On the Shallow Motion Approximations

Abstract

The approximate equations for shallow motions are derived mainly by following the approach of Spiegel and Veronis and the subsequent development of Dutton and Fichtl. Other derivations are also briefly noted. While each derivation assumes shallow flow, the conditions on the time scale and auxiliary assumptions vary between derivations. In the present study, the shallow motion approximations are found to be valid for a wider range of conditions than included in earlier derivations. The more restrictive Boussinesq or ?shallow convection? approximations form a subclass motions. Existing derivations of the full Boussinesq approximations do not apply to near-neutral conditions even though they are often applied to such conditions. The conditions required for the validity of the Boussinesq approximations are reformulated into criteria that are easier to evaluate. Finally; the use of the shallow motion approximations in concert with Reynolds averaging is examined in some detail. Additional necessary conditions resulting from Reynolds averaging appear to be violated only in rather special situations, at least for atmospheric flows.