We study analytically and numerically small amplitude perturbations of a geostrophically balanced semi-infinite layer of light water having a surface front and lying above a heavier layer of finite vertical thickness which is at rest in the mean. In contrast with previous studies where the latter layer was infinitely deep we find that the equilibrium is always unstable regardless of the distribution of potential vorticity, and the maximum growth rates are generally much larger than in the "one-layer" case. The amplifying ageostrophic wave transfers kinetic energy from the basic shear flow as well as potential energy. Good quantitative agreement is found with the laboratory experiments of Griffiths and Linden (1982), and our model seems to be the simplest one for future investigations of cross frontal mixing processes by finite amplitude waves. The propagation speed of very low frequency and nondispersive frontal waves is also computed and is shown to decrease with increasing bottom layer depth.
Killworth, Peter D., Nathan Paldor, and Melvin E. Stern. 1984. "Wave propagation and growth on a surface front in a two-layer geostrophic current." Journal of Marine Research 42, (4). https://elischolar.library.yale.edu/journal_of_marine_research/1744