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.
2018J. N. Reinaud, M.A. Sokolovskiy, X. Carton. Hetonic quartets in a two-layer quasi-geostrophic flow: V-states and stability. Physics of Fluids, 2018, v. 30, N 5, 056602 (8 pp).
2018M.A. Sokolovskiy, J. Verron, X.J. Carton. The formation of new quasi-stationary vortex patterns from the interaction of two identical vortices in a rotating fluid. Ocean Dynamics, 2018, v. 68, N 6, pp. 723-733.