## On the stability of discrete tripole, quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid.(English)Zbl 1346.76204

Summary: 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.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 $$\mathcal G$$-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.

### MSC:

 76U05 General theory of rotating fluids 76B47 Vortex flows for incompressible inviscid fluids 76E20 Stability and instability of geophysical and astrophysical flows
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