This paper addresses the Rossby adjustment problem for an inviscid uniformly rotating two-layer fluid in the presence of a step escarpment of infinite length. The problem can be solved analytically for the case when the ratio of the step height to the average depth of the lower layer is small. In this case two well-separated adjustment time scales emerge; the rapid, inertial and the slow, topographic vortex-stretching time scales. The fluid is assumed to be at rest initially with imposed step discontinuities in the free surface and interfacial displacements oriented perpendicular to the escarpment. A two time-scale approach shows that during the rapid inertial adjustment the fluid is not influenced by the topography. On the slow vortex-stretching time scale the fluid adjusts via the propagation of topographic Rossby waves, modified by stratification, along the step. A steady state solution is established in which the flow is geostrophically balanced in both layers. Therefore, in this steady state no fluid in the lower layer crosses the escarpment. However, cross-escarpment flow occurs in the upper layer. The volume of fluid in the upper layer that crosses the escarpment, rather than being deflected parallel to the topography, is calculated.
Willmott, A. J., and E. R. Johnson. 1995. "On geostrophic adjustment of a two-layer, uniformly rotating fluid in the presence of a step escarpment." Journal of Marine Research 53, (1). https://elischolar.library.yale.edu/journal_of_marine_research/2130