Irina Ostrovskaya

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.

The stability problem of a moving circular cylinder of radius $R$ and a system of $n$ identical point vortices uniformly distributed on a circle of radius $R_0^{}$, with $n\geqslant 2$, is considered. The center of the vortex polygon coincides with the center of the cylinder. The circulation around the cylinder is zero. There are three parameters in the problem: the number of point vortices $n$, the added mass of the cylinder $a$ and the parameter $q=\frac{R^2}{R_0^2}$.
The linearization matrix and the quadratic part of the Hamiltonian of the problem are studied. As a result, the parameter space of the problem is divided into the instability area and the area of linear stability where nonlinear analysis is required. In the case $n=2,\,3$ two domains of linear stability are found. In the case $n=4,\,5,\,6$ there is just one domain. In the case $n\geqslant 7$ the studied solution is unstable for any value of the problem parameters. The obtained results in the limiting case as $a\to\infty$ agree with the known results for the model of point vortices outside the circular domain.

In this paper the two-layer geostrophic model of the rotating fluid and the model of Bessel vortices are considered. Kirchhoff's model of vortices in a homogeneous fluid is the limiting case of both of these models. Part of the study is performed for an arbitrary Hamiltonian depending on the distances between point vortices.
The review of the stability problem of stationary rotation of regular Thomson's vortex $N$-gon of identical vortices is given for ${N\geqslant 2}$. The stability problem of the vortex tripole/quadrupole is also considered. This axisymmetric vortex structure consists of a~central vortex of an arbitrary intensity and two/three identical peripheral vortices. In the model of a two-layer fluid, peripheral vortices belong to one of the layers and the central vortex can belong to either another layer or the same.
The stability of the stationary rotation is interpreted as orbital stability (the stability of a one-parameter orbit of a stationary rotation of a vortex system). The instability of the stationary rotation is instability of equilibrium of the reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied.
The parameter space is divided into three parts: $\bf A$ is the domain of stability in an exact nonlinear setting, $\bf B$ is the linear stability domain, where the stability problem requires nonlinear analysis, and $\bf C$ is the instability domain.
In the stability problem of a vortex multipole, another definition of stability is used; it is the stability of an invariant three-parametric set of all trajectories of the families of stationary orbits. It is shown that in the case of non zero total intensity, the stability of the invariant set implies orbital stability.