  We consider direct statistical simulation (DSS) of a paradigm system of
convection interacting with mean flows. In the Busse Annulus model zonal jets
are generated through the interaction of convectively driven turbulence and
rotation; non-trivial dynamics including the emergence of multiple jets and
bursting `predator-prey' type dynamics can be found. We formulate the DSS by
expanding around the mean flow in terms of equal-time cumulants and arrive at a
closed set of equations of motion for the cumulants. Here, we present results
using an expansion terminated at the second cumulant (CE2); it is fundamentally
a quasi-linear theory. We focus on particular cases including bursting and
bistable multiple jets and demonstrate that CE2 can reproduce the results of
direct numerical simulation if particular attention is given to symmetry
considerations.
