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. GQL systematically interpolates between QL and fully non-linear dynamics by employing a generalised Reynolds decomposition. Here, we examine 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. GCE2 is shown to yield improved performance over statistical representations of purely QL dynamics (CE2) and thus enables Direct Statistical Simulation (DSS) of complex mean flows that do not entirely fall within the remit of pure QL theory. Despite the existence of an exact closure, GCE2 like CE2 admits the possibility of a rank instability that leads to differences with statistics obtained from GQL. Recognition of this instability is a necessary step before further progress can be made with the GCE2 statistical closure.