ H .(u ′ 1,ageos p ′ 1 η -H φ 2 1 dz) ∼ β f U ′ P ′ H 1 L ǫU ′ P ′ H = βL f ǫ = βL 2 U = L 2 Rh 2 = 10 -2 (5.19) with u ′ 1,geos ∼ U ′ , u ′ 1,ageos ∼ ǫU ′ , p ′ 1 ∼ P ′ , η -H φ 2 1 dz ∼ H, ∇ H ∼ 1 L ,

Is the 1 st baroclinic mode the main contribution to the mesoscale eddypressure work term ?

The vertical structure of mesoscale eddies is described by the barotropic and the 1 st baroclinic mode [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF][START_REF] Venaille | Baroclinic turbulence in the ocean: Analysis with primitive equation and quasigeostrophic simulations[END_REF]Tedesco et al., submitted). However, the approximation of their vertical structure by the 1 st baroclinic mode alone is supported by the use of altimetry data which is usually interpreted as primarily reflecting this surface-intensified mode [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF]. We first evaluate the interpretation of altimetry data primarily reflecting the 1 st baroclinic mode by determining the vertical modes (n = 0, n = 1, n = 2 -9) contribution to the eddy surface signal using the modal sea surface height variance (η ′2 n ). We then evaluate the approximation of the 1 st baroclinic mode representing alone the mesoscale eddy vertical structure by comparing the eddy-pressure work diagnosed from the simulated velocity field (Eq. 5.6) for the same vertical modes categories.

The vertical modes contribution to the eddy surface signal

The sea surface height is a 2D field and therefore does not project onto the vertical modes, but its modal coefficient can be inferred from the modal pressure at the surface such as : η n (x, t) = 1 gρ 0 p n (x, t)φ n (x, z = 0). The modal expansion of the sea surface height variance, subsequently referred as η ′2 n , expresses as follows :

η ′2 = ∞ n=0 η ′ n ∞ m=0 η ′ m = ∞ n=0 (η ′2 n + ∞ m =n η ′ n η ′ m Intermodal coupling (Cnm) ) = ∞ n=0 (η ′2 n + C nm ) (5.20)
The η ′2 n expression (5.20) includes a coupling term between modal coefficients (C nm ). It originates from η ′2 n being a 2D variable independent from the vertical dimension which therefore does not involve the orthogonality condition (5.4).

The vertical modes categories (n = 0, n = 1, n = 2 -9) signal on η ′2 ) shows the predominance of the 1 st baroclinic mode (O(60) %) (Figure 5.4a). This is consistent with the usual interpretation of the surface dynamics as primarily reflecting the 1 st baroclinic mode in the open ocean [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF]. The main contribution of the 1 st baroclinic mode to the η ′2 signal is mostly uniformly distributed, but shows local maximum (O(70) %) along the continental shelf and minimum (O(30) %) over the Agulhas Plateau (Figure 5.4a). The barotropic mode has a non-negligible contribution (O(30) %) with a contrasted distribution. It is higher (O(40-60) %) at the Agulhas Retroflection, Agulhas Plateau and in the Subgyre (Figure 5.4b). The other baroclinic modes (n = 2 -9) have an uniformly low contribution to the η ′2 signal (O(10) %) (Figure 5.4c). This result indicates that approximating the η ′ field by the 1 st baroclinic mode alone leads to an overestimation of this vertical mode signal. This therefore explains the magnitude difference between the geostrophic and simulated η field eddy-pressure work terms (Figures 5.3b and 5.2b). The geostrophic term is explicitly diagnosed for the 1 st baroclinic mode whereas for the η field term, the mode is approximated by the η field, which accounts for this mode and in a lesser extend for the barotropic mode.

The vertical modes contribution to the eddy-pressure work term

The eddy-pressure work term for the barotropic (n = 0) and the 1 st baroclinic mode (n = 1) have several patterns of opposite signs along the Agulhas Current, in the Agulhas Retroflection, the Benguela area and at the Agulhas Plateau (Figures 5. 5a andb). These compensating patterns emphasize the regions where energy is transferred between the barotropic and the 1 st baroclinic modes by the barotropization and Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

Along the Western Boundary, the mesoscale eddy-energy sources and sinks are shaped by the topography characteristics (Figure 5.5c) consistently with the documented Agulhas Current variability [START_REF] Lutjeharms | The Agulhas Current[END_REF]. Mesoscale eddies gain energy where the shelf curvature widens (at the Natal Bight : ∼ 31 • E) and where the current separates from the shelf (at the Eastern Agulhas Bank Bight and off the Agulhas Bank tip : ∼ 23 • E) and lose energy where the shelf is narrow and the shelf slope is steep (31

• E -26 • E).
The eddy-pressure work term for the other baroclinic modes (n = 2 -9) is an order of magnitude lower than the mesoscale eddies one (n = 0 -1) over the entire Agulhas Current region (Figure 5.5d). However, the Western Boundary region corresponds to a higher eddy-energy sink for these modes than for the mesoscale eddies (n = 2 -9 : -3.5 10 -3 W m -2 and n = 0 -1 : -6.10 10 -4 W m -2 ). The Interior region also corresponds to an energy sink for these baroclinic mode (-1.58 10 -5 W m -2 ), but this is negligible compared to the Western Boundary one.

Approximating the mesoscale eddy vertical structure by the 1 st baroclinic mode alone does not account for the energy transfer within the mesoscale vertical structure range (barotropic and 1 st baroclinic vertical mode), which are not negligible at and away from the shelf in the Agulhas Current region.

Is the mesoscale eddy-pressure work the main contribution to the mesoscale eddy-energy fluxes divergence ?

Our study focuses on the mesoscale eddy dynamics in the Agulhas Current region, pointing toward the non-linear part of the dynamics. According to the Rhines scale, the mesoscale range (L < Rh) is in fact characterized by the non-linear eddy dynamics and the Agulhas Current is characterized by an intense variability at mesoscale [START_REF] Lutjeharms | The Agulhas Current[END_REF]. However, the approximation of the eddy-energy fluxes divergence by its linear component is supported by the mesoscale eddies propagating westward like long baroclinic waves (Chelton et al., 2011b). We evaluate the approximation of the divergence by its linear component by comparing the mesoscale eddy-energy sources and sinks diagnosed from the linear (eddy-pressure work in Eq. 5.6) and non-linear (advection of EKE n in Eq. 5.6) divergences.

The mesoscale (n = 0 -1) eddy-pressure work and advection of EKE n have comparable magnitudes over the entire Agulhas Current region (Figures 5. 6b andc). The advection of EKE n is less contrasted and shows intense mesoscale eddy sources where the current separates from the shelf (at the Eastern Agulhas Bank Bight and off the Agulhas Bank tip) that add up to the eddy-pressure work ones. In the Western Boundary and Interior regions, both divergences result in net contributions of the same sign, but of different magnitudes (Figure 5.6d). In the Western Boundary region, the advection of EKE n represents 96 % of the mesoscale eddy-energy source and in the Interior region, the eddy-pressure work represents 75 % of the energy sink. This indicates that for the regions of intense mesoscale variability the non-linear divergence represents the main net contribution to the mesoscale eddy-energy sources and sinks and conversely for the low mesoscale variability regions.

Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

The approximation of the divergence by one of its component is not valid everywhere in the Agulhas Current region, it is dynamically and thus spatially-dependent and requires the knowledge of the local eddy dynamics.

Summary and Discussion

In this study, we use a regional numerical simulation to evaluate the significance of the altimetry-based energy budgets for the mesoscale eddy sources and sinks in the Agulhas Current region. We especially address the following questions : (1) Is the 1 st baroclinic mode linear divergence (eddy-pressure work) diagnosed from altimetry data is representative of the mesoscale eddy-energy sources and sinks ? (2) Which approximation, among the ones implied by the use of satellite altimetry data, limit the evaluation of the mesoscale eddy-energy sources and sinks ?

The approximation of the total velocity field by the geostrophic one is the most limiting. The eddypressure work diagnosed from geostrophic-velocities (Eq. 5.17) is not null only in the β-plan approximation and corresponds then to a residual meridional EKE 1 -flux. In the mesoscale eddy range, it has a negligible contribution to the eddy-energy sources and sinks compared to the one of the ageostrophic dynamics. This latter corresponds to scale of O(0.1-10) km which are not measured by the AVISO data which have an effective resolution of about O(100) km (Chelton et al., 2011b). This results does not necessarily discard the satellite altimetry data for estimating the mesoscale eddy-energy sources and sinks. The coming satellite altimetry SWOT mission aims to measure the sea surface height fields at resolutions of O(15-30) km [START_REF] Morrow | Global observations of fine-scale ocean surface topography with the Surface Water and Ocean Topography (SWOT) mission[END_REF]d'Ovidio et al., 2019). This would account for the geostrophic and partially for the ageostrophic surface dynamics. SWOT mission might represent an alternative to AVISO data for estimating the mesoscale eddy-energy sources and sinks, managing the dissociation between the balanced from unbalanced velocities from the sea surface height fields and the incorporation of our two last points (surface reflecting the barotropic and 1 st baroclinic modes and the non-linear divergence). The Doppler satellite oceanography directly measures the surface velocity field and may also represent an alternative dataset.

The approximation of the mesoscale eddy vertical structure by the 1 st baroclinic mode alone, supported by the usual interpretation of the surface primarily reflecting this mode [START_REF] Wunsch | The vertical partition of oceanic horizontal kinetic energy[END_REF][START_REF] Smith | The scales and equilibration of midocean eddies: Freely evolving flow[END_REF], is also limiting. The barotropic and 1 st baroclinic modes eddy-pressure work terms have patterns that compensate each other. Their sum results in an opposite contribution than the 1 st baroclinic mode one for the eddy-energy sources and sinks (Figures 5.5a,5.5b and 5.5c). In addition, the surface eddy signal mostly corresponds to the 1 st baroclinic mode (spatial average of O(60) % Figure 5.4b) and in a lesser extend to the barotropic mode (spatial average of O(30) % Figure 5.4a). Even though altimetry data account for both vertical modes, the data do not account for the ageostrophic dynamics which significantly contribute to the energy transfers between these modes. This present result supports then the significance Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

of both vertical modes and of the ageostrophic part of the dynamics to evaluate the mesoscale eddy-energy sources and sinks.

The approximation of the eddy-energy fluxes divergence by its linear component is also often limiting. This approximation is reliable at 75 % in the Interior region, while the approximation by its non-linear component is reliable at 96 % in the Western Boundary region. The diagnostic of mesoscale eddy-energy sources and sinks by the divergence (Eq. 5.6) requires then the knowledge of the local mesoscale eddy dynamics or the computation of both divergence components.

Diagnosing the complete divergence term in a regional numerical simulation (dx ∼ 2.5 km), we characterize the Agulhas Current Western Boundary region net contribution to the mesoscale eddy-energy sources and sinks. The Western Boundary corresponds to a local generation region of mesoscale eddies dominated by the non-linear ageostrophic dynamics. While the Interior corresponds to a local dissipation region of remotely-generated mesoscale eddies dominated by the linear ageostrophic dynamics. Our study suggests that upon the South Indian Ocean western boundary, the local generation of mesoscale eddies by the Agulhas Current would overcome the local dissipation of remotely-generated mesoscale eddies evaluated by [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF]. This calls for further investigations to reconcile these two visions of the mesoscale eddy dynamics in western boundary current areas. A first step would be to diagnose a "local" and "remote" eddy-energy fluxes divergence, in order to explicitly quantify the remotely-generated mesoscale eddy decay upon the South Indian Ocean western boundary. vertical modes. Our EKE budget diagnostics are based on a set of realistic numerical simulations of the Agulhas Current region built from the Coastal and Regional COmmunity (CROCO) model [START_REF] Shchepetkin | The Regional Oceanic Modeling System (ROMS): A splitexplicit, free-surface, topography-following-coordinate ocean model[END_REF][START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF].

The contribution of Chapters 2 and 3 to the knowledge of the Agulhas Current meso and submesoscale variability is first discussed. The contribution of Chapter 3 to the phenomenological understanding of the energy transfers between the different vertical structures in the Agulhas Current region is then discussed. Finally the contribution of Chapters 3 and 4 to the knowledge and understanding of the Agulhas Current region net contribution to the mesoscale eddy-energy sources and sinks is discussed.

Overall discussion and Perspectives

The meso and submesoscale variability of the Agulhas Current

In Chapters 2 and 3, we completed the historic description of the Agulhas Current variability into a Northern and a Southern Agulhas Current branches connecting around Port Elizabeth (∼ 26 • ) [START_REF] Lutjeharms | The Agulhas Current[END_REF][START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF].

In the Northern Agulhas Current, we evidenced the contribution of the topographic constraint to the current stability along the 28 • E-31 • E current portion (Figure 6.1). At this location, the mesoscale eddies lose energy by bottom dissipation and by transferring energy to the mean current, by the barotropic instability (blue area in Figure 6.1) and unforced non-linear interactions (eddy-mean triadic interactions, dotted line in the blue area in Figure 6.1), and to smaller vertical scales by forced non-linear interactions (stratification-topographic gradients). We confirmed that the Natal Pulses (NP in Figure 6.1) are generated by the barotropic instability at the Natal Bight (red area at ∼ 31 • E in Figure 6.1), where the shelf curvature widens, and that they are the main source of variability of this Agulhas Current branch.

At the transition between the two Agulhas Current branches (∼ 26 • E), we evidenced that the current frontal variability is dominated by submesoscale cyclonic eddies (Subm. frontal eddies in Figure 6.1). They are generated as a vortex street along the 28 • E-26 • E current portion by the barotropic instability (red area in Figure 6.1). This process is recurrent, with a periodicity about 10 days, in the absence of mesoscale meandering (Natal Pulses). The instability triggering is shaped by a frontogenetic background strain which intensifies the inner front shear and therefore sets the barotropic instability favorable conditions. However, the connection between the topographic constraint and the frontogenetic background strain remains an open question.

In the Southern Agulhas Current, we related the current intense variability to the relaxation of the topographic constraint. At the Eastern Agulhas Bank Bight (downstream of 26 • E) and off the Agulhas Bank tip (∼ 23 • E), the current separates from the shelf and undergoes mixed barotropic-baroclinic instabilities (red areas in Figure 6.1). At the first location, mesoscale eddies are reinforced by forced energy transfers from smaller vertical scale eddies (stratification-topographic gradients, red triangular areas in Figure 6.1). This partially confirms the Shear-edge eddies (SE in Figure 6.1) generated by a horizontal shear process controlled by topography (Lutjeharms et al., 2003b,a). This region is also characterized by the generation of internal-gravity waves by mesoscale eddy scattering over topographic roughness (blue oscillating arrows in Figure 6.1).

In offshore regions -i.e., away from the shelf and eddy generation area -, we evidenced that the energy transfers between mesoscale eddy and smaller vertical scales eddy are dominated by unforced non-linear interactions realizing an inverse vertical cascade (triadic interactions, red open arrowhead disk in Figure 6.1). In the Northern part of the Subgyre, the variability is intense at small vertical scales confined at the surface (grey small disks in Figure 6.1). These small vertical scale eddies are generated by the baroclinic instability and subsequently reinforced the mesoscale eddies by an inverse vertical cascade (blue open arrowhead small disks in Figure 6.1). This chain of event is fairly consistent with the submesoscale inverse vertical cascade characterized in this region [START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF]. The Benguela, the Agulhas Retroflection and the Western Agulhas Plateau are regions where the mesoscale eddies generated at the Agulhas Retroflection propagate and where they are reinforced by an inverse vertical cascade (triadic interactions, red open arrowhead disk in Figure 6.1).

Additional investigations to improve our understanding of the Agulhas Current variability directly arise from our results :

• Spatial and Time variability of the submesoscale barotropic instability. The mechanisms driving the spatial and temporal variability of the submesoscale barotropic instability triggering remain an open question. The barotropic instability is related to a frontogenetic strain upstream of the instability triggering. The frontogenetic strain results from a combination of the background mesoscale advection field and of the topographic constraint. Several studies related the frontogenetic strain to the topographic drag and the submesoscale instability triggering to the strain weakening downstream of an abrupt current separation from the topography [START_REF] Molemaker | Submesoscale instability and generation of mesoscale anticyclones near a separation of the California Undercurrent[END_REF]Gula et al., 2015b). However in the Agulhas Current, the submesoscale barotropic instability triggers at a location where the current flows closely to the topography. The control of the frontogenetic strain and of the instability triggering location by the background advection field remain then to be evaluated. The submesoscale frontal eddies generation has a periodicity about 10 days similar to the Agulhas Current and Undercurrent one. This and the strong topographic constraint acting on the Agulhas Current at the instability triggering location, suggest the Coastal Trapped Waves -or so-called Rossby Topographic Waves -as a possible process shaping the instability periodicity.

• The fate of submesoscale frontal eddies. Chapter 2 exclusively focuses on the submesoscale frontal eddies generation, but their fate (dissipation, interactions with other features or detachment from the current) would also be of interest for the Agulhas Current frontal dynamics. Preliminary findings from our realistic numerical simulation suggests that they either interact with Shear-Edge eddies or dissipate, likely due to eddy-topographic interactions, along the Eastern Agulhas Bank Bight.

• Cross-shelf exchanges. The Agulhas Current major influence on the shelf circulation has been evidenced at a location where the current flows closely to the shelf [START_REF] Lamont | Observations of northeastward flow on a narrow shelf dominated by the Agulhas Current[END_REF]. Submesoscale frontal eddies develop at the border with the shelf (200 m isobath), suggesting they also influence the shelf dynamics. These eddies and associated meanders likely intrude onto the shelf and may influence its circulation and mixing, realize water masses transport and may therefore be involved in cross-shelf exchanges.

Energy transfers between the different eddy vertical structures

In Chapter 3, we used the traditional vertical modes [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF] to define the eddy vertical structures and the energy transfers between them.

Mesoscale eddies generally have interior (O(1000) m) and upper-layer (0-1000 m) intensified energy vertical structures corresponding to the barotropic and 1 st baroclinic EKE modes and to the 1 st baroclinic EAP E mode. The relative proportions between the barotropic and 1 st baroclinic EKE modes drive the rate of decrease in energy with depth. Smaller vertical scale eddies have surface-intensified (O(100) m) vertical structures corresponding to the EKE or EAP E higher baroclinic modes (n = 2 -9). The variability is the most intense at mesoscale in the paths of the Agulhas Current, the Agulhas Return Current and the Benguela current. While it is the most intense at smaller vertical scales (EAP E 2-9 ) in the Subgyre region, likely related to mixed-layer turbulence (Figure 6.1). The second signature of the smaller vertical scale variability (EKE 2-9 ) is observed in the path of the Agulhas Current at 26 • E, related to mean horizontal shear instabilities (Subm. frontal eddies in Figure 6.1), and at 23 • E.

We derived the energy transfers between the 10 first vertical modes and organised them into 6 terms in the context of a modal EKE (EKE n ) budget. 4 of the energy transfer terms contribute significantly to the EKE n budget in the Northern and Southern Agulhas Current regions (Figures 6.2). These terms highlight the importance of stratification-topographic gradients in driving energy transfers between eddy vertical structures. The advective eddy-eddy and eddy-mean interactions on slopes represent the energy transfers between mesoscale and smaller vertical scale eddies realized by non-linear interactions forced by the topographic slopes. The latter goes along with an energy transfer between the EKE and M KE reservoirs. It is interpreted as eddies driving mean currents over sloping topography when mesoscale and smaller vertical scales eddies transfer EKE in similar proportions to M KE. Indeed, bottom-intensified flows decompose into a barotropic mode, allowing a nonzero bottom velocity, and into baroclinic modes, to cancel or to shape velocity in the upper layers, as tested by [START_REF] Stanley | Vertical structure of ocean mesoscale eddies with implications for parameterizations of tracer transport[END_REF]. The pressure-gradient eddy-eddy interactions on slopes represent the energy transfers between mesoscale and smaller vertical scale eddies realized by linear interactions forced by stratification-topographic gradients. This term points toward the mesoscale eddy scattering over topographic roughness generating internal-gravity waves -such as lee waves - [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF][START_REF] Nikurashin | Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: Theory[END_REF]. The triadic interactions denote the energy transfers between mesoscale and smaller vertical scale eddies realized by the unforced non-linear interactions. It points toward the mesoscale eddy vertical turbulent cascade, but may also involve waves and time-mean currents.

The energy transfers between the vertical structures globally represent a larger energy loss than the one due to dissipation processes (bottom friction and wind) for mesoscale eddies. In the Northern Agulhas Current, the forward vertical cascade is dominated by non-linear interactions forced by stratification-topographic gradients. The mesoscale eddy driving mean current over sloping bottom as well as their scattering over sloping bottom represent therefore significant contributions to the mesoscale eddy-energy loss in this region. A weak inverse vertical cascade is realized by the triadic interactions (Figure 6.2 top panel). In the presence of a straight and steep sloping bottom, the eddy-eddy linear interactions forced by stratification-topographic slopes realize a weak energy transfer from smaller vertical scales toward mesoscale eddy. This energy transfer is opposite to the one of the mesoscale eddy scattering occurring over topographic roughness [START_REF] Nikurashin | Global energy conversion rate from geostrophic flows into internal lee waves in the deep ocean[END_REF][START_REF] Nikurashin | Routes to energy dissipation for geostrophic flows in the Southern Ocean[END_REF]. In the Southern Agulhas Current, the forward vertical cascade is dominated by linear interactions forced by stratification-topographic gradients. The mesoscale eddy scattering over topographic roughness represent therefore a significant contribution to the mesoscale eddy-energy loss in this region. The only route and into the sum of higher baroclinic modes (top line : EKE 2-9 ), corresponding to smaller vertical scale eddy. The red/dark blue arrows denote the mesoscale eddy generation/dissipation processes, the light blue arrows denote the energy transfers between mesoscale eddy and smaller vertical scales eddy, realized by the processes noted in grey. The width of the arrows denote the significance of the energy transfer within the region. Schemes inspired from [START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF].

Higher Baroclinic modes

of the inverse vertical cascade is represented by the triadic interactions (Figure 6.2 bottom panel).

Additional issues relative to the energy transfers between vertical structures and to the EKE n budget would require further investigations :

• Sensitivity to the vertical modes definition. Our present study is based on the traditional vertical modes [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF] which are defined for no vertical velocities at the bottom. Even though this condition is relaxed in the context of our realistic numerical simulation, we evidenced the validity of this vertical mode decomposition in our study context. An alternative vertical modes basis has been defined for sloping bottom assuming no horizontal velocity at the bottom (LaCasce, 2017). In the continuation of our study, we could test the sensitivity of the eddy-energy vertical partitioning and of the EKE n budget to the vertical mode definition. It would allow to evaluate the limits within which the traditional vertical modes are defined in the presence of steep topographic slopes. It would also give an alternative insight on the mesoscale eddy dynamics because this vertical mode basis does not account for bottom-intensified flows. Comparing the two mesoscale EKE budget based on different vertical mode definitions could allow to point toward the bottom-intensified flows contribution to the mesoscale EKE budget and the regions where the process of eddy driving mean currents over sloping bottom occur.

• Eddy -wave -mean flow interactions. Our present study focuses on the energy transfers between different vertical structures of the eddy and mean KE reservoirs. Dissociating the wave contribution from the eddy one would characterize in a more inclusive way the energy transfers of mesoscale eddy.

• Energy transfers between the barotropic and 1 st baroclinic modes. Chapter 3 exclusively focuses on the energy transfers between mesoscale (barotropic and 1 st baroclinic vertical modes) and smaller vertical scales eddies. Characterizing the energy transfers within the mesoscale range, i.e., between the barotropic and 1 st baroclinic modes, such as barotropization and scattering would complete our understanding of the energy transfers between eddy vertical structures in the Agulhas Current region.

The Agulhas Current net contribution to the mesoscale eddy-energy sources and sinks

In Chapters 3 and 4, we characterize the Agulhas Current region net contribution to the mesoscale eddyenergy sources and sinks using a realistic numerical simulation (dx ∼ 2.5 km) and put our results in perspective with [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF]'s ones.

The eddy-energy sources and sinks correspond to the eddy-energy fluxes divergence term in the EKE n equation. This term mainly account for the l.h.s. of the EKE n equation which equals the sum of the local EKE n sources and sinks. According to its sign, it indicates a local net EKE n source (> 0) or a local net EKE n sink (< 0). This term can also be interpreted as the budget of incoming and outgoing fluxes. The EKE n fluxes have two components : the eddy pressure work and the advection of EKE n by the total flow, which are respectively related to the linear and non-linear parts of the dynamics.

Our diagnostic of the eddy-energy fluxes divergence in the dx ∼ 2.5 km numerical simulation characterizes the Northern and Southern Agulhas Current as moderate and intense generation regions of mesoscale eddies, respectively. This result is not consistent with the vision of the mesoscale eddy decaying upon western boundaries supported by [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF]; Chelton et al. (2011b); [START_REF] Evans | Mesoscale eddy dissipation by a 'zoo' of submesoscale processes at a western boundary[END_REF]. It especially contradicts [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] who suggested that western boundaries are ubiquitous mesoscale eddy-energy sinks and found the largest one at the South Indian Ocean western boundary, including the Agulhas Current region. [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] diagnostic does not account for the mesoscale eddy-energy source in the Agulhas Current region, because the divergence term is sensitive to the approximations required by their study dynamical context. They evaluated the mesoscale eddy-energy sources and sinks by diagnosing the linear component of the divergence (eddy-pressure work) from satellite altimetry data (AVISO). This requires the approximations of the velocity field by the geostrophic one, of the mesoscale eddy vertical structure by the 1 st baroclinic mode alone and of the dynamics by its linear component. The approximation of the velocity field by the geostrophic one is the most limiting. For geostrophic velocities, the eddy pressure-work results in a residual meridional EKE 1 -flux. In the mesoscale eddy range, this represents a negligible contribution to the eddy-energy sources and sinks compared to the ageostrophic dynamics one. The approximation of the mesoscale eddy vertical structure by the 1 st baroclinic mode alone is also limiting. First, the barotropic mode has a non-negligible contribution (O(30) %) on the eddy surface signal. The approximation of the surface only reflecting the 1 st baroclinic mode [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] leads then to an overestimation of this mode. Second, the energy transfers between the two vertical modes describing the mesoscale eddy vertical structure (barotropic and 1 st baroclinic mode) are not negligible over the entire Agulhas Current region. However, the 1 st baroclinic mode eddy-pressure work term points toward the eddy-energy sinks for this mode, but does not indicate the energy transfers direction, i.e., toward the barotropic or higher baroclinic modes, at the origins of the energy sink. Therefore, the eddy-energy sources and sinks of the 1 st baroclinic mode alone are not representative of the mesoscale eddy ones. The approximation of the divergence by its linear component is also limiting. It leads to a large underestimation (96 %) of the mesoscale eddy-energy source in the Agulhas Current region. The Interior region is where this component dominates and it corresponds to an energy sink of remotely-generated mesoscale eddies.

Our results suggest that the use of the altimetry dataset (AVISO) leads to strong biases in the estimation of the mesoscale eddy-energy sources and sinks in the Agulhas Current region. They also suggest that mesoscale eddy dynamics in region of western boundary currents is more complex that the mesoscale eddy decay's picture. The local generation of mesoscale eddies by the Agulhas Current would overcome the local dissipation of the remotely-generated ones.

Additional issues relative to the interpretation of our regional EKE n budget in the context of the basin-scale dynamics and Western Boundary Current regions generic dynamics would require further investigations :

• Dissociation of the local and remote eddy-energy fluxes. In the continuation of our study, dissociating the local and remote eddy-energy fluxes to characterize the mesoscale EKE budget for the locally-and remotely-generated eddies would be a first step for developing a complete vision of the mesoscale eddy dynamics upon western boundary current regions.

• A generic EKE n budget upon the Western Boundary Currents ? [START_REF] Zhai | Significant sink of ocean-eddy energy near western boundaries[END_REF] study suggests a generic eddy dynamics upon Western Boundaries at the origin of energy sinks for the 1 st baroclinic mode eddies. However, our study evidenced the South Indian Ocean western boundary as a net local energy source for mesoscale eddy. In the continuation of our study, we could use our mesoscale EKE budget for characterizing the mesoscale eddy dynamics in another Western Boundary Current region. This would allow to test the net local mesoscale eddy-energy source found over the Agulhas Current region as a generic eddy dynamics over Western Boundary Current regions. -3 ] computed over the simulation period (1994)(1995)(1996)(1997)(1998)(1999). All the terms are diagnosed from the online momentum diagnostics. The blue contours denote the 0.1 and 0.5 10 -4 m 3 s -3 residual isolines, the yellow contours denote the 1 10 -4 m 3 s -3 residual isolines and the black contours denote the 1000 and 3000 m isobaths. The residuals are defined as the difference between the total terms and their modal expansions. With residuals located around topographical features and of small magnitude (about O(0.1-0.5) 10 -4 m 3 s -3 and punctually peaking at 1 10 -4 m 3 s -3 for the advection and pressure gradient terms), the 10 first vertical modes accurately represent the modeled eddy dynamics. (iii) Our modal budget allows to characterize the Agulhas Current region as a net energy source for mesoscale eddies in contradiction with one inferred from altimetry data. The discrepancies come from altimetry data not accounting for the main contribution of the dynamics to mesoscale eddies energy sources and sinks (ageostrophic linear part of the dynamics of the barotropic and 1 st baroclinic modes).

  in Figure 1.3.

Figure 1 . 1 :

 11 Figure 1.1: Scheme of the Indian Ocean wind-driven large-scale mean circulation. Note the western boundary current (Agulhas Current) closing the South Indian Ocean subtropical gyre. Scheme from Tomczak and Godfrey (2013).

Figure

  Figure 1.2: Snapshot (28-Aug-1996) of the global-scale eddy SSH field [cm] from AVISO constructed from the merged T/P and ERS-1 data following a procedure described in Chelton et al. (2011b). Note the ubiquity of the mesoscale eddies. Figure from Chelton et al. (2011b).

Figure 1 . 3 :

 13 Figure 1.3: Snapshot of the surface normalized vertical vorticity ζf computed from a high-resolution numerical simulation (dx ∼ 2 km) of the Agulhas Current region. Note that in most regions, the vorticity is intense over a wide scales range : from mesoscale meanders and eddies to submesoscale eddies within mesoscale eddies, submesoscale fronts, filaments and eddies in the "submesoscale soup". In few regions, the vorticity is weaker and is dominated by mesoscale eddies and submesoscale fronts and filaments. Figure from[START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF]

Figure 1

 1 Figure 1.4: Surface EKE from AVISO and 500 m depths EAP E from the World Ocean Atlas statistics inferred from ARGO (Roullet, 2020) [J m -3 ]. The green contours denote the 0.1, 0.25, 0.5 and 1 m isolines of the mean sea surface height. The overbar denotes a time average over the 2008-2018 period. Note the ubiquitous large EKE and EAP E at the western boundary currents locations (Gulf Stream, Brazil Current, Agulhas Current, Kuroshio, East Australian Current) and of the Antarctic Circumpolar Current.

Figure 1 .

 1 Figure 1.6: Mean eddy-energy fluxes for the 1 st baroclinic mode [W m -2 ] inferred from the AVISO dataset and the WOCE climatology over the 1995-2008 period. Areas shallower than 300 m are shaded and the equatorial band (10 • S -10 • N) is discarded. Note the ubiquitous mesoscale eddies KE sink near the western boundaries. Figure from Zhai et al. (2010).

Figure 1 . 8 :

 18 Figure 1.8: Snapshots of Sea Surface Temperature (SST) [ • C] from the denoised Moderate-Resolution Imaging Spectroradiometer (MODIS). The black isolines denote the 500 (thin), 1000 (bold) and 3000 (thin) m isobaths. Note the Natal Pulse (05-May-2015), the Shear-edge eddy (26-Nov-2011) which led to a cyclonic lee eddy generation (29-Dec-2011) one month later.

Figure 1 .

 1 Figure 1.9: Snapshot (07-July-2016) of Sea Surface Temperature (SST) [ • C] from the denoised Moderate-Resolution Imaging Spectroradiometer (MODIS). The black isolines denote the 500 (thin), 1000 (bold) and 3000 (thin) m isobaths. Note the frontal submesoscale eddies (24-28.5 • E) in the wake of a Shear-edge eddy (23-25 • E).

Figure 2 . 1 :

 21 Figure 2.1: Mean surface current intensity (||u|| z=η ) [m s-1 ] in the Agulhas Current region as simulated by CROCO for the four nested grids : dx ∼ 22.5, 7.5, 2.5 and 0.75 km and time averaged over the 1994-1999 period. The dx ∼ 22.5 km grid simulates the Agulhas Current as a part of the global circulation (grey features), the dx ∼ 7.5 km grid simulates the current as a part of the South Indian Ocean subtropical gyre (white features), the dx ∼ 2.5 km grid focuses on the Agulhas Current (purple features) and the dx ∼ 0.75 km grid is centered on the current portion around Port Elizabeth (purple dot). The circulation scheme is largely inspired from[START_REF] Lutjeharms | The Agulhas Current[END_REF];[START_REF] Talley | Descriptive Physical Oceanography: An Introduction[END_REF] and EMC : East Madagascar Current, NEMC : North East Madagascar Current and EACC : East African Coastal Current.

Figure 2 . 2 :

 22 Figure 2.2: Snapshot of the surface temperature [ • C] in the Agulhas Current region as simulated by CROCO for the four nested grids : dx ∼ 22.5, 7.5, 2.5 and 0.75 km. The surface temperature inside each domains is plotted from the corresponding resolutions numerical outputs. The grey contours denote the 1 st Rossby deformation radius [km] diagnosed from the Brunt-Väisälä frequency simulated for the dx ∼ 22.5 km grid and time averaged over the 1994-1999 period. With an effective resolution about 10dx ∼ 25 km (Soufflet et al., 2016), the dx ∼ 2.5 km resolves the 1 st Rossby deformation radius (O(30) km) in this region.

Figure 2 . 3 :

 23 Figure 2.3: Sections around 28.5 • of the mean along-shore velocity [cm s -1 ] from (top left) the ACT data set[START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF], from (top right) the dx ∼ 1.5 km NEMO simulation[START_REF] Schubert | Submesoscale impacts on mesoscale Agulhas dynamics[END_REF] and from (bottom panels) the dx ∼ 0.75 and 2.5 km CROCO simulations[START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF]. The red contours denote the "equatorward" current with a contours interval of 10 cm s -1 and the black and blue contours denote the "poleward" current with a contours interval respectively of 5 and 10 cm s -1 . The cyan contour denotes the 5 cm s -1 isotach from the ACT data set. The bold black contour denotes the 0 cm s -1 isotach. The mean along-front velocity is defined for the May 2010-February 2013 period (top and bottom right panles) and over the 1996 1-year period the dx ∼ 0.75 km CROCO (bottom left panel).

(

  A) EKE n evolution : -Time rate (A.1) : Evolution of the EKE storage within the vertical mode n over time, which tends Partie , Chapter 2 -Methods : Presentation of the Approaches toward zero for a long time average.

  and corresponds to the following vertical modes combinations : (n = m = l), (n=l = m), (n = m = l) and (n = m = l). These triadic interactions point toward the contribution of the vertical advection to the turbulent energy cascades and to the unforced energy transfers across scales and reservoirs, as a complement to the horizontal triadic interactions (C.1.1).

Figure 2 . 4 :

 24 Figure 2.4: Profiles of the Brunt-Väisälä frequency (N 2 [s 2 ]), averaged over the 1994-1999 period, used for defining the 10 first vertical modes of the horizontal velocity and dynamical pressure (φ n ) (top panels) and of the vertical velocity and dynamical buoyancy (Φ n [m]) (bottom panels). The red dots locate the vertical grid.

  1 and B.2), the vertical advection of EKE n (component of A.3), the horizontal and vertical triadic interactions (C.1.1 and C.1.2), the advective eddy-eddy and eddy-mean interactions on slopes (C.2 and C.3) and the contribution of the modal continuity (C.6) are explicitly diagnosed offline whereas the horizontal advection of EKE n (component of A.

Figure 2 . 5 :

 25 Figure 2.5: Maps of the KE n terms [m 3 s -3 ] (Equation 2.31), for the 1 st baroclinic mode (n = 1) and computed over the 1994-1999 period. All the terms are diagnosed from the online momentum diagnostics. The overbar denotes a time average (over the 1994-1999 period) and the prime denotes fluctuations relative to the time average. The black contours denote the 1000 and 3000 m isobaths. The budget closure is evaluated by the root mean square of the residual, defined as the difference between the Time rate (top left panel) and the sum of the rest of the terms (top right, middle and bottom panels), normalized by the root mean square of the Time rate and converted into percentages [%]. The KE 1 budget closes with a residual of 1.1 10 -4 %.

Figure 2 Figure 2

 22 Figure 2.6: Maps of the M KE n terms [m 3 s -3 ] (Equation 2.32), for the 1 st baroclinic mode (n = 1) and computed over the 1994-1999 period. All the terms are diagnosed from the online momentum diagnostics. The overbar denotes a time average (over the 1994-1999 period) and the prime denotes fluctuations relative to the time average. The black contours denote the 1000 and 3000 m isobaths. The budget closure is evaluated by the root mean square of the residual, defined as the difference between the Time rate (top left panel) and the sum of the rest of the terms (top right, middle and bottom panels), normalized by the root mean square of the Time rate and converted into percentages [%]. The M KE 1 budget closes with a residual of 1.2 10 -4 %.

Figure 2

 2 Figure 2.9: Maps of the EKE n terms [m 3 s -3 ] (Equation2.41), for the 1 st baroclinic mode (n = 1) and computed over the 1994-1999 period. The overbar denotes a time average (over the 1994-1999 period) and the prime denotes fluctuations relative to the time average. All the terms are diagnosed from the online momentum diagnostics, except the advection and the pressure gradient terms. The advection and pressure gradient terms are the sum of their respective contributions which are explicitly diagnosed offline, except one contribution which is defined as a residual using the online advection and pressure gradient diagnostics. The black contours denote the 1000 and 3000 m isobaths.

  te xt et ex te te xt et ex te te xt et ex te te xt et ex te te xt et ex te te xt et ex te A g u l h a s C u r r e n t SST [ C]

Figure 3 . 1 :

 31 Figure 3.1: Snapshot of the Sea Surface Temperature (SST) [ °C] on the 04-June-2014 from the Odyssea Regional South African product, showing the warm Agulhas Current extending from the southern boundary of the Mozambique Channel to the retroflection and returning in the South Indian gyre as the Agulhas Return Current (black legends). The main topographic characteristics along the Agulhas path are displayed (grey legends). The thin black lines follow the 200 m, 1000 m and 3000 m isobaths. Two perturbations, a Natal Pulse and a Shear-Edge eddy (white arrows) are developing along the Agulhas cyclonic front.

Figure 3

 3 Figure 3.2: (a) Black circles track two Seagliders when they sampled two submesoscale cyclonic features between the 19-April and the 20-May 2015 (Krug et al., 2017). The first and last measures inside these features are respectively marked with green and blue circles. The background colors represent the bathymetry. The thin black lines follow the 200 m, 500 m and 2000 m isobaths. The bold black line is the 1000 m isobath, which corresponds to the Agulhas front location during the SAGE observations. (b) A snapshot of Sea Surface Temperature (SST) [ °C], from the denoised Moderate-Resolution Imaging Spectroradiometer (MODIS) on the 19-July-2010. The black lines represent the same isobaths than in (a), except for the 200 m which is not represented. Submesoscale meanders develop upstream of Port Elizabeth, between 28.5°E and 26°E.

Figure 3

 3 Figure 3.3: Snapshot of surface temperature [°C] simulated by CROCO for the four nested domains : dx ∼ 22, 7.5, 2.5 and 0.75 km.

Figure 3

 3 Figure 3.4: Maps of the mean surface EKE [cm 2 s -2 ] for (a) AVISO (1993-2003 period) and (b) CROCO at dx ∼ 2.5 km. Black contours are the 1000 m and 3000 m isobaths and green contours are the isolines of mean sea surface height between -2 and 2 m with a lag of 0.25 m. The black segment in Figure 3.4b shows the location of the vertical sections plotted along the ACT array in Figures 3.5. Both observed and modeled low (O(0.5-2) 10 3 cm 2 s -2 ) and large (> O(3.5) 10 3 cm 2 s -2 ) EKE values along the upstream stable and downstream unstable Agulhas branches. The topography plays a role on the current dynamics and small scales (10-70 km) can locally drive the front variability.

  Figure 3.5: Sections of the mean along-front velocity (u(y) [m s -1 ]) and mean normalized relative vorticity (-∂yu f ) around 28.5°E derived from the ACT dataset (a and b) and the model at dx ∼ 0.75 km (c and d). The location of the ACT array is shown by the black segment in Figure 3.4c. The black dashed lines locate the seven full-depths current meter moorings. The black cross indicates the 150 kHz ADCP at 300 m at the location of the first mooring. On the mean along-front velocity panels, the blue contours are the -0.25, -0.5, -1, -1.25 and -1.5 m s -1 isolines, the red contours are the 0.03 and 0.08 m s -1 isolines and the bold black contour is the 0 m s -1 isoline. The observed and modeled mean vertical structures of the Agulhas Current show an intense cyclonic front expanding from the surface to 500 m.

Figure 3

 3 Figure 3.6: Cross-front sections of the mean along-front velocities (u(y) [m s -1 ], on the left panel) and relative vorticity (-∂yu(y) f , on the right panel) at the SAGE location (25.75°E, shown in Figure 3.2a) for the SAGE glider data (Krug et al., 2017) (grey and dark red lines) and for the four model nests at dx∼ 22.5 (plain lines), 7.5 (dashed-dotted lines), 2.5 (dashed lines) and 0.75 km (dotted lines) resolution. The brackets denote the along-front average. The bottom x-axis represents the cross-front distance from the 1000 m isobath, which represents the front position for the SAGE period. The highest-resolution nest has a frontal structure close to the SAGE dataset.

Figure 3 Figure 3

 33 Figure 3.7: Snapshots of the surface normalized vertical relative vorticity ( ζ z f , on the left panel) and of the temperature (T[ • C], on the right panel) when submesoscale eddies are present. Black contours are the 200 m and 1000 m isobaths. The white dashed line represents the location of the vertical sections plotted along the vortex street in Figures 3.8 and 3.10 bottom panel. The vortex street develops along the front between 28°E and 26°E.

  .2, the terms from the left hand side correspond to the Eulerian time derivative and to the boundary transport of EKE, due to pressure fluctuations and advection by the mean and fluctuative current components, whereas the terms from the right hand side correspond to the local sources and sinks. The two terms M KE → EKE and EP E → EKE are possible EKE sources while vertical mixing and lateral diffusion are the main sinks. The barotropic instability signature on energy transfer is a positive M KE → EKE transfer through its horizontal component: HRS. It corresponds to an energy extraction from the mean horizontal shear toward the growing perturbations.

Figure 3 .

 3 Figure 3.10: Top four panels : Maps of surface EKE [cm 2 s -2 ] and energy conversion rates HRS, V RS and V BF [m -3 s -3 ] integrated over the entire water column. The overbar denotes a week-long average. This average encompasses the Figure (3.4) snapshot (09/29 06:00) and corresponds to a full event of submesoscale eddies generation. Black contours represent the 200 m and 1000 m isobaths. When submesoscale eddies are presents, HRS is the only term accounting for the energy transfer M KE → EKE. Bottom panel : Section along the vortex street, between 29.5 and 25 °E, of the energy conversion rate HRS [m 2 s -3 ]. The overbar denotes the same time average than in the top four panels. Black contours follow the 1 and 2 m 2 s -3 HRS isolines. The location of the section along the vortex street is shown as the white dashed line in Figure 3.7. The energy transfer M KE → EKE occurs between the surface and 50 m deep. 89

Figure 3

 3 Figure 3.11: Cross-front sections at the SAGE location (25.75°E, shown in Figure 3.2a) for surface observed (red) and modeled (blue) partial and full HRS rate. The partial HRS rate, only composed of the crossfront derivatives subterms (-u ′ v ′ ∂ y uv ′ v ′ ∂ y v), is shown for the SAGE dataset as a plain red line and for the model in plain blue lines. The full HRS rate, composed of the cross and along-front derivatives subterms (-u ′ u ′ ∂ x uv ′ u ′ ∂ y u) is shown in dashed blue lines. The modeled partial and full HRS rates are computed for different time periods : a one-month long (light blue lines) and five consecutive-month without Natal Pulses (dark blue lines). The bottom x-axis represents the cross-front distance from the 1000 m isobath, which represents the front position for the SAGE period. The comparison between the modeled partial and full HRS rates shows that the cross-front derivatives are a good proxy of the full HRS rate. The comparison between the modeled 1 and 5 month averages of the HRS rates shows that the submesoscale eddies generation is a locally recurrent process.

Figure 3

 3 Figure 3.12: Map of energy conversion rate HRS [m -3 s -3 ], integrated in the vertical. The overbar denotes a 5 month-long time average, when no Natal Pulse is present. Black contours are the 200 m and 1000 m isobaths. The black dashed box represents the area used for the computations of the along-front distributions (Figures 3.13) and of the time serie (Figure 3.15). The one-week (Figure 3.6) and 5 monthlong time averages of HRS show a similar pattern and confirm the hypothesis that submesoscale eddy generation is a locally recurrent process.

  and 3.11), the EKE equation (Eq. 3.2) reduces at first order to : DEKE Dt ≡ u ∂ x EKE ≡ HRS (3.3) with u the along-front velocity component, x the corresponding direction and the overbar denoting a fivemonth time average when no Natal Pulse is present. The contribution of HRS to the along-front evolution of EKE is estimated from x 0 HRS u dx and compared to the full EKE (Figure 3.13a). Even though the mean energy conversion rate HRS is positive over the Agulhas inshore front from 28°E to 23°E (Figures 3.10 and 3.11), the submesoscale eddies develop preferentially over a specific portion: between 28°E and 26°E (Figures 3.7a and b). This specific portion is also visible from the along-front distributions of EKE and of the energy conversion rates HRS and V BF (Figure 3.13a). Between 29°E and 25.5°E, HRS is the dominant term responsible for the EKE generation and drives its along-front variations. The approximation: EKE ≈ x 0 HRS u dx is here valid. HRS has positive values almost monotonously increasing and the EKE is strongly increasing. Both patterns reflect the recurrent formation of vortex streets between 28°E and 26°E (Figures 3.7a and b) and their downstream advection.

Figure 3

 3 Figure 3.13: Along-front distributions of vertically integrated time mean energy conversion rate, HRS [m 3 s -3 ] of time mean EKE, EKE [m 3 s -2 ] and of x 0 HRS u

Figure 3

 3 Figure 3.14: Map of the mean normalized background strain, S f at 25 m. The overbar denotes a 5 month time average, when no Natal Pulse is present. Grey segments represent the strain axis and black contours the 200 m and 1000 m isobaths. High relative vorticity, found upstream of the barotropic instability triggering (28°E), corresponds to where the background strain is the most efficient at intensifying the front. The location of the barotropic instability triggering (between 28°E and 26°E, Figure 3.7) corresponds to where the background strain relaxes its constraint on the front.

  Figure3.15: Time series of energy conversion rate HRS over a 5 month-long period when no Natal Pulse is present. The time series results from a volume average : on the vertical, HRS is averaged over the whole water column and on the horizontal over the 100 km wide black dashed box centered on the Agulhas front between 29°E and 26°E shown in Figure3.12. The thin dashed and bold plain lines represent the raw and the smoothed (over a 10-point window) time series. The grey-shaded areas correspond to submesoscale eddies generation events identified from a movie a surface relative vorticity. Submesoscale eddies are generated when HRS is peaking, but the reverse is not true. This shows that a large HRS is not a sufficient condition to trigger instabilities and that submesoscale eddies generation result from a combination of several factors.

Figure 4 . 1 :

 41 Figure 4.1: Snapshot of the vertical component of the normalized surface relative vorticity (| ζ z f | z=η ) in the Agulhas Current region as simulated by CROCO for the four nested domains (black boxes) : dx ∼ 22, 7.5, 2.5, and 0.75 km. The green contours denote the -1, -0.75, -0.5, -0.25, 0.25, 0.5, 0.75 and 1 m isolines of sea surface height. The relative vorticity and the sea surface height plotted inside each domains are computed from the corresponding resolutions numerical outputs.

Figure 4

 4 Figure 4.2: Maps of the surface EKE [m 2 s-2 ] for AVISO and for CROCO dx ∼ 2.5 km (top panels) and 500 m depth (EAP E) [m 2 s -2 ] for ARGO[START_REF] Roullet | World ocean atlas of argo inferred statistics[END_REF] and for CROCO dx ∼ 2.5 km (bottom panels). Both EKE are defined from geostrophic velocities diagnosed from sea surface height fluctuations and the time fluctuations are defined relatively to a 1-year rolling mean with a 1-month stepping over the 1994-1999 period. The green contours denote the 0.25, 0.5, 0.75 and 1 m isolines of sea surface height. For the modeled EKE (top right panel), the black boxes delimit the Northern (NAC) and Southern (SAC) Agulhas Currents and the shelf-following black area delimits the Alongshore Agulhas Current (areas discussed in Figure4.3). Both EAP E are defined following[START_REF] Roullet | Global interior eddy available potential energy diagnosed from Argo floats[END_REF], for ARGO the time fluctuations are defined over the whole ARGO dataset[START_REF] Roullet | World ocean atlas of argo inferred statistics[END_REF] and for the model they are defined relatively to the 1-year time rolling mean. The black contours denote the 500, 1000 and 3000 m isobaths (all panels). The modeled eddy energies denote the highest mesoscale variability along the Southern Agulhas Current (downstream of 26 • E) and the Agulhas Return Current (EKE : O(0.05-0.5) m 2 s -2 and EAP E : O(0.05-0.3) m 2 s -2 ) and the lowest one in the South West Indian Ocean Subgyre (EKE and EAP E : O(0-0.05) m 2 s -2 ), consistently with the observations.

Figure 4 . 3 :

 43 Figure 4.3: EKE (φ n ) and EAP E (Φ n[m]) barotropic modes (n = 0), 1 st baroclinic modes (n = 1) and higher baroclinic modes (n = 2, 9) computed from the mean modeled Brunt-Väisälä frequency (N 2 [s -2 ]). The (φ 0 ,Φ 1 ) categories represent interior-intensified vertical structures, the φ 1 represents upper layerintensified vertical structures, the (φ 2-9 ,Φ 0,2-9 ) categories represent surface-intensified vertical structures and the Φ 0 represents surface-intensified linearly decreasing vertical structures.

  Figure 4.2 top right panel). Most of the EKE is in the barotropic mode at the lowest resolution (dx ∼ 22.5 km, plain black lines

Figure 4 . 4 :

 44 Figure 4.4: Probability density functions (pdf) of the EKE partitioning [%]between the three vertical modes categories : φ 0 (plain lines), φ 1 (dashed lines) and φ 2-9 (dashed-dotted lines), for the NAC, Alongshore Agulhas Current and SAC areas (located on Figure4.2) and for the four grids : dx ∼ 22.5 (black lines), 7.5 (grey lines), 2.5 (green lines) and 0.75 km (blue lines, only for the Alongshore Agulhas Current). The overbar denotes a 1-year rolling mean with a 1-month time stepping over the 1994-1999 period. The EKE partitioning is sensitive to the numerical resolution in the NAC and Alongshore Agulhas Current areas, contrary to the SAC, and it shows a convergence for the dx ∼ 2.5 km grid in the Alongshore Agulhas Current.

  23) and the modal hydrostatic equation (4.22) :

Figure 4 . 5 :

 45 Figure 4.5: Maps of the EKE (top panels) and EAP E (bottom panels) partitioning [%] between the vertical mode categories (n = 0, n = 1 and n = 2 -9). The overbar denotes a 1-year rolling mean with a 1-month stepping over the 1994-1999 period. The grey contours denote the 0.25, 0.5 and 0.75 m isolines of sea surface height and the black contours denote the 1000 and 3000 m isobaths. The 8 (1 • × 1 • ) boxes denote the (A -H) areas discussed in Figure 4.6 and in Table 4.1. The mesoscale eddies reservoir is represented by the barotropic and 1 st baroclinic EKE modes and the EAP E 1 st baroclinic mode.

Figure 4 . 6 :

 46 Figure 4.6: Vertical profiles of EKE (black lines) and EAP E (blue lines) spatially averaged over the (1• × 1 • ) boxes located in Figure4.5 (the spatial averages are surrounded by their standard deviation within the box in shaded colors). The overbar denotes a 1-year rolling mean with a 1-month stepping over the 1994-1999 period. The A, B, C, D and E areas have interior and upper layer-intensified vertical profiles of eddy energies, typical from mesoscale variability (φ 0 , φ 1 , Φ 1 ) and the F, G and H areas mainly have surface-intensified vertical profiles (Φ 2-9 ) of eddy energies denoting a low mesoscale variability.

Figure 4

 4 Figure 4.7: Mesoscale eddy-energy (EKE 0-1 ) fluxes divergences [m 3 s -3 ] (top panel) constituted of the eddy-pressure work (bottom left panel) and of the EKE 0-1 advection by the total flow (bottom right panel). Vector fields denote the EKE 0-1 fluxes (note that there is a factor 10 between the vector fields of the bottom right panel and of the others). The NAC and SAC boxes colors denote the sign of the spatially averaged divergences (red/blue colors : positive/negative) (Figure 4.8). The black isolines denote the 1000 and 3000 m isobaths. The NAC and SAC are respectively moderate (0.17 10 -5 m 3 s -3 ) and high (2.06 10 -5 m 3 s -3 ) mesoscale eddy generation regions. The mesoscale eddy transport by the linear (eddy-pressure work, A.2) and non-linear (advection of EKE 0-1 by the total flow, A.3) dynamics follows the Agulhas Current circulation and denotes inter-oceans exchanges in the SAC area (toward the South-East Atlantic Ocean and the Southern Ocean for the non-linear transport).

  interactions on slopes (C. 2) Advective eddy mean interactions on slopes (C. 3) Pressure gradient eddy eddy interactions on slopes (C. 4) Intermodal coupling at the free surface (C

Figure 4 Figure 4

 44 Figure 4.8: Spatial averages of the EKE 0-1 terms (Eq. 4.29) [m 3 s -3 ] over the NAC and SAC regions combined (top panel), the NAC (bottom left panel) and the SAC (bottom right panel) regions. The bars color denotes the sign of the spatially averaged EKE 0-1 terms (red/blue : positive/negative) and the grey lines denote the standard deviations of the values within each box (there is a factor 10 between the standard deviations and the spatial averages). The black horizontal lines mark the organisation of the EKE 0-1 terms into the four categories : B, C and D (cf. section 4.3.3). For the C.2 and C.3 terms, the dark (light) blue color denotes the n = m = l, n = l = m, n = m = l, n = m = l (n = m = l) cases. For the NAC and SAC regions combined, the main EKE 0-1 sources are instability processes (B.1 + B.3) and the main EKE 0-1 sinks are vertical mixing (D.1) and interactions canalized by topography (C.2 + C.3 + C.4). Transfer between mesoscale and smaller vertical scales eddies (terms C) differ for the two regions.

4. 5 .

 5 Results II : Mesoscale Eddy Kinetic Energy (EKE 0-1 ) budget in the Northern (NAC) and Southern Agulhas Current (SAC) vective eddy-eddy interactions on slopes (C.2) and the vertical mixing (D.1) (Figures 4.9 and 4.10). This indicates that mesoscale eddies lose energy to the mean current due to Reynolds stress (B.1) and to non-linear interactions between energy reservoirs and vertical scales (eddy-mean C.1.1). They lose energy to eddies of smaller vertical structures (C.2) and their barotropic component, which accounts for 60 % of the EKE at the Northern Agulhas Current branch (Figure 4.5), is partially dissipated by the bottom drag (D.1). The B.1 and D.1 contributions are typical of currents strongly constrained by topography. Strong topographic slopes can set favourable conditions for barotropic instability (B.1

  interactions (C.1.1)

Figure 4 .

 4 Figure 4.11: Maps of the main EKE 0-1 sources and sinks [m 3 s -3 ] in the SAC, identified from the spatial averaged EKE 0-1 budget (Figure 4.8). The black isolines denote the 1000 and 3000 m isobaths. The Eastern Agulhas Bank Bight (light green area) and Agulhas Bank tip (dark green) EKE 0-1 sources are constituted of the positive contributions of B.1 and B.3, whereas the C.3 term has a positive and negative contributions respectively in both areas (Figure 4.9). The eddy-eddy C.1.1 term represents a positive contribution away from the mesoscale eddies generation locations and the C.4 and D.1 negative contributions are distributed across the SAC region.

1 . 1 )

 11 Off the Agulhas Bank tip, the mesoscale eddies are reinforced by the eddy-mean triadic interactions (eddy-mean C.1.1 > 0) while the eddy-eddy triadic interactions (eddy-eddy C.1.1 < 0) and the advective eddy-mean interactions on slopes (C.3 < 0) partially compensate the mesoscale eddies generation. The eddy-eddy triadic interactions suggest that mesoscale eddies lose a fraction of their energy to eddies of smaller vertical scales locally where they are generated. The advective eddy-mean interactions on slopes are mainly interpreted as a transfer between energy reservoirs, as in the Natal Bight (in the NAC), because energy is equally transferred from EKE to M KE at the mesoscale and from EKE at mesoscale to M KE at smaller vertical scales (respectively light and dark blues in Figure4.9). Off the Agulhas Bank tip, mesoscale eddies are generated by processes extracting energy from the mean currents at mesoscale (B.1 + B.3) and smaller vertical scales (eddy-mean C.1.1) and lose energy to smaller vertical scales (eddy-eddy C.and to the mean current (C.3).

  + Sinks](5.6) with ρ 0 = 1025 kg m -3 the reference density value, the overbar denotes a time average and the prime denotes fluctuations relative to this average. The l.h.s of the EKE n equation (5.6) equals the sum of the local EKE n sources and sinks. According to its sign, it indicates a net local EKE n source (> 0) or a net local EKE n sink (< 0). The l.h.s of the EKE n equation (5.6) mainly accounts for the divergence of the EKE n fluxes, because the time rate tend toward zero for a long time average, and can also be interpreted as the budget of incoming and outgoing fluxes when integrated over a finite domain. The EKE n fluxes have two components : the eddy-pressure work and the advection of EKE n , which are respectively related to the linear and non-linear part of the dynamics. The r.h.s of the EKE n equation (5.6) gathers the local eddy-energy sources and sinks, either coming from the energy transfers between reservoirs, between vertical modes or from the local energy generation and dissipation. The derivation of these terms and their interpretations, in terms of physical processes, are detailed inTedesco et al. (submitted).

Figure 5 . 3 :

 53 Figure 5.3: Maps of the 1 st baroclinic mode eddy-pressure work terms [W m -2 ], averaged over the 1995-2004 period, diagnosed from simulated total-velocity (Eq. 5.6) (a) and geostrophic-velocity fields (Eq. 5.17) (b). Terms on a and b panels are smoothed with a 50 km-radius Gaussian kernel. The grey areas delimit the Western Boundary and Interior regions and spatial averages over both regions are noted. The black isolines denote the 1000 and 3000 m isobaths.

  ǫ denoting the classic quasi-geostrophic scaling of the Rossby number : ǫ = U f L = O(10 -1 ) and the Rhines scale : Rh = U β , L = O(10 4 ), β = O(10 -11 ) and f = O(10 -4 ).

5. 5 .

 5 Result II : Contribution of the altimetry-based eddy-pressure work term to the mesoscale eddy-energy sources and sinks

5. 5 .Figure 5 . 6 :

 556 Figure 5.6: Maps of the mesoscale (n = 0 -1) eddy-energy fluxes divergence (a), of its linear component (eddy-pressure work in b) and non-linear component (advection of EKE 0-1 in c) [W m -2], averaged over the 1995-2004 period, diagnosed using the simulated velocity and pressure fields. Terms in all panels are smoothed with a 50-km Gaussian kernel. The grey areas delimit the Western Boundary and Interior regions and spatial averages over both regions are noted and summarized in a bar plot(d). The black isolines denote the 1000 and 3000 m isobaths.

Figure 6 . 2 :

 62 Figure6.2: Schemes of the energy pathways in the Northern (top panel) and Southern Agulhas Currents (bottom panel). The EKE vertical structures are decomposed into the sum of the barotropic and 1 st baroclinic vertical modes, corresponding to mesoscale eddy vertical structure, (bottom line : EKE 0-1 ) and into the sum of higher baroclinic modes (top line : EKE 2-9 ), corresponding to smaller vertical scale eddy. The red/dark blue arrows denote the mesoscale eddy generation/dissipation processes, the light blue arrows denote the energy transfers between mesoscale eddy and smaller vertical scales eddy, realized by the processes noted in grey. The width of the arrows denote the significance of the energy transfer within the region. Schemes inspired from[START_REF] Ferrari | Ocean circulation kinetic energy: Reservoirs, sources, and sinks[END_REF].

Figure 6

 6 Figure 6.3: Map of the vertically integrated EKE and EAP E [m 3 s -2 ] averaged over the 1994-1999 period.The green contours denote the 1 % (dark), 2 % (light) and 5 % (extra light) isolines of EKE residual, the blue contours denote the 5 % (dark), 10 % (light) and 20 % (extra light) isolines of EAP E residual and the black contours denote the 1000 and 3000 m isobaths. Residuals of both modal energy expansions using the 10 first vertical modes are defined as the difference between the total energies and their modal expansions normalized by the total energies and converted into percentages[%]. With the EKE and EAP E residuals locally peaking at 5 and 20 %, respectively, over limited areas, the 10 first vertical modes accurately represents the modeled eddy energies.

Figure 6 . 4 :Figure 6 . 5 :

 6465 Figure 6.4: Maps of the KE terms [m 3 s-3 ] computed over the 1994-1999 period. All the terms are diagnosed from the online momentum diagnostics. The blue contours denote the 0.1 and 0.5 10 -4 m 3 s -3 residual isolines, the yellow contours denote the 1 10 -4 m 3 s -3 residual isolines and the black contours denote the 1000 and 3000 m isobaths. The residuals are defined as the difference between the total terms and their modal expansions. With residuals located around topographical features and of small magnitude (about O(0.1-0.5) 10 -4 m 3 s -3 and punctually peaking at 1 10 -4 m 3 s -3 for the advection and pressure gradient terms), the 10 first vertical modes accurately represent the modeled dynamics.

Figure 6 . 6 :

 66 Figure 6.6: Maps of the EKE terms [m 3 s-3 ] computed over the simulation period(1994)(1995)(1996)(1997)(1998)(1999). All the terms are diagnosed from the online momentum diagnostics. The blue contours denote the 0.1 and 0.5 10 -4 m 3 s -3 residual isolines, the yellow contours denote the 1 10 -4 m 3 s -3 residual isolines and the black contours denote the 1000 and 3000 m isobaths. The residuals are defined as the difference between the total terms and their modal expansions. With residuals located around topographical features and of small magnitude (about O(0.1-0.5) 10 -4 m 3 s -3 and punctually peaking at 1 10 -4 m 3 s -3 for the advection and pressure gradient terms), the 10 first vertical modes accurately represent the modeled eddy dynamics.

Titre:

  Etude régionale des transferts d'Energie Cinétique pour les réservoirs des Tourbillons de Méso et Sous-mésoéchelle dans l'Océan Mot clés : Courant des Aiguilles, modélisation, tourbillons, budget d'énergie, modes verticaux, intéractions topographiques Résumé : Cette thèse est basée sur des simulations numériques régionales et aborde, d'un point de vue énergétique, la variabilité à Méso (O(10-100)km) et Sous-mésoéchelle (O(0.1-10)km) dans la région du Courant des Aiguilles. (i) La variabilité de sous-mésoéchelle à la transition entre les deux branches du Courant des Aiguilles (28 • E-26 • E) est dominée, en l'absence de méandres de mésoéchelle, par des tourbillons cycloniques frontaux formant un 'vortex street'. La tension ambiante frontogénétique intensifie le cisaillement frontal qui déclenche l'instabilité barotrope menant à la génération de tourbillons de sous-mésoéchelle. (ii) Un budget modal d'Energie Cinétique des Tourbillons est développé pour caractériser les transferts d'énergie entre les différentes structures verticales. Les interactions canalisées par la topographie (3 processus) résultent globalement en une perte d'énergie pour les tourbillons de mésoéchelle plus grande que les processus de dissipation (friction au fond et vent) et une cascade verticale inverse (intéractions triadiques) renforce les tourbillons de mésoéchelle dans les zones au large. (iii) Notre budget modal permet de caractériser la région du Courant des Aiguilles comme une source nette d'énergie pour les tourbillons de mésoéchelle en contradiction avec celui estimé à partir de données d'altimétrie. Cette différence vient des données d'altimétrie ne tenant pas compte de la contribution principale de la dynamique aux sources et puits d'énergie des tourbillons de mésoéchelle (partie linéaire de la dynamique agéostrophique du mode barotrope et du 1 er mode barocline). Title: A regional study on the Eddy Kinetic Energy transfers of the Meso and Submesoscale reservoirs in the Ocean Keywords: Agulhas Current, modeling, eddies, energy budget, vertical modes, topographic interactions Abstract: This dissertation is based on regional numerical simulations and addresses, from an energetic perspective, the Meso (O(10-100)km) and Submesoscale (O(0.1-10)km) variability in the Agulhas Current region. (i) Submesoscale variability at the transition between the two Agulhas Current branches (28 • E-26 • E) is dominated, in the absence of mesoscale meanders, by cyclonic frontal eddies forming a 'vortex street'. The frontogenetic background strain intensifies the frontal shear which triggers the barotropic instability leading to submesoscale eddies generation. (ii) A modal Eddy Kinetic Energy budget is derived to characterize the energy transfers between the different vertical structures. Interac-tions canalized by topography (3 processes) globally result in a larger energy loss for mesoscale eddies than dissipation processes (bottom friction and wind) and an inverse vertical cascade (triadic interactions) reinforces mesoscale eddies in offshore areas.

  the mean Kinetic Energy equation of the total flow for a vertical mode n, M KE n = η ) dz the mean Eddy Energy Equation for a vertical mode n. The first step, closely following[START_REF] Kelly | The vertical mode decomposition of surface and internal tides in the presence of a free surface and arbitrary topography[END_REF], is the projection of the momentum equations onto a vertical mode n. Secondly and thirdly the KE n and M KE n equations are derived to finally obtain an equation for EKE n .

	-H	1 2 (u 2 n + v 2 n ) dz the Kinetic Energy equation of the mean flow for a vertical mode n and
	EKE n = η -H n It must be noted that the vertical modes are defined in a Quasi-Geostrophic framework where the 1 2 (u ′2 n + v ′2
	density and pressure correspond to the dynamic part of the variables (2.26) while the momentum equations
	in CROCO are expressed using the Boussinesq approximation (2.27) :
		ρ =	ρ(z)	+	ρ(x, z)	+	ρ ′ (x, z, t)	(2.26)
			static density			mean dynamic density		f luctuating dynamic density
		and
		ρ = ρ 0 + ρ(x, z)	+ ρ ′ (x, z, t)
			mean density		f luctuating density

  diffusivity and D the horizontal dissipation. The momentum equations are first projected into a vertical mode n following Kelly (2016). Equations (4.18) and (4.20) are multiplied by φ n and vertically integrated and Equation (4.19) is multiplied by Φ n

	4.3. Methods
	and vertically integrated, leading to :
	120

  The final expression of the EKE n equation is obtained by replacing the different contributions originating from the horizontal and vertical advection terms (4.25, 4.26) and from the pressure gradient term (4.27) into the former expression of the EKE n equation (4.24). The terms of the final expression of the EKE n equation are organised into the 4 following categories :

	Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a
	Western Boundary Current
	u ′ n .(h(	∂ ∂t	u) n )						(4.29)
	T ime rate (A.1)
	+	∇ H .(hu ′ n p ′ n )
	Eddy-pressure work (A.2)
					P ressure-gradient eddy-eddy interactions on slopes
				-	p ′ n ∇ H η.(u ′ φ n ) z=η
					Contribution of the f ree-surf ace <<1
			A	=	B	+	C	+	D	(4.28)
			EKEn Evolution	T ransf ers between the EKEn, M KEn and EP En	T ransf ers between vertical modes		EKEn generation and dissipation
					reservoirs
	leading to the final expression of the EKE n equation :
					123

m dz

  Partie , Chapter 4 -Study II : Mesoscale Eddy Kinetic Energy budget and transfers between vertical modes in a Western Boundary Current

Table 4 .

 4 1: Tables of the EKE (top) and EAP E (bottom) vertical partitionings [%] in the vertical modes categories

  Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

  Partie , Chapter 5 -Study III : Can mesoscale eddy-energy sources and sinks be inferred from satellite altimetry data ?

The mesoscale eddies is a terminology used in the literature referring to geostrophic eddies or balanced turbulence

The Rossby number Ro = u.∇ H u f k∧u with u the velocity horizontal components, ∇ H the horizontal gradient operator and f the Coriolis parameter, compares the horizontal advective/inertia term (u.∇ H u) to the Coriolis term (f k ∧ u). It is a common measure of the geostrophic balance (Cushman-Roisin and Beckers, 2011).

The Richardson number (Ri = N 2 ∂zu ) with N 2 the Brunt-Väisälä frequency and ∂ z the derivative in the vertical direction, compares the stratification (N 2 ) to the horizontal velocity shear (∂ z u). It is a common measure of the flow stability to the vertical shear instabilities (Cushman-Roisin and Beckers, 2011).

The SWOT project aims to build a global time serie of sea surface height at the O(15-30) km resolution[START_REF] Morrow | Global observations of fine-scale ocean surface topography with the Surface Water and Ocean Topography (SWOT) mission[END_REF] d'Ovidio et al., 2019).

Partie , Chapter 2 -Methods : Presentation of the Approaches

2.2. Mean Eddy Kinetic Energy equation (EKE)

The Agulhas Current is described as spatially "bi-modal"[START_REF] Paldor | Why is the stability of the Agulhas Current geographically bi-modal[END_REF][START_REF] Lutjeharms | The Agulhas Current[END_REF] with a stable branch upstream of Port Elizabeth (26°E) and an unstable branch downstream.

Partie , Chapter 3 -Study I : Generation of Frontal Submesoscale Eddies in the Agulhas Current

www.nodc.noaa.gov/OC5/woa18/

www.aviso.altimetry.fr

Acknowledgments

This work was granted access to the HPC resources of IDRIS under the allocation A0040107630 made by GENCI at Paris, France, and of the HPC facilities DATARMOR of "Pôle de Calcul Intensif pour la Mer" at Ifremer, Brest, France. This work was supported by the Ifremer and the Brittany region for PhD funding. Model outputs are available upon request. The AVISO data are available at www.aviso.altimetry.fr.

Acknowledgments

This work was granted access to the HPC resources of IDRIS under the allocation A0040107630 made by GENCI at Paris, France, and of the HPC facilities DATARMOR of "Pôle de Calcul Intensif pour la Mer" at Ifremer, Brest, France. This work was supported by the Ifremer and the Brittany region for PhD funding. We also gratefully acknowledge support from the French National Agency for Research (ANR) through the projects DEEPER (ANR-19-CE01-0002-01) and ISblue "Interdisciplinary graduate school for the blue planet" (ANR-17-EURE-0015). WOES36 model outputs are available online at http://dap.saeon.ac.za/thredds/catalog/SAEON.EGAGASINI/2019.Penven/DAILY_MEANS/1_ 36_degree/catalog.html (DOI: 10.15493/SAEON.EGAGASINI.10000096). The AVISO data are available at www.aviso.altimetry.fr and the Atlas of [START_REF] Roullet | World ocean atlas of argo inferred statistics[END_REF] derived from ARGO data are available at https://doi.org/10.17882/72432.

Acknowledgments

This work was granted access to the HPC resources of IDRIS under the allocation A0040107630 made by GENCI at Paris, France and of the HPC facilities DATARMOR of "Pôle de Calcul Intensif pour la Mer" at Ifremer, Brest, France. Model outputs are available upon request. This work was supported by the RADII funding as well as Ifremer and the Brittany region for PhD funding. Odyssea Regional South Africa products are available at http://www.ifremer.fr/opendap/cerdap1/ghrsst/l4/ saf/odyssea-nrt/data and MODIS SST data are available (Level 4) at http://cersat.ifremer.fr/ data/products/catalogue. The AVISO data are available at ftp.aviso.altimetry.fr. Gridded crosssectional velocities are based on in situ current meter measurements from the Agulhas Current Time-series (ACT) experiment [START_REF] Beal | Capturing the transport variability of a western boundary jet: Results from the Agulhas Current Time-Series Experiment (ACT)[END_REF] and can be downloaded at https://beal-agulhas.rsmas.miami. edu/data-and-products/index.html. We, as well, want to especially thank Heather Reagan for her precious help in correcting the manuscript language.

Chapter 4

STUDY II : MESOSCALE EDDY KINETIC ENERGY BUDGET AND TRANSFERS BETWEEN VERTICAL MODES IN A WESTERN BOUNDARY CURRENT

In this Chapter, I present the study addressing the second question of this dissertation :

Question II : What are the processes driving the mesoscale eddy generation, dissipation and energy transfers with small vertical structures in the Northern and Southern Agulhas Currents ?

This Chapter corresponds to a scientific publication draft, referred as Tedesco et al. (submitted) in this dissertation, which will be soon submitted to the Journal of Physical Oceanography. We characterize in this section the mesoscale eddy dynamics in the NAC and SAC areas using the mesoscale EKE n budget (Eq. 4.29) defined as the sum of the n = 0 and n = 1 modes (cf. section 4.4). We first evaluate the EKE 0-1 fluxes divergences (terms A.2 and A.3 in Eq. 4.29) and then we evaluate the leading-order EKE 0-1 sources and sinks (terms B, C and D in Eq. 4.29) and interpret them in terms of physical processes for different areas.

Divergence of the EKE 0-1 fluxes (terms A.2 and A.3)

The spatially averaged EKE 0-1 fluxes divergences (A.2 + A.3 term in Eq. 4.29) are positive in the NAC (0.17 10 -5 m 3 s -3 ) and SAC (2.06 10 -5 m 3 s -3 ) (Figure 4.7). This denotes that both areas are locations of energy generation for mesoscale eddies, which are then advected out of the areas. In term of EKE 0-1 fluxes, this means that the outgoing fluxes are larger than the incoming fluxes.

In the NAC (Northern box in

The NAC is characterized by contrasted mesoscale eddy dynamics along and away of the slope, whereas the SAC is more homogeneous.

The Northern Agulhas Current (NAC)

The EKE 0-1 sink along the 1000 m isobath (Alongshore Agulhas Current, Figure 4.9) results from the sum of the negative contributions of HRS 0-1 (B.1), the eddy-mean triadic interactions (C.1.1), the ad-

Summary and Discussion

In this study, we have characterized the mesoscale EKE budget in the region of the Agulhas Current by addressing the following questions : (1) What is the vertical structure of eddy energy in the Agulhas Current region ? (2) What is the mesoscale EKE budget in the NAC and SAC ? (3) What are the processes driving the mesoscale eddies generation, dissipation and energy transfers toward smaller vertical structures in the NAC and SAC ?

The eddy vertical structures are evaluated through the energy vertical partitioning (EKE n and EAP E n ) into the 3 vertical modes categories : n = 0, n = 1 and n = 2 -9 (Figures 4. 5 and 4.6). The mesoscale eddies have interior (EKE 0 and EAP E 1 ) and upper layers-intensified (EKE 1 ) vertical structures and the relative proportions of EKE 0 and EKE 1 are interpreted as a measure of the mesoscale variability intensity. Mesoscale eddies dominate the variability in the path of intense mean currents (the Agulhas Current, the Agulhas Return Current and the Benguela Current). The smaller vertical scales eddies have surface-intensified vertical structures (n = 2 -9). They sign the EKE in the path of the Agulhas Current, where the current separates from the shelf (∼ 26 • E and 23 • E) and where submesoscale frontal eddies develop (∼ 26 • E) [START_REF] Tedesco | Generation of submesoscale frontal eddies in the Agulhas Current[END_REF]. They sign the EAP E in the Subgyre area, away from intense mean circulation, where mixed-layer related turbulence dominates the variability [START_REF] Schubert | The Submesoscale Kinetic Energy Cascade: Mesoscale Absorption of submesoscale Mixed Layer Eddies and Frontal Downscale Fluxes[END_REF]. Similar EAP E vertical profiles for intense and low mesoscale variability areas are found in the Gulf Stream [START_REF] Vic | Dispersion of deep-sea hydrothermal vent effluents and larvae by submesoscale and tidal currents[END_REF], suggesting that these EAP E vertical structures are generic.

The mesoscale EKE budget in the NAC and SAC is evaluated by the mesoscale eddy-energy (EKE 0-1 ) fluxes divergence (A.2 + A.3), which is constituted of a linear (eddy-pressure work A.2) and a non-linear (advection of EKE 0-1 by the total flow A.3) components (Figure 4.7). Both NAC and SAC are generation regions for the mesoscale eddies, which are then advected outside of the regions (Figure 4.7). The contributions of the linear and non-linear components of the mesoscale eddy-energy fluxes divergence differ over the Agulhas Current region. In low mesoscale variability areas (Alongshore Agulhas Current and Northern Subgyre in NAC), the linear component (A.2) is the main contribution, whereas in high mesoscale variability areas (Natal Bight in NAC and Eastern Agulhas Bank Bight and Agulhas Bank tip in SAC), the non-linear component (A.3) has a comparable or a larger contribution than the linear one (Figure 4.9).

The processes driving the mesoscale eddies generation, dissipation and energy transfers toward smaller This Chapter corresponds to a collection of results in preparation for a scientific publication which will be submitted to a journal after my doctorate.

Methods

The regional numerical simulation

We present in the following the regional numerical simulation of the Agulhas Current region realized with the Coastal and Regional COmmunity (CROCO) model. It is a free surface model, based on ROMS [START_REF] Shchepetkin | The Regional Oceanic Modeling System (ROMS): A splitexplicit, free-surface, topography-following-coordinate ocean model[END_REF], which solves the primitive equations in the Boussinesq and hydrostatic approximations using a terrain following coordinate system.

The dx ∼ 2.5 km numerical simulation (Figure 5.1) is the last of three nested grids built using the online 2-way nesting procedure based on AGRIF [START_REF] Debreu | Two-way nesting in split-explicit ocean models: Algorithms, implementation and validation[END_REF] with successive horizontal grid refinements : dx ∼ 22.5 km, 7.5 km and 2.5 km. Its vertical grid has 60 levels stretched at the surface following [START_REF] Haidvogel | Numerical ocean circulation modeling[END_REF]'s method. The dx ∼ 2.5 km grid is forced at the boundaries by the dx In this section, we assess the 1 st baroclinic mode eddy-energy sources and sinks diagnosed from the altimetry-based diagnostics by comparing the 1 st baroclinic mode eddy-pressure work diagnosed from either simulated or altimetry η (Eq. 5.15) to the one diagnosed from the simulated total-velocity field (eddy-pressure work in Eq. 5.6).

The simulated eddy-pressure work terms are computed using the vertical modes diagnosed from the mean modeled buoyancy frequency and the pressure field diagnosed from the modeled density following the relation from hydrostatic (Eq. 5.7) : p(x, z, t) = g η -H ρ(x, z, t) dz +gρ 0 η. The altimetry eddy-pressure work term is computed using the vertical modes diagnosed from the mean buoyancy frequency derived from the 1 • -resolution WOA18 climatology 1 (World Ocean Atlas 2018) data and the sea surface height anomalies from the 0.25 • -resolution AVISO dataset 2 . For all eddy-pressure work terms, velocity, pressure and sea surface height anomalies are defined relative to a time average over the 1995-2004 period.

The total-velocity eddy-pressure work has a noisy signal with an alternation of large amplitude patterns of opposite signs around topographic accidents (Figure 5.2c). It is smoothed with a 50 km-radius Gaussian kernel to evidence the large-scale signal net contributions (Figure 5.2d). The patterns around topographic accidents mostly compensate each other and do not have a net contribution on the 1 st baroclinic mode eddy-energy sources and sinks. The total-velocity eddy-pressure work is -1.19 10 -2 W m -2 for the smoothed term and -1.91 10 -2 W m -2 for the unsmoothed term in the Western Boundary region and it is 6.93 10 -3 W m -2 for the smoothed term and 5.40 10 -3 W m -2 for the unsmoothed term in the Interior region. The simulated and altimetry-η eddy-pressure works are also smoothed, even though their not smoothed (not shown) and smoothed (Figures 5.2a and 5.2b) patterns and magnitudes are very similar.

Both simulated and altimetry-η eddy-pressure work terms are in good agreement (Figure 5.2a and 5.2b). They have large-scale patterns characterizing the Western Boundary region as an uniform energy sink for the 1 st baroclinic eddies. The term diagnosed from altimetry data has larger magnitudes, especially in the Western Boundary region (AVISO : -1.07 10 -2 W m -2 and CROCO : -5.01 10 -3 W m -2 ). However, this region is located in a coastal area where the altimetry measure is less accurate. The good agreement be- scattering processes. The sum of these two vertical modes results in lower magnitude patterns dominated by the barotropic mode (Figure 5.5c). The Western Boundary (Subgyre) regions correspond to an energy sink (source) for the 1 st baroclinic mode eddies (Western Boundary : -1.19 10 -2 W m -2 and Subgyre : 6.93 10 -3 W m -2 in Figure 5.5b) while it corresponds to a low energy source (sink) for the mesoscale (n = 0 -1) eddies (Western Boundary : 6.06 10 -4 W m -2 and Subgyre : -1.85 10 -3 W m -2 , Figure 5.5c). The negative net contribution of the 1 st baroclinic eddy-pressure work term in the Western Boundary region, indicates that the 1 st baroclinic mode loses energy in this region by transfer toward other vertical modes and/or by dissipation. The partial compensation between a barotropic mode energy source and a 1 st baroclinic mode energy sink, suggests that the latter partially transfers its energy toward larger vertical scales (barotropization). However, this energy loss for the 1 st baroclinic mode eddies does not correspond to an energy loss for the mesoscale eddies. Compensation between both eddy-pressure work terms at the Eastern Agulhas Bank Bight (downstream of 26 • E), additionally suggests a barotropization process controlled by topography.

Chapter 6

CONCLUSIONS AND PERSPECTIVES

In this section, I recall the problematics addressed in this dissertation, the approaches used and the main results. I then discuss the implications of these results and the perspectives that could be addressed in future works.

Problematics and Approaches

This present dissertation aimed to improve our knowledge and phenomenological understanding of the meso and submesoscale dynamics in the Agulhas Current region. It is structured around the studies presented in Chapters 2, 3 and 4 that address the following questions respectively : We tackled these questions from the perspective of the Eddy Kinetic Energy (EKE) budget which gathers all the contributions to the EKE content and describes its evolution. This includes the EKE generation, dissipation and transfers between energy reservoirs (M KE : Mean Kinetic Energy and EAP E : Eddy Available Potential Energy) [START_REF] Harrison | Energy analysis of open regions of turbulent flows-Mean eddy energetics of a numerical ocean circulation experiment[END_REF]Gula et al., 2016b). We were particularly interested in characterizing the energy transfers between the different vertical structures. To this purpose, we derived a modal EKE equation, using the vertical modes traditional definition [START_REF] Gill | Atmosphere-Ocean dynamics[END_REF]. It gathers the classical contributions to EKE, within a vertical mode n, and the EKE transfer between 2019) and results of this study. The major mean currents (thick black line), the background circulation (dotted arrows) and meso and submesoscale eddies (large and small grey disks) are coarsely located and abbreviated this way (in the current direction) : Natal Pulse (NP), Submesoscale frontal eddies (Subm. frontal eddies), Shear-edge Eddy (SE), Cyclonic Lee Eddy (CLE), Agulhas Ring (AR) and South-West Indian Subgyre (Subgyre). The red (blue) areas denote the mesoscale eddies generation (energy loss) by barotropic instability (shaded areas), by mixed baroclinicbarotropic instability (hatched areas) and by triadic interactions transferring energy between reservoirs and vertical scales (dotted areas). The blue (red) triangular areas denote the mesoscale eddies energy loss (gain) by transfers to mean currents canalized by stratification-topographic horizontal gradients. The blue (red) open arrowhead disks denote the mesoscale eddy forward (inverse) vertical turbulent cascade. The blue oscillating arrows denote the generation of internal-gravity waves by the mesoscale eddy scattering over topographic accident and their radiation. The grey isolines denote the 1000 and 3000 m isobaths. Scheme inspired from [START_REF] Lutjeharms | The Agulhas Current[END_REF].

APPENDICES Appendix A -Evaluation of the EKE and EAP E modal expansions

The modal expansions of the modeled EKE and EAP E are used to evaluate the validity of the vertical modes decomposition of the dx ∼ 2.5 km regional numerical simulation, for which the flat-bottom condition is relaxed, and quantify how much of the variability is represented by the first 10 vertical modes. The modal expansions accuracy of the modeled EKE and EAP E is measured by the modal expansions residuals (Figure 6.3). The residuals are defined as the difference between the total energies and their modal expansions ( η

normalized by the total energies and converted into percentage.

The agreement between the eddy energy and their modal expansions using the 10 first vertical modes is weaker near-bottom for the EKE and near-surface for the EAP E (not shown). However, the EKE and EAP E residuals do not exceed 5 % and 20 % locally. This supports the traditional vertical modes approach as valid in the frame of the dx ∼ 2.5 km regional simulation and the 10 first vertical modes as accurately representing the modeled dynamics.

Appendix B -Evaluation of the terms of the KE, M KE and

EKE equations modal expansions

The modal expansions accuracy of the KE, M KE and EKE terms (Eq. 2.31, 2.32 and 2.34 of Chapter 2) using the 10 first vertical modes is measured by the modal expansions residuals. The residuals are defined as the difference between the total terms and their modal expansions (for the KE n terms : The residuals of the KE, M KE and EKE terms modal expansions are concentrated around the topographical features and have small magnitudes compared to the total terms (O(0.1-0.5) 10 -4 m 3 s -3 and locally peaking at 1 10 -4 m 3 s -3 for the advection and pressure gradient terms) (Figures 6.4,6.5 and 6.6). This supports that the 10 first vertical modes accurately represent the modeled (mean and eddy) dynamics.