On the Stability of Simple Solution of Shallow Water Equations on a Rotating Gravitating Sphere

The system of equations of the theory of shallow water on a rotating gravitating sphere admits a simple solution. The problem of stability of this solution is discussed. Small perturbations of the above-mentioned solution are described by the system of equations obtained by linearization of the original system in the present solution. After the complete separation of variables, the problem is reduced to finding the eigenvalues of singular boundary value problem for the system of four ordinary differential equations. In the special case when the perturbations depend only on the latitude variable, one can prove that the eigenvalues are purely imaginary. Thus, the analyzed solution is stable with respect to such perturbations. In the general case, the numerical methods are used. A numerical experiment shows that in the general case eigenvalues are purely imaginary as well, i.e., the solution is stable with respect to small perturbations of arbitrary form.