Internal solitary waves with a weakly stratified critical layer

Motivated by observations of solitary waves in the ocean and atmosphere, this paper considers the evolution of long weakly nonlinear internal waves in an incompressible Boussinesq fluid. The motion is restricted to the vertical plane. The basic state consists of stable horizontal shear flow and density stratification. On a long time scale, the waves evolve and reach a quasi-steady régime where weak nonlinearity and weak dispersion are in balance. In many circumstances, this régime is described by a Korteweg-de-Vries equation. However, when the linear long-wave speed equals the basic flow velocity at a certain height, the critical level, the traditional assumption of weak nonlinearity breaks down due to the appearance of a singularity in the leading-order modal equation, implying a strong modification of the flow in the so-called critical layer. Since the relevant geophysical flows have high Reynolds and Péclet numbers, we invoke nonlinear effects to resolve this singularity. Viscosity and thermal conductivity are considered small but finite. Their presence renders the nonlinear-critical-layer solution unique. Crucially, the density stratification degree is assumed small at the critical level; this has the consequence that the leading-order singularity is then identical to that in an unstratified flow. Thus the asymptotic methodology employed previously for that case can be adapted to this present study. In this critical layer, the flow is fully nonlinear but laminar and quasi-steady, with a strong rearrangement of the buoyancy and vorticity contours. This inner flow is matched at the edges of the critical layer with the outer flow. The final outcome for spatially localized solutions is an integro-differential evolution equation, whose form depends on the critical-layer shape, and especially on the wave polarity, that is, depression or elevation. For a steady travelling wave, this evolution equation when expressed in terms of the streamfunction amplitude is not a Korteweg-de Vries equation, as it contains additional nonlinear terms necessary at a certain order of the asymptotic expansion when matching with the inner flow. However, this steady evolution equation can be transformed with an appropriate change of variables into a Korteweg-de-Vries equation. An analysis of the wave mean flow interaction is given. The horizontal basic stable flow is altered at the critical level at a slow viscous time scale by the nonlinear D-wave in the quasi-steady state régime. In most cases, the mean kinetic energy is likely to decay at the same time scale. The D-wave critical-layer thickness is found inversely proportional to the amplitude of the leading-order viscous outer flow. The D-wave critical-layer symmetry vis-à-vis the critical level is broken; the departure is proportional to the local Richardson number, and so is greater than the analogous asymmetry encountered in an unstratified shear flow. The mean flow horizontal momentum and kinetic energy are unchanged by the nonlinear E-wave in the quasi-steady régime, the exchanges taking place only during the formation of the critical layer and in the transition régime when the nonlinear E-wave is unsteady.