A general method for determining upstream effects in stratified flow of finite depth over long two-dimensional obstacles

Abstract

A procedure for determining the flow which results from introducing a long two-dimensional obstacle of finite height into a unidirectional, two-dimensional, stable, but otherwise arbitrary stratified shear flow of finite depth is described. The method is based on a generalization of the known results for two-layer flows, described in Baines (1984). The flow is assumed to be hydrostatic with negligible mixing, and the stratified flow is represented by an arbitrary number of discrete layers, so that the model is hydraulic in character. The procedure involves the calculation of the changes to the steady-state flow resulting from successive increases in the height of the topography from zero. For a given initial flow, introduction of an obstacle only alters the flow in its vicinity for obstacle heights hm less than a height hc, where the flow is critical (implying zero wave speed) at the obstacle crest for some particular internal wave mode. Increasing the obstacle height further causes the flow to adjust to maintain a critical condition at the obstacle crest, and this causes disturbances with the structure of the critical mode to be propagated upstream. These may take the form of an upstream hydraulic jump or of a time-dependent rarefaction (implying a disturbance which becomes increasingly spread out with time), or both, depending on the nonlinear dispersive properties of the system. Their passage past a given upstream location results in a permanent change to the local velocity and density profiles. As the obstacle height is further increased these processes will continue until the flow becomes critical just upstream of the obstacle, or a fluid layer becomes blocked. For greater obstacle heights the above phenomena may be repeated with other modes. A numerical procedure which implements these processes has been developed, and examples of applicatons to two- and three-layer systems are given.