A Balanced Diagnostic System Compatible with a Barotropic Prognostic Model

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Abstract
We derive a system of diagnostic equations for the velocity field, or “wind laws,” for a barotropic primitive-equation model of large-scale atmospheric flow. The derivation is mathematically exact and does not involve any physical assumptions, such as nondivergence or vanishing of derivatives of the divergence, which are not already present in the prognostic equations. Therefore, initial states computed by solving these diagnostic equations should be compatible with the type of motion described by the prognostic equations of the model, and should not generate initialization shocks when inserted into the prognostic model. Based on the diagnostic system obtained, we are able to give precise meaning to the question whether the wind field is determined by the mass field and by its time history. The answer to this important question is affirmative, in the precise formulation we provide. The diagnostic system corresponding to the chosen barotropic model is a generalization of the classical balance equation. The ellipticity condition for this system is derived and given a physical interpretation. Numerical solutions of the diagnostic system are exhibited, including cases in-which the system is of mixed elliptic-hyperbolic type. Such diagnostic systems can be obtained for other primitive equation models. They are valid for all atmospheric scales and regions for which the prognostic models from which they are derived hold. Some problems concerning the possibility of implementing such a system in operational numerical weather prediction are discussed.

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