Improving the Anelastic Approximation

Dale R. Durran Department of Atmospheric Sciences, University of Washington, Seattle, Washington

Search for other papers by Dale R. Durran in
Current site
Google Scholar
PubMed
Close
Restricted access

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 “pseudo-incompressible 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 mean-state pressure. No assumption about the magnitude of the perturbation potential temperature or the strength of the mean-state 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 mass-balance in the “pseudo-incompressible 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 pseudo-incompressible equation is identical to the anelastic continuity equation when the mean stratification is adiabatic. As the stability increases, the pseudo-incompressible approximation gives a more accurate result. The pseudo-incompressible equation, together with the unapproximated momentum and thermodynamic equations, forms a closed system of governing equations that filters sound waves. The pseudo-incompressible system conserves an energy form that is directly analogous to the total energy conserved by the complete compressible system.

The pseudo-incompressible approximation yields a system of equations suitable for use in nonhydrostatic numerical models. The pseudo-incompressible equation also permits the diagnostic calculation of the vertical velocity in adiabatic flow. The pseudo-incompressible equation might also be used to compute the net heating rate in a diabatic flow from extremely accurate observations of the three-dimensional velocity field and very coarse resolution (single sounding) thermodynamic data.

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 “pseudo-incompressible 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 mean-state pressure. No assumption about the magnitude of the perturbation potential temperature or the strength of the mean-state 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 mass-balance in the “pseudo-incompressible 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 pseudo-incompressible equation is identical to the anelastic continuity equation when the mean stratification is adiabatic. As the stability increases, the pseudo-incompressible approximation gives a more accurate result. The pseudo-incompressible equation, together with the unapproximated momentum and thermodynamic equations, forms a closed system of governing equations that filters sound waves. The pseudo-incompressible system conserves an energy form that is directly analogous to the total energy conserved by the complete compressible system.

The pseudo-incompressible approximation yields a system of equations suitable for use in nonhydrostatic numerical models. The pseudo-incompressible equation also permits the diagnostic calculation of the vertical velocity in adiabatic flow. The pseudo-incompressible equation might also be used to compute the net heating rate in a diabatic flow from extremely accurate observations of the three-dimensional velocity field and very coarse resolution (single sounding) thermodynamic data.

Save