We examine the critical merging distance between two equal-volume, equal-potentialvorticity 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 present work discusses the most commonly occurring shape of the coherent vortical structures in rapidly rotating stably stratified turbulence, under the quasigeostrophic 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.

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.

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.