The Lower Boundary Condition and Energy Consistency in Primitive and Filtered Models

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Abstract
The work of Lorenz (1960) and Charney (1962, 1973) on the use of scale and energy considerations for assessing approximations to the equations governing atmospheric motion is extended to cover the case where the correct kinematic boundary condition applies at the surface of the earth, using pressure as vertical coordinate. Scaled forms of the basic equations and corresponding energy equations are obtained as expansions in terms of the Rossby number as a small parameter, and reflecting an atmosphere which is quasi-hydrostatic, adiabatic, frictionless, quasi-geostrophic, stratified, of limited static stability and of moderate horizontal scale, and where the lower boundary is a suitably smoothed version of the earth's surface. The requirement of either strict conservation of total energy or conservation to a chosen order in the Rossby number is presented as a criterion for finding consistent approximations to the basic equations. For the strict lower boundary condition, the requirement (where differentiated momentum equations are used) that terms in the same Lorenz class must be omitted or retained together, must be replaced by other requirements which yield other sets of terms. When the simpler boundary condition used by Lorenz is applied, these sets generally reduce to Lorenz classes, but this analysis succeeds in providing greater freedom for certain Coriolis and thermodynamic equation terms. In the undifferentiated case, strict conservation with the strict lower boundary condition implies that the “primitive” equations constitute the only (p-coordinate) set which is meaningful in terms of both scale and energy consistency. If conservation is required only to a chosen order, further simplification is possible in the undifferentiated momentum equation, while the simplifications obtained when using the simpler boundary condition with the differentiated momentum equations are extended to apply also with the strict boundary condition. Various examples are presented to illustrate the application of the method.

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