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| In the weather research and forecasting models of certain hurricanes, vortex crystals are found within a polygonal-shaped eyewall. These special configurations can be interpreted as relative equilibria (rigidly rotating solutions) of the point vortex problem introduced by Helmholtz. Their stability is thus of
considerable importance. Adapting Moeckel's approach for the $n$-body problem, we present some theory and results on the linear and nonlinear stability of relative equilibria in the planar $n$-vortex problem. A topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed, with stable solutions found in each case.
As time permits, we will discuss how some of these ideas, at least numerically, can be applied to the $n$-body problem. |
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