L.G. Kurakin, I.V. Ostrovskaya, M.A. Sokolovskiy. On the stability of discrete tripole, quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid. Regular and Chaotic Dynamics, 2016, v. 21, N 3, pp. 291–334.
A two-layer quasigeostrophic model is considered in the
f-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case when the structure consists of a central vortex of arbitrary intensity Γ
and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters (R,Γ,α), where α is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered. The theory of stability of steady-state motions of dynamic systems with a continuous symmetry group G is applied. The two definitions of stability used in the study are Routh stability and G-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the G-stability is the stability of a three-parameter invariant set OG,
formed by the orbits of a continuous family of steady-state rotations
of a multipole. The problem of Routh stability is reduced to the problem
of stability of a family of equilibria of a Hamiltonian system. The
quadratic part of the Hamiltonian and the eigenvalues of the
linearization matrix are studied analytically.
The cases of zero total intensity of a tripole and a quadrupole are
studied separately. Also, the Routh stability of a Thomson vortex
triangle and square was proved at all possible values of problem
parameters. The results of theoretical analysis are sustained by
numerical calculations of vortex trajectories.