The normal mode approach to investigating the stability of a parallel shear flow involves the superposition of a small wavelike perturbation on the basic flow. Its evolution in space and/or time is then determined. In the linear inviscid theory, if u(y) is the basic velocity profile, then a singularity occurs at critical points y c , wherē u = c, the perturbation phase speed. This is plausible intuitively because energy can be exchanged most efficiently where the wave and mean flow are travelling at the same speed. The problem is of the singular perturbation type; when viscosity or nonlinearity, for example, are restored to the governing equations, the singularity is removed. In this lecture, the classical viscous theory is first outlined before presenting a newer perturbation approach using a nonlinear critical layer (i.e., nonlinear terms are restored within a thin layer). The application to the case of a density stratified shear flow is discussed and, finally, the results are compared qualitatively with radar observations and also with recent numerical simulations of the full equations.

The propagation of finite-amplitude internal waves in a shear flow is considered for wavelengths that are long compared to the shear-layer thickness. Both singular and regular modes are investigated, and the equation governing the amplitude evolution is derived. The theory is generalized to allow for a radiation condition when the region outside the stratified shear layer is unbounded and weakly stratified. In this case, the evolution equation contains a damping term describing energy loss by radiation which can be used to estimate the persistence of solitary waves or nonlinear wave packets in realistic environments. A continuous three-layer model is studied in detail and closed-form expressions are obtained for the phase speed and the coefficients of the nonlinear and dispersive terms in the amplitude equation as a function of Richardson number.

It is shown, using a quite general formulation, that the amplitude evolution equation for slowly varying finite amplitude waves is usually first order in both space and time. One advantage of the present formulation is that it becomes possible to easily identify, from their linear eigensolutions, interesting exceptional cases in which the amplitude evolves according to a partial differential equation that is second order in either space or time. The theory is applied to a number of specific problems, including flows with broken line profiles, and inviscid shear flows having nonlinear critical layers.

In 1967, Stuart [J. Fluid Mech. 29, 417 (1967)] found an exact nonlinear solution of the inviscid, incompressible two-dimensional Navier–Stokes equations, representing an infinite row of identical vortices which are now known as Stuart vortices. In this Brief Communication, the corresponding result for an infinite row of counter-rotating vortices, i.e., a row of vortices of alternating sign, is presented. While for Stuart’s solution, the streamfunction satisfied Liouville’s equation, the streamfunction presented here satisfies the sinh–Gordon equation [Solitons: An Introduction (Cambridge U.P., London, 1989)]. The connection with Stuart’s solution is discussed.

The structure of the critical layer in a stratified shear flow is investigated for finite‐amplitude waves at high Reynolds numbers. Under such conditions, which are characteristic of the Clear Air Turbulence environment, nonlinear effects will dominate over diffusive effects. Nevertheless, it is shown that viscosity and heat‐conduction still play a significant role in the evolution of such waves. The reason is that buoyancy leads to the formation of thin diffusive shear layers within the critical layer. The local Richardson number is greatly reduced in these layers and they are, therefore, likely to break down into turbulence. A nonlinear mechanism is thus revealed for producing localized instabilities in flows that are stable on a linear basis. The analysis is developed for arbitrary values of the mean flow Richardson number and results are obtained numerically.