Direct Statistical Simulation using generalised cumulant expansions

In recent years, the Generalised Quasilinear (GQL) approximation has been developed and its efficacy tested against purely quasilinear (QL) approximations (Marston et al. 2016). GQL systematically interpolates between QL and fully non-linear dynamics by employing a generalised Reynolds decomposition for dynamics in one spatial direction. Here, we present an exact statistical closure for the GQL equations on the doubly periodic $\beta$-plane. Closure is achieved at second order using a generalised cumulant approach which we term GCE2. We demonstrate that the GCE2 equations systematically interpolate the statistics between those corresponding to CE2 and fully non-linear systems. GCE2 is shown to yield improved performance over statistical representations of purely QL dynamics (CE2). Hence, GCE2 enables Direct Statistical Simulation of complex mean flows that do not entirely fall within the remit of pure QL theory.