Jean Reinaud

The baroclinic secondary instability of the two-dimensional shear layer

The focus of this study is on the numerical investigation of two-dimensional, isovolume, high Reynolds and Froude numbers, variable-density mixing layers. Lagrangian simulations, of both the temporal and the spatial models, are performed. They reveal the breaking-up of the strained vorticity and density-gradient braids, connecting two neighboring primary structures. The secondary instability arises where the vorticity has been intensified by the baroclinic torque. A simplified model of the braid of the variable-density mixing layer, consisting of a strained vorticity and density-gradient filament, is analyzed. It is concluded that the physical mechanism responsible for the secondary instability is the forcing of the vorticity field by the baroclinic torque, itself sensitive to perturbations. This mechanism suggests a rapid route to turbulence for the variable-density mixing layer.

The stability and the nonlinear evolution of quasi-geostrophic hetons

We analyse the linear stability and nonlinear evolutions of circular hetons under the quasi-geostrophic approximation. We compare results obtained with a three-layer model and with a model based on a continuous density stratification. Though the models also differ by the vertical boundary conditions, they show a remarkable similarity in the stability properties of the hetons (threshold values of vortex radius for baroclinic instability, dominant modes, growth rates, etc.), and in their nonlinear evolutions (spatial reorganization of potential vorticity by nonlinear processes, end-states of the simulations). The hetons prone to baroclinic instability often break into two hetons drifting in opposite directions, and in more hetons, for wider initial structures. In both models, instability is quite sensitive to the vertical gap between the opposite-signed vortices: as it increases, the instability decreases and shifts to lower azimuthal modes. Finally, though modes l ≥ 2 (i.e. elliptical and shorter wave deformations) prevail in most of the parameter space, the mode l = 1 perturbation (a vertical tilt of the vortex column) exists for hetons with small vertical gaps. Such perturbations are concentrated vertically near the gap, and can only be evidenced in the continuously stratified model.

Strong interactions between two corotating quasi-geostrophic vortices

In this paper we investigate the interaction between two corotating quasi-geostrophic vortices. The initially ellipsoidal vortices are separated horizontally by a distance corresponding to the margin of stability, as determined from an ellipsoidal analysis. The subsequent interaction depends on four parameters: the vortex volume ratio, the vertical centroid separation, and the height-to-width aspect ratios of each vortex. The most commonly observed strong interaction is partial merger, where only part of the weaker vortex is incorporated into the stronger one or cast into filamentary debris. Despite the proliferation of small-scale filamentary structure during many vortex interactions, on average the self-induced vortex energy exhibits an ‘inverse cascade’ to larger scales, broadly consistent with spectral theories of turbulence. Curiously, we observe that a range of intermediate-scale vortices are preferentially sheared out during the interactions, leaving two main populations of large and small vortices.

Interaction between two quasi-geostrophic vortices of unequal potential vorticity

In this paper we systematically investigate strong interactions between two like-signed quasi-geostrophic vortices containing different uniform potential vorticity. The interaction depends on six parameters: the potential vorticity ratio between the two vortices, their volume ratio, their individual height-to-width aspect ratio, their vertical offset, and their horizontal separation distance. We first determine the conditions under which a strong interaction may occur. To that end, we calculate equilibrium states using an asymptotic approach which models the vortices as ellipsoids and we additionally assess their linear stability. It is found that vortices having similar potential vorticity interact strongly (e.g. merge) at closer separation distances than do vortices with a dissimilar potential vorticity. This implies that interactions between vortices having significantly different potential vorticity may be more destructive, for a given separation distance. This is confirmed by investigating the nonlinear evolution of the vortices over a subset of the full parameter space, solving the full dynamical quasi-geostrophic equations. Many forms of interaction occur, but merger or partial merger (where the largest vortex grows in volume) is mostly observed for interactions between vortices of similar potential vorticity.

The stability of quasi-geostrophic ellipsoidal vortices

A vertically standing freely-rotating ellipsoidal vortex of uniform anomalous potential vorticity in a rotating stratified fluid under quasi-geostrophic conditions of small Rossby and Froude numbers steadily rotates without change of form. The vortex can have arbitrary axis lengths, but must have one axis parallel to the vertical

Fractal Kelvin–Helmholtz breakups

z

Homostrophic Vortex Interaction under External Strain, in a Coupled QG-SQG Model

-axis along the direction of gravity. The rotation rate is proportional to the potential vorticity anomaly but otherwise depends on only two independent aspect ratios characterizing the shape of the vortex. The linear stability of this class of vortex equilibria was first determined semi-analytically more than a decade ago. It was found that vortices are unstable over a wide range of the parameter space and are stable only when strongly oblate and of nearly circular cross-section. New results, presented here, using a complementary approach and backed by nonlinear simulations of the full quasi-geostrophic equations indicate that these ellipsoidal vortices are in fact stable over a much wider range of parameter space. In particular, a mode previously thought to be unstable over much of the parameter space is evidently stable. Moreover, we have determined that this mode is just the difference between two neighbouring equilibrium states having slightly different horizontal aspect ratios; hence, this mode must be neutrally stable. Agreement is found for all other modes. However, by an independent analysis considering only ellipsoidal (though time-varying) disturbances, we have identified one unstable mode as purely ellipsoidal, i.e. it does not change the form of the ellipsoid, only its shape. Under this instability, the vortex quasi-periodically tilts over while undergoing mild changes in shape. The range of parameters leading to non-ellipsoidal instabilities turns out to be narrow, with instability principally occurring for highly eccentric (horizontally squashed, prolate) vortices. The long-term fate of these instabilities is examined by nonlinear contour-dynamical simulations. These reveal a wealth of complex phenomena such as the production of a sea of small-scale vortices, yet, remarkably, the dominant vortex often tends to relax to a stable rotating ellipsoid.

Destructive interactions between two counter-rotating quasi-geostrophic vortices

The Kelvin–Helmholtz billow developing in an infinite- Schmidt number mixing layer at Re=1500 between two density-contrasted fluids experiences a two-dimensional shear instability. Secondary Kelvin–Helmholtz billows are seen to emerge on the light side of the primary structure, and then are advected towards the core of the main billow as the wave overturns. Due to the inertial baroclinic vorticity production, the braid region turns into a sharp vorticity ridge holding high shear levels and is thus sensitized to the Kelvin–Helmholtz instability. We carry out numerical simulations of the temporal development of the secondary mode when the flow is seeded at t=18 with the perturbation obtained from a linear stability analysis of the primary billow.

Interaction between a surface quasi-geostrophic buoyancy filament and an internal vortex

The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic — Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius R at depth −2H interacting with a disk of uniform density anomaly, with radius R, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.

Vortex merger in surface quasi-geostrophy

This paper illustrates the linear stability and the nonlinear evolution of two opposite-signed quasi-geostrophic vortices. We investigate the influence of the volume ratio between the two vortices as well as the influence of their vertical offset. Instability is always found for sufficiently close vortices. A convenient measure of the separation distance between the two vortices at their margin of stability is the horizontal gap between their two outermost edges. When the vortex volume ratio is very close to unity, the critical gap at the margin of stability tends to increase with the vertical offset. However, for volume ratios greater than 1.1, it decreases with the vertical offset. This is due to differences in the magnitude of the tilt angle of the vortices. The nonlinear evolution of unstable equilibria tends to be destructive, often breaking one vortex or both vortices into smaller vortices.