Jean Noel Reinaud

The shape of vortices in quasi-geostrophic turbulence

The present work discusses the most commonly occurring shape of the coherent vortical structures in rapidly rotating stably stratified turbulence, under the quasi-geostrophic approximation. In decaying turbulence, these vortices - coherent regions of the materially-invariant potential vorticity-dominate the flow evolution, and indeed the flow evolution is governed by their interactions. An analysis of several exceptionally high-resolution simulations of quasi-geostrophic turbulence is performed. The results indicate that the population of vortices exhibits a mean height-to-width aspect ratio less than unity, in fact close to 0.8. This finding is justified here by a simple model, in which vortices are taken to be ellipsoids of uniform potential vorticity. The model focuses on steady ellipsoids within a uniform background strain flow. This background flow approximates the effects of surrounding vortices in a turbulent flow on a given vortex. It is argued that the vortices which are able to withstand the highest levels of strain are those most likely to be found in the actual turbulent flow. Our calculations confirm that the optimal height-to-width aspect ratio is close to 0.8 for a wide range of background straining flows.

The merger of vertically offset quasi-geostrophic vortices

We examine the critical merging distance between two equal-volume, equal-potential-vorticity quasi-geostrophic vortices. We focus on how this distance depends on the vertical offset between the two vortices, each having a unit mean height-to-width aspect ratio. The vertical direction is special in the quasi-geostrophic model (used to capture the leading-order dynamical features of stably stratified and rapidly rotating geophysical flows) since vertical advection is absent. Nevertheless vortex merger may still occur by horizontal advection. In this paper, we first investigate the equilibrium states for the two vortices as a function of their vertical and horizontal separation. We examine their basic properties together with their linear stability. These findings are next compared to numerical simulations of the nonlinear evolution of two spheres of potential vorticity. Three different regimes of interaction are identified, depending on the vertical offset. For a small offset, the interaction differs little from the case when the two vortices are horizontally aligned. On the other hand, when the vertical offset is comparable to the mean vortex radius, strong interaction occurs for greater horizontal gaps than in the horizontally aligned case, and therefore at significantly greater full separation distances. This perhaps surprising result is consistent with the linear stability analysis and appears to be a consequence of the anisotropy of the quasi-geostrophic equations. Finally, for large vertical offsets, vortex merger results in the formation of a metastable tilted dumbbell vortex.

The critical merger distance between two co-rotating quasi-geostrophic vortices

This paper examines the critical merger or strong interaction distance between two equal-potential-vorticity quasi-geostrophic vortices. The interaction between the two vortices depends on five parameters: their volume ratio, their height-to-width aspect ratios and their vertical and horizontal offsets. Due to the size of the parameter space, a direct investigation solving the full quasi-geostrophic equations is impossible. We instead determine the critical merger distance approximately using an asymptotic approach. We associate the merger distance with the margin of stability for a family of equilibrium states having prescribed aspect and volume ratios, and vertical offset. The equilibrium states are obtained using an asymptotic solution method which models vortices by ellipsoids. The margin itself is determined by a linear stability analysis. We focus on the interaction between oblate to moderately prolate vortices, the shapes most commonly found in turbulence. Here, a new unexpected instability is found and discussed for prolate vortices which is manifested by the tilting of vortices toward each other. It implies than tall vortices may merge starting from greater separation distances than previously thought.

The quasi-geostrophic ellipsoidal vortex model

We present a simple approximate model for studying general aspects of vortex interactions in a rotating stably-stratified fluid. The model idealizes vortices by ellipsoidal volumes of uniform potential vorticity, a materially conserved quantity in an inviscid, adiabatic fluid. Each vortex thus possesses 9 degrees of freedom, 3 for the centroid and 6 for the shape and orientation. Here, we develop equations for the time evolution of these quantities for a general system of interacting vortices. An isolated ellipsoidal vortex is well known to remain ellipsoidal in a fluid with constant background rotation and uniform stratification, as considered here. However, the interaction between any two ellipsoids in general induces weak non-ellipsoidal perturbations. We develop a unique projection method, which follows directly from the Hamiltonian structure of the system, that effectively retains just the part of the interaction which preserves ellipsoidal shapes. This method does not use a moment expansion, e.g. local expansions of the flow in a Taylor series. It is in fact more general, and consequently more accurate. Comparisons of the new model with the full equations of motion prove remarkably close.

On the stability of continuously stratified quasi-geostrophic hetons

In this paper we examine the stability of quasi-geostrophic hetons in a stably, continuously stratified fluid. To this purpose we first determinate numerically equilibrium states. Equilibrium hetons consist of two vortices of equal and opposite strength lying at different depths that are steadily translating without deforming. The situation is studied through a parameter space comprising of the vertical offset between the vortices, their horizontal separation distance and their aspect ratio. The study first shows that the equilibrium vortices are not only strongly deformed in the vertical but that their instability modes are also varying within the height of the structures. The main purpose of the present contribution is to study families of equilibria which stem from the case of two vertically aligned cylindrical vortices. It is however shown that other branches of solutions exist with different properties. The paper concludes that hetons may be sensitive to baroclinic instabilities provided the separation distance between the poles of the hetons is moderate both in the horizontal and in the vertical directions. The hetons become stable and efficient ways to transport properties as far as the poles are distant from one another. The critical separation distance in a non-trivial function of the radius-to-height aspect ratio of the poles.

Geostrophic tripolar vortices in a two-layer fluid: Linear stability and nonlinear evolution of equilibria

We investigate equilibrium solutions for tripolar vortices in a two-layer quasi-geostrophic flow. Two of the vortices are like-signed and lie in one layer. An opposite-signed vortex lies in the other layer. The families of equilibria can be spanned by the distance (called separation) between the two like-signed vortices. Two equilibrium configurations are possible when the opposite-signed vortex lies between the two other vortices. In the first configuration (called ordinary roundabout), the opposite signed vortex is equidistant to the two other vortices. In the second configuration (eccentric roundabouts), the distances are unequal. We determine the equilibria numerically and describe their characteristics for various internal deformation radii. The two branches of equilibria can co-exist and intersect for small deformation radii. Then, the eccentric roundabouts are stable while unstable ordinary roundabouts can be found. Indeed, ordinary roundabouts exist at smaller separations than eccentric roundabout...

Vortex merger near a topographic slope in a homogeneous rotating fluid

The effect of a bottom slope on the merger of two identical Rankine vortices is investigated in a two-dimensional, quasi-geostrophic, incompressible fluid.When two cyclones initially lie parallel to the slope, and more than two vortex diameters away from the slope, the critical merger distance is unchanged. When the cyclones are closer to the slope, they can merge at larger distances, but they lose more mass into filaments, thus weakening the efficiency of merger. Several effects account for this: the topographic Rossby wave advects the cyclones, reduces their mutual distance and deforms them. This alongshelf wave breaks into filaments and into secondary vortices which shear out the initial cyclones. The global motion of fluid towards the shallow domain and the erosion of the two cyclones are confirmed by the evolution of particles seeded both in the cyclones and near the topographic slope. The addition of tracer to the flow indicates that diffusion is ballistic at early times.For two anticyclones, merger is also facilitated because one vortex is ejected offshore towards the other, via coupling with a topographic cyclone. Again two anticyclones can merge at large distance but they are eroded in the process.Finally, for taller topographies, the critical merger distance is again increased and the topographic influence can scatter or completely erode one of the two initial cyclones.Conclusions are drawn on possible improvements of the model configuration for an application to the ocean.

Interaction between a surface quasi-geostrophic buoyancy anomaly jet and internal vortices

This paper addresses the dynamical coupling of the ocean’s surface and the ocean’s interior. In particular, we investigate the dynamics of an oceanic surface jet and its interaction with vortices at depth. The jet is induced by buoyancy (density) anomalies at the surface. We first focus on the jet alone. The linear stability indicates there are two modes of instability: the sinuous and the varicose modes. When a vortex in present below the jet, it interacts with it. The velocity field induced by the vortex perturbs the jet and triggers its destabilisation. The jet also influences the vortex by pushing it under a region of co-operative shear. Strong jets may also partially shear out the vortex. We also investigate the interaction between a surface jet and a vortex dipole in the interior. Again, strong jets may partially shear out the vortex structure. The jet also modifies the trajectory of the dipole. Dipoles travelling towards the jet at shallow incidence angles may be reflected by the jet. Vortices travelling at moderate incidence angles normally cross below the jet. This is related to the displacement of the two vortices of the dipole by the shear induced by the jet. Intense jets may also destabilise early and form streets of billows. These billows can pair with the vortices and separate the dipole.This paper addresses the dynamical coupling of the ocean’s surface and the ocean’s interior. In particular, we investigate the dynamics of an oceanic surface jet and its interaction with vortices at depth. The jet is induced by buoyancy (density) anomalies at the surface. We first focus on the jet alone. The linear stability indicates there are two modes of instability: the sinuous and the varicose modes. When a vortex in present below the jet, it interacts with it. The velocity field induced by the vortex perturbs the jet and triggers its destabilisation. The jet also influences the vortex by pushing it under a region of co-operative shear. Strong jets may also partially shear out the vortex. We also investigate the interaction between a surface jet and a vortex dipole in the interior. Again, strong jets may partially shear out the vortex structure. The jet also modifies the trajectory of the dipole. Dipoles travelling towards the jet at shallow incidence angles may be reflected by the jet. Vortices trav...

Hetonic quartets in a two-layer quasi-geostrophic flow: V-states and stability

We investigate families of finite core vortex quartets in mutual equilibrium in a two-layer quasi-geostrophic flow. The finite core solutions stem from known solutions for discrete (singular) vortex quartets. Two vortices lie in the top layer and two vortices lie in the bottom layer. Two vortices have a positive potential vorticity anomaly, while the two others have negative potential vorticity anomaly. The vortex configurations are therefore related to the baroclinic dipoles known in the literature as hetons. Two main branches of solutions exist depending on the arrangement of the vortices: the translating zigzag-shaped hetonic quartets and the rotating zigzag-shaped hetonic quartets. By addressing their linear stability, we show that while the rotating quartets can be unstable over a large range of the parameter space, most translating quartets are stable. This has implications on the longevity of such vortex equilibria in the oceans.

The interaction of two co-rotating quasi-geostrophic vortices in the vicinity of a surface buoyancy filament

Abstract In this paper, we investigate the interaction between two like-signed quasi-geostrophic uniform potential vorticity internal vortices in the vicinity of a surface buoyancy anomaly filament in a three dimensional, stably stratified and rapidly rotating fluid. The surface buoyancy distribution locally modifies the pressure fields and generates a shear flow. We start the study by first considering the effects of a uniform linear horizontal shear on the binary vortex interaction. We confirm that a cooperative shear facilitates the merger of a pair of vortices while an adverse shear has the opposite effect. We next investigate the binary vortex interaction in the vicinity of the surface buoyancy filament explicitly. Here, not only the filament generates a shear flow, but it also responds dynamically to the forcing by the vortex pair. The filament destabilises and forms buoyancy billows at the surface. These billows interact with the internal vortices. In particular, a surface billow may pair with one of the internal vortices. In such cases, the like-signed internal vortex pair may separate if they are initially moderately distant from each other.