Statistical Corrections to Numerical Prediction Equations. II

Abstract

A procedure for statistical correction of numerical prediction equations at the end of each predictive time step is described and tested with a two-dimensional prediction model. The model equations are modified Burgers' equations which contain space and velocity dependent sources of energy to maintain the flow against dissipation. A detailed flow is calculated from a fine-grid numerical integration on a rectangular region with periodic boundary conditions. Coarse-grid values, for initiating and testing various coarse-grid prediction equations, are obtained by space-time mesh-box averages. Since the coarse-grid equations cannot represent subgrid-scale motions, statistical corrections are added in the form of parametric terms as a pragmatic substitute for the missing subgrid-scale effects. Tests with different forms of the equations show that substantial improvements can be obtained when the coefficients of the parametric terms are determined by multiple regression. Still further improvements are found when separate regression equations are calculated for each grid point and when the coefficients are adjusted in time to stabilize the calculations. The model equations are found to give periodic behavior in time despite somewhat complex sources of energy dependent upon the model geography and upon the flow itself. In addition, the predictions are stable to small perturbations of the initial conditions. It is concluded, therefore, that the coarse-grid prediction errors, which grow rapidly with time in the manner of real atmospheric prediction errors, are due entirely to the truncation errors of the coarse-grid equations.