A method is developed for statistical prediction of turbulent geophysical flows that
is more efficient than ensemble integrations. We consider the evolution of low-order
moments for inviscid quasi-geostrophic turbulence. Guided by statistical mechanics,
equations are developed for predicting the mean and the variance about the mean
as functions of position and time. These equations are consistent with the exact
moment equations and contain irreversible (entropy producing) fluxes that must be
specified in terms of known moments. Using simple choices for these dependences, the
resulting scheme, involving just two spatial fields, typically outperforms 100-realization
ensembles.
Statistical mechanics of two-dimensional and geophysical flows