On the Choice of Momentum Control Variables and Covariance Modeling for Mesoscale Data Assimilation

Abstract

For mesoscale variational data assimilation with high-resolution observations, there has been an issue concerning the choice of momentum control variables and related covariance modeling. This paper addresses the theoretical aspect of this issue. First, relationships between background error covariance functions for differently chosen momentum control variables are derived, and different choices of momentum control variables are proven to be theoretically equivalent in the sense that they lead to the same optimally analyzed incremental wind field in the limit of infinitely high spatial resolution provided their error covariance functions satisfy the derived relationships. It is then shown that when the velocity potential ? and streamfunction ? are used as momentum control variables with their background error autocovariance functions modeled by single-Gaussian functions, the derived velocity autocovariance functions contain significant negative sidelobes. These negative sidelobes can represent background wind error structures associated with baroclinic waves on the synoptic scale but become unrepresentative on the mesoscale. To reduce or remove these negative sidelobes for mesoscale variational data assimilation, Gaussian functions are used with two types of modifications to model the velocity covariance functions in consistency with the assumed homogeneity and isotropy in variational data assimilation. In this case, the random (?, ?) background error fields have no classically valid homogeneous and isotropic covariance functions, but generalized (?, ?) covariance functions can be derived from the modified velocity covariance functions for choosing (?, ?) as momentum control variables. Mathematical properties of generalized covariance functions are explored with physical interpretations. Their important implications are discussed for mesoscale data assimilation.