Properties and baroclinic instability of stratified thermal upper-ocean flow

We study the properties of, and investigate the stability of a baroclinic zonal current in, a thermal rotating shallow-water model, sometimes called \emph{Ripa's model}, featuring stratification for quasigeostrophic upper-ocean dynamics. The model has Lie--Poisson Hamiltonian structure. In addition to Casimirs, the model supports weak Casimirs forming the kernel of the Lie--Poisson bracket for the potential vorticity evolution independent of the details of the buoyancy as this is advected under the flow. The model sustains Rossby waves and a neutral model, whose spurious growth is prevented by a positive-definite integral, quadratic on the deviation from the motionless state. A baroclinic zonal jet with vertical curvature is found to be spectrally stable for specific configurations of the gradients of layer thickness, vertically averaged buoyancy, and buoyancy frequency. Only a subset of such states was found Lyapunov stable using the available integrals, except the weak Casimirs, whose role in constraining stratified thermal flow remains to be understood. The existence of Lyapunov-stable states enabled us to \emph{a priori} bound the nonlinear growth of perturbations to spectrally unstable states. Our results do not support the generality of earlier numerical evidence on the suppression of submesoscale wave activity as a result of the inclusion of stratification in thermal shallow-water theory, which we supported with direct numerical simulations.