GENERALIZED INVERSION OF A GLOBAL NUMERICAL WEATHER PREDICTION MODEL .2. ANALYSIS AND IMPLEMENTATION

This is a sequel to Bennett, Chua and Leslie (1996), concerning weak-c onstraint, four-dimensional variational assimilation of reprocessed cl oud-track wind observations (Velden, 1992) into a global, primitive-eq uation numerical weather prediction model. The assimilation is perform ed by solving the Euler-Lagrange equations associated with the variati onal principle. Bennett ct al. (1996) assimilate 2436 scalar wind comp onents into their model over a 24-hour interval, yielding a substantia lly improved estimate of the state of the atmosphere at the end of the interval. This improvement is still in evidence in forecasts for the next 48 hours. The model and variational equations are nonlinear, but are solved as sequence of linear equations. It is shown here that each linear solution is precisely equivalent to optimal or statistical int erpolation using a background error covariance derived from the linear ized dynamics, from the forcing error covariance, and from the initial error covariance. Bennett et al. (1996) control small-scale flow dive rgence using divergence dissipation (Talagrand, 1972). It is shown her e that this approach is virtually equivalent to including a penalty, f or the gradient of divergence, in the variational principle. The linea rized variational equations are salved in terms of the representer fun ctions for the wind observations. Diagonalizing the representer matrix yields rotation vectors. The rotated representers are the ''array mod es'' of the entire system of the model, prior covariances and observat ions. The modes are the ''observable'' degrees of freedom of the atmos phere. Several leading array modes are presented here. Finally, append ices discuss a number of technical implementation issues: time convolu tions, convergence in the presence of planetary shear instability, and preconditioning the essential inverse problem.