Abstract

We regard as a starting point barotropic vorticity equation, for two types of boundary conditions: first, on a rotating sphere, second, in a rectangular basin. Our main goal is to study the dynamical systems describing small clusters of nonlinear resonances, the smallest one being a triad. For both types of boundary conditions, the geometrical and topological structures of the clusters are presented. It is shown that most frequently met clusters are integrable. It is also shown that dynamical phases, usually regarded as equal to zero or constants, play a substantial role in the dynamics of resonance clusters. Indeed, their effects are: (i) To reduce the period of energy exchange $ au$ within a cluster by 20% and more. (ii) To diminish, at time scales of the order of $ au$, the variability of wave energies by 25% and more. (iii) To generate a new time scale $T >> au$ in which we observe a considerable energy exchange within a cluster, as well as an increase in the variability of wave energies for each and every mode.