Fractal Kelvin-Helmholtz breakups

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

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.Publisher PDFPeer reviewe

The baroclinic forcing of the shear-layer three-dimensional instability

It has been demonstrated that, within the context of variable-density shear flows, the generation-destruction of vorticity by the baroclinic torque may substantially alter the transition dynamics of shear flows. The focus of the present contribution is on baroclinic effects beyond the Boussinesq approximation but uncorrelated to compressibility. The baroclinic torque results from the inertial component of the pressure gradient only. The vorticity evolves within a quasi-solenoidal velocity field without suffering from strong dilatationnal effects that scale with any relevant Mach number. This purely inertial influence of density variations is likely to occur in high Reynolds number mixing of fluids of different densities or in thermal mixing. The vorticity is redistributed to the benefits of the light-side vorticity braid, the other being vorticity depleted in a first stage and feeded with an opposite sign vorticity afterwards, as stressed by Reinaud et al. (1999). These two opposite-sign vorticity sheets are lying around the vanishing primary structure core, still figuring the center of this two-layers system. In three-dimensions the vorticity dynamics is also affected by the vortex stretching mechanism that enable circulation to travel among vorticity components through 3D instability modes. The consequences of the baroclinic redistribution of spanwise vorticity on the development of three-dimensionnal modes is the focus point of the present proposition. The interference with the pairing process and further subharmonics emergence is not yet considered

The shape of vortices in quasi-geostrophic turbulence

Partially supported by the UK EPSRC (Grant GR/N11711)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.Publisher PDFPeer reviewe

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.Publisher PDFPeer reviewe

Piecewise uniform potential vorticity pancake shielded vortices

Shielded vortices consist of a core of potential vorticity of a given sign surrounded (or shielded) by a layer of opposite-signed potential vorticity. Such vortices have specific properties and have been the focus of numerous studies, first in two dimensional geometries (where potential vorticity is just the vertical component of the vorticity vector) and in geophysical applications (mostly in layered models). The present paper focuses on three-dimensional, spheroidal shielded vortices. In particular, we focus on vortical structures whose overall volume-integrated potential vorticity is zero. We restrict attention to vortices of piecewise uniform potential vorticity in the present research. We first revisit the problem within the quasi-geostrophic model, then we extend the results to the non-hydrostatic regime. We show that the stability of the structure depends on the ratio of potential vorticity between the inner core and the outer shield. In particular it depends on the polarity of the core and of the wavenumber of the azimuthal mode perturbed.PostprintPeer reviewe

Self-similar collapse of three geophysical vortices

The self-similar collapse of three vortices is the motion of three vortices colliding at a single point at finite time. Such a motion has first been shown to exist for two-dimensional, planar, point vortices. In this paper, we show that the concept generalises naturally to three-dimensional quasi-geostrophic vortices as well as to surface quasi-geostrophic vortices. We first determine the conditions that lead to the collapse for these singular vortices. We then show how these conditions precipitate the merger of finite core vortices both in a three-dimensional quasi-geostrophic flow and in a surface quasi-geostrophic flow.PostprintPeer reviewe

The merger of two-dimensional radially stratified high-Froude-number vortices

We investigate the influence of density inhomogeneities on the merger of two corotating two-dimensional vortices at infinite Froude number. In this situation, buoyancy effects are negligible, yet density variations still affect the flow by pure inertial effects through the baroclinic torque. We first re-address the effects of a finite Reynolds number on the interaction between two identical Gaussian vortices. Then, by means of direct numerical simulations, we show that vortices transporting light fluid in a heavier counterpart merge from further distances than vortices in a uniform density medium. On the other hand, heavy vortices only merge from small separation distances. We measure the critical distance a/b0 of the vortex radii to their initial separation distance. It departs from the homogeneous threshold of 0.22 in response to increasing density contrasts between the vortices and their surroundings. An analysis of the contribution of the baroclinic vorticity to the dynamics of the flow is detailed and explains the observed behaviour. This analysis is completed by a simple model based on point vortices that mimics the flow. It is concluded that vortices carrying light fluid are more likely to generate large-scale structures than heavy ones in an inhomogeneous fluid

Finite Froude and Rossby numbers counter-rotating vortex pairs

We investigate the nonlinear evolution of pairs of three-dimensional, equal-sized and opposite-signed vortices at finite Froude and Rossby number. The two vortices may be offset in the vertical direction. The initial conditions stem from numerically obtained relative equilibria in the quasi-geostrophic regime, for vanishing Froude and Rossby number. We first address the linear stability of the quasi-geostrophic opposite-signed pairs of vortices and show that for all vertical offsets, the vortices are sensitive to an instability when close enough together. In the nonlinear regime, the instability may lead to the partial destruction of the vortices. We then address the nonlinear interaction of the vortices for various values of the Rossby number. We show that as the Rossby number increases, destructive interactions, where the vortices break into pieces, may occur for a larger separation between the vortices, compared to the quasi-geostrophic case. We also show that, for well-separated vortices, the interaction is non-destructive and ageostrophic effects lead to the deviation of the trajectory of the pair of vortices, as the anticyclonic vortex dominates the interaction. Finally, we show that the flow remains remarkably close to a balanced state, only emitting waves containing negligible energy, even when the interaction leads to the destruction of the vortices.Publisher PDFPeer reviewe

Three-dimensional quasi-geostrophic staggered vortex arrays

We determine and characterise relative equilibria for arrays of point vortices in a three-dimensional quasi-geostrophic flow. The vortices are equally spaced along two horizontal rings whose centre lies on the same vertical axis. An additional vortex may be placed along this vertical axis. Depending on the parameters defining the array, the vortices on the two rings are of equal or opposite sign. We address the linear stability of the point vortex arrays. We find both stable equilibria and unstable equilibria, depending on the geometry of the array. For unstable arrays, the instability may lead to the quasi-regular or to the chaotic motion of the point vortices. The linear stability of the vortex arrays depends on the number of vortices in the array, on the radius ratio between the two rings, on the vertical offset between the rings and on the vertical offset between the rings and the central vortex, when the latter is present. In this case the linear stability also depends on the strength of the central vortex. The nonlinear evolution of a selection of unstable cases is presented exhibiting examples of quasi-regular motion and of chaotic motion.PostprintPeer reviewe