Irina Ostrovskaya

A system of four point vortices on a plane is considered. Its motion is described by the Kirchhoff equations. Three vortices have unit intensity and one vortex has arbitrary intensity $\varkappa$. We study the stability problem for the stationary rotation of a vortex quadrupole consisting of three identical vortices located uniformly on a circle around a fourth vortex. It is known that for $ \varkappa> 1 $ the regime under study is unstable, and in the case of $ \varkappa <-3 $ and $ 0 <\varkappa <1 $ the orbital stability takes place. New results are obtained for $ -3 <\varkappa <0 $. It is found that, for all values of $ \varkappa $ in the stability problem, there is a resonance $1:1$ (diagonalizable case). Some other resonances through order four are found and investigated: double zero resonance (diagonalizable case), resonances 1:2 and 1:3, occurring with isolated values of $\varkappa $. The stability of the equilibrium of the system reduced by one degree of freedom with the involvement of the terms in the Hamiltonian through degree four is proved for all $ \varkappa \in (-3,0) $.

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 $\Gamma$ 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 $(R, \Gamma, \alpha)$, where $\alpha$ 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 $\mathcal{G}$ is applied. The two definitions of stability used in the study are Routh stability and $\mathcal{G}$-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the $\mathcal{G}$-stability is the stability of a three-parameter invariant set $O_\mathcal{G}$, 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.

A nonlinear stability analysis of the stationary rotation of a system of five identical point vortices lying uniformly on a circle of radius $R_0$ outside a circular domain of radius $R$ is performed. The problem is reduced to the problem of stability of an equilibrium position of a Hamiltonian system with a cyclic variable. The stability of stationary motion is interpreted as Routh stability. Conditions for stability, formal stability and instability are obtained depending on the values of the parameter $q = R^2/R_0^2$.