Unification of the Anelastic and Quasi-Hydrostatic Systems of Equations

Abstract

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 b 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.

Multi-scale waves in sound-proof global simulations with EULAG

“…We use the f -plane approximation, eq. (6.17) of Arakawa and Konor (2009), which is identical to eq. (5.25) in Davies et al (2003).…”

“…To estimate this effect, the β-plane approximation, eq. (6.26) of Arakawa and Konor (2009) is utilized with horizontal and vertical wavelengths of ≈ 4000 and 15 km, respectively, 40°lat-itude, H ρ ≈ 7.7 km, and T = 273 K. This comparison (i.e., their eq. (6.26) with and without the missing terms of the LH anelastic equations) shows that relative to the fully compressible model, LH anelastic baroclinic modes have intrinsic frequency and phase errors of ≈ 0.1%, and intrinsic group velocity errors of ≈ 0.2% and 0.6% in the horizontal and vertical, respectively.…”

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

“…There are different solutions proposed in the literature, such as a priori filtering of acoustic modes, i.e. sound-proof models, or by solving the compressible Euler equations and treating the acoustic waves via implicit-explicit or fully-implicit time-integrators [4,10,30,90,92]. While filtering the acoustic modes is attractive, the numerical solution procedure of the resulting filtered equations may be more difficult, hence there is still no consensus in the weather community towards one particular model [69], although the unfiltered compressible Euler equations with a numerical handling of acoustic modes is currently favored.…”

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations – Part 1: Nonhydrostatic inertia–gravity modes

“…The anelastic system also excludes the physically insignificant acoustic waves and the Lamb wave, so that we can focus on the much more important inertia-gravity and Rossby waves. Although the anelastic system has inaccuracies, as pointed out by Davies et al (2003), Arakawa and Konor (2009), Dukowicz (2013) and Dubos and Voitus (2014), a wide range of nonhydrostatic phenomena are accurately described by the anelastic system. Arakawa and Konor (2009) show that among the features that can be well simulated by the anelastic system are the frequency of the inertia-gravity modes for all horizontal scales and the frequency of the middle-latitude Rossby modes for medium and small horizontal scales.…”