A Statistical Closure for Turbulent Zonal Jets based on the Generalized Quasilinear Approximation

The generalized quasilinear (GQL) approximation was applied recently to the simulation of zonal jets on the spherical surface and on the beta plane. GQL allows for a systematic improvement over quasilinear equations by generalizing the conventional Reynolds decomposition: rather than decomposing fields into a mean and fluctuation, GQL projects them into large- and small-scale zonal modes. In this framework, considerable accuracy can be achieved in regimes with strong driving even when few large-scale zonal modes are retained. In the present work, we derive a statistical closure based on cumulant expansions starting from the GQL approximation. The resulting generalised cumulant expansion (GCE2) equations allow for direct statistical simulation of the first and second cumulants. Thus, GCE2 retains the generalisation embodied in GQL and also precludes the need for Direct Numerical Simulation in order to collect statistics. We use GCE2 to simulate a deterministically-forced barotropic jet on the spherical surface and on the beta plane, and evaluate its performance in comparison to GQL. Plausible applications of GCE2 to other astro/geophysical problems are discussed.