Stratified separated flow around a mountain with an inversion layer below the mountain top

Abstract

This paper presents analytical and numerical results for separated stratified inviscid flow over and around an isolated mountain in the limit of small Froude number. The vertical density profile consists of a lower strongly stratified layer whose depth is just less than that of the mountain. It is separated from a semi-infinite upper stably stratified layer by a thin, highly stable, inversion layer. The paper aims to provide, for this particular profile, a thorough analysis of the three-dimensional separated flow over a mountain top with strong stratification. The Froude numbers $F$ and $F_I$ of the lower layer and the interface are small with $F_I\,{\ll}\, F\,{\ll}\,1$, but the upper-layer Froude number is arbitrary. The flow at each height in the lower layer is governed by the two-dimensional Euler equations and moves horizontally around the mountain. It is given by a modification of a previous model using Kirchhoff free-streamline theory for the separated flow region downstream of the mountain. The pressure variations associated with the lower-layer flow are of the same order as the dynamic head and induce significant displacements of the inversion layer. When the inversion is near the top of the mountain these deflections are of the same order as the height of the projecting part of the mountain top and combine with the flow over the mountain top to excite vertically propagating internal waves in the upper layer. The resultant pressure field, vertical stream surface displacements, and surface streamlines in the upper layer are described consistently in the hydrostatic limit. Many of the features of the upper flow, including the perturbations of the critical dividing streamlines, are similar to those in flows with uniform stable stratification at low Froude number. Comparisons are made with experiments and approximate models for these summit flows based on the assumption that the dividing streamlines have small vertical displacement.