Several coherent structures encoutered in the oceans or in the atmosphere of giant planets have characteristic scales larger than the local deformation radius. The latter is an intrinsic scale that controls the dynamics of rotating and stratified flows. The deformation radius depends on the vertical structure of the flows and may vary from a few tens of kilometers in the ocean to a few thousand of kilometers in the Earth's atmosphere or in that of giant planets. Large-scale flows correspond to a wide variety of geophysical flows having their characteristic scale larger than the local deformation radius. In such cases, an asymmetry is often observed in the dynamics and in the morphology of cyclonic and anticyclonic structures. For instance, the famous long-lived eddies formed in the zonal circulation of outer planets such as the Great Red Spot or White Oval of Jupiter are anticyclones, whereas cyclones are generally smaller and more elongated. In the Earth's ocean, large-scale eddies such as submesoscale coherent vortex, meddies and swoddies are mostly anticyclonic. The predominance of anticyclones has been observed in simple models, such as rotating shallow water model in the f-plane. As far as shallow-water decaying turbulence is concerned the departure from quasi-geostrophy leads to a significant asymmetry for which anticyclonic vortices are more circular and robust than cyclones. Stability analysis of isolated vortices show that anticyclones are more stable than cyclones as soon as the scale of vortices become large than the deformation radius.
        However, this large-scale asymmetry was never studied for a Karman wake. In such a case, both global (the wake) and local (the vortices) structures could be affected. Hence, the objective of this study is to determine how a cyclone-anticyclone asymmetry could affect the dynamics of large-scale wakes.  
 

In a classical non rotating two-dimensionnal flow, the dynamics of the wake is controled by a single parameter:  the Reynolds number which represents the importance of viscous effects. For Reynolds number larger than a critical value, Re~48, the boundary layers formed on the obstacle detach and roll-up alternatively on each side of the cylinder leading to a classical von Karman street. The vortex street is characterised by the shedding frequency of the vortices, called the Strouhal number when non dimensionalised. The Strouhal number is an intrinsic parameter of the flow, it grows with the Reynolds number and tend ansymptotically to a constant value about 0.2 (Fig. 1).
 

    

 Fig. 1 Von Karman streets behind a cylinder in a non rotating 2D flow for Re=140, fluoresceine visualisation. Strouhal number as a function of the Reynolds number.

In order to characterise the dynamics of large-scale wakes, we use three different methods : numerical simulations, laboratory experiments and theoretical study for a wide parameter range varying from 2D regime, quasi-geostrophic regime (geostrophic flows with characteristic length scale comparable to the deformation radius) and frontal regime (large-scale, geostrophic flows).

   As far as numerical simulations are concerned, shallow-water equations have been integrated with a pseudo-spectral code. The obstalce was represented with a penalisation method. This method considers the whole domain, fluid and solid, as a same porous media whose porosity could vary in space and time. As the penalisation method has never been used with a free surface model, numerical simulations are compared with laboratory experiments.

    Laboratory experiments have been performed in a rotating tank with a two layers stratified fluid, the bottom layer being thick and dense and the upper one shallow and light. A cylinder is translated only in the upper layer.  Thanks to a small depth aspect ratio between the upper and the bottom layer, we can reasonnably neglect the dynamics in the bottom layer and consider the dynamics of one shallow layer  with a free surface.

             In a frontal regime, the structure of large-scale wakes is totally different from the classical von Karman streets observed generally in the wake of a cylinder. This particular structure of the wakes has been observed in both numerical simulations and laboratory experiments. When the surface deviation increases,
a strong cyclone-anticyclone asymmetry appears, anticyclones are circular whereas cyclones are stretched and deformed. For a specific range of parameters, only anticyclones are formed in the wake. Besides, vortices are not shedded from the obstacle like in the classical von Karman wake, but instead, two parallel shear layers separate from the cylinder, extend up to several diameters behind the cylinder and only anticyclonic vortices roll up far dwonstream in the wake. Moreover, the Strouhal number reaches values three times larger than the standard value reached for the same Reynolds number in a classical Karman street. The variation of the Strouhal number is governed by a single parameter, the relative interface deviation (Fig. 2).  The resulting flow pattern ressembles more that of a mixing layer than a von Karman street.

    

Fig. 2. Vorticity field  obtained by numerical simulations performed in a frontal regime ( cyclones are in red and anticyclones in blue). Evolution of the Strouhal number (numerical simulations and laboratory experiments) as a function of the relative surface deviation.

In order to determinate the mechanisms responsible for the cyclone-anticyclone asymmetry in the vortex street and the nature of the instability, we performed a local stability analysis of parallel experimental wakes.  In the frontal regime (i.e. large geopotential deviations) the linear stability induces a selective destabilization of anticyclonic vorticity region. Indeed, in this regime, the most unstable mode is always localized in the anticyclonic shear region. Full nonlinear simulations show that cyclonic vortices are stretched and strongly deformed, in a frontal regime, in comparison with the anticyclonic vortices which remain robust and more circular. We recover with a local analysis the vortex street pattern observed experimentally in the frontal regime with a comparable Strouhal number. This indicate a change in the nature of the wake instability when the geopotential fluctuations become large enough.  This change is confirmed by the spatio-temporal analysis of the asymmetric wake in a frontal regime. In this case, we found that the wake flow measured behind the cylinder is convectively unnstable. A small wavepacket perturbation propagates downstream in the flow but does not contaminate the entire flow as it is the case for the classical 2D von Karman wake. Hence, this shows a transition in the nature  of geophysical wake instability in rotating shallow water flows when the frontal regime is reached.

We have shown that the wake instability, in a frontal regime, favours the production of anticyclones. In order to determine whether horizontal shear instability is a general mechanism for the production of anticyclonic vortices, we extend the stability analysis to various flows : jets, localized shear and wakes. On the one hand, the frontal regime strongly stabilizes jet flows but no cyclone-anticyclone asymmetry occurs in the linear phase. On the other hand, the stability property of cyclonic and anticyclonic shear is different in a frontal regime. The anticyclonic shear is much more unstable than the cyclonic one and lead more rapidly to the formation of anticyclones. Yet, according to the nonlinear evolution of perturbed shear flows, both cyclonic and anticyclonic localised shear lead to the formation of axisymmetric vortices at the end. In the case of wake flows where cyclonic and anticyclonic shear are correlated, the most unstable mode is localised in the anticyclonic shear zone only. Therefore, anticyclones are formed first  and create a strain that stretches the cyclonic shear. When cyclones appear, they are deformed and the wake present a strong cyclone-anticyclone asymmetry. Therefore, barotropic instability of parallel shear flows, in a frontal regime, favours the production of strong circular anticyclones. This mechanism may be a cause of the predominance of large-scale anticyclones in the ocean.
          Moreover, the predominance and the robustess of anticyclones may have an important impact on mixing and transport of passive tracers such that phytoplancto or polluting agents shedded along the coasts.