The nature of upstream blocking in uniformly stratified flow over long obstacles

Abstract

The general method described in Baines 1988 has been applied to stratified flows of finite depth over long obstacles where the flow initially has uniform horizontal velocity. The fluid consists of a finite number of homogeneous layers of equal thickness and with equal density increments. This represents the state of continuous stratification with constant density gradient as closely as possible, for a given number of layers. Two-, three-, four- and sixty-four-layered models are studied in detail. The results are expressed in terms of the initial Froude number F0 (F0 = U/ĉ1 where U is the fluid speed and ĉ1 is the speed of the fastest long internal wave mode in the fluid at rest) and the obstacle height. In general, introduction of an obstacle into the flow causes disturbances to propagate upstream (columnar disturbance modes) which alter the velocity and density profiles there. These may accumulate to cause upstream blocking of some of the fluid layers if F0 is sufficiently small. As the number of fluid layers increases, so does the range of F0 for which this upstream blocked flow occurs. There are no upstream disturbances for F0 > 1, and for F0 < 1 the upstream disturbances are of the rarefaction type if upstream blocking does not occur. The results for three and four layers show how several coexisting modes may interact to affect the upstream profiles. The results for sixty-four-layers provide theoretical support for the observational criterion (Baines 1979b) that blocking in initially uniformly stratified flow occurs when Nhm/U > 2 (N is the Brunt—Väisälä frequency and hm the obstacle height), provided that more than two modes are present. In some situations, layered models are found to be inadequate as a representation of continuous stratification when one or more layers thicken to the extent that their discreteness is significant.