Stochastic-Dynamic Prediction of Three-Dimensional Quasi-Geostrophic Flow

Abstract

This paper concerns the problem of predicting the variance of a large ensemble of predictions evolving from an ensemble of slightly incorrect initial states, randomly distributed around a ?most probable? initial state. Starting with the vorticity and thermodynamic energy equations for three-dimensional quasi-geostrophic flow, we derive evolution equations for the ensemble-averaged potential vorticity, the variance of deviations from that average, and the mean transport of those deviations. Under the assumption that the ensemble of initial error fields is statistically isotropic and homogeneous in horizontal planes, the resulting stochastic equations comprise a complete system with second-moment closure. Those equations provide the basis for a simple method of predicting the spatial distribution of the probable error of predictions. Whether the local error-variance grows or decays is crucially dependent on the detailed structure and ?local? scale of the ?true? field of potential vorticity, relative to the characteristic scale of the error fields. If the local scale of the vorticity field is very large, the error-variance grows very slowly or may even decrease; if it is very small, the error variance grows rapidly. The proposed method of stochastic-dynamic prediction involves only one inversion per time step, and thus requires only marginally more computation than deterministic prediction.