Direct Statistical Simulation of a General Circulation Model by a Cumulant Expansion

Low-order statistics of the large-scale circulation of planetary atmospheres may be directly accessed by solving the equations of motion (EOM) for the equal-time cumulants. We implement a cumulant expansion, truncated at second order, within a GFDL Flexible Modeling System (FMS) general circulation model dry dynamical core. The Held-Suarez physical parameterizations consist of Newtonian relaxation toward a prescribed radiative-convective equilibrium temperature profile and surface Rayleigh friction. The first cumulant is the zonally averaged vorticity, divergence, and temperature as a function of latitude and height, and the second cumulant contains information about fluctuations and teleconnections. The second order expansion retains the eddy -- mean-flow interaction but neglects eddy-eddy interactions, and is equivalent to a quasi-linear approximation and thus realizable. To implement the cumulant expansion, the contributions to the spectral EOM tendencies are organized into constant terms, and terms that are linear, quadratic, or cubic in the fields. We compare equal-time statistics described by the fixed point of the cumulant expansion to those accumulated by time-integration of the full non-linear EOM, and the quasi-linear approximation to the EOM.