A physical model for merging in two-dimensional decaying turbulence

Abstract

Some aspects of the simulation by point vortices of two-dimensional decaying turbulence are discussed, together with their most relevant consequences for the decay of the number of vortices. Two different merging models are proposed to simulate dissipative interactions between corotating vorticity structures within a point vortex simulation. Both models are based on a statistical approach to assign the number of vortices resulting from the merging of two vortices of different size. Probability distributions are defined for the production of one, two, or three vortices, and they are assigned as functions of the circulation ratio for the two interacting vortices. These functions are determined by performing a large number of Contour Dynamics simulations of vortex interactions, each under a given strain field. A simple rule to reset the vorticity field, at the end of the merging process, is discussed in terms of conserved quantities for the formation of one or three vortices. To complete the merging model, a criterion that indicates the onset of a merging event between two close vortices of like sign is required. The first model is defined by adopting a critical distance as merging criterion. In this way, the effect of the strain on the merging process is taken into account only in a statistical manner, i.e., by the probability law for the merging products. The use of this merging model into a point vortex simulation gives a surprisingly good agreement with the results for the vortex number decay obtained by Dritschel via Contour Surgery. Nevertheless, a merging criterion based on the critical distance appears questionable, since it disregards the effect of the strain on the merging conditions. To this aim, we introduce a more sophisticated merging model that uses the dynamics of elliptical patches under a given strain field to select the merging events. This second model accounts also directly, and not only in a statistical sense, for the strain influence over the merging conditions, revising thoroughly the critical distance concept. The results show that the vortex number decay is not strongly sensitive to a detailed description of each individual merging process and, if the interest is focused on the vortex number decay, further improvements of the merging model are not strictly required.