Aimie Moulin
email: aimie.moulin@legi.grenoble-inp.fr

Achim Wirth

Keywords: air-sea interaction, numerical modelling, turbulence

The air-sea interaction at the (atmospheric) synoptic and the mesoscale due to momentum transfer only, is considered. Two superposed one-layer fineresolution shallow-water models are numerically integrated. The upper layer represents the atmosphere and the lower layer the ocean. The frictional force between the two-layers is implemented using the quadratic drag law. Experiments with different values of the surface friction drag coefficient are performed. The actual energy loss of the atmosphere and the energy gain by the ocean, due to the interfacial shear, is determined and compared to the estimates based on average speeds. The correlation between the vorticity in the atmosphere and the ocean is determined. Results differ from previous investigations where the exchange of momentum was considered at basin scale.

It is shown that the ocean has a passive role, absorbing kinetic energy at nearly all times and locations. Our results show that the energy input to the ocean increases almost quadratically with the value of the drag coefficient.

Due to the feeble velocities in the ocean, the energy transfer depends only weakly on the ocean velocity. The ocean dynamics leaves nevertheless its imprint in the atmospheric dynamics leading to a quenched disordered state of the atmosphere-ocean system, for the highest value of the friction coefficient considered. This finding questions the ergodic hypothesis for the idealised configuration studied here. The ergodic hypothesis is at the basis of a large number of experimental, observational and numerical results in ocean, atmosphere and climate dynamics.

PACS 92.60.Cc

Introduction

Today's theory of the basin-scale ocean dynamics is based on the conservation and fluxes of (potential) vorticity (see e.g., [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics[END_REF]). The large scale features of the ocean dynamics can be explained to leading order by a windstress forcing that is averaged in space and time. In numerical simulations of the ocean dynamics, performed during the past four decades, the wind-stress field was commonly averaged over one month in time and a few degrees in both longitude and latitude. There has more recently been a growing interest in the ocean circulation kinetic energy: reservoirs, sources and sinks [START_REF] Ferrari | Ocean circulation kinetic energy: reservoirs, sources, and sinks[END_REF]. The major input of mechanical energy into the ocean is through the wind-stress, that is the friction caused by the different horizontal velocities of the atmospheric winds and the ocean currents near the ocean surface. A theory of how kinetic energy is injected by the large-scale windstress into the basin-scale geostrophic ocean circulation is given in [START_REF] Roquet | On the patterns of wind-power input to the ocean circulation[END_REF]. At smaller scales the energy injection is, so far, less studied and understood. Furthermore, the ocean currents were commonly not taken into account when the wind-stress field was calculated, because the ocean surface currents velocities are small as compared to atmospheric winds over much of the ocean. Calculating the wind-stress using the difference between the ocean currents and the atmospheric wind, rather than the wind only, was found to lead to a twenty percent reduction of energy input in the large scale doublegyre circulation [START_REF] Duhaut | Wind stress dependence on ocean surface velocity: implication for mechanical energy input to ocean circulation[END_REF].

As the resolution of (satellite) observations becomes ever finer in space and shorter in time, air-sea interactions at smaller scales are found to leave their imprint in the atmosphere and ocean dynamics [START_REF] Db | Satellite measurements reveal persistent small-scale features in ocean winds[END_REF]) leading to persistent small-scale structures in the atmospheric winds. The air-sea heat flux influencing the marine atmospheric boundary layer is found in [START_REF] Db | Satellite measurements reveal persistent small-scale features in ocean winds[END_REF] to be the dominant source of the small-scale structures in the atmospheric winds. In the present work the influence of momentum transfer due to ocean dynamics is considered.

The importance of these small scale interactions for the atmosphere and ocean dynamics is confirmed using numerical simulations in idealised configurations [START_REF] Hogg | The effects of mesoscale ocean-atmosphere coupling on the large-scale ocean circulation[END_REF]) and when applied to regional configurations [START_REF] Seo | Modeling of mesoscale coupled ocean-atmosphere interaction and its feedback to ocean in the Western Arabian Sea[END_REF]. As the resolution of regional and global ocean models becomes ever finer in space and shorter in time, they become increasingly sensitive to more accurate and finer-resolution higher-frequency surface-windstress forcing. The importance of realistic amplitude, fine resolution and highwavenumber, surface-wind-forcing has been demonstrated in a variety of ocean modelling and climate applications. Recent fine scale satellite observations of the sea surface show an abundance of dynamical features at the oceanic meso and sub-meso scale. The explanation of the origin, turbulent dynamics and fate of these structures represents a formidable problem of geophysical fluid dynamics.

In the present work we exclusively focus on the exchange of momentum to evaluate its potential as a source of small scales structures in the ocean and the atmosphere. The most conspicuous feature of the atmosphere-ocean system are the different densities of the two phases, leading to a large difference in the inertia of the atmosphere and the ocean layer. At large times the atmosphere is influenced by the persistent ocean dynamics, while the short time fluctuations appear to be independent of the ocean. On the other side, short time fluctuations of the atmospheric winds have only a negligible influence on the ocean dynamics, which however responds to a persistent atmospheric forcing. This is discussed in detail by [START_REF] Esau | Indirect air-sea interactions simulated with a coupled turbulence-resolving model[END_REF] for the case of the heat exchange, the case of the momentum exchange is considered here. As noted in [START_REF] Esau | Indirect air-sea interactions simulated with a coupled turbulence-resolving model[END_REF] the difference in density leads in numerical models to imposing different types of boundary conditions to the atmosphere, the value is imposed (Dirichlet boundary conditions) and the ocean, the flux is imposed (von Neumann boundary conditions). This is not only the case for the heat flux (as discussed in Esau ( 2014)) but the same is true for the flux of inertia.

Estimates of energy fluxes are usually based on coarse-grained variables, that is variables averaged in time and space. Variables from (satellite) observations and numerical models are always coarse-grained representations of the real dynamics. When energy fluxes are estimated based on coarse-grained variables, it is often not clear how good the estimate is, or even if the true value is over-or under-estimated. The energy input and cycle is however key to understanding the dynamics of the atmosphere, ocean and climate system.

The fluxes at smaller scales contribute to the turbulent mixing of substances and momentum, in the atmosphere and the ocean, without which we can not fully understand the respective dynamics. Furthermore, in the two-dimensional turbulent dynamics of the atmosphere and the ocean, energy cascades from smaller to larger scales (see e.g. [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics[END_REF]), emphasising the importance of energy exchanges at small scales for the large scale dynamics.

In the here presented study, two superposed one-layer fine-resolution shallowwater models are used. The upper layer represents the atmosphere and the lower layer the ocean. Their only interaction is through a frictional force at the interface, which is parameterised by the prominent quadratic drag law.

The characteristic horizontal length scales in the atmosphere and the ocean are the respective first baroclinic Rossby radii of deformation. The layer thicknesses of the atmosphere and the ocean in our model are chosen, so that the (barotropic) Rossby radii of deformation correspond to the respective length scales. The first baroclinic atmospheric Rossby radius of deformation (a few hundreds of kilometres) is usually referred to as the synoptic scale and in oceanography the first baroclinic oceanic Rossby radius of deformation (a few tenths of kilometres) is referred to as the mesoscale. All structures with a horizontal extension smaller than the synoptic scale will be called small scale, for convenience. Synoptic structures in the atmosphere rarely take more than a few days until they have moved away from a specific location, while the corresponding time scale for mesoscale eddies in the ocean is of a few weeks.

A fully coupled 3D-atmosphere and ocean model was used by [START_REF] Esau | Indirect air-sea interactions simulated with a coupled turbulence-resolving model[END_REF] to consider heat fluxes and the induced turbulent convective dynamics in the mixed layers at the meso and sub-meso scales. He, however, considered a stressfree interface, that is without transfer of momentum. We consider momentum transfer, without including buoyancy or buoyancy fluxes. The two studies are complementary and the combination of the two effects is left for future research.

In the present work we focus on the local exchange of momentum between the ocean and the atmosphere at the synoptic, and the mesoscale. The forcing has no large scale component. The situation is very different from basin scale forcing, dynamics and flow. A basin wide forcing and circulation can create strong boundary currents which can separate from the boundary and penetrate into the interior of the domain, as is the case for the double gyre circulation [START_REF] Duhaut | Wind stress dependence on ocean surface velocity: implication for mechanical energy input to ocean circulation[END_REF][START_REF] Hogg | The effects of mesoscale ocean-atmosphere coupling on the large-scale ocean circulation[END_REF], leading to strong ocean currents.

To the best of our knowledge the local exchange of momentum, only, between a turbulent atmosphere and ocean, at the small scale has never been considered before.

In the present work, the qualitative dynamics of air sea-interaction is considered and the existence of two different regimes is exposed. We further assemble quantitative results on the energy fluxes and vorticity correlations between the atmosphere and the ocean for different values of the atmospheric friction coefficient.

Exchanges of heat and chemical substances are of paramount importance for the dynamics of the atmosphere, the ocean and the climate of our planet but they are not considered here as we exclusively focus on the exchange of momentum. As models of the atmosphere, the ocean and the climate system become ever more refined, further processes have to be explored and their possible impact on the dynamics has to be evaluated. It is therefore important to investigate and quantify the exchange of momentum between the synoptic and the mesoscale, the scales where the atmospheric and oceanic speeds are high.

The two superposed shallow water models and their coupling are introduced in the next section. Experiments are performed for different values of the drag coefficient given in Sect. 3, where the results concerning the energy transfer and the correlation between the vorticity dynamics in the ocean and the atmosphere are presented. The results are discussed in Sect. 4 and conclusion presented in Sect. 5.

Shallow Water Model

Physical model

The model consists of two superposed homogeneous fluid layers, a shallow layer for the lower atmosphere above an ocean surface layer. The average thicknesses are respectively H a = 1000 m and H o = 200 m. The actual layer thicknesses h a (x, y, t), h o (x, y, t) vary over time and space. The ocean surface layer superposes a motionless layer of higher density and of infinite depth.

Similarly a motionless layer of air of a lesser density superposes the shallow atmosphere layer. Layers have an average density of ρ a = 1 kg m -3 and ρ o = 1000 kg m -3 . The fluid motion considered extends over a period of many days and so, the model must take into account the Earth's rotation. Using the f-plane approximation we set the Coriolis parameter f = 10 -4 s -1 , a typical value at mid-latitudes.

Mathematical model

This physical model can be described by two coupled reduced-gravity shallow water equations:

∂ t u k + u k ∂ x u k + v k ∂ y u k -f v k + g k ∂ x h k = ν k ∇ 2 u k + F k x (1) ∂ t v k + u k ∂ x v k + v k ∂ y v k + f u k + g k ∂ y h k = ν k ∇ 2 v k + F k y (2) ∂ t h k + ∂ x [h k u k ] + ∂ y [h k v k ] = F k h , (3)
where k = a, o stands for the atmosphere and the ocean, respectively. The parameters g a and g o are the reduced gravity of the atmosphere and of the ocean (i.e., the gravitational acceleration multiplied by the fractional density difference between the two layers). They are respectively set to 0.8 m s -2 and 2.10 -2 m s -2 . The restoring force F k h in the atmosphere and ocean acts on the layer-thickness.

The typical horizontal scale is the Rossby radius of deformation Rd k = g k H k /f . It is about one order of magnitude smaller in the ocean where

Rd o = 20 km than in the atmosphere where Rd a 283 km. The domain size is L x = L y = 4000 km and there are periodic boundary conditions in both horizontal directions. In the absence of forcing and friction the potential vorticity (PV):

q k = ζ k + f h k with ζ k = ∂ x v k -∂ y u k , (4)
is conserved by the flow.

The initial oceanic height variation is defined by the sum of the 2π periodic cosinus in the x-and y-direction.

η a o (x, y) = 50m × (cos( 2π(x -7/24) L x ) + cos( 2π(y -7/24) L y )) (5)
The initial velocity field is calculated using the geostrophic equilibrium.

The initial atmospheric height variation is defined by periodic alternating cyclones and anticyclones in the x and y direction, with a period of 1000 km.

η a o (x, y) = 350m× [ cos(8πx/L x ) + cos(16πx/L x ) + cos(24πx/L x ) + cos(8πy/L y )) + cos(16πy/L y ) + cos(24πy/L y ) + cos(8π(x + y)/L x ) + cos(16π(x + y)/L x )] (6)
The initial velocity field is calculated using the geostrophic equilibrium.

So four cyclonic and anti-cyclonic alternating structures are imposed on the x and on the y direction, leading to 16 identical tiles in the forcing field.

A restoring acts to force the average of the atmospheric layer thickness projected on the forcing profile (Eq.6) towards its initial value. To this end, the projection is compared to its initial value and a multiple of the initial profile is added or subtracted to restore towards the initial amplitude of the projected mode. Such kind of restoring directly affects only the forced mode without directly influencing the other modes which can evolve more freely.

The restoring time is two days. The variation of layer thickness in the ocean layer is locally and linearly damped to zero with a long damping time of 1000 days, in order to not disturb the air-sea interaction.

When parameterizing the effect of small-scale turbulent friction at a solid boundary a quadratic drag law is used. Such a drag law dates back to the work of [START_REF] Prandtl | Das Widerstandsgesetz rauher Platten, Werft, Reederei[END_REF] and [START_REF] Schlichting | Boundary-Layer Theory[END_REF] and has been extensively studied since then (see Schlichting Gertsen ( 2000)). All these investigations essentially confirm its robustness and its applicability to all fields of fluid dynamics. When the motion of the atmosphere and the ocean are considered a large volume of research is dedicated to the determination of the drag coefficient over various surfaces [START_REF] Stull | An Introduction to boundary layer meteorology[END_REF]. When the sea surface is considered the drag coefficient depends on the sea-state, which itself is a function of various parameters (see e.g. [START_REF] Högström | Air-sea interaction features in the Baltic Sea and at a Pacific trade-wind site: an inter-comparison study[END_REF]. The robustness of the law itself seems above any doubt.

The two layers are only linked by frictional forces at the interface, parameterised by a quadratic drag law. The frictional acceleration between the two layers (see eqs. 1 and 2) is defined by:

F k x F k y = ± 1 ρ k h k f k x f k y , (7)
where f x and f y are the surface forces varying in x, y and time. They are calculated using the velocity difference between wind and ocean current,

f o x f o y = ρ a C d |u| u o -u a v o -v a , (8)
with the square of the relative speed

u 2 = (u o -u a ) 2 + (v o -v a ) 2 .
The shear applied to the atmosphere is the opposite to the shear applied to the ocean.

The drag coefficient C d is constant within each experiment.

Numerical model

The A second order centred finite difference method is used for the space discretization and a second-order Runge-Kutta scheme is used for the time discretization. The timestep is constraint by the Courant-Friedrichs-Lewy condition (see [START_REF] Courant | Uber die partiellen Differenzengleichungen der mathematischen Physik Mathematische Annalen[END_REF]. It states that the timestep must be less than the time for the wave to travel to adjacent grid points. As atmospheric waves are ten times faster than oceanic waves, it is the Courant-Friedrichs-Lewy condition for the atmosphere that sets the minimum timestep ∆t = 15 s to well resolve the temporal evolution of the atmospheric dynamics.

The model has been already employed in [START_REF] Moulin | A drag-induced barotropic instability in air-sea interaction[END_REF] at lower resolution in a non-parallelised version.

3 Results

Experiments Performed

We present results from four numerical experiments which only vary by the drag coefficient used. The drag coefficient C d is constant for each experiment.

In the four simulations discussed here, the drag coefficient is C d = 1 × 10 -4 , 2 × 10 -4 , 4 × 10 -4 and 8 × 10 -4 . The values used are within the typical values for ocean dynamics.

The drag coefficient depends on a variety of parameters as the ocean state and the density stratification of the atmosphere. When the motion of the atmosphere and the ocean are considered a large volume of research is dedicated to the determination of the drag coefficient over various surfaces (e.g. [START_REF] Stull | An Introduction to boundary layer meteorology[END_REF][START_REF] Högström | Air-sea interaction features in the Baltic Sea and at a Pacific trade-wind site: an inter-comparison study[END_REF]) and atmospheric conditions. In the shallow water equations the velocity vertically averaged over the whole layer thickness, is represented. This explains, why these geostrophic drag coefficients are lower than the drag coefficients based on velocities at 10m above the ocean surface (e.g., [START_REF] Stull | An Introduction to boundary layer meteorology[END_REF]). According to [START_REF] Stull | An Introduction to boundary layer meteorology[END_REF] the drag coefficient for a reference level of about 500m is typically between 1 × 10 -4 for stable atmosphere and 8 × 10 -4 for a neutral atmosphere.

Integrations are performed over 2700 days. The last 1000 days are considered for temporal averages. This ensures that the calculations have converged close to a statistically stationary state.

Qualitative Description

In all the four experiments a nonlinear dynamics is observed in the atmosphere and the ocean. The speeds and the absolute vorticity values of the ocean increase with the drag coefficient, the opposite is true for the atmosphere, as expected. The structures in the ocean are smaller than those in the atmosphere,

showing that both developed their own dynamics. The difference in size is readily explained by the different Rossby radii of deformation. For all values of the drag coefficient, a correspondence between the vorticity in the atmosphere and the ocean is observed, showing that the dynamics in the atmosphere and the ocean is co-organised.

For the smaller values of the drag coefficient the co-organisation is only visible in the averages (see Fig. 1, lower panels), as the atmospheric perturbations are strong and evolve on a much smaller time scale than the ocean. The shortterm instantaneous perturbations in the atmosphere are almost transparent to the ocean, which only adapts to a time-integrated atmospheric dynamics.

The atmospheric dynamics appears independent of the ocean dynamics and the external forcing on the atmosphere, when instantaneous values are considered (see Fig. 1 upper panels). In the instantaneous vorticity field of the experiment with the lowest drag coefficient, the 16 cells of the space periodic forcing are visible in the ocean but not in the atmosphere, although the forcing is applied to the atmosphere and not to the ocean, which experiences it only through the atmospheric dynamics. When long-time averages are considered the forcing is also the dominant signal in the atmosphere. This shows that the ocean adapts to a time-averaged atmosphere. For the smaller drag coefficients there are only a few coherent structures in the ocean. The velocities are too small for the non-linear terms to become important, and their dynamics is hindered by the strong atmospheric turbulence representing a high frequency forcing on the ocean dynamics. Inspection of movies show that at times, the atmospheric eddies are attracted by like-sign ocean vorticity structures and they superpose during few days. This injects further vorticity in the ocean and leads to stronger eddies during a laps of time. These transient structures in the ocean disappear after the atmospheric eddy has moved and the highfrequency forcing of to the atmosphere disintegrates the structures.

For the largest value of the drag coefficient the situation is qualitatively different. A variety of coherent structures dominate the vorticity filed in the ocean and the atmosphere and they appear co-located (see Fig. 2). The temporal vorticity anomaly in the atmosphere is strongly damped (see Fig. 2 upper-left panel) and the atmosphere and ocean dynamics is in a quenched disordered state. That is, a spatial disorder with a very slow evolution in time, in both the atmosphere and the ocean is observed. Such type of dynamics is found and extensively studied in spin-glasses but has so far not been mentioned in the context of air-sea interaction, to the best of our knowledge. The atmosphere is in a quasi-stationary state. This allows the ocean to develop larger eddies (Fig. 2). After a few years, the ocean and the atmosphere have a similar dynamics at large scale with many co-located features. As we see in Fig. 2, the mean vorticity, averaged over 1000 days, is strong as compared to the temporal vorticity anomalies, in both the atmosphere and the ocean. In the atmosphere the vorticity anomaly shows the formation of dipole structures, where there is a strong mean vorticity. They are the signature of the displacement of vortices. Many of the atmospheric eddies have a ring-structure, due to opposite vorticity in the ocean at the centre of the atmospheric eddy. When no opposite vorticity is present in the ocean, the atmospheric eddies have stronger vorticity.

Kinetic energy transfer

The density differences between the air and the water masses leads to a large discrepancy between the speeds in the ocean and the atmosphere as shown in Table 1. When increasing the drag coefficient eight fold, the atmospheric r.m.s velocity decreases by about 40%, while the oceanic r.m.s velocity increases almost an order of magnitude. This is also visible in the ratio between the kinetic energy in the ocean and the atmosphere, which is over 200 times higher with the highest drag coefficient as compared to the lowest (Table 1).

The mechanical power input per unit surface area to the ocean is the vector-product of the wind-stress shear τ and the ocean velocity u o , that is: [START_REF] Duhaut | Wind stress dependence on ocean surface velocity: implication for mechanical energy input to ocean circulation[END_REF]. The power input using the correct representation of the shear force, is based on the velocity difference between the atmospheric and the oceanic velocity

P = τ • u o (e.g.,
τ c = ρ a C d |u a -u o |(u a -u o ).
The approximation where the effect of the ocean current is neglected when calculating the shear τ app = ρ a C d |u a |u a is often used. It is justified by the fact, that the ocean currents are feeble as compared to the wind speeds (u o u a ). [START_REF] Duhaut | Wind stress dependence on ocean surface velocity: implication for mechanical energy input to ocean circulation[END_REF] revealed an overestimation of about 20% of the power input to the ocean, in a double gyre configuration with a prescribed time-independent wind velocity, when the approximation is used. It can be shown that the correct values of the power input P c are always smaller than the approximated values P app . In our results, there is almost no difference between P c and P app for the smaller drag coefficients, because the ocean speeds are very small as compared to the atmospheric speed. For the highest drag coefficient the energy input estimated with the approximation overestimates the power input by 5%. The difference is accentuated for larger drag coefficients as higher velocities are induced in the ocean, and the ocean velocities are more aligned with the atmospheric velocities (Fig. 2). The fact that on average the difference is low, does not mean, that locally it is not important, in areas where oceanic currents are strong and aligned to the atmospheric winds.

On average, the atmosphere loses energy over more than 95 percent of its surface area for all drag coefficient (Table 1). The percentage increases with a decreasing drag coefficients, attaining 99% for the smallest drag coefficient.

This shows, that for the atmospheric kinetic energy, the ocean has a passive role. The loss of kinetic energy is of the same order of magnitude for all drag coefficients. The main part of the kinetic energy lost by the atmosphere in P a(ua t ) (W m -2 ) -3.6 × 10 -3 -4.9 × 10 -3 -6.3 × 10 -3 -6.4 × 10 -3 P a(ua)

t (W m -2 ) -4.6 × 10 -3 -6.1 × 10 -3 -7.5 × 10 -3 -7.0 × 10 -3 P o(ua t , uo t ) (W m -2 )
2.6 × 10 -6 7.9 × 10 -6 2.4 × 10 -5 7.0 × 10 the air-sea interaction is dissipated. Indeed, only 3 percent is transferred to the ocean for the highest drag coefficient, it decreases to 0.06 percent for the smallest drag coefficient. This can again be explained by the lower speeds in the ocean as compared to the atmosphere and by the non-alignment between the local velocity vectors.

The ocean gains energy from the atmosphere, whenever the absolute value of the angle between the ocean current and the shear force exerted by the atmospheric wind is smaller than 90 • . Note that the direction of the shear force is aligned with the difference of the atmospheric wind vector and the ocean current vector. For the smaller drag coefficient the ocean gains energy from the atmosphere over 58% of its surface area, against 77% for the highest drag coefficient (Table 1). This is explained by the stronger coupling between the atmosphere and ocean dynamics at higher drag coefficients.

Figure 3 shows histograms of the wind speeds (left) and the ocean speeds (right). Superposed is the percentage of energy lost by the atmosphere for the corresponding speed range. The percentage of the energy gained by the ocean for the corresponding speed range is shown by the blue curve. The thick vertical line is the mean velocity. Due to the cubic law for the energy loss, the atmosphere loses its power at its high atmospheric speeds (Fig. 3).

A comparison of the real energy loss to its approximation calculated with the mean velocity (P a = ρ a C d u a 3 ), shows that it underestimates the energy loss by almost a factor of one-and-a-half for the lowest drag coefficients and provides a good estimate for the largest (Table 1).

The power input to the ocean is principally at the same atmospheric velocities at which the atmosphere loses energy, as is seen in Fig. 3, where the red full line and the blue dashed line are similar in the figures corresponding to the atmospheric speeds (left column).

The power input to the ocean (Fig. 3, right column) shows a strong dependence on the drag coefficient C d . For the lowest C d value, 14 % of the fastest ocean speeds contribute half of the power input. For the largest C d , this is achieved by only the fastest 3 % of ocean speeds. An explanation of this is an increased correlation between oceanic and atmospheric velocities for higher drag coefficients (see below). This result questions the significance of mean wind and current speeds when the power input is estimated. Indeed, as a rule of thump, the power input from the atmosphere to the ocean is often estimated based on the mean wind and ocean speeds. In this estimation not only the correlation of the local wind and ocean speeds are neglected, but also the angle between the corresponding velocity vectors. The estimated value is compared to the exact value in Table 1. For the highest drag coefficient the estimated value is 16% lower than the true value, while it is 3 times higher for the smallest drag coefficient (Table 1). The higher value of the estimate is explained by the fact that it neglects the non-alignment between atmospheric wind and the ocean currents. The non-alignment is highest for the lower values of the drag coefficient. The lower value of the estimate is explained by the correlation of the magnitudes of the atmospheric and oceanic speeds.

Performing a log-log plot of the power-input to the ocean (averaged in space and time) as a function of the drag coefficient (Fig. 4) we see, that it is close to a quadratic law. Note, that the drag coefficient has a stronger than linear influence on the power input to the ocean as a higher drag coefficient increases the shear force and the velocity in the ocean and the power-input is the (vector) product of the two. A higher correlation between atmosphere and ocean velocities for higher drag coefficients further increase the power input.

Vorticity correlation between the atmosphere and the ocean

The normalised bi-variate probability density function for the atmospheric and the oceanic vorticity is used, to study the correlation between the dynamics in the ocean and the atmosphere. It is obtained by first subtracting the mean-value and then normalising the atmospheric and oceanic vorticity by their respective standard deviation, before considering the correlations. For an instantaneous snapshot after 7 years, and the smallest drag coefficient, the isolines of the probability density function and the overlaid correlation ellipse appear almost circular (Fig. 5). Indeed the ratio between the two principal correlations is 1.48 (Table 2), close to 1 which is the value for a perfect decorrelation. This shows that there is only a feeble correlation between the atmospheric and oceanic vorticities. The correlation increases for the drag coefficients C d = 2 × 10 -4 and C d = 4 × 10 -4 before slightly dropping to 2.24 for -4 . For the vorticities averaged over 1000 days the cross-correlation ratio between the big and the small half axes of the ellipse 3.66 4.91 5.45 2.57 slope of principal axis 5.17 × 10 -3 7.45 × 10 -3 1.09 × 10 -2 3.52 × 10 -2 is higher than the instantaneous values. The ratio between the two principal correlations is more than 2 times higher than for the instantaneous snapshot (Table 2) for the lower three drag coefficients, confirming the above mentioned observation that, when the fast variations of the atmosphere are filtered out the correlation between the ocean and the atmosphere increases. For the highest drag coefficient the dynamics is qualitatively different: the time variability, in the atmosphere and the ocean, is low as the dynamics is in a quenched state (see next subsection) and the cross correlation increases only by 15% when averages, rather than instantaneous values are considered.

C d = 8×10
When the normalised bi-variate probability density function is considered the slope of the principal axis, which is the ratio of the standard deviation of the ocean and the atmosphere, is diagonal by definition (see Figs. 5 and6). In the non normalised case it establishes a correspondence between the vorticity in the two media. The slope of the first principal axis is about two times stronger for the temporal mean than for the snapshot, for the three smaller drag coefficients. For the largest drag coefficient the two slopes are similar as there is almost no variation in time.

Spatial versus Temporal Variability of the Vorticity

In this part we study the variability in space and time of the vorticity field in the atmosphere and the ocean as a function of the drag coefficient. We need to define two different types of average: a spatial average and a temporal average. The temporal average of the vorticity has already been defined in the Table 3 Variability in space (varSpat) and time (vartemp) for the four drag coefficient for the ocean and the atmosphere.

C d 1 × 10 -4 2 × 10 -4 4 × 10 -4 8 × 10 -4 varTota (s -2 )
3.48 × 10 -10 2.77 × 10 -10 1.76 × 10 -10 9.57 × 10 -11 varToto (s -2 ) 8.64 × 10 -16 1.69 × 10 -15 4.86 × 10 -15 2.16 × 10 -13 varSpata (s -2 ) 6.17 × 10 -12 3.89 × 10 -12 5.69 × 10 -12 8.32 × 10 -11 varSpato (s -2 ) 3.46 × 10 -16 7.65 × 10 -16 2.25 × 10 -15 1.79 × 10 -13 vartempa (s -2 ) 3.43 × 10 -10 2.73 × 10 -10 1.70 × 10 -10 1.27 × 10 -11 vartempo (s -2 ) 5.18 × 10 -16 9.30 × 10 -16 2.60 × 10 -15 3.63 × 10 -14 first part. It is the average of the vorticity over 1000 days denoted by ζ k 1000 .

Where k corresponds to the ocean or the atmosphere.

The forcing applied to our periodic domain repeats itself 4 times in the

x-and in the y-direction, leading to 16 identical cells. Furthermore, rotations of angles 90, 180 and 270 degrees do not change the forcing. The discrete symmetry group of the forcing has therefore 64 members. The dynamics in the atmosphere and the ocean, does not exhibit this periodicity, as it is not present in the initial conditions and as the system is chaotic. There are 4 × 4 identically forced squares present in the domain and we expect to find the periodicity of the forcing in the averages of dynamical quantities as there is a priory no reason to suppose that averages differ from one square to another once the initial conditions are "forgotten". That is, we suppose the system to be ergodic, meaning that ensemble averages and time averages coincide.

For the calculation of the spatial average we use the full symmetry group with 64 members. We calculate the average over all members of size 1000 km× 1000 km, and continue it periodically over the large domain of size 4000 km × 4000 km.

This discrete spatial average is denoted by ζ k 64 .

To calculate the variability we also need to define the spatial variance:

var(a) = 1 n x n y nx,ny i,j=0 (a i,j ) 2 , ( 9
)
where a is a matrix of size n x × n y .

With all these terms the spatial variability can be defined by:

V arS k = var( ζ k 1000 -ζ k 1000 64
).

(10)

The space variability reveals the difference obtained when we reproduced the experiment with a little initial perturbation.

The temporal variability represents the evolution of the variable in time, it can be defined by:

V art k = var(ζ k -ζ k 1000 ) 1000 . (11)
The total variability is written:

V arT ot k = var(ζ k -ζ k 1000 64 ) 1000 . (12)
It can be shown that the total variability is the sum of the spatial and the temporal variability:

V arT ot k = V arS k + V art k . ( 13
)
The results, for the four drag coefficient, are plotted in an histogram for the atmosphere (Fig. 7 (left)) and for the ocean (Fig. 7 (right)). In the atmosphere, the total variability decreases when the drag coefficient increases. The spatial variability is small ( 3%) for the the three lower drag coefficients but skyrockets to 86% for C d = 8 × 10 -4 of the total variability, showing that the dynamics changes qualitatively when the drag coefficient is increased from -4 to 8 × 10 -4 . In the ocean, the total variability is very low for the smallest three drag coefficient (order of 10 -15 s -2 ) with again a strong dominance of the the temporal variability. The case with the strongest drag coefficient is again very different from the lower ones. It has a total variability around hundred times larger with more than three quarters due to the spatial variation. The disorder in the highest drag coefficient case is dominated by spatial disorder (quenched disorder), whereas the three other cases, the temporal disorder prevails (annealed disorder).

C d = 4 × 10

Discussion

From an kinetic-energy point of view the atmosphere-ocean system is driven by the injection of kinetic energy into the atmosphere. The system losses energy due to friction at the atmosphere-ocean interface (and to a lesser extent by horizontal friction, not discussed here). A small fraction of the kinetic-energy lost by the atmosphere drives the ocean. The ocean has an almost perfectly passive role, absorbing energy from the atmosphere. Concluding that the ocean has no influence on the dynamics of the atmosphere is however wrong, as demonstrated above. The amount of energy taken out of the atmosphere at its lower interface depends only slightly on the ocean dynamics, but due to the persistence of the ocean dynamics it leaves an imprint in the long-time dynamics of the atmosphere. In this respect it is difficult to dissect the atmosphere dynamics, the ocean dynamics and their interaction. One should rather see the combined system, as interactions and correlations depend on the time scale considered.

The difference of the power input estimated with the wind speeds, only, to the consistent calculation, based on the shear between the atmosphere and the ocean, is found to be small. This is different to the work by [START_REF] Duhaut | Wind stress dependence on ocean surface velocity: implication for mechanical energy input to ocean circulation[END_REF], who treated a large scale double gyre circulation. They have a constant-in-time wind velocity and the double gyre configuration that leads to a strong boundary current that separates from the boundary and penetrates into the interior of the domain with large ocean velocities of the order of 1m/s.

In our calculations periodic boundary conditions are used and no accumulation of energy at a boundary is possible. Sverdrup theory, based on the conservation of vorticity predicts velocities for an ocean subject to a constant in time large scale wind-forcing of only a few centimetres per second. In our calculations the forcing by the atmosphere on the ocean is (for the three lower values of the drag coefficient) constantly changing magnitude and direction, in time-andspace. This temporal and spatial incoherence in the forcing explains the feeble ocean speeds in our model and thus also the smaller differences between the correct implementation and its approximation.

For the highest drag coefficient, the atmosphere injects more energy into the ocean, this leads to stronger ocean currents. Stronger ocean currents together with the higher drag coefficient have an increased influence on the atmosphere, leading to an alignment of the instantaneous atmospheric winds with the ocean currents. This not only leads to stronger ocean currents but also to a reduction of temporal variability in the atmosphere, which furthermore reduces the temporal variability in the ocean. This positive feedback finally causes a strong oceanic current and a strong collocation of atmospheric and oceanic features. The disorder in the highest drag coefficient case is dominated by spatial disorder (quenched disorder), whereas in the three other cases, the temporal disorder prevails (annealed disorder). An astonishing result of our calculations is that the change between the two kinds of disorders seems to happen abruptly and it can not be excluded that it resembles a phase change.

The transition happens when the non-dimensional stability parameter (see e.g.

Chen and Jirka 1997)

S = C d L a H a (14)
comparing the drag force to the advection increases towards unity. Even more striking is that the dynamics in the ocean and atmosphere does not converge to the purely periodic state imposed by the forcing but to a spatial disordered state. Such kind of behaviour is commonly observed in condensed matter physics, where it is called glass transition (see e.g. [START_REF] Stillinger | Glass transition Thermodynamics and Kinetics[END_REF][START_REF] Stillinger | Glass transition Thermodynamics and Kinetics[END_REF][START_REF] Berthier | Theoretical perspective on glass transition and amorphous materials[END_REF][START_REF] Berthier | Theoretical perspective on glass transition and amorphous materials[END_REF], it occurs around a glass transition temperature, at which the internal relaxation times begin suddenly to exceed the practical measurement times and, furthermore, the state becomes protocol dependent. When the material is performing a glass transition it does not, or only very very slowly, converges to a periodic (crystalline) state. In our case the drag coefficient plays the role of the control parameter.

The quadratic dependence of the power input to the ocean is important for ocean dynamics as this small-scale energy contributes not only to the mixing in the ocean at the scale where it is injected, but will cascade to larger scales in an inverse energy cascade.

Conclusions

The majority of our results concerning the air-sea interaction were expected:

the passive role of the ocean in the energy transfer; the weak dependence of the air-sea energy transfer on the ocean velocities and its increase with the drag coefficient; and the higher correlation of the averaged vorticities as compared to instantaneous part. The value of the present work lies in the quantification of the air sea interaction at the synoptic-and mesoscale.

The increased correlation of the vorticity between the atmosphere and the ocean (Figs. 5 and6) when a temporal average rather than instantaneous values are considered shows, that even when such a correlation is hidden on instantaneous snapshots due to the fast turbulent dynamics of the atmosphere it emerges when averages are considered.

The most striking result is the glass transition in the dynamics of the atmosphere-ocean system when the drag coefficient is increased from C d = 4×10 -4 to 8 × 10 -4 . For the higher drag coefficient, the time variability is largely reduced and the system is in a quenched disordered state. This is similar to the what happens in a glass transition, when the temperature is reduced below critical. The consequence of this is, that time averages are different from spatial averages, such that the system is no-longer ergodic. In studies of the earth system based on observations, ergodicity is often assumed, as there is only one planet. The ergodic hypothesis has been proven not to apply to the inviscid dynamics on the β-plane and the sphere subject to strong rotation by [START_REF] Shepherd | Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere[END_REF] and questioned in other idealised models. When studies are based on laboratory experiments or numerical simulations, experiments are continued until statistical quantities have converged, using again the ergodic hypothesis. To further investigate such transition analytically and/or numerically a simpler model is necessary.

In the calculations with the high drag coefficient the high instantaneous correlation between the vorticity in the atmosphere and the ocean (Table 3 see also Fig. 2), shows that the ocean currents leave a definite imprint in the atmospheric winds leading to fine scale structures in the atmosphere. This suggest, that the fine scale structures of the ocean dynamics which have recently been exposed by fine resolution satellite observations can influence the atmospheric dynamics through momentum transfer, even when heat-fluxes are excluded. To evaluate the robustness of our results observational studies similar to Chelton et al. (2004) should also consider the correlation between persistent winds in the atmospheric boundary layer and currents in the oceanic boundary layer, at small scales.

The genesis of hurricanes is strongly influenced by air-sea interaction, mostly through the heat exchange but also by the frictionally induced convergence [START_REF] Charney | On the growth of the hurricane depression[END_REF]. The near surface processes such as mesoscale positive vorticity anomalies are important in the bottom-up scenarios of hurricane genesis (Fang and Zhang 2008). The capture of atmospheric eddies by oceanic structures and their co-evolution can be an important ingredient in this context. To further investigate the co-evolution of atmospheric and oceanic vorticity anomalies and their statistical properties, a study with a simplified point vortex model is currently undertaken.

  ocean and the atmosphere basins are represented by a rectangle, of L x × L y . Periodic boundary conditions are used in both horizontal directions. The numerical grid is regular and contains n x × n y points. Fine spacial resolutions ∆x = L x /n x = ∆y = L y /n y are employed to well resolved the horizontal scales. We choose n x = n y = 2048 and L x = L y = 4000 km for the 2D shallow water model. The horizontal components of the velocity u k , v k , and height variations η k are calculated on each grid point. The eddy viscosity of the layers are ν a = 500 m 2 s -1 and ν o = 0.1 m 2 s -1 , which are constants in space and time.

Fig. 1

 1 Fig. 1 For C d = 1 × 10 -4 . Map of the average vorticity, averaged over 1000 days, starting from day 1700 (bottom). Map of the vorticity anomaly, averaged over 20 days, starting from day 2680, with respect to the 1000 days average (middle). Map of the instantaneous vorticity anomaly at day 2675, with respect to the 1000 days average (top).

Fig. 3

 3 Fig. 3 Percentage of surface area for a range of speed (histogram) for the atmosphere (left row) and the ocean (right row) for different values of the drag coefficient C d = 1 × 10 -4 at the top, C d = 4 × 10 -4 in the middle, and C d = 8 × 10 -4 at the bottom. The curves superposed give the power input calculated with the consistent formulation for the ocean (red-full) and lost for the atmosphere (blue-dashed) corresponding to the specific range of speed.

Fig. 4

 4 Fig.4Log-log plot of the mean power input (P) as a function of the drag coefficient.

Fig. 5

 5 Fig. 5 Normalised probability density function for C d =1 × 10 -4 for the instantaneous snapshot after 2675 days (left) and for the mean temporal vorticity (right).

Fig. 6

 6 Fig. 6 Normalised probability density function for C d =8 × 10 -4 for the instantaneous snapshot after 2675 days (left) and for the mean temporal vorticity (right).

Fig. 7

 7 Fig. 7 Space variability (black) and time variability (white) for four values of the drag coefficient in the atmosphere (right) and in the ocean (left). For the ocean the variability are multiplied by 100 for the three lower drag coefficients.

Table 1

 1 Averaged quantities for the four values of the drag coefficient. Mean power lost/gained by the atmosphere/ocean for the consistent calculation (P a(ua) lost/gained based on the mean atmospheric/oceanic velocity (P a(ua t ), P o(ua t , uo t )).

	C d	1 × 10 -4	2 × 10 -4	4 × 10 -4	8 × 10 -4
	u 2 a (m s -1 )	3.3	2.9	2.5	2.0
	u 2 o (m s -1 )	3.4 × 10 -3	6.4 × 10 -3	1.3 × 10 -2	3.1 × 10 -2
	Ea/Eo	4.6 × 10 3	1.1 × 10 3	2.2 × 10 2	22
	surface fraction where	99	98	97	95.5
	the atmosph. losses en-
	ergy(%)
	surface fraction where	58	70	82	77
	the ocean gains en-
	ergy(%)

t , P o(ua, uo) t ) and power

Table 2

 2 Variance and spatial average of the vorticity in the ocean and in the atmosphere for an instantaneous snapshot (after 2675 days) above and for a temporal average over 1000 days (below).

	C d			1 × 10 -4	2 × 10 -4	4 × 10 -4	8 × 10 -4
				Instantaneous
	variance	of	atmo-	3.28 × 10 -10	3.35 × 10 -10	2.77 × 10 -10	1.71 × 10 -10
	spheric vorticity (s -2 )
	variance of oceanic vor-	1.35 × 10 -15	4.43 × 10 -15	1.49 × 10 -14	2.37 × 10 -13
	ticity (s -2 )
	ratio between the big	1.48	1.74	2.28	2.24
	and the small half axes
	of the ellipse
	slope of principal axis	2.03 × 10 -3	3.64 × 10 -3	7.33 × 10 -3	3.73 × 10 -2
				1000 days average
	variance	of	atmo-	3.60 × 10 -11	5.85 × 10 -11	9.90 × 10 -11	1.60 × 10 -10
	spheric vorticity (s -2 )
	variance of oceanic vor-	9.58 × 10 -16	3.25 × 10 -15	1.18 × 10 -14	1.97 × 10 -13
	ticity (s -2 )


A Wirth

J B Flor

Pressure drag of two-layer shallow-water flow at an inclined oceanic ridge, independent of velocity

The drag in geophysical flows over topography is commonly modeled by a quadratic law. For a rotating fluid, we show that in the two-layer shallow-water approach, the pressure drag of a sub-critical flow around an inclined ridge is in good approximation independent of the flow speed.

Numerical simulations results confirm this for a large range of Rossby numbers for both, barotropic and baroclinic flows approaching the ridge. The behavior is explained by the observation that for larger speeds the flow crosses the ridge at lower depth leading to a shorter path-length. As the frictional head loss is a product of the velocity and the path length, both compensate.

I. INTRODUCTION

In 1786 Du Buat [START_REF] Du Buat | Principes D'Hydraulique[END_REF] measured the pressure over the surface of an obstacle in a moving fluid and observed that the upstream pressure exceeds the pressure at the down-stream side of the obstacle. The difference was eventually explained with the pressure drag, a result then much used for navigation industry. More recently the role of topography and pressure drag in oceanic and atmospheric flows has been subject of discussion. For the dependence of the drag on the flow velocity, generally, the simple fact that the inertia is the product of mass and velocity is used. Since the mass of fluid encountering an obstacle is itself a linear function of velocity, a quadratic-drag law follows [START_REF] Stull | An Introduction to Boundary Layer Meteorology Kluwer Academic Publishers[END_REF]. The law applies also to the frictional forces in a turbulent boundary layer. A considerable effort in all fields of fluid dynamics is devoted to the determination of the friction coefficient, taking the quadratic law for granted.

In a homogeneous fluid, a current over topography is known to generate a vertical shear in the horizontal velocity field over a large depth. In rotating fluid flows that are dominated by the Coriolis force, however, this shear is confined to a thin boundary layer while the interior flow is depth independent and shows the formation of Taylor columns in accordance with the theory of Taylor Proudman [START_REF] Proudman | On the motion of solids in a liquid possessing vorticity[END_REF], [START_REF] Taylor | Motion of solids in fluids when the flow is not irrotational[END_REF]. Supposing a dominantly horizontal motion, we question how the drag varies with the dominance of the Coriolis force expressed by the Rossby number. This drag depends on the large scale motion, and in addition, when multiple layers or a stratification are present, the baroclinic interaction of this large scale flow with other layers.

The understanding of drag and its parametrization is relevant for the modeling of many geophysical flows and in particular climate models. Some examples are the Antarctic Circumpolar Current (ACC) and its interaction with topography [START_REF] Olbers | Comments on On the Obscurantist Physics of Form Drag in Theorizing about the Circumpolar Current[END_REF], [START_REF] Olbers | The dynamical balance, transport and circulation of the Antarctic Circumpolar Current[END_REF]. But also smaller scale flows around ridges are relevant [START_REF] Warner | Dissecting the Pressure Field in Tidal Flow past a Headland: When Is Form Drag Real?[END_REF]. A ridge which is believed to have a strong influence on the thermohaline circulation in the ocean is the Reykjanes Ridge. It is a major topographic feature of the North-Atlantic Ocean south of Iceland, where the lower and the upper limp of the meridional overturning circulation (MOC) superpose. It influences the main paths followed by the two limbs of the MOC and is the gate towards the deep convection areas for the warm water masses of the upper limb. However, the temporal mean and variability of the structure of the flow along and over the ridge as well as the prevailing dynamical regimes are still poorly understood, and the governing dynamical processes involved are not fully identified. About 50% of the warm to cold conversion associated with the MOC occurs in the Reykjanes Ridge area (see [START_REF] Desbruyères | Simulated decadal variability of the meridional overturning circulation across the A25-Ovide section[END_REF]), thus highlighting the importance of the flow dynamics around this ridge for climate. Despite this clear influence, the horizontal and vertical structure of the flow along and over the ridge and their variability are still incompletely documented and sometimes subject to controversy [START_REF] Lherminier | The Atlantic Meridional Overturning Circulation and the subpolar gyre observed at the A25-OVIDE section in June 2002 and 2004[END_REF]. Recently, the atmospheric flow patterns around a ridge have been investigated to explain the increase of the summer near-surface temperatures over the northeast coast of the Antarctic Peninsula There is a large body of literature about stratified flow over idealized topography in a rotating fluid, either in the context of wind over mountains or ocean flows over ridges, of which some are reviewed by [START_REF] Boyer | Laboratory Studies of Orographic Effects in Rotating and Stratified Flows[END_REF] and [START_REF] Van Heijst | Laboratory Modeling of Geophysical Vortices[END_REF]. The Shallow Water Model (SWM) has been used for the study of the flow around a spherical topography [START_REF] Schär | Shallow-water flow past isolated topography. Part I: Vorticity production and wake formation[END_REF], [START_REF] Schär | Shallow-water flow past isolated topography. Part II: Transition to vortex shedding[END_REF], and continued with the effect of bottom friction [START_REF] Grubisic | The effect of bottom friction on shallow-water flow past an isolated obstacle[END_REF]. A single-layer flow intersected a circular topography. The fore-aft asymmetry was broken by a hydraulic control over the topography followed by a hydraulic jump or flow separation at the topography. Both lead to eddy shedding in the wake. Though sub-critical flow was considered and frictional processes were discussed [START_REF] Grubisic | The effect of bottom friction on shallow-water flow past an isolated obstacle[END_REF], the drag was not evaluated. Most related to the present study are the experiments on a uniform flow over a uniform ridge on the f-plane [START_REF] Maxworthy | Topographic Effects in Rapidly-rotating Fluids: Flow Over a Transverse Ridge[END_REF] reporting on the derivation of single layer flow over topography as a function of the Rossby number. To our knowledge, this type of flow has not further been investigated in very much detail.

In this paper, we consider two-layer flow over an inclined ridge that is extended in part of the domain (see Fig. 1), and as diagnostic, investigate the loss of momentum due to pressure drag. We use a two-layer Shallow Water Model (SWM) in the rotating frame (f-plane), and restrict our investigation to sub-critical flow, i.e. we eliminate hydraulic jumps and flow separation. Since the veracity of the integration of super-critical flow and hydraulic jumps depends on the parameterization of the non-hydrostatic process that are not explicitly resolved in the shallow water equations, the flow is no longer depth independent and vertical velocities are no longer small as compared to their horizontal counter parts, i.e.

the basis of the shallow water approximation itself breaks down. For a strict investigation of super-critical flow and hydraulic jumps, fully three-dimensional models based on the Navier-Stokes equations should be used, which is out of the scope of the present study. The physical situation put forward to study the flow, the mathematical model adapted to its study and the corresponding numerical model are pre-sented in the next section, where we also discuss the conservation properties derived from the mathematical model. The numerical simulations and the obtained results are given in section III and discussed in section IV.

II. THE MODEL A. The Physical Model

The physical model chosen to study the dynamics is a canal configuration which is L x = 2400km long and L y = 1000km wide. The depth is 2km plus/minus a topography, which is a continental shelf with a ridge as seen in Fig. 1. The half-length of the ridge is /L r = 125km, the e-folding thickness is D r = 60km and the height is B r = 800m. The analytic form of the topography is given by:

b(x, y) = B r [tanh (2(y -1)) (1 -e(x )) + tanh (2(y + 1)) e(x )] (1)
with e(x ) = exp -x 2 and x = x/D r , y = y/L r Two layers of different density fill the canal, with a reduced gravity g = g(ρ 2 -ρ 1 )/ρ 2 = 2.0 • 10 -2 , where g is the earth's gravitational constant. The subscript 1 and 2 denote the upper and the lower layer, respectively. When the fluid is at rest the heavy water (ρ 2 ) extends from the bottom to a depth of 1km. The upper 1km is filled with the lighter water mass (ρ 1 ).

The dynamics is forced by damping towards a constant current along the topography in sponge layers at both ends of the channel. This forcing induces a current along the topographic slope. The mathematical detail of the forcing is given in the next subsection.

Note that no topographic feature penetrates into the upper layer, or equivalently the topography is always submerged by the lower layer. The vanishing of the lower layers thickness leads to super-critical flow, breaking of waves on the interface and increased mixing. Such phenomena are not explicitly foreseen in the shallow water model used here (see subsection II B). The dynamics of never vanishing layers allows to forgo implementing a parameterization of this process in the shallow water equations and facilitates the numerical solution (see subsection II D).

B. The Mathematical Model

The mathematical model we employ for this flow is the classical two-dimensional two-layer shallow water (SW) equations on the f-plane. Considering the model hierarchy, the SW equations include vortex stretching and Poincaré waves and are therewith more realistic than quasi-geostrophic or the 2D Navier Stokes equations, which neither include Poincaré waves.

The reference thicknesses in the two layers are h 0 i . The horizontal components of the vertically-averaged velocity vector are u i (t, x, y) (x-direction) and v i (t, x, y) (y-direction), where the index (i = 1, 2) refers to respectively the upper and the lower layer. The deviation from a flat surface and interface is denoted by η 1 (t, x, y) and η 2 (t, x, y), respectively. Note that the total thickness of the upper layer is h 1 = h 0 1 + η 1 -η 2 and for the lower layer the thickness is

h 2 = h 0 2 + η 2 -b.
The writing of the equations is largely simplified if both symbols, h and η are used:

∂ t u 1 + u 1 ∂ x u 1 + v 1 ∂ y u 1 + g∂ x η 1 -f v 1 = ν∇ 2 u 1 (2)
∂ t v 1 + u 1 ∂ x v 1 + v 1 ∂ y v 1 + g∂ y η 1 + f u 1 = ν∇ 2 v 1 (3)
∂ t η 1 +∂ x [h 1 u 1 ] + ∂ y [h 1 v 1 ] + ∂ x [h 2 u 2 ] + ∂ y [h 2 v 2 ] = κ∇ 2 η 1 (4) ∂ t u 2 +u 2 ∂ x u 2 + v 2 ∂ y u 2 + g ∂ x η 1 + g ∂ x η 2 -f v 2 = ν∇ 2 u 2 (5)
∂ t v 2 + u 2 ∂ x v 2 + v 2 ∂ y v 2 + g ∂ y η 1 + g ∂ y η 2 + f u 2 = ν∇ 2 v 2 (6)
∂ t η 2 + ∂ x [h 2 u 2 ] + ∂ y [h 2 v 2 )] = κ∇ 2 η 2 (7)
Where g = 10ms -2 is gravity, g = g(ρ 2 -ρ 1 )/ρ 2 = g -g and g = gρ There are sponge layers extending 200km in the x-direction at both ends of the canal.

The dynamics is forced by restoring the free surface and the interface towards a prescribed value in the sponge layer:

η sponge i = l i b B r ( 8
)
where b is the topography as defined in eq. 1 and the parameters l i are varied between experiments. When l 1 = l 2 the forcing is barotropic.

The Reynolds number based on the lateral viscous dissipation is:

Re = uD r ν . (9)
An upstream baroclinic Froude number (based on the velocity in the lower layer relative to the maximal wave speed at the interface) is given by:

F r = u g 2 g h 0 e . ( 10
)
where we used the equivalent depth:

h 0 e = h 0 1 h 0 2 h 0 1 + h 0 2 (11)
and u g 2 is the maximal geostrophic velocity imposed in the sponge layer. The barotropic Froude number is small in all experiments. The Rossby number is given by:

= u g 2 D r f . ( 12
)
It compares the relative vorticity of the flow around the ridge to the planetary vorticity.

Note that the vorticity around the ridge depends mainly on its width. A Burger number, which compares the stratification to rotation is Bu = ( /F r) 2 .

Due to the bottom topography the layer thickness of the lower layer varies from 200m to 1800m (when the fluid is at rest) so that the barotopic Rossby radius of deformation varies between 1100km ≤ r bt = g(h 0 1 + h 0 2 )/f ≤ 1700km and a baroclinic radius of deformation between 18km ≤ r bc = √ g h e /f ≤ 32km. In experiments eX05 (see table I) the Coriolis parameter and therefore also the Rossby radii are varied. A non-dimensional height of the

topography given by b = √ g b 0 /c bc = (h 0 1 + h 0 2 )b 0 /h 0 1 h 0 2 ≈ 1.
26 exceeds unity which means that the topography is high according to [START_REF] Orr | Characteristics of Summer Airflow over the Antarctic Peninsula in Response to Recent Strengthening of Westerly Circumpolar Winds[END_REF].

C. Pressure Drag

This subsection clarifies the notion of pressure drag in the frame of the SW equations and can easily be skipped by a reader familiar with this concept.

For the analysis of the influence of the ridge we consider the momentum M i in the xdirection in each vertical layer in a rectangular domain that spans the entire width of the canal in the y-direction and from

x A to x B in the x-direction (A = [x A , x B ] × [y A , y B ]): M i = ρ i A u i h i da. (13)
The momentum balance in the x-direction in the two layers, derived from eqs. ( 2) -( 7), is

given by:

1 ρ 1 ∂ t M 1 = A ∂ t (u 1 h 1 )da = y B y A u 2 1 h 1 dy x B x A + A -gh 1 ∂ x η 1 + f h 1 v 1 da - A νh 1 ∇ 2 u 1 + κu 1 ∇ 2 h 1 da (14)
and

1 ρ 2 ∂ t M 2 = A ∂ t (u 2 h 2 )da = - y B y A u 2 2 h 2 dy x B x A + A -h 2 ∂ x (g η 2 + g η 1 ) + f h 2 v 2 da - A νh 2 ∇ 2 u 2 + κu 2 ∇ 2 h 2 da (16)
The first terms on the r.h.s of eqs. ( 14) and ( 16) represents the difference between the momentum transport entering the domain at x A and leaving at x B . The second line in both equations contains the accelerated mass due to the pressure gradient induced by variations of the surface (η 1 ) and the interface (η 2 ). They form a total integral if there exists a function so that h 1 = F 1 (η 1 ) and a function so that h 2 = F 2 (g η 2 + g η 1 ), which is the case when Bernoulli potentials exist. Note that when the flow is in geostrophic equilibrium we have:

v 1 = g f ∂ x η 1 and v 2 = g f ∂ x η 1 + g f ∂ x η 2 (17)
and the Coriolis term (the second term of the surface integral) in both equations cancels the potential terms. This is explained by the fact that in a geostrophic flow differences in surface and isopycnal height do not lead to acceleration, as the pressure gradient is balanced by the Coriolis force. The value of the domain integral in the second line of eqs. ( 14) and ( 16) can thus be used to measure the departure from geostrophy in the domain. In the above equations the lower boundaries of the two layers do not appear explicitly and one might falsely conclude that they are not important for the momentum balance. The lower boundaries do modify both, the transport in the layers and shape of the upper boundary.

This becomes apparent when hydrostacy is used and the momentum balance is evaluated based on the pressure at the lower boundaries of each layer. In this case the pressure drag exerted by the upper layer on the interface is

D i = gρ 1 A h 1 ∂ x η 2 da, (18)
and the pressure drag exerted by the upper layer and the lower layer on the topography:

D b = g A (ρ 1 h 1 + ρ 2 h 2 )∂ x b da = D b1 + D b2 (19)
appears in the equations. The pressure terms used in eqs. ( 14) and ( 16) can be expressed as:

g A h 1 ∂ x η 1 da = g y B y A h 2 1 2 x B x A dy + D i (20) ρ 2 A h 2 ∂ x (g η 2 + g η 1 )da = g y B y A (ρ 2 h 2 2 dy + ρ 1 h 1 )h 2 dy x B x A -D i + D b . (21)
The first terms in the above equations represent the acceleration due to differences in the layer thickness at the two boundaries of the canal. The second terms represent the pressure drag (also called "form drag") on the interface and third term in the second equation represents the bottom drag due to the surface and interface inclination. Note that if the densities and velocities in the two layers are equal we obtain the one-layer solution.

As in the case of d'Alembert's paradox [START_REF] D'alembert | Paradoxe proposé aux géomètres sur la résistance des fluides[END_REF] it is friction and time-dependence that bring the pressure gradient out of phase with the topography and allow for the existence of a pressure drag.

D. Numerical Model

The numerical model chosen to solve the above detailed mathematical model is a secondorder-in-time Runge-Kutta scheme. The spatial discretisation is a centered second order in space scheme (first order at the boundary). The resolution is ∆x = ∆y = 2km (1200 × 500 points) in space and ∆t = 1s in time. The short time-step is imposed by the barotropic gravity wave speed. The spatial resolution is 2km in both horizontal directions. The friction parameters adapted for this resolution and the time-stepping scheme are ν = κ (m 2 s -1 ).

Using even higher resolutions might not add much to our understanding of the physics involved as the approximations (e.g. hydrostatic-approximation) at the basis of the shallow water model are no longer valid for dynamical phenomena that have horizontal extensions comparable to the water depth (2km). Even higher resolution might teach us things about the mathematics of the SW model but not about the physics of the ocean.

Note also, that thethickness of the layers never vanishes, meaning that no numerical scheme to treat the processe is needed.

During spin-up of 100days the flow is forced not only in the sponge layers, but also in the interior. After that, the dynamics rapidly converges to an almost stationary state which is analyzed after 150days.

Using an explicit free surface model clearly slows down the numerical integration as compared to rigid-lid or implicit-free-surface schemes. But in the present work we investigate the pressure due to the combined surface and interface dynamics and they should therefore be solved in the most veracious way.

E. Experiments performed

Results from 20 numerical experiments are presented here. The experiments vary by strength of forcing, baroclinicity, horizontal viscosity and the Coriolis parameter (see table I).

As mentioned above, the time and space dependent (barotropic and baroclinic) Froude number is sub critical (F < 1) at all locations and times.

III. RESULTS

A. Description

All flows discussed here are almost time independent and close to geostrophy. A flow which is in geostrophic equilibrium is completely described by the pressure gradient created by the free surface and the interface. A strong (almost perfect) fore-aft symmetry is observed for both layers. The ridge leaves a strong imprint also in the dynamics of the upper layer, although the ridge is always submerged in the lower layer only, it modifies the interface between the two layers. In Fig. 2 we see that the path in the lower layer extends further along the ridge showing that the dynamics above the ridge is baroclinic.

There is a localized signal of weak Poincaré waves above the ridge. These Poincaré waves are modified by: (i) the variation in the layers thicknesses due to the topography and the variation of the interface; (ii) the vortical current in both layers. Above the topography the relative vorticity is negative and so the absolute vorticity is lower than the Coriolis parameter allowing for Poincaré waves with a frequency below the Coriolis parameter. Away from the ridge the waves are evanescent.

The dynamics is close to time independent. As the diffusive terms are small, potential vorticity, defined as q i = (ζ i + f )/h i is almost conserved along the flows in both layers so that:

u i • ∇q i = 0. ( 22
)
For small Rossby number (eq. ( 12)) (i.e. the relative vorticity is small compared to the Coriolis parameter) the flow turns around the topography in the lower layer. For larger Rossby numbers the relative vorticity becomes more important and is negative at the tip of the topography. To compensate the reduction of total vorticity (ζ i + f ) the layer thickness decreases as the flow crosses the ridge further above the topography. Since the flow performs a less pronounced bend around the topography this leads at the same time to a decrease of the relative vorticity. This is clearly visible in Fig. 2 where the path of the centre of the current (y = c(x)) around the ridge in the two layers is shown for different Rossby numbers.

The center of the current is determined by the constant potential vorticity q i (x, c(x)) = f /h 0 i in both layers. In Fig. 3 the y-coordinate of the center of the current above the ridge (c(0)) is given as a function of the Rossby number for both layers.

Since the variation in the topography influences the interface deviation the Froude number also plays a role for the path of the current. The effect of different Froude number on the path is found to be small for the lower layer. When comparing the ridge crossing for experiments with the same Rossby number and different Froude number (Fig. 3, right) only a small difference is seen (dotted and straight line almost superpose). The same does not apply to the current around the ridge in the upper layer (Fig. 3, left), for which the Froude number has a strong effect. The interface deviation in all experiments with a barotropic forcing shows almost the same spatial distribution, except that the amplitude changes.

Comparing the series of experiments e00X to e01X and exp025, we see that viscosity, varied within the range chosen, has to leading order no influence on the path of the current (•, and in Fig. 3 fall on almost the same line).

When the Coriolis parameter is increased, the path in the upper-layer approaches the path in the lower layer, which is a manifestation of Taylor-Proudman-Poincaré Theorem. This is clearly visible in Fig. 3 where the crossing of the ridge in the upper layer approaches the location of the crossing in the lower layer when the Burger number (= ( /F r) 2 ) decreases, and the flow becomes more barotropic.

B. Pressure Drag

We have seen in the previous subsection, that to leading order the velocity field does not depend on the value of the chosen viscosity. In a non frictional and stationary case a Bernoulli potential exists and the flow around the symmetrical ridge is completely symmetrical and the pressure drag therefore vanishes. This is d'Alembert's paradox [START_REF] Grimberg | Genesis of dAlemberts paradox and analytical elaboration of the drag problem[END_REF], [START_REF] Darrigol | Worlds of flow: A history of hydrodynamics from the Bernoullis to Prandtl[END_REF] applied to the flow around a symmetric ridge.

When considering the path of the flow around the ridge we can observe a slight asymmetry across the ridge, as depicted in Fig. 4, which is proportional to the viscosity. It is a measure of the frictional head-loss and is almost proportional to the viscosity value, the velocity (linear friction) and the path length. As we have seen in the previous section, a stronger flow leads to a shorter path length so that the pressure drag is, in the case considered here, almost independent of the Rossby number for values of > 0.1.

For our analysis we consider the quantities defined in eq. ( 19) divided by the surface area and the density in the upper layer (ρ 1 )(units m 2 s -2 ). The total pressure drag (D b ) and the pressure drag due to the interface (D b2 ) for the experiments e00X are given in Fig. 5, the difference between the two curves is the pressure drag due to the surface (D b1 ) . Results of e025 and e01X (not shown) perfectly superpose on these results when all are divided by the corresponding viscosity value, demonstrating that the pressure drag is a linear function of the friction parameter for the range of friction parameters considered here. Note, that the pressure drag due to the interface slightly decreases with increasing Rossby number, for Rossby numbers higher then = 0.12. For the experiments with baroclinic forcing (e100X) the total pressure drag is almost identical to the barotropic forcing when plotted as a function of the Rossby number in the lower layer (see Fig. 5. The pressure drag due to the interface is however five-fold higher but it is then compensated by a negative pressure drag due to the surface elevation.

IV. DISCUSSION AND CONCLUSION

We have studied a sub-critical flow around a ridge in a two layer configuration, with small dissipative processes having only a weak effect on the cross-stream structure of the flow along its path. The path of the current at the ridge in both layers is well described by the conservation of potential vorticity in both layers.

It is found that even though the dissipation is weak, it leads to a fore-aft asymmetry of the flow with respect to the ridge. This results in a pressure drag which depends only weakly (see Fig. 5) on the Rossby number for a substantial range of Rossby numbers ( > 0.1). This can be explained by the decreasing path-length of the fluid across the ridge, with increasing Rossby number, and is a consequence of the (to leading order) conservation of potential vorticity in the weak dissipative cases considered here. Since the total drag is the sum of the frictional drag and the pressure drag, the former depends linearly on the path-length and the fluid velocity, i.e. a longer-path and a larger velocity results in a larger frictional drag across the ridge.

A velocity independent pressure drag has so far not been noticed in the context of flowtopography interaction, probably since previous investigations focused on strait ridges. The behavior is remarkably robust: for a baroclinicly forced flow almost identical results are obtained for the total pressure drag. This and the fact that the findings are explained by the conservation of potential vorticity suggest that our results are more than a curiosity in geophysical fluid dynamics. Previous investigations [START_REF] Maxworthy | Topographic Effects in Rapidly-rotating Fluids: Flow Over a Transverse Ridge[END_REF] have focused on strait ridges, we show that the case of an inclined ridge gives rise to new behaviors at large scales. Our results based on the (almost) conservation of potential vorticity also suggest that the drag depends on the precise representation of the inclination of the ridge. Other inclinations and forms will not only change the friction parameters, but also the friction law.

Similar results with the drag being independent of the flow speed, are obtained for flows over flexible objects such as plants [START_REF] Barois | Flexible body with drag independent of the flow velocity[END_REF], low Reynolds number flow of DNA molecules [START_REF] Wirtz | Direct measurement of the transport properties of a single DNA molecule[END_REF] and for objects moving in granular media [START_REF] Albert | Slow drag in a granular medium[END_REF]. Here the reduction of the path-length across the ridge with increasing speed can be considered as an analogue to the increased bending of a flexible object, both reducing the influence of the obstruction in the flow and lessen the drag.

Although the configuration is highly idealized, analytic solutions are not straightforward to deduce, as the presence of very low-amplitude trapped-Poincaré-waves confirms.

Our results apply also to atmospheric flows around mountain ridges, if they are large enough so that the Rossby number is around unity or smaller. The width of our ridge (120km) is small compared to the grid-size of most numerical models used to simulate the ocean dynamics in climate models. Its important effect on the flow dynamics suggests therefore that its parameterization in those models is relevant.

[ 15 ]

 15 . Their results from numerical and laboratory experiments suggests that a change from blocked flow around the ridge to flow over the ridge can explain the strong anomaly in local warming. Low amplitude climatic changes in weather patterns can lead to flow transitions around topographic features resulting in strong variations in the local climate. They suggest that such mechanisms could also be at work in other mountainous areas as for example the Alps.

FIG. 1 .

 1 FIG. 1. Function b(x,y) from eq. (1) defining the bottom topography of the domain (m)

  1 /ρ 2 . The kinematic viscosity is given by ν. The boundary conditions at y = ±L y /2 are free-slip (u = ∂ y v = 0) for the velocity and there are no diffusive fluxes of layer thickness across the boundary. The boundary conditions at x = ±L x /2 are periodic. The Gent-McWilliams-parameter κ (Gent & McWilliams 1990, diffusion of layer thickness) is used in the same spirit as the viscosity parameter. Their role is to prevent the accumulation of energy/enstrophy at the smallest scales that are resolved numerically (see Frisch et al. 2008, for a detailed discussion of this bottleneck phenomena). The explicit value of the thickness-diffusion and the viscosity parameter depend on the resolution of the numerical model and the numerical scheme used.

  Parameters varied between numerical experiments with the values of l i present the forcing in the sponge layers, see eq. (8), λ = g/g = 500 and non-dimensional parameters (based on the values in the sponge layer) associated.

FIG. 2 .

 2 FIG. 2. Paths of the center of the current in the upper layer (left) and the lower layer (right) for e001 (-), e002 (• • •), e003 (---), e004 (• -•), e005 ( -• •-), e006 (• --•), e007 (--). Higher Rossby-number flows show a smaller deviation of the flow by the topography.

FIG. 3

 3 FIG. 3. y-coordinate of ridge crossing as a function of the Rossby number ( ) for the upper layer (left) and the lower layer (right), for different series of experiments: e00X (•), e01X ( ), e025 ( ), eX05 ( )

FIG. 4 .

 4 FIG. 4. Asymmetry in the y-component of path of the current across the ridge (∆Y (x) = Y (x) -Y (-x)) for e005 (-), e015(• • •) and e025 (---). When devided by the respective viscosity value (see Tab. I curves superpose.

FIG. 5 .

 5 FIG. 5. Pressure drag (divided by the surface area and the density in the upper layer ρ 1 , units m 2 s -2 integrated from x = -120km to x = 120km, see text) as a function of the Rossby number ( ). Total pressure drag (D b ) of exp00X (•) and exp100X ( ) and pressure drag due to the interface (D b2 ) of exp00X ( ) and exp100X ( ).

ACKNOWLEDGMENTS

We are grateful to the members of the RREX group for discussion. This paper is a contribution to the RREX project supported by IFREMER, CNRS and the French National Program LEFE coordinated by INSU and by a grant from Labex OSUG@2020 (Investissements d'avenir ANR10 LABX56).