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

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 differ...