Resonantly Forced Rossby Waves

Abstract

A shallow, rotating layer of fluid that supports Rossby waves is subjected to turbulent friction through an Ekman layer at the bottom and is driven by a wave that exerts a shear stress on the upper boundary and for which the phase approximate that of a Rossby wave. The steady-state response may be either 1) a single wave that is synchronous with the driving wave or 2) a resonant triad of waves, one of which is synchronous with the driving wave. The triadic solutions constitute a one-parameter family, of which not more than one member is stable. The evolution equations for the slowly varying complex amplitudes of the responding waves, the fixed points of which correspond to the solutions 1) and 2), are established. The stability of these fixed points, and hence the stability boundaries for 1) and 2), are determined. There are no Hopf bifurcations of the fixed-point solutions, and the evolution equations apparently do not admit periodic (limit cycle), multiply periodic or chaotic solutions.