A Semihydrostatic Theory of Gravity-Dominated Compressible Flow

From Hamilton's least-action principle, compressible equations of motion with density diagnosed from potential temperature through hydrostatic balance are derived. Slaving density to potential temperature suppresses the degrees of freedom supporting the propagation of acoustic waves and results in a soundproof system. The linear normal modes and dispersion relationship for an isothermal state of rest on f and beta planes are accurate from hydrostatic to nonhydrostatic scales, except for deep internal gravity waves. Specifically, the Lamb wave and long Rossby waves are not distorted, unlike with anelastic or pseudoincompressible systems. Compared to similar equations derived by A. Arakawa and C. S. Konor, the semihydrostatic system derived here possesses an additional term in the horizontal momentum budget. This term is an apparent force resulting from the vertical coordinate not being the actual height of an air parcel but its hydrostatic height (the hypothetical height it would have after the atmospheric column it belongs to has reached hydrostatic balance through adiabatic vertical displacements of air parcels). The Lagrange multiplier lambda introduced in Hamilton's principle to slave density to potential temperature is identified as the nonhydrostatic vertical displacement (i.e., the difference between the actual and hydrostatic heights of an air parcel). The expression of nonhydrostatic pressure and apparent force from lambda allow the derivation of a well-defined linear symmetric positive definite problem for lambda. As with hydrostatic equations, vertical velocity is diagnosed through Richardson's equation. The semihydrostatic system has therefore precisely the same degrees of freedom as the hydrostatic primitive equations, while retaining much of the accuracy of the fully compressible Euler equations.