Motion of an elliptical vortex under rotating strain: conditions for asymmetric merging

Abstract

The analysis of the motion of a uniform vortex ( patch) of elliptical shape under a rotating strain field is employed to investigate the conditions leading to merging for two co-rotating elliptical patches of equal vorticity but different circulation. The motion of the patch having smaller circulation under the rotating strain field induced by the other vortex is analyzed, by considering both the intensity and the rotation rate of the strain constant. Under this assumption, we may adopt a first integral of motion which has been already used to discuss the different kinds of motion experienced by the elliptical patch. In the present paper, the dependence of the motion on the initial conditions and on the strain parameters is analyzed in further detail, to provide an overall picture of the elliptical patch dynamics. These results are employed in the analysis of the merging conditions. To this aim, the strain parameters, written in terms of the circulation of the larger vortex and of the distance between the two vortices, are kept frozen at their initial values. For a given circulation of the larger vortex and for a given vorticity and initial configuration of the smaller one, it is possible to find a particular distance between the vortices – transitional distance, d t – to which a significant change in the dynamics of the smaller elliptical patch is associated. It is always slightly lower than the critical distance and the difference between the two values decreases for vanishing size of the smaller vortex. The analysis of the first integral shows that, for an initial distance larger than d t, the motion of the vortex leads to small periodic variations of its second-order moment. On the contrary, when the distance is smaller than d t, the motion implies a large growth of its second-order moment, that is able to force the merging for frozen strain parameters.