Sergey Lysenko

The stability problem of a vortex tripole/quadrupole in the Alfvén model of a two-fluid plasma is considered. These axisymmetric vortex structures consist of a central vortex of arbitrary intensity and $N$ identical peripheral vortices (cases $N = 2, 3$) rotating around the central vortex with constant angular velocity.
In addition to $N$, this problem has 3 more parameters: $R$ is the radius of a circle on which $N$ identical vortices are uniformly located around the central one, $c$ is the parameter that characterizes the type of vortices in a two-fluid plasma, and $\gamma$ is the ratio of the intensities of the central vortex and the identical vortices.
Stability is understood as the stability of a one-parameter orbit of stationary rotation of a vortex system, that is, orbital stability. Instability of stationary rotation is interpreted as a spectral instability. The eigenvalues of the linearization matrix are investigated analytically. Sufficient conditions for orbital stability are obtained by constructing the first integral, which has a strict local minimum in the orbit of the regime under study. As a result, the parameter space $(N, R, c, \Gamma)$ is divided into three different areas: the orbital stability domain in an exact nonlinear formulation, the spectral instability domain, and the spectral stability domain, where additional nonlinear analysis is required.