Nonlinear saturation of thermal instabilities

Recent low-frequency simulations of a one-layer model with lateral buoyancy inhomogeneity have revealed circulatory motions resembling quite closely submesoscale features on the ocean surface often visible in satellite observations. This model is known to lack a high-wavenumber instability cutoff and, thus, to possibly undergo ultraviolet catastrophe. However, the numerically observed instabilities, referred to as “thermal” due the ability of the above inhomogeneous-layer model to incorporate thermodynamic processes, are not seen to grow indefinitely. In this note, I show that the presence of a convex pseudo-energy–momentum integral of motion for the inviscid, unforced dynamics can arrest their nonlinear grow in the zonally symmetric case. Our result is an application of Arnold and Shepherd's methods.