The Constraints of Energy-Conserving Vertical Finite Difference on the Hydrostatic Equations in a NWP Model

Samuel Y. K. Yee Air Force Geophysics Laboratory, Hanscom AFB, MA 01731

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Abstract

In NWP models using energy-conserving finite-difference approximations in the vertical, the imposition of different constraints of the discrete energy equation leads to different forms of the hydrostatic equation. This paper shows, using the National Meteorological Center spectral model as a specific example, that the consistent application of all the constraints suggested by Phillips (1974) on the discrete energy equation leads to an algebraic hydrostatic system of (3K–1) unknowns and equations, K being the number of layers in the model. It is emphasized that in relating the vertical structure between the mass and thermal fields, these and only these equations must be satisfied. The introduction of any additional equations without introducing an equal number of unknowns may defeat the purpose of an energy-conserving finite-difference scheme.

Abstract

In NWP models using energy-conserving finite-difference approximations in the vertical, the imposition of different constraints of the discrete energy equation leads to different forms of the hydrostatic equation. This paper shows, using the National Meteorological Center spectral model as a specific example, that the consistent application of all the constraints suggested by Phillips (1974) on the discrete energy equation leads to an algebraic hydrostatic system of (3K–1) unknowns and equations, K being the number of layers in the model. It is emphasized that in relating the vertical structure between the mass and thermal fields, these and only these equations must be satisfied. The introduction of any additional equations without introducing an equal number of unknowns may defeat the purpose of an energy-conserving finite-difference scheme.

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