On the steady-state relations between disturbances above and below a critical level

Abstract

In studying the behaviour of a density-stratified shear flow difficulties are encountered at the ‘critical’ level where wave velocity equals fluid velocity.Here a stratified shear layer of finite thickness is considered and a two-dimensional nonlinear steady-state problem is studied. It is assumed that blocking creates separate pockets of trapped fluid, each mixed to uniform density. These pockets are not in static equilibrium with the surrounding stratified fluid. They must be supported either by pressures dynamically developed in the curved flow along continuous streamlines outside the pockets or by centrifugal forces resulting from circulation within the pockets. The latter effect is considered only through evaluation of a crude ‘factor of importance’, FR, for the rotational effects and the pockets are assumed to be stagnant in the primary analysis.For small but finite disturbance amplitude FR approaches zero, indicating that no correction of the primary analysis is required. A limiting Richardson number of unity appears. Above this limit the primary analysis gives no solutions and apparently the separate pockets of stagnant fluid merge to form a continuous stagnant insulating layer. This behaviour of the critical level (as a barrier to communication) resembles earlier results from transient linearized investigations although the two analyses have little in common except the existence of a critical level separating two fluid regions.For moderate-to-large disturbance amplitudes the geometry of the flow pattern suggests Kelvin–Helmholtz billows. Rotational effects increase as the amplitude increases and may become significant at this stage. The primary analysis then becomes less accurate and cannot be used to exclude Kelvin–Helmholtz billows at Richardson numbers somewhat greater than unity.