Continuous and discontinuous transitions in geophysical turbulence

In geophysical turbulent flows, it is customary to have two or more attractors (typically responsible for low-frequency variability). Such a phenomenon is investigated in the dynamics of two-dimensional and quasi-geostrophic turbulence. In the inertial limit, with a time scale separation (between spin-up time and inertial time), the attractors are concentrated near a set of steady states of the inviscid equations. Statistical mechanical approaches help us determine which of these steady states are eligible. Then, we can establish phase diagrams, where phase transition lines delimitate regions corresponding to different regimes, i.e., different flow structures. The latter can be related to the different (say, two) regimes, which the Kuroshio alternatively finds itself into (bistability). Our present aim is to classify the bifurcations -given the two inertial equilibria - when tuning an external parameter. We give a general theory predicting whether the transitions are continuous (second-order) or discontinuous (first-order). The continuity or discontinuity depends on the value of some moments of the potential vorticity. We discuss applications to oceanic (Kuroshio) and experimental geophysical flows.