Nonlinear interactions of spherical Rossby modes

Abstract

Weakly nonlinear triad interactions between spherical Rossby harmonics are studied. Each of the harmonics has the form AP n m (sin θ)exp[ i( mλ − σt)], where A is an amplitude and P n m is the associated Legendre function. Equations for the amplitudes are derived and resonance conditions are analysed. The resonance conditions differ substantially from the usual resonance conditions on a β-plane and include a Diophantine equation and a few inequalities for the integer wavenumbers n and m of the interacting modes. Particular analytical series of solutions to the resonance conditions are constructed, which show that modes with arbitrary large wavenumbers can participate in the interactions. A numerical study of the resonance conditions reveals that no more than 21% of the Rossby harmonics can participate in the triad interactions and that chains of the interacting triads soon break. Thus precise interactions (for which the resonance conditions hold exactly) do not result in any significant redistribution of energy over the spectrum. On the other hand, approximate interactions (for which the resonance conditions hold approximately) generate an intensive energy redistribution among short Rossby modes with typical scales much smaller than the Earth's radius. Thus the energy cascade is concentrated mainly in the short-wave part of the spectrum and is very weak in the large-scale domain. The efficiency of the triad interaction of Rossby modes with scales much smaller than the Earth's radius depends strongly on the existence of the so-called interaction latitude at which the local wave-vectors and frequencies of the interacting modes satisfy resonance conditions for plane Rossby waves on the β-plane approximating the neighbourhood of the latitude. If the interaction latitude exists, the interaction is intensive; in the opposite case, it is weak.