The evolution of a small inviscid disturbance to a marginally unstable stratified shear flow; stage two

Abstract

In this paper we examine the evolution of the critical layer in a stratified shear flow when the Richardson number Ri = 1/4. We postulate an initial state consisting of a hyperbolic tangent profile for both velocity and density, together with a disturbance periodic in x . We suggest that, at large values of a non-dimensional time t , all Fourier components of this distur­bance but one have died out, and we are concerned with the subsequent development of the remaining one. The magnitude of this component is characterized by a small parameter ϵ , and the next stage of its evolution is forced by the nonlinear terms in a critical layer where the streamwise com­ponent of velocity is initially O ( t 1/2 ). This stage takes place on a time scale O ( ϵ -2/3 ) when the thickness of the critical layer is O ( ϵ 2/3 ). Outside this layer the evolution is linear except in that it is forced by the dynamics of the critical layer, and we find that it has three interesting features. The first is the generation throughout the shear flow of another disturbance of the same wavelength as the original one, but out of phase with it by 1/2π, whose amplitude increases algebraically with time. The second is the appearance of higher harmonics throughout the shear layer, and the third is the occurrence of Stuart-Landau resonance which requires a subtle amendment to the matching procedure.