Resumen
Stability of the Rossby-Haurwitz wave of subspace H1 Hn and two types of Verkley's modons is analyzed within the vorticity equation of an ideal incompressible fluid on a rotating sphere. Here Hn is the eigen subspace of the Laplace operator on a sphere corresponding to the eigenvalue n(n + 1). It is shown that arbitrary perturbations of the Rossby-Haurwitz wave can be divided into three invariant sets one of which contains a stable invariant subset Hn. Three invariant sets of small perturbations of the stationary Verkley modon are also found. The- separation of perturbations have been performed with the help of a conservation law for perturbations. Formulas for determining the distance between any two solutions from the whole set of modons and Rossby-Haurwitz waves are derived through the energy and enstrophy of the corresponding perturbation. Necessary and sufficient conditions for the distance between these solutions to be constant are obtained. It is shown that any super-rotation flow on a sphere (belonging to H1) is stable independently of choice of the rotation axis. Liapunov instability of any non-zonal Rossby-Haurwitz wave from H1 Hn where n = 2 as well as of any dipole modon on a sphere is proved. It is shown that the Liapunov instability is caused by the algebraic growth of perturbations due to asynchronous oscillations of waves and has nothing in common with the orbital instability. It is proved that any monopole Verkley (1987) modon, as well as any Legendre polynomial, is linearly Liapunov stable with respect to invariant subsets of perturbations of sufficiently small scale.