## Exact solutions for rotating vortex arrays with finite-area cores.(English)Zbl 1019.76011

Summary: We construct a class of explicit solutions of two-dimensional Euler equations consisting of a finite-area patch of uniform vorticity surrounded by a finite distribution of co-rotating satellite line vortices. The results generalize the classic study of co-rotating vortex arrays by J. J. Thomson. For $$N$$ satellite line vortices $$(N\geq 3)$$, a continuous one-parameter family of rotating vortical equilibria is derived in which different values of continuous parameter correspond to different shapes and areas of the central patch. In an appropriate limit, vortex patch equilibria with cusped boundaries are found. A study of the linear stability is performed, and a wide range of the solutions is found to be linearly stable. Contour dynamics methods are used to calculate the typical nonlinear evolution of configurations. The results are believed to provide the only known exact solutions involving rotating vortex patches besides the classical Kirchhoff ellipse.

### MSC:

 76B47 Vortex flows for incompressible inviscid fluids 76U05 General theory of rotating fluids 76E99 Hydrodynamic stability
Full Text: