On Resonant Interactions Between Unstable and Neutral Baroclinic Waves

Abstract

Resonant interactions between a marginally unstable baroclinic wave and one or two pairs of neutral waves in a rotating, two-layer zonal Row model are studied. Analyses based on a multiple scaling technique are carried out for inviscid flow on a beta plane, inviscid flow on an f-plane, and slightly viscous flow on an f-plane. It is assumed that the neutral waves are of smaller order in amplitude than the unstable wave, and the flow domain is taken to be zonally periodic so that zonal wavelengths are discrete. In this and other respects, the theory differs from an earlier study by Loesch (1974a,b) and avoids a number of difficulties inherent in that work. For an inviscid beta plane flow, the evolution of a wave triad is governed by a certain parameter N2N3/U. If this is negative, the neutral waves are unable to grow substantially larger in amplitude than their respective initial amplitudes, despite which they have a disproportionately large effect on the evolution of the unstable wave. If O<N2N3/U;<0.125, the neutral waves are largely unrestricted by their initial amplitudes and interact strongly with the unstable wave. Finally, if 0.125<N2N3/U, all three waves grow rapidly without bound and the scaling assumption is soon violated. For the flows on an f-plane, there exist two resonant triads with the unstable wave common to both. In this case, the neutral waves are always constrained by the size of their initial amplitudes. When the flow is inviscid, wave evolution is much the same as in the beta-plane problem with N2N3/U?0, whereas the addition of a small amount of viscosity results in the neutral waves being damped. There is no intermediate viscous regime in which the neutral waves are undamped, but ?forget? their initial conditions.