  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.
