Meta-stability of equilibrium statistical structures for prototype geophysical flows with damping and driving

The most-probable states of an equilibrium statistical theory, which consist of monopole vortices, dipole vortex streets, and zonal shear flows for various parameter regimes, are shown to be meta-stable with respect to damped and driven quasigeostrophic dynamics in a periodic β-plane channel. Through a series of numerical experiments that include (1) pure decay, (2) both damping and driving, and (3) both direct and inverse cascades of energy, we demonstrate that statistically most-probable states evolve into other most-probable states with high accuracy, even as the energy changes substantially and the flow undergoes topological transitions from vortex to shear flow, or vice versa. The predictions of the equilibrium statistical theory are calculated by an algorithm, which we call an “approximate dynamics”, that constructs the most-probable states from the instantaneous values of a few quantities in the evolving flow. Qualitatively, the approximate dynamics predicts the correct topological structure — whether vortex flow or zonal shear — in the evolving flow. Quantitatively, the predictions are evaluated by measuring the relative errors between the velocity fields and vorticity fields of the evolving flow and the most-probable states. For evolving monopole vortices we find that errors in the velocity field are generally near 5% and errors in the potential vorticity field are generally near 15%. For evolving dipole vortex streets, the magnitude of the relative errors depends on the amplitude of the localized forcing. For pure decay, the errors in the velocity field are generally near 5% and errors in the vorticity field are generally near 12%; for runs in which the flow is strongly forced by small-scale vortices whose amplitude is nearly 310 the peak vorticity in the initial flow, the errors in the velocity field can rise to 20% and the errors in the vorticity field rise to 40%.