Unification of the Anelastic and Quasi-Hydrostatic Systems of Equations

A system of equations is presented that unifies the nonhydrostatic anelastic system and the quasi-hydrostatic compressible system for use in global cloud-resolving models. By using a properly defined quasi-hydrostatic density in the continuity equation, the system is fully compressible for quasi-hyd...

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Bibliographic Details
Published in:Monthly weather review 2009-02, Vol.137 (2), p.710-726
Main Authors: ARAKAWA, Akio, KONOR, Celal S
Format: Article
Language:English
Subjects:
Acoustics
Anelasticity
Approximation
Barotropic mode
Compressibility
Continuity equation
Density
Divergence
Earth, ocean, space
Energy conservation
Exact sciences and technology
External geophysics
Flow control
Mathematical analysis
Meteorology
Momentum
Numerical analysis
Planetary waves
Pressure
Rossby waves
Scale models
Sound propagation
Sound waves
Surface pressure
Thermodynamics
Turbulence
Wave propagation
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Description
Summary:A system of equations is presented that unifies the nonhydrostatic anelastic system and the quasi-hydrostatic compressible system for use in global cloud-resolving models. By using a properly defined quasi-hydrostatic density in the continuity equation, the system is fully compressible for quasi-hydrostatic motion and anelastic for purely nonhydrostatic motion. In this way, the system can cover a wide range of horizontal scales from turbulence to planetary waves while filtering vertically propagating sound waves of all scales. The continuity equation is primarily diagnostic because the time derivative of density is calculated from the thermodynamic (and surface pressure tendency) equations as a correction to the anelastic continuity equation. No reference state is used and no approximations are made in the momentum and thermodynamic equations. An equation that governs the time change of total energy is also derived. Normal-mode analysis on an f plane without the quasigeostrophic approximation and on a midlatitude β plane with the quasigeostrophic approximation is performed to compare the unified system with other systems. It is shown that the unified system reduces the westward retrogression speed of the ultra-long barotropic Rossby waves through the inclusion of horizontal divergence due to compressibility.
ISSN:0027-0644
1520-0493
DOI:10.1175/2008MWR2520.1