A class of solutions for steady stratified flows

Abstract

Under the assumption that the horizontal scales of the flow of a stratified fluid are much greater than the vertical scale, it can be shown that the pressure distribution in the fluid is nearly hydrostatic and that the solution for steady flows can be reduced to the solution of a non-linear partial differential equation with only horizontal co-ordinates as the space variables. The theory built on the basic assumption is the shallow-water theory for stratified fluids. Transformations are explicitly given with which a class of solutions for steady three-dimensional flows of a fluid of arbitrary stratification, continuous or discontinuous, issuing from a large reservoir can be found from a corresponding solution for a homogeneous fluid, provided a free surface is present and the shallow-water theory is applicable. A few examples of exact solutions according to the shallow-water theory are given and the parallel flow in a horizontal canal issuing from a large reservoir with the same horizontal bottom, which has some bearing on previous works on stratified flows, is discussed. But it is emphasized that the class is a very special one and that there are other solutions not belonging to this class. The conditions under which a solution belonging to this class is valid are discussed.