NONLINEAR-INTERACTIONS OF SPHERICAL ROSSBY MODES

Weakly nonlinear triad interactions between spherical Rossby harmonics are studied. Each of the harmonics has the form AP-n(m)(sin phi)exp[i (mlambda - sigmat)], where A is an amplitude and P-n(m) is the associa ted Legendre function. Equations for the amplitudes are derived and re sonance conditions are analysed. The resonance conditions differ subst antially from the usual resonance conditions on a beta-plane and inclu de a Diophantine equation and a few inequalities for the integer waven umbers n and m of the interacting modes. Particular analytical series of solutions to the resonance conditions are constructed, which show t hat modes with arbitrary large wavenumbers can participate in the inte ractions. A numerical study of the resonance conditions reveals that n o more than 21% of the Rossby harmonics can participate in the triad i nteractions 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 s pectrum. On the other hand, approximate interactions (for which the re sonance conditions hold approximately) generate an intensive energy re distribution among short Rossby modes with typical scales much smaller than the Earth's radius. Thus the energy cascade is concentrated main ly in the short-wave part of the spectrum and is very weak in the larg e-scale domain. The efficiency of the triad interaction of Rossby mode s with scales much smaller than the Earth's radius depends strongly on the existence of the so-called interaction latitude at which the loca l wave-vectors and frequencies of the interacting modes satisfy resona nce conditions for plane Rossby waves on the beta-plane approximating the neighbourhood of the latitude. If the interaction latitude exists, the interaction is intensive; in the opposite case, it is weak.