DYNAMICAL STRUCTURE FUNCTIONS IN A 4-DIMENSIONAL VARIATIONAL ASSIMILATION - A CASE-STUDY

This paper contributes to the understanding of the structure functions used implicitly in the four-dimensional variational assimilation (4D- Var) developed at the European Centre for Medium-Range Weather Forecas ts in the last few years. The theoretical equivalence between 4D-Var a nd the Kalman filter allows us to interpret (after normalization by th e error standard deviations) the analysis increments produced by one s ingle observation as the structure functions used implicitly in 4D-Var . The shape of the analysis increments provides a three-dimensional pi cture of the covariances of the background errors, modified by the dyn amics. We study a baroclinic situation and observations have been regu larly distributed along a latitude circle crossing the baroclinic wave . Eight standard pressure levels have been considered to sample the ve rtical. The forecast error standard deviations and the structure funct ions implied in 4D-Var may differ considerably from those used in the 3D-Var analysis. Unlike 3D-Var, the structure functions are flow depen dent: the effective background error standard deviation can be four ti mes larger and the correlation length scale twice as short in the vici nity of a low. A meridional extension of the experimentation at the su rface shows that the effective background error standard deviations at 1000 hPa are largest in the areas of strong pressure gradient. We qua ntify the link between the analysis increments produced by 4D-Var and the fastest growing perturbations over the same time interval. In the depression, the explained variance of the analysis increments by the f irst 13 singular vectors reaches 30%. The impact of the temporal dimen sion is assessed. A period of 24 hours seems a minimum for the increme nts to develop fully baroclinic structures.