On the stability of two-layer geostrophic point-vortex multipoles

A point-vortex multipole is an ensemble of m+1 vortices (m = 2, 3,...) possessing a m-fold symmetry, with the "core vortex" being located at the centre and m identical "satellite vortices" located at the vertices of an equilateral m-sided polygon (at m > 2) or at the ends of a straight-line segment (at m = 2). At m = 2, m = 3, and m = 4, the multipole is commonly termed a tripole, a quadrupole, and a pentapole, respectively, and the distance from the core vortex to the satellite vortices, the multipole leg. A multipole is said to be stable if, in response to sufficiently small initial perturbations in the distances between the vortices, the variations in the distances remain small for all times. The main issue of this article is an analytical study of the nonlinear stability of point-vortex tripoles characterized by that their core and satellite vortices reside in different layers of a two-layer f-plane quasigeostrophic model. Also the stability of pentapoles and quadrupoles is discussed. The parameters affecting the stability properties of a multipole are the length of its leg and the intensity of the core vortex relative to the satellite vortices. Among the invariants of the dynamical system that describes the motion of an ensemble of m+1 vortices, there are two ones depending on the distances between the vortices only. To establish the stability/instability of a multipole, we consider the restriction of one of the two invariants to the sheet (in the phase space) constituted by the states at which the second invariant takes the same value as at the multipole equilibrium state. Two versions of the method are presented and employed to analyze the stability of collinear states (m = 2) and non-collinear states (m > 2). For tripoles, complete stability analysis is performed resulting in the determination of the regions of stability/instability in the parameter plane. Depending on the parameters, a multipole can rotate clockwise or counterclockwise, and also can be static. A stable static tripole is shown to minimize the energy of interaction between the vortices. Rigorous stability analysis of pentapoles and quadrupoles is carried out under certain constrains on the permissible perturbations: in pentapoles the perturbations should preserve the central symmetry, and in quadrupoles, the zero linear momentum; only quadrupoles with zero total intensity are considered.