Two-dimensional asymmetric vortex merger: merger dynamics and critical merger distance

Two-dimensional asymmetric merger of two like-signed vorticity monopoles with different sizes and vorticities is examined by combining simplified analytical models and contour dynamics experiments. The model results can capture the key dynamics and hence allow the prediction of the critical merger distance in a number of the situations. The models ignore deformation of one of the two vortices, replacing it with a point vortex, and employ a corotating frame of reference with a rotation rate estimated by point vortices. Thus, the two vortex problem becomes two separate problems of a single vortex in a background shear flow. Vortex merger is found to happen when the vortex cannot resist the background shear flow. Vortex merger and merging processes depend on the centroid distance d, the circulation ratio, (q_i and r_i are the vorticity and radius, respectively) and initial conditions. In the lowest order, the background flow is approximated by a uniform shear field, and the behavior of an elliptical vortex can be described by the Kida (1981) equation supplemented with one describing the time evolution of the centroid distance. This model reveals that merger takes place because the natural rotation of an elliptical vortex is overcome by the background uniform shear flow; the ellipse inversely rotates and is drawn out by the background straining field. The vortex deformation in a background flow field induces an inward flow at the position of the other vortex; as a result, the centroid distance decreases and two vortices merge. The critical merger distance from this model agrees quite well with the results from contour dynamics experiments for two vortices. Inclusion of higher order non-uniform shear in the background flow extends the critical merger distance, which gives almost perfect estimates for the experiment. In the non-uniform shear flow, partial merger occurs, where the vortex sheds off a filament, but the remaining part of the vortex resumes its natural rotation.