Statistical Mechanics of a Geophysical Jet

We investigate the equal-time statistics of an equatorial jet in a two-dimensional quasi-geostrophic model of a planetary atmosphere on a rotating sphere\footnote{R. S. Lindzen, A. J. Rosenthal, and R. Farrell, J. Atmos. Sci. {\bf 40}, 1029 (1983).}. Potential vorticity is advected by the barotropic flow and at the same time relaxes towards the zonal shear flow of an underlying equatorial jet. A transition to turbulence occurs at sufficiently slow relaxation rates. Statistics accumulated by direct numerical simulation\footnote{Akio Arakawa, J. Comp. Phys. {\bf 1}, 119 (1966).} are compared to those obtained by a simple cumulant expansion. We study rigorous upper bounds on the instability size\footnote{T. G. Shepherd, J. Fluid. Mech. {\bf 196}, 291 (1988).} and discuss the limitations of the cumulant expansion.