Aperiodic Trajectories and Stationary Points in a Three-Component Spectral Model of Atmospheric Flow

Abstract

Aperiodic solutions to spectrally truncated models based on the vorticity equation are considered for the case of a zonal flow interacting nonlinearly with two other components both having the same zonal wavenumber. It is shown that all such aperiodic trajectories proceed asymptotically to either a stationary point in the phase space of coefficients or to a periodic solution with steady amplitudes. It is also shown that the set of such solutions is of measure zero on surfaces of constant energy in phase space. Thus if the initial coefficients for a nonlinear, three-component flow are selected at random, then the resulting flow will in all probability be periodic.