De Szoeke (1986) developed an asymptotic solution for the nonlinear evolution of a type of baroclinic instability of a two-layer quasi-geostrophic model over topography. He found that under certain conditions pairs of hybrid modes interacting with topography could become unstable in the linearized model. He also found that the addition of the nonlinearity stabilized the flow which he analyzed using an expansion in small ε, a measure of the topographic height. This work is extended and modified by first considering a slightly varying mean flow (time dependent parametric forcing) which produces chaotic behavior, and then considering friction which allows chaos on a strange attractor. This chaos is examined in various ways including using Melnikov's method. The original unforced system without friction can be called "critically nonchaotic" in that only a very small amount of forcing produces significant chaotic behavior. We then investigate whether the original system can be chaotic without any variation in the mean flow. An additional term is included in the asymptotic system to form a six variable system which can become chaotic. We also look at a more general, nonasymptotic, initial value problem consisting of sixteen variables, assuming small amplitudes, which also can become chaotic. Finally, we consider an asymptotic expansion in small α, the aspect ratio of the top to the bottom layer, assuming ε = O (1), which is often more realistic in the ocean. It is found that at the conditions of greatest initial growth of the amplitudes the system can be chaotic with the largest amplitude in the top layer.
Kroll, John. 1999. "On the chaotic evolution of baroclinic instability of wave-wave interactions with topography." Journal of Marine Research 57, (1). https://elischolar.library.yale.edu/journal_of_marine_research/2310