Nonlinear resonance interactions and index cycles in the atmosphere

A barotropic spectral model is used to study the planetary-scale motions of an atmosphere whose wave ensemble modes are externally driven. Pertubations are induced by a barotropic analogue of thermal driving and by Ekman friction, bottom topography, and the vanished internal dissipation. The use of complete spectral expansions without truncation leads to that the nonlinear coupling equations between the low-index mode and the high-index mode are obtained by means of the random phase approximation and the projection operator techniques. The nonlinear coupling equations are entirely equivalent to theVolterra systems in ecology. In the phase-plane, the orbits of the nonlinear coupling equations are the family of closed curves, indicating a bound, and periodic motion. The qualitative behaviors of low-index and high-index modes as functions of time picture the motion of atmospheric flows, with exchanges of energy between the low-index mode and the high-index mode by nonlinear resonance interaction. It is suggested that the phenomenon of blocking be exponentially grown of the low-index mode, the atmospheric motion then evolved to the high-index mode due to relaxation process. The results therefore lead to a plausible hypothesis concerning index cycles in the atmosphere discussed by Lorenz’s early works.