The interaction of an elliptical patch with a point vortex

An integrable model is proposed to analyze the two-dimensional asymmetrical interaction of two vortices, either co-rotating or counter-rotating, in th e absence of viscosity. To this purpose two assumptions are made: one vorte x 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 t he motion are known: the excess energy and the second-order moment of the v orticity field. By considering the latter as a parameter, the two remaining degrees of freedom are combined into a complex variable z, hence the isoli nes of the excess energy may be analyzed in the z-plane, to study the motio n of the system. In particular, the number of the extremal points of the ex cess energy field, which identify the stationary configurations of the syst em, is calculated in different regions of the parameter space. The excess e nergy field, associated to each of these regions, leads to the specificatio n of the system dynamics for any possible initial condition. Depending on t he values of the parameters and on the initial conditions, we find differen t types of motion, corresponding to periodic, merging (also for counter-rot ating vortices) and diverging solutions. Diverging interactions lead to a k ind of straining out of the patch and they are possible only for counter-ro tating 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 t he interactions leading to merging, where the analysis in terms of an ellip tical patch under rotating strain provides an useful physical interpretatio n. (C) 2000 The Japan Society of Fluid Mechanics and Elsevier Science B.V. All rights reserved.