Weakly nonlinear evolution of a wave packet in a zonal mixing layer

A nonlinear integrodifferential equation governing the amplitude evolution of a wavepacket near the critical value of the beta parameter is derived. The basic velocity profile is a hyperbolic tangent shear layer and although the neutral eigensolution is regular, all higher-order terms in the expansion of the stream function are singular at the critical point. The analysis is inviscid and in the critical layer both wave packet effects and nonlinearity are present, but the former are taken to be slightly larger. Unlike the Stuart-Watson theory, the critical layer analysis dictates the form of the amplitude equation, the outer expansion being relatively passive. A secondary instability analysis shows that the packet is unstable to sideband perturbations, but the instability is weak so its main consequence would be to produce some modulation of the packet without destroying its coherence.