Improving the Anelastic Approximation.
Abstract
A new diagnostic equation is presented which exhibits many advantages over the conventional forms of the anelastic continuity equation. Scale analysis suggests that use of this `pseudoincompressible equation' is justified if the Lagrangian time scale of the disturbance is large compared with the time scale for sound wave propagation and the perturbation pressure is small compared to the vertically varying meanstate pressure. No assumption about the magnitude of the perturbation potential temperature or the strength of the meanstate stratification is required.In the various anelastic approximations, the influence of the perturbation density field on the mass balance is entirely neglected. In contrast, the massbalance in the `pseudoincompressible approximation' accounts for those density perturbations associated (through the equation of state) with perturbations in the temperature field. Density fluctuations associated with perturbations in the pressure field are neglected.The pseudoincompressible equation is identical to the anelastic continuity equation when the mean stratification is adiabatic. As the stability increases, the pseudoincompressible approximation gives a more accurate result. The pseudoincompressible equation, together with the unapproximated momentum and thermodynamic equations, forms a closed system of governing equations that filters sound waves. The pseudoincompressible system conserves an energy form that is directly analogous to the total energy conserved by the complete compressible system.The pseudoincompressible approximation yields a system of equations suitable for use in nonhydrostatic numerical models. The pseudoincompressible equation also permits the diagnostic calculation of the vertical velocity in adiabatic flow. The pseudoincompressible equation might also be used to compute the net heating rate in a diabatic flow from extremely accurate observations of the threedimensional velocity field and very coarse resolution (single sounding) thermodynamic data.
 Publication:

Journal of Atmospheric Sciences
 Pub Date:
 June 1989
 DOI:
 10.1175/15200469(1989)046<1453:ITAA>2.0.CO;2
 Bibcode:
 1989JAtS...46.1453D