Asymptotics, structure, and integration of sound-proof atmospheric flow equations

Klein, Rupert

doi:10.1007/s00162-009-0104-y

Cited by 55 publications

(67 citation statements)

References 48 publications

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“…Here we assume that H u 5 H/k is sufficiently larger than H that adopting the anelastic approximation is reasonable (Klein 2009). By assuming an isothermal atmosphere, the corresponding stratification is uniform in that the squared buoyancy frequency…”

Section: B Equations Of Motion and Linear Theorymentioning

confidence: 99%

Anelastic Internal Wave Packet Evolution and Stability

Dosser

Sutherland

2011

Journal of the Atmospheric Sciences

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As upward-propagating anelastic internal gravity wave packets grow in amplitude, nonlinear effects develop as a result of interactions with the horizontal mean flow that they induce. This qualitatively alters the structure of the wave packet. The weakly nonlinear dynamics are well captured by the nonlinear Schrö dinger equation, which is derived here for anelastic waves. In particular, this predicts that strongly nonhydrostatic waves are modulationally unstable and so the wave packet narrows and grows more rapidly in amplitude than the exponential anelastic growth rate. More hydrostatic waves are modulationally stable and so their amplitude grows less rapidly. The marginal case between stability and instability occurs for waves propagating at the fastest vertical group velocity. Extrapolating these results to waves propagating to higher altitudes (hence attaining larger amplitudes), it is anticipated that modulationally unstable waves should break at lower altitudes and modulationally stable waves should break at higher altitudes than predicted by linear theory. This prediction is borne out by fully nonlinear numerical simulations of the anelastic equations. A range of simulations is performed to quantify where overturning actually occurs.

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Section: B Equations Of Motion and Linear Theorymentioning

confidence: 99%

Anelastic Internal Wave Packet Evolution and Stability

Dosser

Sutherland

2011

Journal of the Atmospheric Sciences

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“…The exposition in this section of the three sets of model equations to be analyzed subsequently closely follows Klein (2009). Here, we restrict our considerations to flows under gravity, but without Coriolis effects and nonresolved-scale closures, and present consistent dimensionless forms of the compressible Euler equations and of two soundproof models.…”

Section: Compressible and Soundproof Model Equationsmentioning

confidence: 99%

Regime of Validity of Soundproof Atmospheric Flow Models

Klein

Achatz

Bresch

et al. 2010

Journal of the Atmospheric Sciences

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Ogura and Phillips derived the original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. To arrive at a reduced model that would simultaneously represent internal gravity waves and the effects of advection on the same time scale, they had to adopt a distinguished limit requiring that the dimensionless stability of the background state be on the order of the Mach number squared. For typical flow Mach numbers of M ; 1 /30, this amounts to total variations of potential temperature across the troposphere of less than one Kelvin (i.e., to unrealistically weak stratification). Various generalizations of the original anelastic model have been proposed to remedy this issue. Later, Durran proposed the pseudoincompressible model following the same goals, but via a somewhat different route of argumentation. The present paper provides a scale analysis showing that the regime of validity of two of these extended models covers stratification strengths on the order of (h sc /u)du/dz , M 2/3 , which corresponds to realistic variations of potential temperature u across the pressure scale height h sc of Duj h sc 0 , 30 K. Specifically, it is shown that (i) for (h sc /u)du/dz , M m with 0 , m , 2, the atmosphere features three asymptotically distinct time scales, namely, those of advection, internal gravity waves, and sound waves; (ii) within this range of stratifications, the structures and frequencies of the linearized internal wave modes of the compressible, anelastic, and pseudoincompressible models agree up to the order of M m ; and (iii) if m , 2 /3, the accumulated phase differences of internal waves remain asymptotically small even over the long advective time scale. The argument is completed by observing that the three models agree with respect to the advective nonlinearities and that all other nonlinear terms are of higher order in M.

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“…The anelastic and pseudo‐incompressible systems have been successfully applied in mesoscale models, as both are able to accurately represent small‐scale features of atmospheric flow (Lafore et al . ; Klein, ; Kurowski et al . ; Smolarkiewicz et al .…”

Section: Introductionmentioning

confidence: 99%

Semi‐implicit integration of the unified equations in a mass‐based coordinate: model formulation and numerical testing

Voitus

Bénard

Kühnlein

et al. 2019

Quart J Royal Meteoro Soc

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This paper derives the unified equations of Arakawa and Konor rigorously formulated in a suitable mass‐based sigma coordinate and develops an efficient semi‐implicit integration scheme. The unified equations accurately capture the non‐hydrostatic small‐scale effects and retain the hydrostatic compressibility of the flow at large scales. As with the classical quasi‐hydrostatic equations, the underlying approximations filter vertically propagating acoustic waves. In contrast to the quasi‐hydrostatic equations, however, the filtering property of the unified equations requires that the wind field satisfy a divergence constraint similar to anelastic and pseudo‐incompressible (small‐scale limit) soundproof systems. An efficient semi‐implicit integration scheme for the unified equation system is achieved by combining a constant‐coefficient linear partitioning approach with an iterative implicit treatment of the nonlinear residuals arising from the soundproof divergence constraint. The resulting linear implicit problem to be solved at each iteration may be reduced to a single Helmholtz equation with horizontally homogeneous coefficients, which is akin to the one typically solved in the semi‐implicit integration of the quasi‐hydrostatic equations. The stability and accuracy of the developed semi‐implicit scheme for the unified equations in the mass‐based coordinate is numerically assessed by means of standard vertical plane test cases in linear and nonlinear atmospheric flow regimes. Moreover, in order to ascertain the convergence of the iterative semi‐implicit scheme, the test cases also include a large‐scale 3D configuration that resembles the stiffness typically encountered in global atmospheric models.

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Asymptotics, structure, and integration of sound-proof atmospheric flow equations

Cited by 55 publications

References 48 publications

Anelastic Internal Wave Packet Evolution and Stability

Anelastic Internal Wave Packet Evolution and Stability

Regime of Validity of Soundproof Atmospheric Flow Models

Semi‐implicit integration of the unified equations in a mass‐based coordinate: model formulation and numerical testing

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