Vortex merger in rotating stratified flows

This paper describes the interaction of symmetric vortices in a three-dimensional quasi-geostrophic fluid. The initial vortices are taken to be uniform-potential-vorticity ellipsoids, of height 2h and width 2R, and with centres at ( +/-d/2, 0, 0), embedded within a background flow having constant background rotational and buoyancy frequencies, f/2 and N respectively. This problem was previously studied by von Hardenburg et al. (2000), who determined the dimensionless critical merger distance d/R as a function of the height-to-width aspect ratio h/R (scaled by f/N). Their study, however, was limited to small to moderate values of h/R, as it was anticipated that merger at large h/R would reduce to that for two columnar two-dimensional vortices, i.e. d/R approximate to 3.31. Here, it is shown that no such two-dimensional limit exists; merger is found to occur at any aspect ratio, with d similar to h for h/R much greater than 1.

New results are also found for small to moderate values of h/R. In particular, our numerical simulations reveal that asymmetric merger is predominant, despite the initial conditions, if one includes a small amount of random noise. For small to moderate h/R, decreasing the initial separation distance d first results in a weak exchange of material, with one vortex growing at the expense of the other. As d decreases further, this exchange increases and leads to two dominant but strongly asymmetric vortices. Finally, for yet smaller d, rapid merger into a single dominant vortex occurs-in effect the initial vortices exchange nearly all of their material with one another in a nearly symmetrical fashion.