Dynamical Criterion for a Marginally Unstable, Quasi-linear Behavior in a Two-Layer Model

A two-layer quasi-geostrophic flow forced by meridional variations in heating can be in regimes ranging from radiative equilibrium to forced geostrophic turbulence. Between these extremes is a regime where the time-mean (zonal) flow is marginally unstable. Using scaling arguments, we conclude that such a marginally unstable state should occur when a certain parameter, measuring the strength of wave-wave interactions relative to the beta effect and advection by the thermal wind, is small. Numerical simulations support this proposal. In the last section, we examine a transition from the marginally unstable regime to a more nonlinear regime through numerical simulations with different radiative forcings. In our simulations, we find that transition is not caused by secondary instability of waves in the marginally unstable regime. Instead, the time-mean flow can support a number of marginally unstable normal modes. These normal modes interact with each other, and if they are of sufficient amplitude, the flow enters a more nonlinear regime.