Properties and baroclinic instability of stratified thermal ocean flow

We investigate the properties of, and can carry out a stability analysis of a baroclinic current in, a stratified thermal rotating shallow-water model for the subinertial dynamics of the upper ocean, a key player in the global climate system where most of the ocean variability is concentrated affecting the lateral transport of floating material such as plastic garbage, oil, and Sargassum seaweed.  Unlike the standard thermal model, the model considered here includes linear buoyancy variation in the vertical, still maintaining its two-dimensional structure and that of the adabatic (constant density) model.  Like the standard thermal model, the stratified thermal model produces submesoscale circulations resembling those observed in satellite imagery, yet taking longer to manifest.  Our study is motivated by this numerical observation. The model possesses a Lie--Poisson Hamiltonian structure.  A particular aspect of the model is that it supports motion integrals which neither form the kernel of the corresponding bracket nor are related to any explicit symmetries via Noether's theorem.  Among other things, we investigate the role of these conservation laws in constraining the growth of finite-amplitude perturbations to a zonal flow with quadratic vertical shear. Joint work with Maria J. Olascoaga.