# MFE

## Seminar

### Direct Statistical Simulation of a Two-Layer Primitive Equation Model

Seminar Room 1, Newton Institute

#### Abstract

Co-authors: Wanming Qi (Brown University), Steve Tobias (University of Leeds)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 statistics. We implement such Direct Statistical Simulation of a two-layer primitive equation model by systematic expansion in 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 nonlocal teleconnections. At second order (CE2) the expansion retains the eddy – mean-flow interaction but neglects eddy-eddy interactions and is realizable. Eddy-eddy interactions appear at third (CE3) order, but care must be taken to maintain realizability with a non-negative probability distribution function. The cumulant expansion is conservative, order-by order, in the total angular momentum, total energy, and mean-squared potential temperature. An intermediate approximation, CE2.5, is related to the Edd y-Damped Quasi-Normal Markovian (EDQNM) approximation and maintains realizability at the expense of the introduction of a phenomenological timescale for eddy damping. First and second cumulants accumulated by time-integration of the two-layer primitive equations are compared with those obtained at the fixed points found at CE2, CE2.5, and CE3 levels of approximation, and against statistics obtained from reanalysis of the mid-level atmosphere of the Earth. CE2 reproduces qualitative features of the zonal mean general circulation such as the mid-latitude jets. CE2.5 and CE3 improve quantitative agreement in both the zonal means, and in the teleconnections.

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