Low Froude Number Flow Past Three-Dimensional Obstacles. Part I: Baroclinically Generated Lee Vortices

Abstract

We study the flow of a density-stratified fluid past a three-dimensional obstacle, using a numerical model. Our special concern is the response of the fluid when the Froude number is near or less than unity. Linear theory is inapplicable in this range of Froude number, and the present numerical solutions show the rich variety of phenomena that emerge in this essentially nonlinear flow regime. Two such phenomena, which occupy Parts I and II of this study, are the formation of a pair of vertically oriented vortices on the lee side and a zone of flow reversal on the windward side of the obstacle. The Ice vortices have been explained as a consequence of the separation of the viscous boundary layer from the obstacle however, this boundary layer is absent (by design) in the present experiments and lee vortices still occur. We argue that a vertical component of vorticity develops on the lee side owing to the tilting of horizontally oriented vorticity produced baroclinically as the isentropes deform in response to the flow over the obstacle. This deformation is adequately predicted by linear gravity-wave, which allows one to deduce, using the next-order correction to linen theory, the existence of a vortex pair of the proper sense in the lee of the obstacle. Thus, the lee vortices are closely associated with the dynamics of gravity waves. The generation of the lee vortices may also be understood as a consequence of Ertel's theorem which in the present circumstance demands that vortex lines adhere to isentropic surfaces? since the isentropes are depressed behind the hill, the vortex lines must run upward and downward along the depression implying vertically oriented vorticity.