A Weakly Nonlinear Theory of Amplitude Vacillation and Baroclinic Waves

Abstract

The weakly nonlinear dynamics of quasi-geostrophic Perturbations of various basic states is investigated. The basic states differ slightly from Eady's, i.e., from an inviscid zonal flow of a fluid of uniform Brunt?Väisälä frequency in which the velocity varies linearly with height. Such differences lead to weak critical layers which may, in a certain region of parameter space where the Eady basic state is neutrally stable, render the flows unstable to some modes: the Green modes. Weak dissipation may balance the growth of these modes. In accord with the numerical results of Lindzen, Farrell and Jacqmin, it is found asymptotically that, in that region of parameter space, two modes with the same wavenumbers may grow. The weakly nonlinear interactions of these unstable modes and the basic state am examined in detail. It is found that the modes may equilibrate. Amplitude vacillation is identified as the physical manifestation of this equilibration, because the two finite amplitude waves with different vertical structures alternately reinforce and cancel one another by interference as they propagate zonally with different velocities. The results for a special case are found to illustrate the theory and are compared with experimental observations of a differentially heated rotating annulus.