Direct Statistical Simulation of a Two-Layer Primitive Equation Model

Low-order statistics of the large-scale circulation of planetary atmospheres may be directly accessed by solving the equations of motion for the equal-time cumulants. We implement such Direct Statistical Simulation (DSS) of a two-layer primitive equation model by systematic expansion of the equations of motion for the cumulants. The first cumulant is the zonally averaged vorticity, divergence, and temperature as a function of latitude and level, and the second cumulant contains information about fluctuations and teleconnections. At second order (CE2) the expansion retains the eddy -- mean-flow interaction but neglects eddy-eddy interactions, is equivalent to a quasi-linear approximation, and is realizable. Eddy-eddy interactions appear at third (CE3) and higher orders, but care must be taken to maintain realizabiliy with a non-negative probability distribution function. The cumulant expansion is conservative, order-by-order, in quantities such as the total angular momentum, total energy, and mean-squared potential temperature.

First and second cumulants accumulated by time-integration of the fully non-linear two-layer primitive equations are compared with those obtained at the fixed points found at CE2 and CE3 levels of approximation. CE2 reproduces qualitative features of the zonal mean general circulation such as the mid-latitude jets. CE3 improves quantitative agreement of both the zonal means, and the teleconnections.

Generation and dissipation of large-scale baroclinic eddies at different latitudes is associated with eddy momentum fluxes between latitudes that maintain the general circulation. Wave breaking and absorption in the upper layer involve non-linear eddy-eddy interactions that are not captured at the CE2 level, accounting for some of its shortcomings . We describe numerically, and analyze theoretically, the ability of third order closures to capture flow statistics associated with wave breaking within a one-layer barotropic model. The two-layer primitive equations are interpreted in light of these single-layer results.