Three-dimensional problem of exponentially stratified flow over bottom roughness in the case of finite depth

Abstract

The three-dimensional problem of the flow of an exponentially stratified fluid of finite depth over bottom roughness is considered in the rigid roof approximation and in the presence of a free surface. In the rigid roof approximation the solution is obtained in the form of a Fourier series in the vertical Lagrangian coordinate, and the series coefficients are expressed in terms of single integrals outside a horizontal strip whose sides are parallel to the flow axis and tangential to the projection of the support of the function describing the bottom roughness. This makes it possible to investigate the near field in regions not considered in [1, 2]. The presence of a small parameter in the boundary condition at the free surface makes it possible to find, in the first approximation, the wave motions and nonwave disturbances at the free surface in the near and far fields. In the near field the width of the wave zone is of the order of the flow depth, expands with distance from the bottom and is broadest at the free surface. As distinct from the annular disturbances within the fluid, the pattern of the nonwave disturbances at the free surface depends on the polar angle. The law of similarity for the diverging waves at the free surface is also obtained.