A Semi-Hydrostatic Theory of Gravity-Dominated Compressible Flow

Abstract:

Compressible Euler equations support the propagation of acoustic waves. Although much progress has been achieved towards efficient and accurate solutions to the resulting numerical difficulties, it can still be desirable to identify “unified” equations of motion that would not support acoustic waves while retaining accuracy at large and small scales. Even if such equations are eventually not chosen as the basis of a numerical model, they may help identifying the independent degrees of freedom of the atmospheric flow to be modeled and how the dependent fields are related to the independent fields.

From Hamilton's least action principle (HP), "semi-hydrostatic" compressible equations of motion with density diagnosed from potential temperature through hydrostatic balance are derived. Energy, potential vorticity and momentum are conserved. Slaving density to potential temperature suppresses the degrees of freedom supporting the propagation of acoustic waves and results in a sound-proof system. Scale analysis and linear normal modes analysis for an isothermal state of rest suggest that the semy-hydrostatic system is accurate from hydrostatic to non-hydrostatic scales, except for deep internal gravity waves (Figure : decimal logarithm of relative error of the frequency of internal normal modes of a non-rotating isothermal atmosphere as a function of horizontal and vertical wavenumbers k,m normalized by the scale height H). Especially the Lamb wave and long Rossby waves are not distorted, unlike with anelastic or pseudo-incompressible systems.

Compared to similar equations derived by Arakawa and Konor (2009), the semi-hydrostatic system 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, i.e. the hypothetical height it would have after the atmospheric column it belongs to has reached hydrostatic balance through adiabatic vertical displacements of air parcels. As with hydrostatic prmitive equations (HPE), vertical velocity is diagnosed through Richardson's equation. The semi-hydrostatic system has therefore precisely the same degrees of freedom as the HPE, while retaining much of the accuracy of the fully compressible Euler equations.