Stable elliptical vortices in a circular disk.
We show that large elliptical vortices in a finite disk are stable in a two-dimensional (2D) ideal fluid (this also applies to a column of quasi-2D non-neutral plasma in an axial magnetic field). The stability is established by comparison between the energy of elliptical and symmetrical states to satisfy a sufficient condition, without dynamical eigenanalysis. Analytical small ellipticity expansion of system energy and exact numerical values for arbitrary ellipticity are both obtained for uniform vortices. An approximating calculation is presented for general smooth vortices. Numerical simulations of the 2D Euler equation are also performed. The simulations not only confirm the sufficient condition, but also show that the stability persists to smaller vortex sizes. The reason why decaying l=2 modes were obtained by Briggs, Daugherty, and Levy [Phys. Fluids 13, 421 (1970)] using eigenanalysis is also discussed.
Chen
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Department of Physics and Center for Complex Systems, National Central University, Chungli 320, Taiwan. peilong@toast.phy.ncu.edu.tw