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Chapitre 1 Introduction

The understanding of fluid on earth leads us to the study of geophysical fluids. In general, these fluids are stratified fluids such as the atmosphere and the ocean. Various layers are formed in nature due to the stratification. These layers have important effects on the geophysical movements such as climate changes and ocean currents. The ecosystem is also highly influenced through temperature, salinity and transport of micro-organisms, etc.

In stratified fluids like ocean interior and atmosphere, turbulence mixing is a phenomenon that frequently appears. Moreover, mixing and stratification effects are essential to the global distribution of energy and nutrients. This property makes them quite significant to climate and biogeochemical cycles in nature circulation system [START_REF] Wunsch | Vertical Mixing, Energy, and the General Circulation of the Oceans[END_REF][START_REF] Kuhlbrodt | On the driving processes of the Atlantic meridional overturning circulation[END_REF]. Therefore, their influences and related mechanisms are worth thorough examinations.

Basic concepts about stratified fluid

The stratified fluid is a fluid with density variations in the vertical direction. Two typical examples of stratified geophysical fluid are the Earth's atmosphere and its ocean.

Besides, although not commonly mentioned, the Earth's liquid core and the fluids in the core of other celestial bodies in astrophysics are all stratified.

Stratification is the result of multiple factors. For the Earth, the atmosphere is a gaseous envelope surrounding our planet, and the density stratification mainly results from its compressibility and the Earth's gravity. Figure 1.1 shows the spatial variations of density on the altitude and density decreases as the altitude increases. The density stratification in the atmosphere is also affected by temperature differences, which is the result of solar radiation, greenhouse effects and the presence of clouds. According to the temperature, the atmosphere is usually divided into four layers along elevation : troposphere, stratosphere, mesosphere, and thermosphere. The troposphere contains nearly 80% of the total mass and almost all water vapour. Meteorologists mainly focus on this region as it is the place where most weather occurs.

The density stratification in the ocean is shown in figure 1.2(a) and this stratification results from the temperature and salinity variations, which are demonstrated in figure 1.2(b). The density gradient is the largest in the layer called pycnocline. This region is stably stratified and separates the mixed layer of the surface and deeper ocean. The separation may hinder vertical transport, but [START_REF] Fernando | Turbulent Mixing in Stratified Fluids[END_REF] indicated that vertical mixing still appears due to the turbulence triggered by the shear effects. Such kind of mixing has significant effects on the marine ecosystem and transports nutrients in the ocean.

Buoyancy frequency

Stratification is usually stable in nature, as shown in figure 1.1 and figure 1.2. Under gravity, a stratified fluid tends to arrange itself so that heavier fluid (the fluid with higher density) are below the lighter fluid. If we consider a fluid in static equilibrium, this situation will be an intuitively stable status. However, if a fluid parcel is disturbed vertically and deviates from its equilibrium position, a recalling force will be produced by the buoyancy force and gravity. This process is illustrated in figure 1.3. Hence, the displaced parcel will oscillate around the equilibrium level under the recalling force. The frequency of the oscillation is termed as the buoyancy frequency, which is also named as Brunt-Väisälä frequency or stratification frequency. The buoyancy frequency is defined Source : [START_REF] Gossard | Waves in the atmosphere : atmospheric infrasound and gravity waves-their generation and propagation[END_REF].

as N = -g∂ z ρ/ρ 0 , where z is the vertical direction, g the gravity and ρ 0 the reference density. If ∂ z ρ > 0 in the definition of buoyancy frequency, which corresponds to a state where a heavier fluid is on top of the lighter fluid, overturning will appear in this case.

Furthermore, if a persistent external destabilisation is forced onto the fluid, such as heating on the bottom or cooling on the top, convection will dominate the fluid motion. In the following content of the thesis, our attention mainly focuses on the stably stratified fluid so that the buoyancy frequency is real in our studies.

The oscillation described in this section obviously exists in the atmosphere and ocean, as both of them are stably stratified fluids in terms of density. However, for compressible fluid like the atmosphere, the definition of buoyancy frequency is based on the potential temperature (figure 1.4). The buoyancy frequency in nature are depicted in figure 1.4 and figure 1.5 as examples. The profile in the atmosphere, as shown in figure 1.4, has more variations and a larger average frequency than that of the ocean. The maximum buoyancy frequency shown in figure 1.5 is approximately N ≈ 0.02 rad • s -1 .

The oscillations of a fluid parcel can propagate in the medium and then, waves are formed through propagation. To differentiate the external gravity wave (surface wave), which is generated by a similar but different mechanism, we can particularly call them the internal gravity waves. However, unlike the normal surface waves, internal waves can stretch tens of kilometres in length and propagate throughout the ocean for several hours.

As an example, the buoyancy frequency in the Sulu Sea illustrated in figure 1.5 was [START_REF] Mowbray | A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid[END_REF]. (b) Small-scale mixing processes in ocean interior. Source : [START_REF] Garrett | Oceanography : Mixing with latitude[END_REF] measured by [START_REF] Apel | The Sulu Sea Internal Soliton Experiment[END_REF]. The internal gravity waves appear can also be observed by satellite, as shown in figure 1.6 in the Sulu Sea.

Internal gravity wave

The internal gravity waves are transverse waves as the movement of the fluid parcels is perpendicular to the direction of wave propagation. For a freely-propagating internal gravity wave, there is a special characteristic : the group velocity is perpendicular to the phase velocity. As shown in figure 1.7(a), internal gravity waves can be generated experimentally in a uniformly stratified fluid (constant N ), and the energy is propagating away along surfaces of constant phase [START_REF] Mowbray | A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid[END_REF]. More features and behaviours of internal gravity waves can be described with the help of linear theories, as reviewed by [START_REF] Fritts | Gravity wave dynamics and effects in the middle atmosphere[END_REF]. The influences and effects of these waves are evaluated in these theories.

In nature, internal gravity waves also become visible in the satellite era, for example, figure 1.6 and 1.8. Internal waves exist both in the atmosphere and in the ocean, where the density stratification is stable and continuous. They are significant in vertical transfer and mixing.

In the circulation system of the ocean, the internal gravity waves transport microorganisms, plankton, and nutrients. However, the physics of mixing in ocean interior is not well understood, and the internal mixing problem should be considered [START_REF] Ivey | Density Stratification, Turbulence, but How Much Mixing ?[END_REF].

Although existing in the interior, internal gravity waves can alter the sea surface currents and make a difference. Where these currents converge, the surface is more turbulent and brighter. Where they diverge, the surface is smoother and darker, and thus, creating a zone called "slicks". The slicks in the Sulu sea appear as dark bands in the centre of figure 1.6. Faint but visible ripples of internal gravity waves appear in figure 1.8 with the help of sunlight. In the ocean, the corresponding waves are tens to hundreds of meters beneath the sea surface and propagate slowly there. The comparisons of the scales in the mixing processes are shown in figure 1.7(b). Phenomena that mix the ocean are hard to be resolved in general ocean circulation models because the scales of relevant turbulent eddies are typically smaller than the grid spacing. In both ocean and atmosphere, this kind of small eddies should be examined so that their effects can be parametrized [START_REF] Garrett | Oceanography : Mixing with latitude[END_REF]).

In the atmosphere, internal waves mainly appear in the troposphere where the density is relatively high. Due to a decrease in density, the amplitude of these waves has a quasi-exponential growth when propagating in the vertical direction. Hence, at a certain altitude, the energy and momentum of these waves are transported and spread as turbulence mixing. Although the gravity waves themselves are not visible in the atmosphere, the marks of their propagation are left behind and represented by the cloud. The special structures appear on satellite photos, for example, figure 1.8, where the uncommon clouds are marked as atmospheric waves.

The gravity waves in the atmosphere are usually caused by the mountain waves (Lee waves), which have been studied for a long time through multiple observations and analysis [START_REF] Fritts | Gravity wave dynamics and effects in the middle atmosphere[END_REF]. We often see the waves patterns of the clouds when the dry air moves towards the much moister air. The dry air pushes the moist air higher in the atmosphere, causing water vapour to amass into clouds. When the moister air rises, the gravity pulls it down since it is heavier in density. This procedure repeats and forms the wave patterns in the clouds. However, the source of the atmospheric waves shown in figure 1.8 is not clear. Because the wave patterns are far from the inland, and Western Australia is relatively flat, it is unlikely that these patterns are formed by mountain waves. The studies of internal gravity waves require further attentions because of their importance in global energy circulation and energy budget.

Numerous studies have also examined the effects of stable stratification in the context of atmospheric flows [START_REF] Mahrt | Stably Stratified Atmospheric Boundary Layers[END_REF]). The internal gravity waves are fully three-dimensional in space, and they are relatively small compared to the normal scales in climate models.

With the influence of some instabilities, the exchange of energy and momentum by internal gravity wave can lead to turbulence, mixture and dissipation [START_REF] Staquet | Internal Gravity Waves : From Instabilities to Turbulence[END_REF].

To have a better understanding of mechanisms and mixing effects, we need to take the localized dissipation of energy and small-scale processes into account. The onset of turbulent mixing is related to the instability of fluid. To uncover the basic physical mechanisms of instabilities, we need to simplify the realistic complex context while keeping the essential characteristics of the flow. Researches on the instability start with homogeneous fluids, both experimentally and theoretically.

Modal instability in stratified shear flow

The instability of shear flows has been a classic subject in fluid dynamics since the theoretical studies of [START_REF] Rayleigh | On the stability, or instability, of certain fluid motions[END_REF] and the experiments by [START_REF] Reynolds | An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels[END_REF] at the end of 19th century, and it is covered by several textbooks (e.g. [START_REF] Drazin | Hydrodynamic Stability[END_REF][START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]. For the instability problem, small amplitude perturbations and linearised Navier-Stokes equations are useful approximations and methods in the research field. With the exactly parallel base flow profiles, the linearized instability can be determined by the eigenvalue problem of Orr-Sommerfeld equation and the critical Reynolds number can be defined by the solutions. Some of the well-known instabilities are reviewed in this section.

Kelvin-Helmholtz instability

The instability associated with the inflexion point of the velocity profile (Kelvin-Helmholtz instability) is the one of the first instabilities discovered in fluid mechanics. The Kelvin-Helmholtz instability may appear in the atmosphere, as outlined by the clouds in Wavelike disturbances appear in this kind of unstable flow and grow exponentially with time. [START_REF] Fjørtoft | Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex[END_REF] later complemented that the inflexion point has to correspond to a maximum (rather than a minimum) of the velocity shear. Another general stability condition is called Howard's semicircle theorem [START_REF] Howard | Note on a paper of John W. Miles[END_REF]). This condition bounds the complex phase velocity of the disturbance in a semicircle, whose diameter is the difference between the largest and smallest velocity in the parallel shear flow.

The stable stratification is intuitively believed to have a stabilising effect. In a stably stratified fluid, [START_REF] Miles | On the stability of heterogeneous shear flows[END_REF] and [START_REF] Howard | Note on a paper of John W. Miles[END_REF] showed that the criterion of Kelvin-Helmholtz instability for two-dimensional inviscid parallel shear flow can be determined by the local Richardson number (Ri), which is the square of the ratio between buoyancy frequency and the vertical velocity shear rate. The Miles-Howard theorem states that the flow is stable when the local Richardson number is everywhere greater than or equal to 0.25. The stable stratification is thus believed to have a stabilisation effects on Kelvin-Helmholtz instability and can even eliminate the turbulence when the stratification is strong enough. However, the numerical simulations (e.g. Little work has been done when shear and stratification are not aligned in the same direction, but such kind of cases occur in nature (e.g. Figure 1.10). The Kelvin-Helmholtz instability is modified in such cases as studied by [START_REF] Deloncle | Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid[END_REF][START_REF] Candelier | Shear instability in a stratified fluid when shear and stratification are not aligned[END_REF][START_REF] Arratia | Non-modal instability mechanisms in stratified and homogeneous shear flow[END_REF].

Viscous instability (Tollmien-Schlichting wave)

In the subject of instability in fluids, the viscosity is well-known to have dual roles.

Most studies about viscous instability depend on the exact parallel shear flows. The plane Poiseuille flow is one of the simplest shear flows, but its instability and transition to turbulence have proved to be subtle. As a result, this flow is widely used to illustrate the fundamentals of the stability theory [START_REF] Drazin | Hydrodynamic Stability[END_REF]).

In the context of an unstratified fluid, [START_REF] Heisenberg | Über stabilität und turbulenz von FlÜssigkeitsströmen[END_REF] first demonstrated the viscous instability mechanism for plane Poiseuille flow. As the velocity profile of base flow has no inflexion point, the instability must be related to viscosity. The Reynolds number of the base flow can estimate the effects of viscosity, and instability appears when the Reynolds number exceeds a specific critical value (the critical Reynolds number). The viscous instability is commonly referred to as Tollmien-Schlichting (TS) wave [START_REF] Tollmien | Ein allgemeines Kriterium der Instabilität laminarer Geschwindigkeitsverteilungen[END_REF][START_REF] Schlichting | Zur enstehung der turbulenz bei der plattenströmung[END_REF]) in honour of the researchers who first predicted that Orr-Sommerfeld equation has unstable modes for flows without inflexion points. The Squire's theorem [START_REF] Squire | On the Stability for Three-Dimensional Disturbances of Viscous Fluid Flow between Parallel Walls[END_REF] gives the assertion that it is sufficient to consider two-dimensional modes to determine the critical Reynolds number. In other words, the most unstable mode is always two-dimensional. Experimental observations of TS wave were obtained by [START_REF] Klebanoff | The three-dimensional nature of boundary-layer instability[END_REF].

For unstratified plane Poiseuille flow, [START_REF] Orszag | Accurate solution of the Orr-Sommerfeld stability equation[END_REF] Experimental results of [START_REF] Nishioka | An experimental investigation of the stability of plane Poiseuille flow[END_REF] show good agreement that the small perturbations behave as predicted by the linear stability theory, in a condition of a very low background turbulence level. However, the transition in plane Poiseuille flow is sensitive to ambient disturbances. [START_REF] Carlson | A flow-visualization study of transition in plane Poiseuille flow[END_REF] observed that both natural and artificially triggered transition can occur for Reynolds numbers slightly greater than 1000, and the most amplified wave is three dimensional. More experimental results (e.g. [START_REF] Nishioka | Some observations of the subcritical transition in plane Poiseuille flow[END_REF][START_REF] Alavyoon | Turbulent spots in plane Poiseuille flow-flow visualization[END_REF]) and the considerable variations of different critical Reynold numbers indicate that the three-dimensional effects are important in the transition of plane Poiseuille flow, which is opposed to the Squire's theorem. This reduction of critical

Reynolds number is related to the so-called subcritical instability. The subcritical instability exists in a fluid with a Reynolds number below the critical one. Experiments in boundary layer flows (e.g. [START_REF] Klebanof | Effect of free-stream turbulence on a laminar boundary layer[END_REF][START_REF] Kendall | Experimental Study of Disturbances Produced in a Pre-Transitional Laminar Boundary Layer by Weak Freestream Turbulence[END_REF][START_REF] Matsubara | Disturbance growth in boundary layers subjected to free-stream turbulence[END_REF] also demonstrate that transition is usually preceded by streamwise motion in the form of streaks rather than the TS waves predicted by modal stability analysis. This phenomenon is especially distinct in the presence of natural background disturbance.

The limitation of Squire's theorem was further revealed by [START_REF] Wu | Influence of small imperfections on the stability of plane Poiseuille flow and the limitation of Squire's theorem[END_REF], who examined the influence of small imperfections on the stability of plane Poiseuille flow.

The classical Squire's transformation was also extended by [START_REF] Jerome | Extended Squire's transformation and its consequences for transient growth in a confined shear flow[END_REF].

In a stratified fluid, the Squire's theorem is not applicable, but [START_REF] Ohya | Turbulence structure of stable boundary layers with a near-linear temperature profile[END_REF] experimentally proved the existence of TS waves in stratified boundary layers. However, the effect of stratification on TS waves is less understood, and it could be destabilising as indicated by Wu & Zhang (2008a). Although the Squire's theorem applies strictly to linear normal-mode analysis for a homogeneous viscous fluid, both the 3D and 2D TS waves should be examined in a stably stratified fluid.

Radiative instability

Although it may be true to a degree that stratification can stabilise the fluid [START_REF] Miles | On the stability of heterogeneous shear flows[END_REF][START_REF] Howard | Note on a paper of John W. Miles[END_REF], instability still exist even without inflexion points in the velocity profiles of the flow. For instance, [START_REF] Churilov | Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points[END_REF][START_REF] Churilov | Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points. Part 2. Continuous density variation[END_REF] analysed the stability of the flows with inflection-free velocity profiles in a stratified fluid. Stratified fluids can support internal gravity waves. Together with shear effects, small scale unstable waves are generated and thus can promote turbulent mixing in stratified fluids.

When shear and stratification are not aligned, an inflection-free boundary layer can become unstable. The coupling between shear effects and internal gravity waves becomes the strongest when the inclination angle between shear and stratification is π/2, as demonstrated by [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF]. The responsible instability for this phenomenon is called radiative instability. The word 'radiative' describing this instability is borrowed from the quantum mechanics because the demonstrated phenomenon is analogous to the radioactive decay of nuclei (Le Dizès & Billant 2009).

The radiative instability is a hydrodynamic instability that is much less known than the shear instability. However, it has been a subject of many works for decades, and it has also shown its significant influences with some specific conditions. [START_REF] Broadbent | Acoustic destabilization of vortices[END_REF] first found the radiative instability in the context of vortices in a compressible fluid.

This instability appears in various contexts, in the shallow waters associated with surface gravity waves, in compressible fluids with acoustic waves and in linearly stratified flows with internal gravity waves. The genesis of radiative instability is reviewed in [START_REF] Riedinger | Instabilité radiative d'un tourbillon dans un fluide stratifié[END_REF].

With the help of a WKBJ analysis for large axial wave number, Le Dizès & Billant (2009) and Billant & Le Dizès (2009) showed that the radiative instability could be vie-wed as a result of an over-reflection process at a critical level. The relationship between instability and over-reflection mechanism has been studied for a long time (e.g. [START_REF] Lindzen | Wave Over-reflection and Shear Instability[END_REF][START_REF] Lindzen | Instability and wave over-reflection in stably stratified shear flow[END_REF]. Further details about this mechanism will be explained later in section 1.3.1.

In recent researches, Riedinger et al. (2010a) obtained radiative instability for the Lamb-Oseen vortex and analysed the Froude and Reynolds number effects. Experimental results were first provided in Riedinger et al. (2010b). In figure 1 .11, Riedinger et al. (2011) also demonstrated, both experimentally and numerically, the occurrence of the radiative instability generated by a rotating cylinder in the stratified flow. In Taylor-Couette and Keplerian flows, Le Dizès & Riedinger (2010) analysed the transition of the instability for a finite-gap, which is associated with a mechanism of resonance, into the radiative instability for an infinite gap. Park & Billant (2013b) studied the stability of stratified

Taylor-Couette flow and its relationship to radiative instability. [START_REF] Park | Radiative instability of an anticyclonic vortex in a stratified rotating fluid[END_REF], 2013a) also investigated the radiative instability of a columnar Rankine vortex in a stratified rotating fluid.

In addition to the stratified rotating flows, the radiative instability can also be found in supersonic flows [START_REF] Mack | On the inviscid acoustic-mode instability of supersonic shear flows -Part 1 : Two-dimensional waves[END_REF][START_REF] Parras | Temporal instability modes of supersonic round jets[END_REF], atmospheric jets, boundary layers [START_REF] Candelier | Instabilités radiatives des couches limites atmosphériques[END_REF][START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF]) and stratified shallow waters [START_REF] Satomura | An investigation of shear instability in a shallow water[END_REF][START_REF] Riedinger | Critical layer and radiative instabilities in shallow-water shear flows[END_REF].

Mechanisms of modal instability in stratified fluids

Because of the abundant literature about the radiative instability and its universal characters, the mechanism was rediscovered several times in history by different communities, and was explained in terms of over-reflection phenomenon [START_REF] Grimshaw | On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile[END_REF][START_REF] Lindzen | Instability and wave over-reflection in stably stratified shear flow[END_REF][START_REF] Takehiro | Over-reflection and shear instability in a shallow-water model[END_REF], negative energy waves [START_REF] Kópev | Acoustic instability of an axial vortex[END_REF][START_REF] Schecter | Damping and pumping of a vortex Rossby wave in a monotonic cyclone : Critical layer stirring versus inertia-buoyancy wave emission[END_REF] or spontaneous wave emission [START_REF] Plougonven | Internal gravity wave emission from a pancake vortex : An example of wave-vortex interaction in strongly stratified flows[END_REF]Le Dizès & Billant 2009).

The details of the over-reflection process is explained in this part.

Over-reflection phenomenon

Over-reflection is a phenomenon that the incident waves in a fluid can be reflected with a consequent increase in amplitude. In other words, the reflection coefficient in some circumstances can be greater than 1. This mechanism is found in numerous situations and physical models. The over-reflection phenomenon was first encountered in the field of sound waves by [START_REF] Miles | On the Reflection of Sound at an Interface of Relative[END_REF] and [START_REF] Ribner | Reflection, Transmission, and Amplification of Sound by a Moving Medium[END_REF]. [START_REF] Acheson | On over-reflexion[END_REF] later summarised the conditions for the occurrences of over-reflection. He also attempted to clarify the way in which the excess reflected waves extracted energy from the mean flow motion, and the sense that the transmitted wave can be viewed as a carrier of 'negative energy'. The overreflection process was thus explained through the conservation of wave action [START_REF] Acheson | On over-reflexion[END_REF]).

The summary by [START_REF] Acheson | On over-reflexion[END_REF], however, is not sufficient. The connections between over-reflection and unstable modes are discovered in different context [START_REF] Lindzen | Wave Over-reflection and Shear Instability[END_REF][START_REF] Lindzen | A study of over-reflection in viscous Poiseuille flow[END_REF]. [START_REF] Lindzen | Instability of plane parallel shear flow (toward a mechanistic picture of how it works)[END_REF] summarised and gave the statement that the mode which satisfies 'quantization' conditions can become unstable through the over-reflection process. Then the unstable mode determines the wave geometry.

A quantization zone is an oscillating area confined either by two turning levels or by one turning level together with a perfectly reflecting wall. Perturbations in this area will propagate, be reflected by over-reflection and thus increase. The reflection is always accompanied by a transmitted wave in the external transmission region. This reflection satisfies the conservation law of momentum for three waves : incident, transmitted and reflected waves. However, the conservation law of energy is different in this case, as part of the energy is drawn from the mean flow motion.

A perturbation in the quantization zone can be modelled as a wave propagating in this domain to illustrate the mechanism. A simple example of a wave packet is shown in figure 1.12. The phase velocity and group velocity may be different and can move towards separate directions.

The comparison between a conventional reflection case and an over-reflection case is illustrated in figure 1.13. In the conventional reflection case (figure 1.13 (a)), for example,

the radioactive emission through a potential barrier, there is no critical level inside the evanescent region between two turning levels. The phase velocity and group velocity of the incident wave in the conventional case are in the same direction. The intermediate evanescent region partially reflects the incident wave with an amplitude A i . The amplitudes of the reflected and transmitted waves are A r and A t , respectively. The classical relationship

(A i = A r + A t )
for the amplitudes of these three waves applies to this conventional case.

However, the circumstances are different in the over-reflection case, as illustrated in figure 1.13 (b). In the intermediate evanescent region, there exists a critical level, which has the effect of changing the group velocity's direction. Hence, the directions of the group velocity and phase velocity are opposite in the propagation area of the incident and reflected wave, but not in the field of the transmitted wave. As the group velocity represents the energy of the wave packet, the roles of the incident and reflected waves are 'inverted' from the point of energy conservation. Therefore, the amplitude of the reflected wave is greater than that of the incident wave (i.e. A r > A i ). The transmitted wave has an opposite amplitude (A t < 0) or phase compared to the incident wave, as shown in figure 1.14. According to the amplitudes, the energy associated with the wave packet grows after reflection, and this phenomenon is called 'over-reflection'.

Although the over-reflection process is not a sufficient condition for the appearance of instability in a fluid, many kinds of instabilities are revealed to be associated with this mechanism, for example, barotropic instability [START_REF] Lindzen | Wave Over-reflection and Shear Instability[END_REF] and baroclinic instability [START_REF] Lindzen | The Concept of Wave Overreflection and Its Application to Baroclinic Instability[END_REF]. The generation process of radiative instability is illustrated in figure 1.15 based on [START_REF] Takehiro | Over-reflection and shear instability in a shallow-water model[END_REF]. The wave packet is overreflected by the critical level and perfectly reflected by the rigid wall. The amplitude of the wave is increased by repeating this process. The disturbance propagates far away and does not diminish like in the common cases. The oscillation frequency and the wavelength are determined by the reflection processes, while the growth rate is associated with the over-reflection.

Resonance phenomenon

In the stratified rotating flows with Taylor-Couette and Keplerian velocity profiles, Le Dizès & Riedinger (2010) showed that the linear inviscid instability change into radiative instability as the size of the gap d becomes infinite, and is associated with a mechanism of resonance when d is finite. This instability is called strato-rotational instability in the Taylor-Couette system and is considered to be directly related to the emission of internal gravity flows (Le Dizès & Billant 2009;Billant & Le Dizès 2009).

The resonance mechanism for small gaps is interpreted by [START_REF] Yavneh | Non-axisymmetric instability of centrifugally stable stratified Taylor-Couette flow[END_REF].

For large gaps, results of numerical methods and WKBJ analysis show that when the frequency curves of trapped modes cross, one of the two modes becomes unstable around the corresponding points (Le Dizès & Riedinger 2010). This mechanism of resonance is also described in the context of shallow waters [START_REF] Satomura | An investigation of shear instability in a shallow water[END_REF]. The same phenomenon is to be expected for the stratified plane Poiseuille flow, and we will discuss the resonance mechanism in chapter 4

Transient growth and optimal perturbation

Modal analysis has dominated the field of stability theory for many decades. Based on the eigenvalue analysis, plenty of significant accomplishments were achieved (e.g. [START_REF] Orr | The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid[END_REF][START_REF] Tollmien | Über die Entstehung der Turbulenz[END_REF]. However, the subcritical instability has shown its importance in numerous studies, as mentioned in section 1.2. The limitation of modal approach is recognised, and then a novel method (transient growth theory) has emerged in the 1990s (e.g. [START_REF] Farrell | An Ajoint Method for Obtaining the Most Rapidly Growing Perturbation to Oceanic Flows[END_REF][START_REF] Trefethen | Hydrodynamic Stability Without Eigenvalues[END_REF]). Transient growth theory, which can give a quantitative description of the short-time effects of the subcritical instabilities, has drawn considerable attention since then. Even though all eigenvalues are stable according to the modal analysis, a general initial disturbance may undergo large growth in finite time [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF][START_REF] Reddy | Energy growth in viscous channel flows[END_REF].

Non-normality and finite time intervals

The transient growth theory has achieved several developments and now provided a better understanding of the transition in various flows [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF][START_REF] Schmid | Nonmodal Stability Theory[END_REF].

The finite-time increase of the total perturbation can appear without nonlinear effects. Mathematically, it is because the governing linear operator is not self-adjoint [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF]) and the associated eigenfunctions are non-orthogonal [START_REF] Reddy | Energy growth in viscous channel flows[END_REF]. The linear nature of the transient growth is demonstrated in figure 1.16, and is also revealed in [START_REF] Trefethen | Hydrodynamic Stability Without Eigenvalues[END_REF] and [START_REF] Henningson | On the role of linear mechanisms in transition to turbulence[END_REF]. The basic mechanism can be explained by this simple geometric example [START_REF] Schmid | Nonmodal Stability Theory[END_REF]. Over a finite time interval, the lengths of two component vectors Φ 1 and Φ 2 decrease by 20% and 50%. However, the difference between these two vectors f may grow in length and changes its direction transiently. The vector f can be viewed as an initial condition for a flow at time t = 0, and it may be amplified over a finite time interval, even though f → 0 when t → ∞ in this specific case. As illustrated by the process in figure 1.16, even if each eigenfunction decays exponentially according to its associated eigenvalue, a linear combination of non-orthogonal eigenfunctions may still lead to large transient growth before the time it eventually decreases at the rate of the least stable eigenfunctions. Although the transient growth in many studies is related to degenerate eigenvalues, it is not necessarily limited to them [START_REF] Reddy | Energy growth in viscous channel flows[END_REF].

Optimal perturbation and adjoint equations

Different types of perturbations may experience transient growth, but only the most amplified one can dominate the behaviour and structure of the fluid. The optimal perturbation provides the initial conditions that lead to a maximum energy growth over a finite time interval T , thus is also referred to as optimal initial conditions. The most amplified perturbation at the time t = T can be named as the optimal response. As T → ∞, the optimal response tend towards the eigenfunction associated with most unstable modal instability. Correspondingly, in this case, the optimal initial conditions are determined by the dominant adjoint eigenmode. Even in the flows expected to be stable by the modal analysis, [START_REF] Kerswell | An optimization approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar[END_REF] demonstrated that when the amplified transient perturbations reach finite amplitude, they may be essential to the 'subcritical' transition.

Mechanism of transient growth

In the transient growth analysis, two fundamental mechanisms are commonly identified, the Orr mechanism [START_REF] Orr | The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid[END_REF]) and the lift-up mechanism [START_REF] Ellingsen | Stability of linear flow[END_REF][START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF]). The background of these two mechanisms is introduced in this section.

First works are also performed on homogeneous fluid.

Orr mechanism

The Orr mechanism, also called Reynolds stress mechanism [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF], is associated with a disturbance that can extract energy from mean shear through the Reynolds stress term. As shown in figure 1.17(a), with the effect of Orr mechanism, the optimal initial condition is titled against the mean shear at t = 0. As time evolves, the vortices are sheared by the mean flow and maximise the energy grow when it aligns in the cross-stream direction, as shown in figure 1.17(b) [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]. [START_REF] Ellingsen | Stability of linear flow[END_REF] demonstrated that a finite streamwise-uniform perturbation might lead to instability in a linear flow which has no inflexion point in the velocity profile.

Lift-up mechanism

Even in an exponentially stable flow predicted by the modal stability analysis, the initial disturbance may undergo an important total perturbation energy growth. According to the linear theory, [START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF] showed that such kind of transient growth appears in all inviscid parallel shear flows, and the kinetic energy of perturbation can grow at least linearly with time. [START_REF] Gustavsson | Energy growth of three-dimensional disturbances in plane Poiseuille flow[END_REF] proved that the initial perturbations could determine the onset of transition to turbulence in plane Poiseuille flow, even if the flow is expected to be exponentially stable. [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] and [START_REF] Reddy | Energy growth in viscous channel flows[END_REF] later established that transient growth is important for the amplification of perturbations in a finite time interval. In some stable shear flows, the primary initial perturbations that maximise the energy growth is shown to be streamwise-uniform with non-zero wall normal velocity [START_REF] Luchini | Reynolds-number-independent instability of the boundary layer over a flat surface : optimal perturbations[END_REF][START_REF] Schmid | Nonmodal Stability Theory[END_REF]). For unstratified plane Poiseuille flow, as illustrated in figure 1.17(c,d), the streamwise vortices change little as time evolves. However, the streamwise velocity greatly grows in the process which will, in the end, lead to streaks in the fluid [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF].

This phenomenon is the physical representation of the lift-up mechanism [START_REF] Ellingsen | Stability of linear flow[END_REF][START_REF] Landahl | Wave Breakdown and Turbulence[END_REF], which is also called vortex tilting or vortex stretching process [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF]) because of the three-dimensional deformation.

Combination of Orr and lift-up mechanisms

The two mechanisms described above determines the optimal initial conditions as streamwise vortices (lift-up) or upstream tilting spanwise vortices (Orr). However, for short time intervals, the structures of the most amplified perturbations are titled roll vortices. Farrell & Ioannou (1993a) interpreted this phenomenon as combinations of the two fundamental mechanisms. Farrell & Ioannou (1993b) also showed that this threedimensional structure is a universal character in shear flows and is consistent with the result in [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] and [START_REF] Butler | Optimal perturbations and streak spacing in wall bounded shear flow[END_REF]. In the stratified fluid, Arratia (2011) also studied this synergy effect in the context of horizontal shear layers.

Motivation and purpose

As discussed in section 1.1.2, internal waves are ubiquitous and essential in nature. Thus, their generation mechanisms have drawn lots of research interest. The problem about the sources remains a debate in an extensive literature. Wu & Zhang (2008b) indicated that instability waves could generate internal waves if they approach a boundary.

Le Dizès & Billant (2009) also demonstrated that the radiative instability can be associated with the emission of internal waves in stratified shear flows. [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF] showed that the radiative instability is present in inviscid boundary layers when the shear direction is inclined relative to the stratification. It is reasonable to quantify the viscous effect on the radiative instability, and it is one of the major purposes of the present thesis.

We also aim to evaluate the stratification effect on the transient growth, especially the effects on the related fundamental mechanisms.

In fluid mechanics, idealised models are widely used in theoretical and experimental researches to focus on the important aspects and elements. We use tanh velocity profile as a boundary layer and plane Poiseuille flow as a prototype of channel flow in our studies because of their simple configurations. Modal and non-modal analysis are both conducted in our studies. As one of the first prototypes for transient growth, unstratified plane Poiseuille flow is demonstrated to exhibit large transient growth in finite time intervals [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF][START_REF] Reddy | Energy growth in viscous channel flows[END_REF]. The basic mechanisms are revealed and classified in previous works (Farrell & Ioannou 1993a;[START_REF] Arratia | Non-modal instability mechanisms in stratified and homogeneous shear flow[END_REF]. It becomes natural to ask, what is the effects of vertical stratification on the transient growth ? Which mechanism is dominant in stratified flows ?

Summary of this thesis

The present thesis is composed of 5 chapters.

In chapter 1, basic concepts in stratified fluid have been introduced in a general way.

The motivations and objectives of our study have been presented. We have also briefly introduced some well-known instabilities in parallel shear flows and stratified fluid : viscous instability, Kelvin-Helmholtz instability and the radiative instability. We have mostly focused on the viscous instability and the radiative instability in boundary layers.

The concepts of both modal and non-modal stability analysis have been introduced and described. For the modal instability, we have discussed the over-reflection and resonance phenomena, and for transient growth, we have reviewed the Orr mechanism, lift-up mechanism and the combination of both.

In chapter 2, the governing equations are derived, and the numerical method we used is introduced. We have solved the linearised Navier-Stokes equations under the Boussinesq approximation for the stratified flow. The pseudo-spectral method is used for the spatial discretization. We have also applied a complex path method. The transient growth is evaluated by the 'Direct-Adjoint-Loop' method, and the specific procedure is described in this chapter.

Chapter 3 contains the modal stability results of a boundary layer flow on a vertical wall in the presence of stratification. The viscous instability and radiative instability are compared for various Reynolds numbers and Froude numbers. The viscous effect on radiative instability is also exhibited in this part.

Chapter 4 reveals both the exponential and transient growth in stratified plane Poiseuille flow. The three-dimensional instability due to the resonance mechanism is investigated. The transient growth is computed for both long and short time intervals. The stratification effects on the fundamental mechanisms are examined to determine the dominate one.

Finally, some conclusions and perspectives are provided in chapter 5.

Chapitre 2 Method

In this chapter, the mathematical formulation of two models are derived : a plane Poiseuille flow between two vertical walls and a hyperbolic tangent boundary layer flow on one vertical wall. The fluid in both cases is uniformly stratified.

With the Boussinesq approximation, the governing equations for an incompressible viscous fluid are

∂u ∂t = -(u • ∇)u - 1 ρ 0 ∇p + ν∆u + ρ ρ 0 g, (2.1a) ∇ • u = 0 (2.1b)
where ν is the kinetic viscosity and g is the gravity. For a stratified flow, the density

diffusion equation is ∂ρ ∂t + (u • ∇)ρ = D∆ρ (2.2)
where D is the mass diffusivity.

Base flow and perturbation equations

In this part we consider a horizontally directed flow of a viscous and stably stratified incompressible fluid in a vertical channel and on a wall, as sketched in figure 2.1. The stratification is assumed to be uniform with a constant buoyancy frequency

N = - g ρ 0 ∂ρ ∂z (2.3)
where g is gravity, ρ 0 is a characteristic value of the mean density ρ.

In the channel, the base flow is assumed to be a parallel flow, i.e., a flow in x-direction that only depends on the wall-normal direction y (see figure 2.1 defining the coordinate system and base flow). Using the maximum velocity at the centre of the channel U and half the channel width L for non-dimensionalization, we define the base flow velocity field as For the boundary flow on a wall, we apply a local parallel-flow assumption and assume that the base flow velocity field is given by the tanh profile u = U 0 (y)e x = tanh (y) e x .

U = U 0 (y)e x = (1 -y 2 )e x . -1 y 1 (2.4)
(2.5)

where e x is the unit vector in the streamwise direction x, and where we have used the main stream velocity U and the boundary layer thickness L to non-dimensionalize all the variables.

The characteristics of the base flow are defined by three parameters :

Re = U L ν
The Reynolds number, (2.6a)

Sc = ν D
The Schmidt number, (2.6b)

F = U N L The Froude number, (2.6c)
where ν and D are the kinematic viscosity and the mass diffusivity of the fluid. In the present study, for the plane Poiseuille flow, the Schmidt number is set to Sc = 1. For the boundary layer flow, we assume Sc = ∞.

To study the stability of this flow, we consider a basic state (U i , P , ρ) and a perturbed state (U i + u , P + p , ρ + ρ ), both satisfying the Navier-Stokes equations (2.1) and the density diffusion equation (2.2). The expressions are

u(x, t) = U + u (x, t), (2.7a) P (x, t) = P + p (x, t), (2.7b) ρ(x, t) = ρ + ρ (x,
∂u ∂t = -U 0 ∂u ∂x - dU 0 dy v - ∂p ∂x + 1 Re ∆u , (2.8a) ∂v ∂t = -U 0 ∂v ∂x - ∂p ∂y + 1 Re ∆v , (2.8b) ∂w ∂t = -U 0 ∂w ∂x - ∂p ∂z + 1 Re ∆w + b F 2 , (2.8c) ∂b ∂t = -U 0 ∂b ∂x + 1 ReSc ∆b -w , (2.8d) 0 = ∂u ∂x + ∂v ∂y + ∂w ∂z (2.8e)
where b = ρ /∂ z ρ and the nonlinear terms are neglected.

Optimal perturbations

The method of non-modal analysis is first introduced in this section. Optimal perturbation provides the initial conditions that excite the most energetic perturbation in a given base flow. Recently, [START_REF] Kaminski | Transient growth in strongly stratified shear layers[END_REF] analysed the transient growth in a strongly stratified shear layers depending on the now-conventional 'direct-adjoint looping' (DAL) method, which is summarised by [START_REF] Schmid | Nonmodal Stability Theory[END_REF]. We used a similar method for the non-modal analysis.

The application of adjoint equations has been developed significantly in the field of hydrodynamic stability theory [START_REF] Luchini | Adjoint Equations in Stability Analysis[END_REF]). In the DAL method, it helps to identify the optimal perturbations [START_REF] Schmid | Nonmodal Stability Theory[END_REF]. Variational and optimisation techniques are incorporated in this method. The objective of the optimisation process is the total energy growth over a given time interval (0 t T ), and the constraints are the governing equations, for example, the linearised Navier-Stokes equations, and boundary conditions and initial conditions.

The iterative procedure of 'direct-adjoint-loop' method is shown in figure 2.2. At the start of the loop in DAL method, a general initial perturbation for the 'direct' variables (mostly physical variables) is involved, and the 'direct' equations for the evolution are integrated forward from t = 0 to t = T . At time t = T , the total energy growth is evaluated, and the direct variables are converted to the adjoint variables as the terminal conditions for the 'adjoint' equations. Then the adjoint equations are subsequently integrated backward from t = T to t = 0. At time t = 0, the adjoint variables are recovered to the direct ones and thus the new initial conditions for the direct problem are generated. This process is repeated until the initial perturbation converges to the optimal one and the total energy growth is maximised. It has become an essential purpose of transient growth analysis to evaluate the evolution and amplification quantitatively.

In this part, we introduce the DAL variational method in the context of stratified flows. The total perturbation energy E(T ) is defined as a sum of the kinetic energy and potential energy of the perturbation (2.9),

E = u , u 2 + 1 2 b , b F 2 (2.9)
where angled bracket denote an appropriate inner product, e.g. u, v ≡ 1 V V u • vdV , and V is the domain volume. The weighted 2-norm in equation (2.9) is natural because in this way, the body force terms in equation (2.8c,d) can be cancelled in the expression

dE dt = - 1 V V dU 0 dy v u + 1 Re |∇u | 2 2 + 1 ReScF 2 |∇b | 2 2 dV (2.10)
The boundary conditions in our model are applied to get this expression. In equation (2.10), the first term in the bracket is related to the Reynolds stress, and the other two positive definite terms are respectively associated with the dissipation of kinetic and potential energy.

The objective function of our optimisation procedure is the perturbation energy gain

(2.11)

G(T ) = E(T ) E 0 (2.11)
where

E 0 = u 0 , u 0 2 + 1 2 b 0 , b 0 F 2 .
Specifically, we search for the initial conditions (u 0 (x), b 0 (x), p 0 (x)) that can maximize the perturbation energy gain G(T ) over a finite time T .

The evolution of the initial perturbations is governed by (2.8). The objective functional is defined as

L = E(T ) E(0) - ∂u i ∂t + u j ∂U i ∂x j + U j ∂u i ∂x j + ∂p ∂x i - 1 Re ∆u i - b F 2 δ i3 , u A i - ∂b ∂t + U j ∂b i ∂x j + u i δ i3 - 1 ReSc ∆b , b A - ∂u i ∂x i , p A -u i (0) -u 0i , u A 0i -b (0) -b 0 , b A 0 -p (0) -p 0 , p A 0 (2.12) where [u, v] ≡ T 0 u, v dt.
The first term is the perturbation energy gain, and the penalty terms contain the constraints of governing equations (2.8) and the initial conditions of the flow at t = 0. The Lagrange multipliers (u A (x, t), b A (x, t), p A (x, t)) represent the adjoint variables. The direct evolution equations, i.e. the linearised Navier-Stokes equations, continuity equation, and density distribution equation, can be recovered by taking variations of L with respect to the adjoint variables. Variations of L with respect to

(u A 0 (x), b A 0 (x), p A 0 (x)) set the initial conditions (u 0 (x), b 0 (x), p 0 (x)
) for the flow. The subscript '0' in these variables indicates that these quantities are the initial conditions at time

t = 0.
The adjoint equations, which describe the evolution of the adjoint variables, can be obtained by taking variations of (2.12) with respect to the direct (physical perturbation) variables (u , b , p ), and setting the results equal to zero. These equations read

∂u A ∂t = -U 0 ∂u A ∂x - ∂p A ∂x - 1 Re ∆u A , (2.13a) ∂v A ∂t = dU 0 dy u A -U 0 ∂v A ∂x - ∂p A ∂y - 1 Re ∆v A , (2.13b) ∂w A ∂t = -U 0 ∂w A ∂x - ∂p A ∂z - 1 Re ∆w A + b A , (2.13c) ∂b A ∂t = -U 0 ∂b A ∂x - w A F 2 - 1 ReSc ∆b A , (2.13d) 0 = ∂u A ∂x + ∂v A ∂y + ∂w A ∂z (2.13e)
The adjoint equations (2.13) must be satisfied for all times over the interval [0, T ],

and should be well-posed when integrated backwards in time [START_REF] Schmid | Nonmodal Stability Theory[END_REF][START_REF] Luchini | Adjoint Equations in Stability Analysis[END_REF]. Variations of (2.12) with respect to (u , b , p ) can also give compatibility conditions, which represent the relationships between the direct variables (u , b , p ) and the adjoint variables (u A , b A , p A ) at the target time T :

u i E 0 -u A i t=T = 0, (2.14a) b F 2 E 0 -b A t=T = 0 (2.14b)
The variations of L with respect to the initial perturbations (u 0 , b 0 , p 0 ) should also be set as zero to maximize the objective functional (2.12). This functional can give the compatibility conditions that should be satisfied at the initial time t = 0 :

u A 0i E 2 0 E(T ) -u 0i t=0 = 0, (2.15a) F 2 E 2 0 b A 0 E(T ) -b 0 t=0 = 0 (2.15b)
The perturbations can be further assumed to be in the form of

u (x, t) = u(y, t)e ikxx+ikzz , (2.16a) b (x, t) = b(y, t)e ikxx+ikzz , (2.16b) p (x, t) = p(y, t)e ikxx+ikzz (2.16c)
This form (2.16) is introduced into (2.8), the resulting equations can be written as :

∂u ∂t = -iU 0 k x u - dU 0 dy v -ik x p + 1 Re ∆ y u, (2.17a) ∂v ∂t = -iU 0 k x v - dp dy + 1 Re ∆ y v, (2.17b) ∂w ∂t = -iU 0 k x w -ik z p + b F 2 + 1 Re ∆ y w, (2.17c) ∂b ∂t = -iU 0 k x b -w + 1 ReSc ∆ y b, (2.17d) 0 = ik x u + dv dy + ik z w (2.17e)
where

∆ y = d 2 dy 2 -k 2 x -k 2 z .
The corresponding adjoint equations are :

∂u A ∂t = -iU 0 k x u A -ik x p A - 1 Re ∆ y u A , (2.18a) ∂v A ∂t = dU 0 dy u A -iU 0 k x v A - dp A dy - 1 Re ∆ y v A , (2.18b) ∂w A ∂t = -iU 0 k x w A -ik z p A + b A - 1 Re ∆ y w A , (2.18c) ∂b A ∂t = -iU 0 k x b A - w A F 2 - 1 ReSc ∆ y b A , (2.18d) 0 = ik x u A + dv A dy + ik z w A (2.18e)
In the context of method described above, optimal perturbation is an initial configuration (u 0 , v 0 , w 0 , b 0 , p 0 ) that can maximize the energy gain G(T ) at a prescribed terminal time t = T , for a given set of parameters (k x , k z , Re, F, Sc).

The optimal perturbations can be obtained by applying the iterative optimisation algorithm, which essentially relies on the power iteration method [START_REF] Corbett | Optimal perturbations for boundary layers subject to stream-wise pressure gradient[END_REF][START_REF] Schmid | Nonmodal Stability Theory[END_REF]). The randomly initialized flow marches forward in time from t = 0 to t = T , satisfying the direct governing equations (2.17a-e). At time t = T , the initial adjoint variables are generated with the help of the direct physical variables and the compatibility conditions (2.14). Then the adjoint variables evolve backward in time with the adjoint equations (2.18a-e) until t = 0. The initial conditions can then be updated by adopting the direct variables and the compatibility conditions (2.15). Successive iterations can be scaled to ensure that their magnitudes remain reasonable. This procedure, which is the so-called 'direct-adjoint-loop', is repeated until the initial conditions converge to the optimal ones, and the energy gain G(T ) reaches its maxima. The typical convergence criterion is that the relative error of G(T ) is smaller than 10 -3 .

In the iterative procedure, the evolutions in time are implemented with backward differentiation formula (BDF), which is a family of implicit multi-step method for numerical integration of ordinary differential equations, e.g.

∂f ∂t = Af , f = (u, v, w, p, b) T (2.19)
Depending on the direct equations (2.17), we can get the matrix

A A =          -ik x U 0 + 1 Re ∆ y -dU 0 dy 0 -ik x 0 0 -ik x U 0 + 1 Re ∆ y 0 -d dy 0 0 0 -ik x U 0 + 1 Re ∆ y -ik z 1 F 2 ik x d dy ik z 0 0 0 0 -1 0 -ik x U 0 + 1 ReSc ∆ y         
Correspondingly, the matrix A A for the adjoint equations (2.18) is

A A =          -ik x U 0 -1 Re ∆ y 0 0 -ik x 0 dU 0 dy -ik x U 0 -1 Re ∆ y 0 -d dy 0 0 0 -ik x U 0 -1 Re ∆ y -ik z 1 ik x d dy ik z 0 0 0 0 -1 F 2 0 -ik x U 0 -1 ReSc ∆ y         
and the adjoint equation for (2.19) is

∂f A ∂t = A A f A , f A = (u A , v A , w A , p A , b A ) T .
(2.20)

The specific formulas of s-step BDFs for the time integration are [START_REF] Süli | An introduction to numerical analysis[END_REF] :

BDF1 : f n+1 -f n ∆t = Af n+1 BDF2 : 3f n+2 -4f n+1 + f n 2∆t = Af n+2 , BDF3 : 11f n+3 -18f n+2 + 9f n+1 -2f n 6∆t = Af n+3 BDF4 : 25f n+4 -48f n+3 + 36f n+2 -16f n+1 + 3f n 12∆t = Af n+4 BDF5 : 137f n+5 -300f n+4 + 300f n+3 -200f n+2 + 75f n+1 -12f n 60∆t = Af n+5 BDF6 : 147f n+6 -360f n+5 + 450f n+4 -400f n+3 + 225f n+2 -72f n+1 + 10f n 60∆t = Af n+6
where ∆t is the step size and the s-step formula can achieve s-th order during computation.

Typically, the formula BDF6 is used and the time step size in the present study is set as ∆t = 0.05. However, for larger wavenumbers, a smaller step size should be used to avoid the numerical stability problem in time integration. In the present study, only the transient growth in plane Poiseuille flow is provided (see Chapter 4).

Spatial derivatives are computed with the pseudo-spectral method. We have applied Gauss-Lobatto points and Chebyshev polynomials for the plane Poiseuille flow (see section 2.3.1), and the collocation points of Laguerre polynomials for the boundary layer flow (see section 2.3.2).

Numerical method for eigenvalues

In this part, we provide the framework of linear temporal stability analysis of the base flow (2.4). The perturbations are searched in the form of normal modes

(u , v , w , p , b ) = (u(y), v(y), w(y), p(y), F 2 b(y))e ikxx+ikzz-iωt (2.21)
where k x and k z are real wavenumbers and ω the complex frequency (ω = ω r + iω i ). The equations for (u, v, w, p, b) can be obtained by substituting (2.21) into (2.8)

-iωu = -iU 0 k x u - dU 0 dy v -ik x p + 1 Re ∆ y u, (2.22a) -iωv = -iU 0 k x v - dp dy + 1 Re ∆ y v, (2.22b) -iωw = -iU 0 k x w -ik z p + b + 1 Re ∆ y w, (2.22c) -iωb = -iU 0 k x b - w F 2 + 1 ReSc ∆ y b, (2.22d) 0 = ik x u + dv dy + ik z w (2.22e)
where

∆ y = d 2 dy 2 -k 2 x -k 2 z .
The equation system (2.22) is solved numerically using a pseudo-spectral method. A generalised problem

Af = ωBf , f = (u, v, w, p, b) T (2.23)
can be derived from (2.22) and then be solved with the help of several subroutines in Matlab c .

Eigenvalues for plane Poiseuille flow

For the plane Poiseuille flow, the boundary conditions for (2.22) are set as non-slip boundary conditions

u(±1) = v(±1) = w(±1) = 0.
The equation (2.23) is solved using Gauss-Lobatto points and Chebyshev polynomials.

The advantage of Chebyshev polynomials in solving hydrodynamic stability problems is verified by [START_REF] Orszag | Accurate solution of the Orr-Sommerfeld stability equation[END_REF]. Typically, the number of the points used in the present study is 100.

Illustration of the spectrum associated with (2.22) is shown in figure 2.3. This figure can be compared to the result of figure 3.1 (page 64) in [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF].

The three main branches of Orr-Sommerfeld modes (the eigenvalues of Orr-Sommerfeld equations), which have been label A (ω r → 0), S (ω r ≈ 0.67), P (ω r → 1) by [START_REF] Mack | A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer[END_REF], are all included in figure 2.3. In addition to those Orr-Sommerfeld modes, an extra branch of Squire modes (the eigenvalues of Squire equation) also exists, which is located between the smaller branches of A modes. There is one slightly unstable eigenvalue (ω i > 0) on the A branch in figure 2.3, ω = 0.2375 + 0.00374i, and this value exactly agrees with the 'accurate' solution of [START_REF] Orszag | Accurate solution of the Orr-Sommerfeld stability equation[END_REF]. This unstable mode roots in the destabilising effects of viscosity, and is the so-called TS wave. The eigenfunctions associated with this mode are displayed in figure 2.4. The largest variations of velocities are close to the boundaries, and the pressure is the smallest (p(0) = 0) at the centre.

Eigenvalues for boundary layer flow

In our study of boundary layer flow, the choice of a tanh profile (instead of a Blasius profile) has been made for numerical convenience. Both the viscous and the radiative instabilities that we discuss here are expected to be weakly sensitive to the details of the profile (as long as it remains non-inflectional). In the very large Reynolds number limit, it is indeed known that the characteristics of these instabilities only depend on the velocity derivatives at the boundary (see [START_REF] Drazin | Hydrodynamic Stability[END_REF] for the viscous instability and [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF] for the radiative instability).

In the present study, the Froude and Reynolds numbers are varied in the intervals 100 < Re < 10 7 , 0.01 < F < 20.

(2.24)

We neglect the diffusion of mass and set Sc = ∞. This hypothesis is valid for salty water, for which Sc is around 700.

The goal of the present study is to perform a linear temporal stability analysis of the base flow (2.5). Perturbations are then searched in the form of normal modes. The velocity, pressure and buoyancy amplitudes (u, v, w, p, b) satisfy the equation system (2.22a-e) obtained by linearising the Navier-Stokes and density equations under the Boussinesq approximation.

The no-slip boundary condition gives u(0) = v(0) = w(0) = 0. Far away from the boundary, we apply a condition of radiation which prescribes that the energy should propagate outward.

The system (2.22a-e) is discretized using a pseudo-spectral method on the collocation points of Laguerre polynomials. These polynomials are well-adapted to semi-infinite domain [0, +∞[. This method has already been used in Riedinger et al. (2010b) and [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF]. The system of equations is then transformed into a generalized eigenvalue problem (2.23), which is solved using subroutines of Matlab c .

The eigenfunctions obtained by this method automatically vanish at infinity. As already explained by Riedinger et al. (2010b), this is not an adequate behaviour for all the physical perturbations.

By considering (2.22) for large y, it is easy to show that all solutions to (2.22) are, for large y, a sum of exponentials

e iβz with Λ = k 2 x + k 2 z + β 2 satisfying (Λ -i(ω -k x )Re) i(Λ -k 2 z )Re -ΛF 2 (ω -k x ) (Λ -i(ω -k x )Re) = 0. (2.25)
The condition of radiation prescribes that the solution should expend at infinity on 3 exponentials among the 6 possible (only the exponentials corresponding to waves propagating energy outward are kept). When Im(ω) > 0, this condition of radiation is equivalent to discarding the growing exponentials and keeping the decreasing exponentials. It is therefore equivalent to the vanishing of the solution at infinity. This implies that all unstable modes can a priori be captured by the spectral code. When Im(ω) = 0 or Im(ω) < 0, the conditions of radiation and of vanishing do not necessarily match. There are large regions of the complex ω plane where these two conditions are not equivalent. In these regions, the modes obtained by the spectral code are therefore "unphysical". The boundaries of these regions correspond to curves in the ω plane where one of the wavenumber β satisfying (2.25) becomes real. These curves are the so-called continuous spectrum. An illustration showing the numerical spectrum and the continuous spectrum is displayed in figure 2.5(a)

for a typical example.

In figure 2.5, we have indicated, by a dashed rectangle, the region of the ω plane where the interesting eigenvalues corresponding to the radiative instability are expected.

This region is very close to the continuous spectrum, which means that the eigenfunctions decay slowly at infinity.

As explained in Riedinger et al. (2010b), the exponentially decreasing behaviour of the solution can be improved by introducing a complex mapping z = z e iα where α is a fixed positive angle. The mapping of the complex path is shown in figure 2.6(a). Such a mapping modifies an oscillating behaviour on the z variable, say e iβz with β > 0 into an For a given set of parameters (k x , k z , Re, F ), there may exist several radiative modes, for example, the eigenvalues shown in figure 2.7. The pressures eigenfunctions for the corresponding radiative modes (Mode 1, 2, 3) are displayed in figure 2.8. All modes exhibit a well-defined localised structure close to the wall and an oscillating structure faraway. The higher the mode, the more complex the localised structure close to the wall and the larger the wavelength in the radiative zone. In the following, we mainly focus on the first mode, which has both the simplest wall structure and the largest growth rate.

0 0.5 1 1.5 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Re(ω) Im(ω) U U 0 0.5 1 1.5 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 Re(ω) Im(ω) U U (a) (b)
As soon as an eigenvalue is obtained, its robustness is tested by varying the angle α and the number N of polynomials. We have typically used α = π/20 and N = 100.

Introduction

Boundary layers are ubiquitous in any flow close to boundaries. They are known to be unstable with respect to viscous instabilities for Reynolds numbers above a few thousands.

In the presence of a stable stratification, such a flow on a vertical wall is shown to be unstable with respect to a stronger instability associated with the emission of internal gravity waves. The goal of the present chapter is to analyse the competition between both instabilities for a simple model of boundary layer flow as a function of viscosity and stratification strength.

The stability of boundary layer flows is a subject almost as old as fluid mechanics, which is covered by several textbooks (e.g. [START_REF] Betchov | Stability of Parallel Flows[END_REF]. In particular, it is well-established that an inflection-free boundary layer profile is unstable with respect to a 2D viscous instability that gives rise to the so-called Tollmien-Schlichting (TS) waves.

The effect of a stable stratification has been mainly studied in the context of atmospheric flows on flat horizontal surfaces (see Mahrt 2014, for a review). However, in this context, the velocity profile usually exhibits an inflectional point which makes it unstable with respect to the Kelvin-Helmholtz instability. This instability is modified by stratification as shown in [START_REF] Howard | Note on a paper of John W. Miles[END_REF]; [START_REF] Miles | On the stability of heterogeneous shear flows[END_REF]; [START_REF] Churilov | Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points[END_REF][START_REF] Churilov | Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points. Part 2. Continuous density variation[END_REF]; [START_REF] Candelier | Shear instability in a stratified fluid when shear and stratification are not aligned[END_REF]. The effect of stratification on Tollmien-Schlichting waves is less known. It was studied experimentally in [START_REF] Ohya | Turbulence structure of stable boundary layers with a near-linear temperature profile[END_REF]. Wu & Zhang (2008a) also demonstrated that it could be destabilizing. On a vertical wall, the 2D Tollmien-Schlichting waves are not expected to be affected by stratification. But as Squire theorem is not applicable in the presence of stratification, more unstable 3D TS waves could a priori exist.

The inclination of shear with respect to the direction of stratification is also known to be a source of instability. [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF] showed that an inflection-free boundary layer profile becomes unstable with respect to an inviscid "radiative instability" as soon as there is an angle between the directions of shear and stratification, the instability being the strongest for an angle of π/2, that is for a vertical wall. This instability which results from the coupling between shear and internal waves has been obtained in other contexts : shallow water flows [START_REF] Satomura | An investigation of shear instability in a shallow water[END_REF][START_REF] Balmforth | Shear instability in shallow water[END_REF][START_REF] Riedinger | Critical layer and radiative instabilities in shallow-water shear flows[END_REF], compressible flows [START_REF] Mack | On the inviscid acoustic-mode instability of supersonic shear flows -Part 1 : Two-dimensional waves[END_REF]; Parras & Le Dizès 2010), rotating flows (Riedinger et al. 2010b[START_REF] Riedinger | Radiative instability of the flow around a rotating cylinder in a stratified fluid[END_REF]. It has often been associated with a phenomenon of resonant over-reflection [START_REF] Mcintyre | On Radiating Instabilities and Resonant Overreflection[END_REF][START_REF] Grimshaw | On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile[END_REF][START_REF] Lindzen | Instability and wave over-reflection in stably stratified shear flow[END_REF], negative energy waves [START_REF] Kópev | Acoustic instability of an axial vortex[END_REF], or spontaneous wave emission [START_REF] Plougonven | Internal gravity wave emission from a pancake vortex : An example of wave-vortex interaction in strongly stratified flows[END_REF]Le Dizès & Billant 2009). In the present work, we shall analyse the effect of viscosity on the radiative instability of a boundary layer.

This chapter is organized as follows. In section 3.2.1, we first analyse the effect of stratification on the viscous instability, and then, in section 3.2.2, we study the radiative instability in the presence of viscosity. In section 3.2.3, we provide a summary of the stability results. The last section (section 3.3) is a brief discussion in the context of applications.

Temporal stability results

Boundary layer instability (Tollmien-Schlichting waves)

Considering a boundary layer flow, the viscous instability associated with Tollmien-Schlichting waves is expected to be active. Without stratification (F = ∞), the viscous instability of a boundary layer profile has been known for a long time (see for instance [START_REF] Betchov | Stability of Parallel Flows[END_REF]. In that case, Squire's theorem is applicable : the most unstable mode among all the possible wavenumbers k x and k z is obtained for k z = 0. Being 2D without variation in the stratification direction y, this mode is also expected to exist in the presence of stratification, as the equations for the transverse velocity v and the buoyancy b are decoupled from the other velocity components and the pressure. However, there is no Squire theorem which guarantees that this 2D mode remains the most unstable in the presence of stratification.

The effect of the Froude number on the growth rate curve in the (k x , k z ) plane is illustrated in figure 3.1. In these plots, we do see that for a given Reynolds number, the instability domain associated with the viscous instability grows as the stratification The evidence that Squire's theorem is not applicable in the presence of stratification is given in figure 3.4 which is an extension in a larger (k x , k z ) domain of the growth rate contours. We clearly see the presence of 3D modes which are more unstable than all the 2D modes. These 3D unstable modes are associated with another instability, the so-called radiative instability, analysed in the next section.

Radiative instability

The radiative instability is inviscid in nature and results from the coupling between shear and internal gravity waves associated with the fluid stratification. The inviscid characteristics of this instability have been obtained for the tanh boundary layer profile in [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF].

In figure 3.5, we have plotted the growth rate of the (most unstable) radiative mode in the (k x , k z ) plane for various Froude and Reynolds numbers (Several unstable radiative modes may exist for a given set of parameters (k x , k z , Re, F ). We always keep the mode with the largest growth rate even if we do not systematically mention that it is the most unstable radiative mode). We clearly see that viscous effects reduce the domain of instability. In the inviscid limit, the maximum growth rate is obtained for infinite k x and k z [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF]. As soon as viscous effects are present, the maximum growth rate is reached for finite k x and finite k z . In figure 3.6, we have plotted the maximal value of the growth rate over all k z as a function of k x for various Reynolds numbers and F = 0.5 and 3. The viscous damping of higher wave numbers is also clearly seen on these plots. It should be noticed that there are the two local peaks of the growth rate curve for Re = 10 6 and 10 7 . These peaks have also been indicated by symbols in figure 3.5(a,c).

While the first peak remains almost fixed, the location of the second peak strongly varies with Reynolds number. This strong variation can be associated with the extremely broad character of this peak and the fact that it goes to infinity as Re → ∞.

In figure 3.7, we have plotted the characteristics of the most unstable radiative mode (growth rate maximized over all the possible wavenumbers (k x , k z )) as a function of the Froude number for different Reynolds numbers. As expected, viscous results tend to inviscid results as Re increases. Note however that the convergence is slow, especially for small Froude numbers. This can be explained by the fact that even for Re = 10 7 , the most unstable wavenumbers are still of order 1 (Remember, they should go to infinity in the inviscid limit). For this Reynolds number, the most unstable wavenumbers correspond to the first peak of the growth rate curve in the (k x , k z ) plane for all Froude number F < 1 The critical Reynolds number for the radiative instability is found to be Re (r) c ≈ 1995 and is reached for F (r) c ≈ 1.51. The characteristics of the critical radiative mode are k x ≈ 0.615, k z ≈ 4.236, ω ≈ 0.535. As expected, the Froude number instability range increases with the Reynolds number, but the most unstable growth rates are obtained for strongly stratified configurations for which F is smaller than 1. In figure 3.8(a), the white dotted line gives the Froude number which maximizes the radiative instability for a prescribed Reynolds number.

The behaviour of constant growth rate curves for small Froude and large Reynolds numbers can be obtained by adding the viscous corrections to the non-viscous estimate. In this limit, [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF] have indeed shown that the most unstable inviscid mode was obtained for k z /k x ∼ 4.9/F . This means that for small F , k z is much larger than k x as well as the wavenumber based on the characteristic scale in the normal direction (y) which varies as k x . The viscous damping of the mode is then expected to be just -k 2 z /Re. If we assume that k x is approximatively constant, this gives a viscous growth rate which is constant on the line ReF 2 =Cst, as approximatively observed in figure 3.8(a).

The eigenfunctions of the distinguished radiative modes marked by symbols in figure 3.5 are shown in figures 3.9 and 3.10. Figures 3.9(a,b) show the pressure eigenfunction of the most unstable mode for a weakly stratified configuration (F = 3) at Re = 10 4 and 10 7 . The pressure eigenfunction for Re = 10 7 strongly resembles the inviscid eigenfunction obtained by [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF] for large Froude numbers and large wavenumbers. For smaller Reynolds numbers, the eigenfunction is by contrast different. These differences are not due to a change of instability mode. The modes are on the same instability branch but they are associated with two different local peaks of the growth rate contours in the (k x , k z ) plane. For Re = 10 4 , the most unstable mode corresponds to the first peak, while it is the second peak for Re = 10 7 (see figure 3.5(c)). It should be noted that the mode associated with the first peak is still unstable for Re = 10 7 , and its pressure eigenfunction is the same as for Re = 10 4 , as seen in figure 3.9(c).

In figure 3.10, we have considered a strongly stratified case (F = 0.5). Figures 3.10(a) and 3.10(b) show the pressure eigenfunction of the most unstable radiative mode for two different Reynolds numbers. By comparing these two figures, we observe that the impact of the Reynolds number is much weaker than for the weakly stratified case, as the pressure eigenfunction is almost not modified. Besides, this eigenfunction is very similar to the eigenfunction of the most unstable mode obtained for Re = 10 4 and F = 3. However, it is very different from the most unstable inviscid eigenfunction obtained by Candelier This difference is mainly due to a property already mentioned above : the viscous mode is associated with the first peak and has a small streamwise wavenumber (as for the modes shown in figure 3.9(a,c)). We suspect that this peak could be of viscous nature. It indeed gives a mode with a non-oscillating structure which resembles that of 2D Tollmien-Schlichting waves (see figure 3.3). If we consider the mode associated with the second peak, we obtain an eigenfunction with an oscillatory tail [figure 3.10(c)] which resembles the inviscid eigenfunction obtained for small Froude numbers (see [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF], figure 4). It is important to stress that this mode has almost the same growth rate as the most unstable mode.

Although the pressure eigenfunctions in figure 3.10(b) and (c) are quite different, these eigenfunctions are essentially associated with the same mode but for different wave numbers. Several internal cases between figure 3.10(b) and (c) are tested and their corresponding pressure eigenfunctions are illustrated in figure 3.11(a)-(d). The radiative phenomenon (oscillations far away from the wall) are increasing progressively. This is a gradual transition, rather than a "jump" from one mode to another. (5) Both instabilities present but radiative instability dominates viscous instability.

Competition between radiative instability and viscous instability

As shown above, the viscous instability is present as soon as Re > 3981 for any Froude number, while the radiative instability is active in a domain of the (Re, F ) plane which has been displayed in figure 3.8(a). Both instabilities are then expected to be in competition in a large domain of the parameter space. In figure 3.12, we have provided a summary of this competition by indicating 5 different domains corresponding to a domain of no instability, 2 domains where there is a single instability active, 2 other domains where one instability dominates the other. This figure clearly demonstrates that the radiative instability is the dominant instability in a large domain of the parameter space. It is the first instability to appear as the Reynolds number increases in the range 0.5 < F < 5.3.

For Re = 10 6 , it dominates the viscous instability in the large range 0.03 < F < 30.

Discussion

In this article, we have demonstrated that a stably stratified boundary layer flow on a vertical wall is affected by two instabilities : a classical 2D viscous instability and a threedimensional non-viscous radiative instability. We have shown that the radiative instability is the first instability to appear as the Reynolds number increases for 0.5 < F < 5.3, and is the dominant instability in a large Froude number interval around F = 1 for large Reynolds numbers. In the domain of parameters where the instability growth rate is the largest (large Reynolds numbers, small Froude numbers), the instability is characterized by a streamwise wavelength scaling with the boundary layer thickness and a small transverse wavelength proportional to F . The radiative instability is oscillatory with a frequency close to 1 (that is U/L in dimensional form). The most unstable mode extends up to a few boundary layer thickness.

The work has focused on the temporal stability property of a local boundary layer profile. We have found that the streamwise wavenumbers of the most unstable radiative mode increases with the Reynolds number and remains always larger than 0.1 for F < 16.

For TS waves, we have obtained that the streamwise wavenumber weakly decreases with the Reynolds number but is still larger than 0.05 for Re = 10 7 . For both types of modes, there is then a clear separation of scales between the instability wavelength and the O(Re) The deep ocean is stably stratified. If we assume a buoyancy frequency approximatively equal to 0.01 rad s -1 [START_REF] Alford | Structure, propagation, and mixing of energetic baroclinic tides in Mamala Bay, Oahu, Hawaii[END_REF]) and a kinematic viscosity ν = 10 -6 m 2 s -1 , a boundary layer flow of velocity U = 0.15 m s -1 and thickness L = 10 m would correspond to the parameters Re = 1.5 × 10 6 and F = 0.15. On a vertical ridge or a very steep slope, such a flow would be unstable with respect to the radiative instability by a mode of characteristics k x = 1.23, k z = 33, ω = 1 + 0.02i. This would give a period T ≈ 7 mn, a growth time τ ≈ 1 h, and a spatial structure of wavelength 51 m in the streamwise direction and 1.9 m in the transverse direction.

spatial
We have analysed the stability of a boundary layer flow on a vertical wall. For oceanic applications, it would be more relevant to consider inclined slopes. In that case, the radiative instability is still present but with a weaker growth rate [START_REF] Candelier | Inviscid instability of a stably stratified compressible boundary layer on an inclined surface[END_REF].

However, the radiative instability disappears on a horizontal surface. For the flow on a surface of moderate slope, a stronger competition could exist between viscous and radiative instability, but for large Reynolds numbers, the radiative instability is always expected to dominate as its growth rate does not decrease with the Reynolds number.

Note finally that we have not considered the effect of density diffusion. For both the atmosphere and the ocean, one should consider the temperature diffusion, for which the equivalent Schmidt number (Prandtl number) is of order 1. This stronger diffusion is expected to enhance the damping of the mode but this effect should remain small for the Reynolds numbers relevant to the geophysical applications (typically Re > 10 6 ).

The impact of the radiative instability on the dynamics of the atmosphere or the ocean remains unknown. When it is present, does it only contribute to the local mixing in the boundary layer, or does it induce a mixing further away from the boundary due to the radiative extension of the instability mode are among the interesting questions that have to be addressed. Answering these questions would require an understanding of the nonlinear evolution of the instability. This could constitute a nice objective for the future numerical or experimental works on the subject.

Chapitre 4

Instability of plane Poiseuille flow in a stably stratified fluid

Introduction

The famous pipe flow experiment conducted by [START_REF] Reynolds | An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels[END_REF] showed that a laminar flow could break down and turn into complicated spatial and temporal behaviours.

To illustrate underlying mechanisms in the fluid motions, researchers have used different prototypes in both theoretical and experimental studies. Plane Poiseuille flow is one of the simplest parallel shear flows that satisfy the Navier-Stokes equations and its stability and transition to turbulence is subtle. This flow is proved to be unstable when the Reynolds number exceeds a critical value. However, as stratification is ubiquitous in nature, it is of fundamental interest to investigate the stability and related properties of stratified plane Poiseuille flow.

In this chapter, we examine the modal and non-modal stability of a plane Poiseuille flow in the presence of stable density stratification, under the Boussinesq approximation.

The various mechanisms of exponential and transient growth are identified and characterised. Mathematically, the exponentially growing instability can be investigated through modal stability analysis, and the transient growth can be evaluated with the non-modal analysis [START_REF] Schmid | Nonmodal Stability Theory[END_REF]).

As we have shown in chapter 3, stable stratification can induce radiative instability in the boundary layer, it is natural to study the stratification effects on the stability of plane Poiseuille flow. Both the exponential and transient growth should be examined. It is the main objective of this chapter to illustrate the influence of stratification on the instabilities and fundamental mechanisms.

Modal stability of stratified Poiseuille flow

Modal stability analysis has been developed for a long time and has been used to predict the critical Reynolds number for instability.

The method of normal modes has been adopted to identify the exponential growth in the fluid. The earliest known stability analysis of a homogeneous plane Poiseuille flow is due to [START_REF] Heisenberg | Über stabilität und turbulenz von FlÜssigkeitsströmen[END_REF], who first demonstrated that the instability responsible for transition is a viscous instability, which is also referred to as Tollmien-Schlichting (TS) waves. [START_REF] Orszag | Accurate solution of the Orr-Sommerfeld stability equation[END_REF] solved the Orr-Sommerfeld stability equation and showed that the unstratified plane Poiseuille flow become unstable to the viscous mode if the Reynolds number, non-dimensionalised by the half-width of the channel, exceeds the critical value Re (v) c (≈ 5772.2). The argument of the critical value is also based on the Squire's theorem [START_REF] Squire | On the Stability for Three-Dimensional Disturbances of Viscous Fluid Flow between Parallel Walls[END_REF], which predicts that the most unstable mode is two-dimensional. However, in the presence of stratification, the Squire's theorem is not applicable. [START_REF] Miles | On the stability of heterogeneous shear flows[END_REF][START_REF] Howard | Note on a paper of John W. Miles[END_REF] showed the Kelvin-Helmholtz instability is diminished by stable stratification, but the effect of vertical stratification on a viscous instability is less known.

The results in Chapter 3 have shown that on a vertical wall, the stratification can induce the radiative instability in the boundary layer. The radiative instability is also obtained by Le Dizès & Riedinger (2010) in the Taylor-Couette system in the limit of an infinite gap. When the size of the gap is finite, Le Dizès & Riedinger (2010) showed that the radiative instability changes into a so-called strato-rotational instability, which is associated with a mechanism of resonance. A similar resonance phenomenon will be shown to be active in the present study. The modes involved in the resonance are associated with the stratification, we shall refer to this instability as a "gravity mode".

Non-modal stability of stratified Poiseuille flow

In many experiments on channel flows, the transition to turbulence can occur at a Reynolds numbers which is much smaller than the critical value for instability. In addition to that, experiments about boundary layers [START_REF] Klebanof | Effect of free-stream turbulence on a laminar boundary layer[END_REF][START_REF] Kendall | Experimental Study of Disturbances Produced in a Pre-Transitional Laminar Boundary Layer by Weak Freestream Turbulence[END_REF][START_REF] Matsubara | Disturbance growth in boundary layers subjected to free-stream turbulence[END_REF] show that the transition is usually preceded by the appearance of streamwise motion in the form of streaks rather than the TS waves. The reason for this phenomenon is that any initial disturbance has the potential to grow transiently whether or not the base flow is exponentially unstable.

As explained above, the transient growth derives from the non-normality of the Navier-Stokes operator and can occur without non-linear effects [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF][START_REF] Reddy | Energy growth in viscous channel flows[END_REF]. The combination of non-orthogonal eigenfunctions may lead to large transient growth even if every component decays exponentially at its own growth rate. Although the combination eventually decreases at a rate associated with its least stable component, it could produce large enough transient growth that can excite the transition process. Three basic mechanisms are identified for the transient growth : Orr mechanism [START_REF] Orr | The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid[END_REF], lift-up mechanism [START_REF] Ellingsen | Stability of linear flow[END_REF][START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF], and the combination of them (Farrell & Ioannou 1993a;[START_REF] Arratia | Transient perturbation growth in time-dependent mixing layers[END_REF]. Physically, the increase of the total perturbation energy is related to vortex tilting and stretching in the finite time intervals.

Non-modal stability has proved to be an important complementary part for the modal stability, because the modal stability analysis fails to explain many experimental results [START_REF] Klebanof | Effect of free-stream turbulence on a laminar boundary layer[END_REF][START_REF] Kendall | Experimental Study of Disturbances Produced in a Pre-Transitional Laminar Boundary Layer by Weak Freestream Turbulence[END_REF][START_REF] Matsubara | Disturbance growth in boundary layers subjected to free-stream turbulence[END_REF]. The transition shown in these experiments is preceded by the streamwise streaks instead of the viscous instabilities.

However, in the non-modal analysis, it is illustrated that the streamwise-uniform optimal initial perturbations can lead to large transient growth in streamwise motions through the lift-up mechanism [START_REF] Ellingsen | Stability of linear flow[END_REF][START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF]).

In the unstratified plane Poiseuille flow, [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] showed that the three-dimensional initial disturbance gains the most energy over a finite time interval. [START_REF] Reddy | Energy growth in viscous channel flows[END_REF] also demonstrated that the maximal energy gain can achieve O(Re 2 ) at a time that is proportional to Re. The amplified perturbations can facilitate the transition from laminar to turbulent flow [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF][START_REF] Reddy | Energy growth in viscous channel flows[END_REF].

As the plane Poiseuille flow is one of the representative prototypes which is sensitive to the initial conditions and the background disturbance, it is of fundamental interest to study its linear stability in the presence of vertical stratification.

Effect of horizontal shear and vertical stratification

As discussed by [START_REF] Basak | Dynamics of a stratified shear layer with horizontal shear[END_REF], many observations and various situations in the geophysical background, such as complex topography interacting with ocean currents, motivate the interest to study the stably stratified, horizontal shear flow. Dynamic evolutions including the selection of vertical structures are described in the DNS study of [START_REF] Basak | Dynamics of a stratified shear layer with horizontal shear[END_REF]. [START_REF] Deloncle | Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid[END_REF] showed that the Kelvin-Helmholtz instability is modified by the stratification. Subsequent works about the transient growth are conducted by [START_REF] Arratia | Non-modal instability mechanisms in stratified and homogeneous shear flow[END_REF].

According to the studies of Bakas & Farrell (2009a,b), horizontal shear flow can excite internal gravity waves in the presence of a vertical stratification. They discussed the spontaneous gravity wave generation and gave the analytic expressions for unbounded constant shear in their works. Gravity waves are also found to be generated by large transient growth in inviscid horizontal shear layers [START_REF] Arratia | Inviscid Transient Growth on Horizontal Shear Layers with Strong Vertical Stratification[END_REF].

The previous works mainly focus on the unbounded shear flows, in the present study, we examine the instability of plane Poiseuille flow bounded by two vertical walls in a stably stratified fluid. The remainder of this chapter is organised as follows. In section 4.2, modal stability analysis is conducted on the stratified plane Poiseuille flow. We compare the viscous mode and the gravity mode, and also analyse the resonance mechanism for the gravity mode. In section 4.3, the non-modal stability analysis is performed, and the transient growth is evaluated. In section 4.4, we discuss the stratification effects on exponential and transient growth and draw some conclusion.

Modal stability analysis

We consider a horizontal channel flow in a vertically stratified fluid (see figure 2.1).

As derived in chapter 2, the governing equations of the disturbance fields for the base flow are obtained by linearising the Navier-Stokes and the density equations under the Boussinesq approximation. By solving the general eigenvalue problem (2.23), the complex frequency ω can be obtained as a function of k x , k z , Re and F , namely ω(k x , k z , Re, F ).

Two kinds of exponential growing instabilities are found in the vertically stratified plane Poiseuille flow. First, the viscous mode (TS wave) should be active. Second, a threedimensional mode associated with stratification is found to be unstable, even in the region where Re < Re (v) c . This mode is the "gravity mode", as named in section 4.1.1.

Tollmien-Schlichting waves

The viscous instability associated with TS waves is active in a plane Poiseuille flow.

It corresponds to the unstable mode shown in figure 2.3. The instability in plane Poiseuille flow has been known for a long time (e.g. [START_REF] Reid | The stability of parallel flows[END_REF]). Those early studies are performed in unstratified fluid (F = ∞), wherein the Squire's theorem can draw the conclusion that the two-dimensional (2D) disturbance is the most unstable mode. The 2D mode is obtained for k z = 0 and thus has no variation in the direction of stratification (z-axis).

Although the 2D mode still exists in the fluid in the presence of stratification, it cannot be guaranteed to be the most unstable one, because the Squire's theorem is not applicable in a stratified fluid.

In figure 4.1 and 4.2, the growth rate contours of TS waves are plotted for Re = 10 4

and different Froude numbers. The effect of stratification on the growth rate contours in (k x , k z ) plane is illustrated. The unstable domains associated with viscous instability are also determined. Moreover, from figure 4.1(a-c), we can tell that the instability domain shrinks as the stratification increases for a given Reynolds number in the limit of weak stratification. As shown in figure 4.2(a-c), this trend continues in the strongly stratified limit. However, there is a sudden expansion of the unstable domain between F = 2 and

The gravity mode in the presence of stratification

In the unstratified plane Poiseuille flow (F = ∞), the critical Reynolds number is Re (v) c ≈ 5772. However, in the presence of stratification, the critical Reynolds number is smaller. Examples of eigenvalue spectrum for Re = 5000 are shown in figure 4.5. Without stratification, all the modes shown in figure 4.5(a) are stable (ω i < 0), because the Reynolds number is below the critical one Re (v) c . The spectrum in figure 4.5(a) can be compared to the eigenvalues found previously by [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] (at y = y c1 , ω r -k x U 0 (y c1 ) = 1/F ) can be observed in figure 4.7(c,e). In the inviscid limit, the eigenfunctions of transverse velocity and buoyancy are peaked at the critical layer level. Although these critical layers are believed to be stabilising (Riedinger et al. 2010a), their influence is not strong enough to stabilize the flow.

Another group of critical levels at y = y c2 where ω r -k x U 0 (y c2 ) = 0 are shown in figure 4.7(a,c). As discussed in the introduction, these critical levels forms the 'quantization' conditions. Similar to their roles in the over-reflection process, the critical levels and the 'quantization' zone make the mode become unstable.

This kind of instability results from a mechanism of resonance, as it will be explained in details in Section 4.2.3. With the same transverse wavenumber k z = 8.873, two other modes (Mode 2 and 3 ) are also marked to demonstrate the effect of resonance. As shown in figure 4.10, sharing the same set of parameters (Re, F, k x , k z ), the eigenvalues of mode (1 , 2 , 3 ) are quite different, especially the imaginary parts (i.e. the growth rates). Increased by the resonance phenomenon with the viscous modes, the growth rate of Mode 1 is greatly larger than those of the other modes. Table 4.1 classifies the corresponding parameters of these modes, including the specific transverse wavenumbers k z and eigenvalues ω.

In figure 4.9, the shape of the eigenfunctions clearly explains the essential resonance mechanism. The increasing complexity of the structures for Branch 1-3 comes form the inviscid modes. The pressure structure of Mode 1 in figure 4.9(a) shows that it is a viscous mode and the maximum absolute values is located at the boundaries. The maxima in figure 4.9(c-e) are located at the centre part of the channel and this characteristic also comes from the inviscid modes. In figure 4.9(c,e) the feature of the viscous mode is still kept through the resonance process, especially in the regions close to the walls. Similar behaviours of the pressure eigenfunctions can also be found in figure 4.9(f,h) as these modes are also the results of resonance between viscous and inviscid modes.

The modes (Mode 2 and 3 ) without resonance phenomenon are away from the crossing points of the frequency curves in figure 4.8(a). The shapes of pressure eigenfunctions in figure 4.9(d,g) are similar to those of the modes on the same frequency branch (see figure 4.8(a)). The notable difference is the values at the boundaries. Without resonance phenomenon, the absolute values at the boundaries in figure 4.9(d,g) are less than 0.5 when normalised by the maxima of the eigenfunction. However, in figure 4.9(c,e,f,h), the corresponding values are more than 0.6 with the same normalisation. These facts also indicate that the enhancement of the absolute values at boundaries in figure 4.9(c,e,f,h) derives from the the viscous mode, because this feature is the same as the structures shown in figure 4.9(a). Located on the same frequency branch, the structure of Mode 2 is similar to Mode 2 and 2 (figure 4.9(middle)), and Mode 3 is similar to Mode 3 and 3 (figure 4.9(bottom)).

In figure 4.10, the eigenvalues of Mode (1 , 2 , 3 ) are displayed in the same spectrum with the help of their common parameters (Re, F, k x , k z ). The resonance with the viscous mode enhances the growth rate of Mode 1 . In the spectrum, these three modes are located on the same branch in figure 4.10, representing the same inviscid nature. As for the boundary layer flow, Squire's theorem does not apply for the plane Poiseuille flow in the presence of stratification. This is shown in figure 4.11. In that figure, the unstable region inside the intervals 0.6 k x < 1.2 and 0 k z < 1 relates to the TS waves.

Instability contours in wavenumber plane

The most unstable viscous mode for this Reynolds number Re = 10 4 is : k x = 0.955, k z = 0, ω = 0.2221 + 0.0042i, which is independent of the Froude number. For a moderate stratification (F = 20), the most unstable mode is the "gravity mode" in figure 4.11(c), and the corresponding parameters are : k x = 0.1345, k z = 2.928, ω = 0.0948 + 0.0050i.

This 3D unstable mode is generated by the resonance mechanism discussed in section 4.2.3. The effects of stronger stratifications on the gravity modes are shown in figure 4.12.

In figure 4.12(a), four separate continuous regions represent unstable modes, including the TS waves. Hence, there exist four different local peaks, and each of them stands for the most unstable mode in each region. As the Froude number decreases, the first region (the left-most region with small streamwise wavenumbers 0.1 < k x < 0.7) merges with the area of the TS waves (the region with the smallest transverse wavenumber 0 k z < 0.5).

The eigenfunctions related to the modes in these two areas are similar. In figure 4.12(b), these two regions become connected for F = 1.1. Eventually, in a strongly stratified fluid (F = 0.2), they merge into one continuous region with only one local peak, as shown in With stronger stratification effects, the growth rate of the most unstable mode in figure 4.12(b) is larger than the one shown in figure 4.12(a). Mathematically, the imaginary part of the corresponding eigenvalues grows as the Froude number decreases. However, in a strongly stratified fluid (e.g. F = 0.2), the gravity modes become stable, but the viscous modes can remain unstable in this case, and the most unstable one becomes 2D again.

For example, in figure 4.12(c) where Re = 10 4 and F = 0.2, only the region associated with viscous instability remains unstable in the wavenumber plane.

Depending on the change of Reynolds number and Froude number, the growth rates of the most unstable mode in different cases are summarised in figure 4.13. The growth rate contours are shown for the intervals 400 Re 5000 and 0.1 F 300 in (Re, F ) plane. The critical Froude number for Re = 5000 is F ≈ 0.259 and F ≈ 256, which represent the extent from the strong stratification to the weak stratification. For a fixed Reynolds number, an extremum is achieved around F ≈ 1, as shown in figure 4.13.

Non-modal stability analysis

To have a better understanding of the stability in a stratified Poiseuille flow, we investigate the transient growth in this section. Depending on the equation (2.9), the total energy of the perturbation is measured in the form of

E(t) = 1 -1 1 2 |u| 2 + |v| 2 + |w| 2 + |b| 2 F 2 dy (4.1)
The total energy is defined as a sum of kinetic and potential energy.

By solving equation (2.19), the evolution and transient growth characteristics can be obtained for an initial condition f (0), which can give rise to the total energy E(t) in the process. The growth of the total energy is evaluated by the energy gain G(T ). For a finite time interval (t ∈ [0, T ]), the optimal energy gain G(T ) is defined as equation ( 4.2)

G(T ) = max ∀f (0) =0 E(T ) E(0) . (4.2)
The optimal initial conditions (also called optimal perturbations) are the fastest growing perturbations that achieves the maximal energy gain over a prescribed time T . Usually, they are a linear combination of different eigenfunctions, whether the corresponding eigenmodes are stable or not.

Verification for unstratified fluid

Over finite time intervals, the behaviour of the plane Poiseuille flow is not primarily influenced by the unstable modes we mentioned in section 4.2. The optimal response approaches the dominant eigenfunctions when T → ∞, hence the eigenvalue of the most unstable mode can only determine the long-time optimal response. In a homogeneous fluid, the short-time response is usually governed by the Orr mechanism [START_REF] Orr | The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid[END_REF], liftup mechanism [START_REF] Ellingsen | Stability of linear flow[END_REF][START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF]) and the combination of these two mechanisms (Farrell & Ioannou 1993a;[START_REF] Arratia | Non-modal instability mechanisms in stratified and homogeneous shear flow[END_REF]). These three mechanisms can be differentiated by the wavenumbers, k z = 0 for the Orr mechanism, k x = 0 for the lift-up The lift-up mechanism dominates in this case as the maximum is located on the k z -axis.

The transient growth of perturbation energy is substantial even though the Reynolds number (Re = 5000) is below the critical value. This result agrees with the result in the previous works about 3D unstratified plane Poiseuille flow [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF][START_REF] Reddy | Energy growth in viscous channel flows[END_REF]. Determined by the lift-up mechanism, the initial perturbations are streamwise-uniform vortices (k x = 0) and the streamwise velocity u greatly grows as time evolves. In the evolution process, the variation of v in the wall-normal direction (yaxis) is believed to "lift-up" low-speed fluid parcels from the wall so that the streamwise momentum is conserved (in the direction of x-axis). same region as the least stable eigenmodes.

Combination of

The maximal energy gain results shown in figure 4.14 and 4.15 are consistent with the values obtained by [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF]. The spatial structures of the optimal initial conditions are shown in figure 4.16, which is associated with the maximal energy gain G m in figure 4.14. Moreover, the corresponding optimal response at the terminal time T = 379 are illustrated in figure 4.17.

As illustrated in figure 4.16, the optimal initial conditions contain very small streamwise velocity u when normalized by the initial total perturbation energy E 0 , while the maximum absolute values of normal velocity v and transverse velocity w are of the same order. The velocity field of the initial disturbances represents streamwise-uniform vortices.

However, at terminal time T = 379, the maximum absolute value of u is nearly 200 times larger than v and w (see figure 4.17). The formation of the initial disturbances and large growth in the streamwise velocity agree with the characteristics expected for the lift-up mechanism [START_REF] Ellingsen | Stability of linear flow[END_REF][START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF].

Eigenfunctions and optimal perturbations

The correlation between eigenvalues and the structure of physical perturbations are verified in this part. [START_REF] Farrell | Optimal excitation of perturbations in viscous shear flow[END_REF] showed that the unstable mode does not gain the most perturbation energy and the initial conditions that grow fastest in the short-time transient phase is a linear combination of different eigenfunctions. However, for longtime evolutions in the fluid, the optimal spatial structures of the physical perturbations should be determined by the dominant eigenmode and the adjoint of it. The 'direct' and 'adjoint' eigenfunctions set the optimal response at terminal time (t = T ) and the initial disturbances at the start (t = 0), respectively. |u| max = 138.7, which is approximately 882 times larger than the value at t = 0. The large growth in u could therefore correspond to the formulation of streamwise streaks.

As demonstrated by figure 4.19-4.22, the physical structures of the optimal initial conditions and responses for the long time interval (T = 379) are determined by the adjoint and direct eigenmode. However, the characteristics of the evolution in time present a behaviour that resembles the lift-up mechanism. Although [START_REF] Ellingsen | Stability of linear flow[END_REF] and [START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF] described the optimal initial conditions for the lift-up mechanism as streamwise-uniform vortices, in the presence of stratification, the streamwise wavenumber for the perturbations in figure 4.23 and 4.24 is non-zero (k x = 0.1507). The features of the most unstable mode is similar to that of the lift-up mechanism : nearly streamwise vortices and the greatly amplified velocity u. In the following parts, we have shown that, in the presence of stratification, the eigenmodes play an important role in the transient growth process, even if the time interval is finite. In this section, we focus on a region where viscous modes are stable (Re 5000) in order to highlight the effect of the stratification which causes unstable gravity modes. We know that as T → ∞, the transient growth analysis is equivalent to a modal analysis.

Stratification effects on transient growth

For a finite time interval T = 379, there are differences between the results of modal and non-modal analysis, which are qualified in figure 4.25. The transient growth is thought to approach the exponentially unstable mode as T → ∞, but in the presence of stratification, the unstable modes are shown to be important at finite times much earlier than those expected in the unstratified fluid.

The effect of the stratification on the energy gain can also be analysed by comparing (2011) showed that the self-similar form is

σ m (k x , k z , T, Re, F ) ≈ σ c (k x , F k z , T ) - k 2 z Re (4.4)
where σ m is the mean growth rate defined by To illustrate the effect of viscosity, we examined some highly viscous cases. As we have shown above, the associated eigenvalues highly influence the contours of the energy gain, and there is a 'valley' shown by the growth rate contours in figure 4 shows that the total perturbation energy grows via the lift-up mechanism. The effect of viscosity is shown in figure 4.31(b). As Re grows, the maximal energy gain increases and so does the terminal time for the maximum growth. Although [START_REF] Biau | The effect of stable thermal stratification on shear flow stability[END_REF] and [START_REF] Jerome | Transient growth in Rayleigh-Bénard-Poiseuille/Couette convection[END_REF] showed that the maximum energy gain in the plane Poiseuille flow is proportional to Re 2 and the corresponding time increases as Re under both stable and unstable temperature gradient, the scaling law presented by [START_REF] Gustavsson | Energy growth of three-dimensional disturbances in plane Poiseuille flow[END_REF][START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] and [START_REF] Reddy | Energy growth in viscous channel flows[END_REF]

σ m = ln G 2T . ( 4

Discussion

In this chapter, we have performed a modal and non-modal analysis of a plane Poiseuille flow with stable stratification in the vertical direction. In the modal analysis, we The non-modal analysis has been performed using the conventional 'Direct-Adjoint-Loop' method [START_REF] Schmid | Nonmodal Stability Theory[END_REF]. We have described the effects of stratification on the transient growth and on the selection of optimal initial perturbations. The unstable eigenmodes we found in the modal analysis have been shown to have a significant influence on the transient growth. The optimal perturbations for the unstratified flows presented in this chapter agree with results obtained by [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] and [START_REF] Reddy | Energy growth in viscous channel flows[END_REF]. The three fundamental mechanisms discussed in these papers have been examined

in the presence of stratification. For short time intervals, the Orr mechanism [START_REF] Orr | The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid[END_REF], which leads to the growth of spanwise-uniform vortices through the action of the Reynolds stress, is not affected by the stable stratification in the vertical direction. The expression of equation (2.10) also explains this fact. The lift-up mechanism [START_REF] Ellingsen | Stability of linear flow[END_REF][START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF]) that leads to substantial growth of streamwise velocity perturbations and the formulation of streamwise-uniform vortices is weakened by the stable stratification.

The combination of these two mechanisms (Farrell & Ioannou 1993a;[START_REF] Arratia | Transient perturbation growth in time-dependent mixing layers[END_REF], which leads to the growth of tilted roll vortices, is amplified by the 3D unstable modes No significant energy growth 5000 0.26 G(1.0, 6.8) = 30.9 1000 10 G(0.4, 2.9) = 96.7 686.6 6 G(0, 3.0) = 40.9 The different regimes for the optimal perturbation are summarised in figure 4.32. The corresponding parameters and values are presented in table 4.2. Note that for short time intervals (e.g. T = 25), the transient growth is O(10 -100) in strongly stratified flows or highly viscous flows, even if they are expected to be exponentially stable. The reasons for the growth are the three-dimensional eigenmode (k x k z = 0) or lift-up mechanism (k x = 0). Since the result is not optimised over all the terminal time, the global maximum growth could be larger than the values we have shown.

As demonstrated in figure 4.32, the optimal initial disturbances in most regions are tilted roll vortices (k x k z = 0). The nonlinear effects on these perturbations are expected to be richer [START_REF] Reddy | Energy growth in viscous channel flows[END_REF] than those on the streamwise-uniform perturbations. Moreover, nonlinear effects can be destabilizing and provides even larger energy gain [START_REF] Schmid | Nonmodal Stability Theory[END_REF][START_REF] Luchini | Adjoint Equations in Stability Analysis[END_REF]. It is therefore possible that a faster transition to a strongly nonlinear regime is reached in the the presence of stratification.

Chapitre 5

Conclusion and perspective

Stratification effects have been analysed in the context of boundary layer and channel flows.

In chapter 3, we have examined the stability of a stratified, viscous boundary layer using modal analysis. Various aspects of the instability have been analysed as functions of the Reynolds number and the Froude number. We have shown that there are two kinds of instabilities. The first one is the classical viscous instability and the second is the radiative instability, which is inviscid in nature and closely related to the stratification.

For the Froude number interval 0.5 < F < 5.3, the radiative instability is the first to appear as the Reynolds number increases. It starts to appear at Re > Re (r) c ≈ 1995 for F ≈ 1.5. For large Reynolds numbers, the radiative instability dominates in a large Froude number interval. For example, it is more unstable than the viscous instability in the interval 0.03 < F < 30 for Re = 10 6 . On the other side, the most unstable viscous instability remains two-dimensional in the presence of stratification. The critical Reynolds number for the viscous mode is Re > Re (v) c ≈ 3980. Non-dimensionalised by the main mean velocity U and boundary layer thickness L, the oscillation frequency of the radiative modes is close to 1. The region covered by the most unstable mode can be as large as several times the boundary thickness. We have also shown that the growth rate of the radiative instability scales as ReF 2 in the parameter domain (large Re and small F ) where the growth rate is the largest.

Applied to some practical examples, we believe that this instability could be present in experimental facilities and geophysical flows like ocean and atmosphere. However,as long as this instability has not been evidenced in the atmosphere and ocean, the impact of radiative instability remains unclear. Whether it contributes to the local mixing or the mixing far away from the boundary layer are interesting topics for further studies. The knowledge of nonlinear evolution of the radiative instability is also required to answer these questions and thus can become the subject of future works.

In chapter 4, we have performed modal and non-modal stability analysis of a stratified plane Poiseuille flow. A new exponential growing instability is obtained through modal analysis, which behaves similarly to the radiative instability found in stratified boundary layers. The three-dimensional instability, which is referred to as the gravity mode in the present study, can be viewed as a version of the radiative instability in the finite domain or another version of strato-rotational instability in a parallel shear system. In the Reynolds number range that we have considered (Re 10 4 ), this gravity mode turns out to be the result of a resonance between an inviscid mode and a viscous mode. The critical Reynolds number can be as small as Re (p) c ≈ 494 for F ≈ 2.5. The classical viscous instability preserves its property by remaining two-dimensional and thus independent of F . For Re = 5000, the viscous mode is stable whereas the gravity mode is unstable in a large Froude number interval 0. The nonlinear evolutions of the optimal perturbations can be interesting because the related nonlinear effects of these tilted roll vortices (k x k z = 0) are rich [START_REF] Reddy | Energy growth in viscous channel flows[END_REF]. Experimental research are needed about the transient growth in stratified fluid in order to verify these effects. The influence of the transient growth with stratification is also an interesting topic for geophysical flows like oceans and the atmosphere.

As only the result of the modal analysis is provided in Chapter 3, one potential extension is the non-modal analysis of stratified boundary layer flows. The mathematical framework to search for the optimal perturbations is the same. However, as we discussed in Chapter 2, the standard pseudo-spectral method cannot precisely capture the eigenmodes, further treatment on the spatial discretization operator is required. One possible solution is to extend the method of the complex path, which is the method we have used for the modal analysis. A potential difficulty in this solution is the influence of meaningless numerical eigenmodes, which is a common problem encountered when using the spectral method. Another possible solution is to apply a numerical non-reflection boundary layer conditions. This solution may provide a clean spectrum of the eigenmodes and avoid the weakness of pseudo-spectral method.

Another interesting topic is the Direct Numerical Simulation (DNS). The effects of non-linear terms can indeed be considered by DNS. Based on our primary result of oscillation frequencies and wavenumbers in Chapter 3 and 4, small time steps and fine meshes are required for the simulation. These simulations would contribute to a better understanding of the stability in a stratified fluid.

Although the transient growth is optimised mathematically, background or white noise may select different structures. The perturbations which grow robustly rather than optimally may become dominant in the experiment, as suggested by [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] for the unstratified cases. Experiments about stratified flows can experience same problems, but the transient growth result can give fundamental evaluations and suggestions for these studies.

In the field of geophysical flows, our results can help to explain the generation of instabilities and turbulence mixing in stably stratified fluids, especially in the context of complicated geometry. Internal gravity waves can be produced due to the radiative instability, which can exhibit far away from the boundary layers on the cliffs. The transient growth analysis can also be regarded as a preliminary study about the stabilities of realistic stratified flows in nature.

Figure 1

 1 Figure 1.1 -Density and temperature profiles of the atmosphere (US Standard Atmosphere, 1976). The solid line is the temperature ratio T N = T /T 0 , and the dashed line is the density ratio ρ N = ρ/ρ 0 , where T 0 = 288.15 K and ρ 0 = 1.226 kg/m 3 .

Figure 1 Figure 1

 11 Figure 1.2 -Variations of different quantities for the depth in the ocean : (a) density, (b) salinity and temperature. Pycnocline, haloocline, thermocline are marked for the density, salinity and temperature respectively. Source : URL http : //ocean.stanf ord.edu/courses/bomc/chem/lecture_03.pdf

Figure 1 . 4 -

 14 Figure 1.4 -Distribution of buoyancy frequency N (Hz) for the US standard atmosphere. Source : Gossard & Hooke (1975).

Figure 1 . 5 -Figure 1

 151 Figure1.5 -Observed buoyancy frequency in the Sulu Sea. SS2 and SS3 are two mooring sites near the centre of the Sulu Sea and the mean profiles around these two locations are illustrated here. Source :[START_REF] Apel | The Sulu Sea Internal Soliton Experiment[END_REF]

Figure 1 . 8 -

 18 Figure 1.8 -Marks of gravity waves in the atmosphere and ocean. Source : Acquired on February 10, 2016. NASA image by Jeff Schmaltz, LANCE/EOSDIS Rapid Response. Caption by Mike Carlowicz and Holli Riebeek.

Figure 1 . 9 -

 19 Figure 1.9 -Special cloud structures formed by Kelvin-Helmholtz instability. Source : Terry Robinson, URL : http : //www.telegraph.co.uk/

Figure 1 .

 1 Figure 1.10 -Kelvin-Helmholtz instability when when shear and stratification are not aligned. Source : CSIRO Marine Research, Hobart (2002).

  obtained the critical Reynolds number Re c = 5772.22 from the accurate solution of the Orr-Sommerfeld stability equation. The definition of the Reynolds number is based on the half width of the channel, the centre-plane velocity, and the kinematic viscosity.

Figure 1

 1 Figure 1.11 -Vertical density gradient pattern of the radiative instability in the Taylor-Couette flow. The grey vertical rectangle corresponds to the cylinder (of radius 15mm). (a) Experimental results obtained with synthetic Schlieren. (b) Numerical results. Source : Riedinger et al. (2011)

Figure 1 .Figure 1

 11 Figure 1.12 -A wave packet and its envelope. Solid line : The wave packet. Dashed line : The envelope of the wave packet. The peaks and troughs move at the phase velocity and the envelope moves at the group velocity. Source : Wikipedia

Figure 1 .

 1 Figure 1.14 -Schematic picture of over-reflection. Source : Takehiro & Hayashi (1992).

Figure 1 .Figure 1

 11 Figure 1.15 -Multiple over-reflection of a shallow-water wave. Source : Takehiro & Hayashi (1992).

Figure 1 .

 1 Figure1.17 -Velocity field of optimal perturbations for plane Poiseuille flow. (a,b) Two-dimensional optimal initial conditions at t = 0 and response at t = T due to Orr mechanism (spanwise uniform). (c,d) Three-dimensional optimal initial conditions at t = 0 and response at t = T due to lift-up mechanism (streamwise uniform). Source :[START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]

Figure 2

 2 Figure 2.1 -Sketch of the base flow geometry and coordinate system

Figure 2 Figure 2

 22 Figure 2.3 -Eigenvalues of unstratified (F = ∞) plane Poiseuille flow for Re = 10 4 , k x = 1, k z = 0. The eigenvalue with a positive imaginary part is ω = 0.2375 + 0.00374i.

Figure 2 Figure 2

 22 Figure 2.5 -Continuous spectrum (solid lines) and numerical eigenvalues (stars) for N = 100 and the parameters Re = 10 4 , F = 1, k x = 1, k z = 10. The 'unphysical' domain is indicated by the letter 'U'. (a) Integration on the real axis (α = 0) ; (b) Integration on a complex path z = ze -iα with α = π/20. The eigenvalue domain of interest is indicated by the dashed rectangle.

Figure 2

 2 Figure 2.7 -Continuous spectrum (solid lines) and numerical eigenvalues (stars) for N = 320 and the parameters Re = 10 7 , F = 0.5, k x = 1.8, k z = 15.76. The 'unphysical' domain is indicated by the letter 'U'. Integration on a complex path z = ze -iα with α = π/10. The eigenvalue domain of interest is indicated by the dashed rectangle. Three different radiative modes are indicated by the arrows.

Figure 2

 2 Figure 2.8 -Pressure eigenfunction of distinguished radiative modes for Re = 10 7 , F = 0.5, k x = 1.8, k z = 15.76 (corresponding to the three modes in figure 2.7 but with α = 0). (a) Mode 1 (most unstable), ω = 0.928+0.0209i ; (b) Mode 2, ω = 1.33+0.00957i ; (c) Mode 3, ω = 1.567 + 0.00488i. Thick solid lines, solid lines and dashed lines represent absolute value, real part and imaginary part, respectively.

(

  This chapter is based on the article Chen, J., Bai, Y., & Le Dizès, S. 2016 Instability of a boundary layer flow on a vertical wall in a stably stratified fluid. J. Fluid Mech. 795, 262-277.)

Figure 3 Figure 3

 33 Figure 3.1 -Growth rate contours of the viscous instability in the (k x , k z ) plane for Re = 10 4 and different Froude numbers. (a) F = ∞ (unstratified case) ; (b) F = 1 ; (c) F = 0.1. Contours are every 0.00025 from 0 to 0.0028.

Figure 3

 3 Figure 3.3 -Eigenfunctions of the most unstable TS wave (top : streamwise velocity u ; middle : normal velocity w ; bottom : pressure p) for the critical Reynolds number Re = Re (v) c ≈ 3981 (left) and the most dangerous Reynolds number Re = Re (v) m ≈ 22390 (right). Thick solid lines, solid lines and dashed lines represent absolute value, real part and imaginary part, respectively.

Figure 3 Figure 3

 33 Figure 3.4 -Growth rate contours in the (k x , k z ) plane for Re = 10 4 and F = 1 [enlarged view of figure 3.1(b)]. The two regions correspond to the viscous and radiative instability respectively. Contours are every 0.001 from 0.001 to 0.012.

Figure 3

 3 Figure 3.6 -Maximum growth rate max kz [Im(ω)] of the radiative modes versus k x for F = 0.5 (a) and F = 3 (b). Solid line : Re = 10 7 , Dashed line : Re = 10 6 , Dash-dot line : Re = 10 5 , Dotted line : Re = 10 4 .

[Figure 3 . 7 -Figure 3

 373 Figure 3.7 -Characteristics of the most unstable radiative mode (obtained by maximizing the growth rate over all k x and k z ) as a function of the Froude number F for different Reynolds numbers. Re = 10 4 (dotted line), 10 5 (dash-dot line), 10 6 (dashed line), 10 7 (solid line), ∞ (thick grey line). (a) Growth rate Im(ω) ; (b) Oscillation frequency Re(ω) ; (c) Streamwise wavenumber k x ; (d) Wavenumber ratio k z /k x ; (e) Phase velocity Re(ω)/k x along x ; (f) Phase velocity Re(ω)/k z along z.

Figure 3

 3 Figure 3.9 -Pressure eigenfunction of distinguished radiative modes for F = 3 and two Reynolds numbers (corresponding to the symbols of figure 3.5(c,d)). (a) Most unstable mode for Re = 10 4 : k x = 0.33, k z = 3.13, ω = 0.2971 + 0.0043i ; (b) Most unstable mode for Re = 10 7 : k x = 0.64, k z = 8.75, ω = 0.2817 + 0.0086i ; (c) Less unstable mode corresponding to the first local peak of the growth rate curves for Re = 10 7 : k x = 0.34, k z = 2.75, ω = 0.3023 + 0.0084i. Thick solid lines, solid lines and dashed lines represent absolute value, real part and imaginary part, respectively.

Figure 3

 3 Figure 3.10 -Pressure eigenfunction of distinguished radiative modes for F = 0.5 and two Reynolds numbers (corresponding to the symbols of figure 3.5(a,b)). (a) Most unstable mode for Re = 10 4 : k x = 1.18, k z = 9.34, ω = 0.964 + 0.011i ; (b) Most unstable mode for Re = 10 7 : k x = 1.17, k z = 9.58, ω = 0.9499 + 0.0209i ; (c) Slightly less unstable mode for Re = 10 7 corresponding to the second local peak of the growth rate curves : k x = 1.80, k z = 17.56, ω = 0.928 + 0.0209i. Thick solid lines, solid lines and dashed lines represent absolute value, real part and imaginary part, respectively.

Figure 3 Figure 3

 33 Figure 3.11 -Transition of the pressure eigenfunction of radiative modes for Re = 10 7 , F = 0.5 (the internal cases between figure 3.10(b) and (c)). (a) k x = 1.191, k z = 9.8466, ω = 0.9488 + 0.0206i ; (b) k x = 1.211, k z = 10.08, ω = 0.9493 + 0.0204i ; (c) k x = 1.232, k z = 10.31, ω = 0.9499 + 0.0203i ; (d) k x = 1.342, k z = 11.60, ω = 0.9500 + 0.0205i. Thick solid lines, solid lines and dashed lines represent absolute value, real part and imaginary part, respectively.

  evolution length of the boundary layer flow. This justifies a posteriori the local parallel-flow assumption that we have made in neglecting the spatial development of the flow. Nevertheless, it would be interesting to perform a spatial stability analysis (in which the frequency is fixed, and one of the wavenumbers unknown) to get some information on the spatial development of the instability from a localized excitation, as done inWu & Zhang (2008b).It may be useful to apply the present results to experimental facilities such as the large stratified water flume of CNRM at Toulouse[START_REF] Paci | Topographic internal waves in the laboratory : two recent experiments carried in the CNRM-GAME stratified water tank[END_REF] to determine whether the radiative instability could develop on the side wall. Toulouse facility is 30 m long and 1.6 m deep. In normal conditions of use, it typically generates on the side walls a boundary layer flow of characteristics U = 0.1 m s -1 , L = 10 cm and N = 1 rad s -1 which gives Re = 10000 and F = 1. The most unstable mode of such a flow has the characteristics k x = 0.83, k z = 5.47, ω = 0.7 + 0.012i. It gives a period T = 2πL/(U Re(ω)) ≈ 9 s, a growth time τ = L/(U Im(ω)) ≈ 80 s, a streamwise wavelength 75 cm, and a transverse wavelength 12 cm. Such a perturbation clearly fits within the channel. It is much stronger than the viscous instability of the boundary layer which has a growth time ten times larger for this Reynolds number. We believe that this instability could be present on the side wall of the channel. We suspect that it could affect the quality of the flow within the channel.The radiative instability could also be present in real geophysical flows. Consider a stably stratified nocturnal atmospheric boundary layer, as reported in[START_REF] Frehlich | Measurements of Boundary Layer Profiles with In Situ Sensors and Doppler Lidar[END_REF]. The velocity and thickness of this boundary layer flow is U = 10 m s -1 , and L = 100 m, respectively. The buoyancy frequency associated with the stratification is approximatively N = 0.031 rad s -1 while the kinematic viscosity is ν = 10 -5 m 2 s -1 . This gives F = 3.2, and Re = 10 8 . If such a flow was present on a vertical cliff, it would be unstable with respect to a radiative instability by a mode of characteristics k x = 0.95, k z = 13.84, ω = 0.27 + 0.0083i. This would give a perturbation period T ≈ 3.5 mn and a characteristic growth time τ ≈ 20 mn. The streamwise and transverse wavelengths of this perturbation would be 660 m and 45 m, respectively.

Figure 4 Figure 4

 44 Figure 4.1 -Growth rate contours in (k x , k z ) plane for Re = 10 4 and large and moderate Froude numbers. (a) F = ∞ (unstratified) ; (b) F = 20 ; (c)F = 2. Contours are every 0.0004 from 0 to 0.0042. It should be noted that the k z intervals here are [0, 1].

Figure 4

 4 Figure 4.3 -Viscous instability (TS waves). Characteristics of the most unstable TS wave as a function of Re. (a) Growth rate Im(ω) ; (b) Oscillation frequency Re(ω) ; (c) Wavenumber k x .

Figure 4

 4 Figure 4.4 -Eigenfunctions of TS waves (left : streamwise velocity u ; middle : normal velocity v ; right : pressure p) for the critical Reynolds number Re = Re v c ≈ 5772 (top) and the most dangerous Reynolds number Re = Re v m ≈ 4.75 × 10 4 (bottom). Thick solid lines, thin solid lines and dashed lines represent absolute value, real part and imaginary part, respectively. All eigenfunctions are normalized by v(0).

Figure 4

 4 Figure 4.5 -Eigenvalues of plane Poiseuille flow for Re = 5000. (a) F = ∞, k x = 1.48, k z = 0 ; (c) F = ∞, k x = 0.986, k z = 3.615 ; (c) F = 2, k x = 0.986, k z = 3.615. The horizontal dashed line is Im(ω) = 0.

Figure 4 . 6 -

 46 Figure 4.6 -Eigenfunctions associated with the unstable mode for the parameters Re = 5000, F = 2, k x = 0.986, k z = 3.615, ω = 0.6987 + 0.03906i. (a) streamwise velocity u ; (b) normal velocity v ; (c) transverse velocity w ; (d) pressure p ; (e) buoyancy b ; Thick solid lines, thin solid lines and dashed lines represent absolute value, real part and imaginary part, respectively. All eigenfunctions are normalized by v(0).

Figure 4

 4 Figure 4.7 -Eigenfunctions associated with the unstable mode for the parameters Re = ∞, F = 2, k x = 0.986, k z = 3.615, ω = 0.6930 + 0.0391i. (a) streamwise velocity u ; (b) normal velocity v ; (c) transverse velocity w ; (d) pressure p ; (e) buoyancy b ; Thick solid lines, thin solid lines and dashed lines represent absolute value, real part and imaginary part, respectively. All eigenfunctions are normalized by v(0). The dash-dotted lines in (c,e) describe the critical levels, where ω r -k x U 0 (y c1 ) = 1/F . The vertical dashed lines in (a,c) represent the critical levels ω r -k x U 0 (y c2 ) = 0.

Figure 4 Figure 4 . 9 -Figure 4

 4494 Figure 4.8 -Frequency ω r (a) and growth rate ω i (b) versus transverse wavenumber k z . The streamwise wavenumber is k x = 1, and the other associated parameters are Re = 10 4 , F = 1.1. The thick solid lines are inviscid modes. The dashed lines and the dash-dotted line are viscous modes.

Figure 4

 4 Figure 4.11 -Growth rate contours in (k x , k z ) plane for Re = 10 4 and different Froude numbers. (a) F = ∞ (unstratified) ; (b) F = 100 ; (c) F = 20. Contours are every 5×10 -4 from 0 to 5 × 10 -3 .

Figure 4 Figure 4

 44 Figure 4.12 -Growth rate contours in (k x , k z ) plane for Re = 10 4 and different Froude numbers. (a)F = 2 ; (b)F = 1.1 ; (c) F = 0.2, the same domain as figure 4.2(a). Contours are every 0.005 from 0.001 to 0.053. The dashed line k x = 1 in (b) represents the crosssection made by figure 4.8.

figure 4 .

 4 figure 4.12(c). This phenomenon explains the sudden extension of the unstable domain from figure 4.1(c) to figure 4.2(a) as the associated Froude numbers change from F = 2 to F = 0.2.

Figure 4

 4 Figure 4.14 -Energy gain contours in (k x , k z ) plane for Re = 5000, F = ∞ and T = 379. Contours are every 257 from 10 to 4895.

Figure 4 Figure 4

 44 Figure 4.15 -Energy gain G for unstratified plane Poiseuille flow (Re = 5000, F = ∞). (a) Contours in (k x , k z ) plane for T = 20. The contours are every 56 from 10 to 510. (b) G(T ) at different wavenumbers. The associated wavenumbers are : (k x , k z ) = (1.48, 0) for the dashed line, (k x , k z ) = (0, 7.3) for the dash-dotted line, and (k x , k z ) = (1.48, 7.3) for the solid line.

Figure 4

 4 Figure 4.17 -Structures of the physical variables at terminal time for the parameters Re = 5000, F = ∞, k x = 0, k z = 2.044, T = 379. (a) streamwise velocity u ; (b) normal velocity v ; (c) transverse velocity w ; (d) pressure p ; Thick solid lines, thin solid lines and dashed lines represent absolute value, real part and imaginary part, respectively. All variables are normalized by the total energy E 0 at the initial time.

Figure 4 Figure 4

 44 Figure4.18 -Spectrum of the eigenvalues for the parameters Re = 5000, F = 20, k x = 0.1507, k z = 2.888. (a) 'Direct' spectrum, the eigenvalue marked in the small rectangle is ω = 0.1079+0.0036i ; (b) 'Adjoint' spectrum, the eigenvalue marked in the small rectangle is ω A = 0.1079 -0.0036i. The most unstable 'direct' mode and the 'adjoint' mode marked in rectangles are the modes that determine the structures of the perturbations at terminal time and initial time.

Figure 4

 4 Figure 4.20 -Structures of the physical variables at terminal time, the associated parameters are Re = 5000, F = 20, k x = 0.1507, k z = 2.888, T = 379. (a) streamwise velocity u ; (b) normal velocity v ; (c) transverse velocity w ; (d) pressure p ; (e) buoyancy b/F 2 (the definitions of b for eigenvalues and optimal perturbations are slightly different, see Chapter 2 for details). Thick solid lines, thin solid lines and dashed lines represent absolute value, real part and imaginary part, respectively. All variables are normalized by v(0).

Figure 4

 4 Figure 4.21 -Eigenfunctions associated with the adjoint mode for the parameters Re = 5000, F = 20, k x = 0.1507, k z = 2.888, ω A = 0.1079 -0.0036i. (a) adjoint streamwise velocity u A ; (b) adjoint normal velocity v A ; (c) adjoint transverse velocity w A ; (d) adjoint pressure p A ; (e) adjoint buoyancy b A ; Thick solid lines, thin solid lines and dashed lines represent absolute value, real part and imaginary part, respectively. All eigenfunctions are normalized by v A (0).

Figure 4

 4 Figure 4.22 -Structures of the optimal perturbations at initial time, the associated parameters are Re = 5000, F = 20, k x = 0.1507, k z = 2.888, t = 0, (terminal time T = 379). (a) streamwise velocity u ; (b) normal velocity v ; (c) transverse velocity w ; (d) pressure p ; (e) buoyancy b/F 2 (the definitions of b for eigenvalues and optimal perturbations are slightly different, see Chapter 2 for details). Thick solid lines, thin solid lines and dashed lines represent absolute value, real part and imaginary part, respectively. All variables are normalized by v(0).

Figure 4 Figure 4

 44 Figure 4.23 -Velocities of the optimal perturbations at initial time, the associated parameters are Re = 5000, F = 20, k x = 0.1507, k z = 2.888, t = 0, (terminal time T = 379). (a) Projections on the x -z plane (streamwise velocity u and transverse velocity w on the plane y = 0.5) ; (b) Projections on the y -z plane (normal velocity v and transverse velocity w on the plane x = 0) ; (c) Projections on the x -y plane (streamwise velocity u and normal velocity v on the plane z = 0) ; (d) streamwise velocity u contours on plane x = 0, |u| max = 0.1572. All velocities are normalized by v(0) at initial time t = 0.

Figure 4

 4 Figure 4.25 -Contours in (k x , k z ) plane for Re = 5000 and F = 20. (a) Energy gain G(T ) at T = 379. Contours are every 615 from 1 to 5524. (b) Growth rate ω i . Contours are every 5 × 10 -4 from 0 to 3.6 × 10 -3 .

figure 4 .

 4 figure 4.25(a) and 4.14. The comparison shows that the energy gain in stratified flow is slightly larger than that in the unstratified case for the same Reynolds number and time interval (Re = 5000, T = 379). The maximum in figure 4.25(a) is located at (k x , k z ) = (0.1507, 2.9565) and G m = 5524. On the other side, in figure 4.25(b), the eigenvalue with the largest growth rate is ω = 0.1091 + 0.0036i and the corresponding wavenumbers are (k x , k z ) = (0.1526, 2.848). This set of wavenumbers is very close to that of the maximum energy gain G m in the (k x , k z ) plane, meaning that G m is mainly determined by the most unstable mode. The other local peaks in figure 4.25(a) with non-zero streamwise wavenumbers are also believed to result from an unstable eigenmode visible in figure 4.25(b).

Figure 4 Figure 4 Figure 4 Figure 4 Figure 4

 44444 Figure 4.26 -Contours in (k x , k z ) plane for Re = 5000 and F = 4. (a) Energy gain G(T ) at T = 25. Contours are every 10 from 10 to 180. (b) Growth rate ω i . Contours are every 2.5 × 10 -3 from 0 to 2.5 × 10 -2 .

. 5 )

 5 The equation (4.4) is the corrected form of the self-similarity found by[START_REF] Deloncle | Three-dimensional stability of a horizontally sheared flow in a stably stratified fluid[END_REF] and[START_REF] Arratia | Non-modal instability mechanisms in stratified and homogeneous shear flow[END_REF] for the inviscid flows in the presence of strong stratification (F 1). The corrected mean growth rate σ c is shown as a function of F k z in figure4.30(b). We can see that the curves of F = 0.05, 0.1, 0.2 are close to each other.

Figure 4

 4 Figure 4.31 -Energy gain G for different Reynolds numbers with F = 5.969. (a)Energy gain contours in (k x , k z ) plane for Re = 686.6 at time T = 25. Contours are every 4 from 5 to 41. (b)Energy gain for different Reynolds numbers (from up to down Re = 5000, 3000, 1000, 686.6, 500) at constant F . The associated wavenumbers are (k x , k z ) = (0, 3.05).

  is for the global maximum G global = max ∀kx,kz,T G in unstratified fluid. The growth displayed in figure 4.31(b) is not optimised over all wavenumbers, and only the tendency of the scaling law is kept by the curves of G(T ).

Figure 4

 4 Figure4.32 -Description of the optimal perturbations in the (Re, F ) plane for a short time interval T = 25 (a) and a long time interval T = 379 (b). Three regions are identified according to the wavenumbers k x = 0, k z = 0, k x k z = 0. The regions are delimited by dashed lines. The solid line delimits the region with sufficiently large growth (typically G(T ) > 100). The region in grey is the unstable region predicted by the modal analysis. Markers : 'N' represents 'No significant energy growth'. The symbols '•', ' ', ' ' and ' * ' correspond to the verified cases in different regions.

(

  the 'gravity mode') in the presence of stratification. For long time intervals, the transient growth is mainly dominated by the most unstable eigenmodes.The results of transient growth are consistent with what we have demonstrated in equation (2.10). Viscosity and stratification dissipate the perturbation energy. The contributor to the growth of the total energy is related to the Reynolds stress, and the energy is extracted from the mean shear, which is also universal in inviscid and unstratified fluids[START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF]).

Figure

  Figure 4.32(a) T = 25 Marker Re F G(k x , k z ) Region 5000 4 G(0, 7.2) = 179.6 k x = 0 7000 0.2 G(1.0, 6.3) = 43.0 k x k z = 0 5000 300 G(0.7, 2.9) = 699.5 5000 0.1 G(1.1, 0) = 26.1

Figure

  Figure 4.32(b) T = 379 Marker ReF G(k x , k z ) Region 5000 300 G(0,2.1)=4568 k x = 0 7000 300 G(0, 2.3) = 8412 * 7000 0.2 G(1.0, 0) = 86.6 k z = 0 5000 20 G(0.2, 3.0) = 5523 k x k z = 0 5000 0.2 G m <1 No significant energy growth 686.6 6.65 G m < 10 Table 4.2 -Positions of the markers in figure 4.32

  27 F 258. The plane Poiseuille flow which is stable to viscous instability can then become exponentially unstable in the presence of a very weak stratification.In Chapter 3 and 4, we have demonstrated that in the presence of stratification, stratified flows are more likely to become unstable because of the 3D radiative instability and the gravity modes. A resonance mechanism between the inviscid modes and viscous modes is demonstrated to be a source of modal instability. However, in terms of transient growth, the stably stratified flows are less sensitive. On the other side, the link between the eigenvalues and transient growth is strengthened by the stratification. The fundamental mechanisms of transient growth can also help to describe the growth of unstable mode in finite time intervals.In the presence of stratification, the largest energy gain is associated with non-zero wavenumbers (k x k z = 0) for most cases in the (Re, F ) plane. In exponentially stable flows with small Reynolds number or small Froude number, the transient growth can still reach O(10 -100) for short time intervals (e.g. T = 25). The global maximum can be larger thus possibly exciting the nonlinear effects and transition to turbulence.

Remerciements

I would like to thank all those people who helped me during my studies.

First of all, I deeply thank my supervisor Stéphane Le Dizès, for his instructive advice and encouragement. He has offered me guidance through all stages of this thesis. I am also thankful to Patrice Meunier, who have put considerable time and effort into the drafts and given me constructive suggestions, especially about the scientific expressions.