Critical Layers in Shear Flows

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 ū(y) is the basic velocity profile, then a singularity occurs at critical points yc, where ū = 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. ∗Address for correspondence: Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, Canada. e-mail: [email protected]