Stability of traveling water waves on the sphere Original Research Article

We study the stability of a class of traveling waves in a model of weakly nonlinear water waves on the sphere. The model describes free surface potential flow of a fluid layer surrounding a gravitating sphere, and the evolution equations are Hamiltonian. For small amplitude oscillations the Hamiltonian 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-monochromatic” in space, i.e. have amplitude consisting of one spherical harmonic plus small corrections. We observe that such motions are stable for long times. 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 conditions for the long time nonlinear stability of a certain class of periodic orbits in Hamiltonian systems of resonantly coupled harmonic oscillators.