Abstract
An integrable model is proposed to analyze the two-dimensional asymmetrical interaction of two vortices, either co-rotating or counter-rotating, in the absence of viscosity. To this purpose two assumptions are made: one vortex is uniform and elliptical and the other one is a point vortex. It follows a system with three degrees of freedom, for which first two integrals of the motion are known: the excess energy and the second-order moment of the vorticity field. By considering the latter as a parameter, the two remaining degrees of freedom are combined into a complex variable z, hence the isolines of the excess energy may be analyzed in the z-plane, to study the motion of the system. In particular, the number of the extremal points of the excess energy field, which identify the stationary configurations of the system, is calculated in different regions of the parameter space. The excess energy field, associated to each of these regions, leads to the specification of the system dynamics for any possible initial condition. Depending on the values of the parameters and on the initial conditions, we find different types of motion, corresponding to periodic, merging (also for counter-rotating vortices) and diverging solutions. Diverging interactions lead to a kind of straining out of the patch and they are possible only for counter-rotating vortices, with the ratio between the circulation of the point vortex and the one of the patch equal to −1/2. Particular attention is given to the interactions leading to merging, where the analysis in terms of an elliptical patch under rotating strain provides an useful physical interpretation.