The efficiency of symmetric vortex merger

Abstract

The coalescence of two identical vortices with uniform vorticity is investigated using the numerical method of ‘‘contour surgery,’’ for two-dimensional inviscid, incompressible vortex dynamics (2VD), and quasigeostrophic shallow-water dynamics (QGSW), to quantify the differences and similarities between the two models. High-resolution calculations show that the evolution of the vortices may fall into three different regimes depending on the initial intercentroid separation. The efficiency of the merger of two vortices into a single elliptical-like vortex is determined by calculating the loss of globally conserved quantities to filaments and small-scale structures. In the 2VD model, it is found that the resultant central vortex always has at least 56% more circulation than either of the initial vortices, showing that the merger process indeed forms larger scales. From energetics, the initial separation for an inviscid transition from two elliptical vortices to a single elliptical vortex is determined, and is shown to be close to the critical merger separation obtained from full numerical calculations. Confirming the results of Polvani et al. [J. Fluid Mech. 205, 215 (1989)], the calculations in the QGSW model show that as the radius of deformation LR decreases, the critical merger separations decrease, the resultant vortex is nonelliptical, fewer filaments are shed, and those filaments shed show an increased propensity to roll up into small vortices (consistent with previous analysis of the stability of strips of potential vorticity with imposed shear). Furthermore, the efficiency of the merger process increases as LR decreases (to perfect efficiency in the limit of small LR).