The nonlinear critical layer resulting from the spatial or temporal evolution of weakly unstable disturbances in shear flows

A study is made of the formation of a nonlinear critical layer (first found by Benney & Bergeron 1969 and Davis 1969) in homogeneous and weakly stratified incompressible shear flows as initially small unstable disturbances develop, whose growth rate is so small that all the evolution proceeds in the 'quasi-steady' regime. It is shown that such an evolution can be described from start to finish analytically using a pair of evolution equations for the wave amplitude and phase which involve universal functions of the familiar Haberman (1972) parameter, lambda, that characterizes the relative importance of the dissipation and nonlinearity. In addition to the function of a 'logarithmic phase jump' that was introduced and investigated by Haberman (1972), the evolution equations generally also contain other functions of lambda. In this paper we introduce and study (numerically and analytically) another three such functions.