Stability of traveling water waves on the sphere

We study the stability of a class of traveling waves in a model of weakly n onlinear water waves on the sphere. The model describes free surface potent ial flow of a fluid layer surrounding a gravitating sphere, and the evoluti on equations are Hamiltonian. For small amplitude oscillations the Hamilton ian can be expanded in powers of the wave amplitude, yielding simpler model equations. We integrate numerically Galerkin truncations of such a model, focusing on a class of traveling and standing waves that are "near-monochro matic" in space, i.e. have amplitude consisting of one spherical harmonic p lus small corrections. We observe that such motions are stable for long tim es. To explain the observed behavior we use methods of Hamiltonian dynamics , first showing that decay to all but a small number of modes must be very slow. To understand the interaction between these modes we obtain general c onditions for the long time nonlinear stability of a certain class of perio dic orbits in Hamiltonian systems of resonantly coupled harmonic oscillator s. (C) 2001 Published by Elsevier Science B.V. on behalf of IMACS.