An analytical solution for linear gravity and sound waves on the sphere as a test for compressible, non-hydrostatic numerical models

Abstract

An analytical solution for the expansion of gravity and sound waves for the linearized form of the fully compressible, non‐hydrostatic, shallow atmosphere Euler equations on the sphere is derived. The waves are generated by a weak initial temperature and density perturbation of an isothermal atmosphere. The derived analytical solution can be used as a benchmark to assess dynamical cores of global models based on the above‐mentioned (in general nonlinear) equation system. Three different test configurations, with or without Coriolis force or additional advection, are discussed. Convergence studies of the newly developed ICOsahedral Non‐hydrostatic global model (ICON) against this solution are performed.

The ICON (ICOsahedral Non‐hydrostatic) modelling framework of DWD and MPI‐M: Description of the non‐hydrostatic dynamical core

“…The related parameter δ 1 (with δ 2 = 1 − δ 1 ) for and has a default value of 0.4, which has been found to optimize the numerical stability for combinations of coarse horizontal resolution (mesh sizes of 100 km or more) with very high model tops (75 km or more). In dynamical core tests considering sound wave propagation, like that proposed by Baldauf et al (2014), α = 0, β 1 = 0.5 and δ 1 = 0.5 deliver the closest agreement with the analytical solution, with the caveat that non‐zero large‐scale winds require β 1 to be slightly larger than 0.5 in order to avoid numerical instabilities near the pentagon points of the icosahedron. Note that, for efficiency reasons, and are used instead of and in the predictor step.…”

“…To demonstrate the functionality and the quality of the dynamical core described in the preceding section, results from a hierarchy of tests of varying complexity will be presented in the following. In addition, we refer the reader to the sound-wave-gravity-wave test of Baldauf et al (2014), where the ICON dynamical core is shown to achieve second-order convergence against the analytical solution derived in that work, provided that the above-mentioned damping mechanisms for sound waves (section 2.4) are turned off.…”

“…The time level ñ , at which the horizontal pressure gradient is taken, is an extrapolated time level using levels n and n − 1: with α typically ranging between a third for mesh sizes smaller than about 40 km and two‐thirds for very coarse resolutions. As revealed by the combined sound‐wave–gravity‐wave test developed by Baldauf et al (2014), the extrapolation in time damps horizontally propagating sound waves without exerting a significant impact on gravity waves, which increases the numerical stability range with respect to evaluating the pressure gradient only at time level n . It can also be interpreted as an approximation of divergence damping by reformulating the terms ∼ α as a time derivative of π and combining them with Eq.…”

A moist aquaplanet variant of the Held–Suarez test for atmospheric model dynamical cores

“…Differences in GCMs are apparent when comparing models with different parameterizations, computational grids, and numerical methods (e.g., Lauritzen et al, 2010;Blackburn et al, 2013). However, the interactions and feedbacks between the dynamical core and physical parameterizations make it difficult to diagnose sources of error or clearly distin-Published by Copernicus Publications on behalf of the European Geosciences Union.…”

“…In an ideal situation, dynamical core tests should evaluate the fluid flow by directly comparing the model results to a known analytic solution. However, analytical solutions are not available for complex simulations and can only be used to evaluate very idealized flow conditions, such as steady states (Jablonowski and Williamson, 2006), linear flow regimes (Baldauf et al, 2014), or the advection of passive tracers with prescribed wind fields (Kent et al, 2014). Dynamical core tests for more complex, nonlinear flow scenarios without a known solution rely on the premises that models tend to converge toward a high-resolution reference solution and the results of multiple dynamical cores closely resemble each other within some uncertainty limit (Jablonowski and Williamson, 2006).…”