Editorial Type:
Article
Article Type:
Research Article

A Low Mach Number Model for Moist Atmospheric Flows

Max Duarte Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California

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Ann S. Almgren Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California

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John B. Bell Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, California

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Print Publication:
01 Apr 2015
DOI:
https://doi.org/10.1175/JAS-D-14-0248.1
Page(s):
1605–1620
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Abstract

A low Mach number model for moist atmospheric flows is introduced that accurately incorporates reversible moist processes in flows whose features of interest occur on advective rather than acoustic time scales. Total water is used as a prognostic variable, so that water vapor and liquid water are diagnostically recovered as needed from an exact Clausius–Clapeyron formula for moist thermodynamics. Low Mach number models can be computationally more efficient than a fully compressible model, but the low Mach number formulation introduces additional mathematical and computational complexity because of the divergence constraint imposed on the velocity field. Here, latent heat release is accounted for in the source term of the constraint by estimating the rate of phase change based on the time variation of saturated water vapor subject to the thermodynamic equilibrium constraint. The authors numerically assess the validity of the low Mach number approximation for moist atmospheric flows by contrasting the low Mach number solution to reference solutions computed with a fully compressible formulation for a variety of test problems.

Corresponding author address: Max Duarte, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd. MS 50A-1148, Berkeley, CA 94720. E-mail: mdgonzalez@lbl.gov

Abstract

A low Mach number model for moist atmospheric flows is introduced that accurately incorporates reversible moist processes in flows whose features of interest occur on advective rather than acoustic time scales. Total water is used as a prognostic variable, so that water vapor and liquid water are diagnostically recovered as needed from an exact Clausius–Clapeyron formula for moist thermodynamics. Low Mach number models can be computationally more efficient than a fully compressible model, but the low Mach number formulation introduces additional mathematical and computational complexity because of the divergence constraint imposed on the velocity field. Here, latent heat release is accounted for in the source term of the constraint by estimating the rate of phase change based on the time variation of saturated water vapor subject to the thermodynamic equilibrium constraint. The authors numerically assess the validity of the low Mach number approximation for moist atmospheric flows by contrasting the low Mach number solution to reference solutions computed with a fully compressible formulation for a variety of test problems.

Corresponding author address: Max Duarte, Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd. MS 50A-1148, Berkeley, CA 94720. E-mail: mdgonzalez@lbl.gov
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  • Almgren, A. S., 2000: A new look at the pseudo-incompressible solution to Lamb’s problem of hydrostatic adjustment. J. Atmos. Sci., 57, 995–998, doi:10.1175/1520-0469(2000)057<0995:ANLATP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Almgren, A. S., J. B. Bell, C. A. Rendleman, and M. Zingale, 2006a: Low Mach number modeling of Type Ia supernovae. I. Hydrodynamics. Astrophys. J., 637, 922–936, doi:10.1086/498426.

    • Search Google Scholar
    • Export Citation

  • Almgren, A. S., J. B. Bell, C. A. Rendleman, and M. Zingale, 2006b: Low Mach number modeling of Type Ia supernovae. II. Energy evolution. Astrophys. J., 649, 927–938, doi:10.1086/507089.

    • Search Google Scholar
    • Export Citation

  • Almgren, A. S., J. B. Bell, A. Nonaka, and M. Zingale, 2008: Low Mach number modeling of Type Ia supernovae. III. Reactions. Astrophys. J., 684, 449–470, doi:10.1086/590321.

    • Search Google Scholar
    • Export Citation

  • Almgren, A. S., J. B. Bell, A. Nonaka, and M. Zingale, 2014: Low Mach number modeling of stratified flows. Finite Volumes for Complex Applications VII—Methods and Theoretical Aspects, J. Fuhrmann, M. Ohlberger, and C. Rohde, Eds., Springer Proceedings in Mathematics and Statistics, Vol. 77, Springer, 3–15, doi:10.1007/978-3-319-05684-5_1.

  • Batchelor, G. K., 1953: The conditions for dynamical similarity of motions of a frictionless perfect-gas atmosphere. Quart. J. Roy. Meteor. Soc., 79, 224–235, doi:10.1002/qj.49707934004.

    • Search Google Scholar
    • Export Citation

  • Betts, A. K., 1973: Non–precipitating cumulus convection and its parameterization. Quart. J. Roy. Meteor. Soc., 99, 178–196, doi:10.1002/qj.49709941915.

    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 2917–2928, doi:10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Clark, T. L., 1979: Numerical simulations with a three-dimensional cloud model: Lateral boundary condition experiments and multicellular severe storm simulations. J. Atmos. Sci., 36, 2191–2215, doi:10.1175/1520-0469(1979)036<2191:NSWATD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Clark, T. L., and R. D. Farley, 1984: Severe downslope windstorm calculations in two and three spatial dimensions using anelastic interactive grid nesting: A possible mechanism for gustiness. J. Atmos. Sci., 41, 329–350, doi:10.1175/1520-0469(1984)041<0329:SDWCIT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Duarte, M., A. S. Almgren, K. Balakrishnan, J. B. Bell, and D. M. Romps, 2014: A numerical study of methods for moist atmospheric flows: Compressible equations. Mon. Wea. Rev., 142, 4269–4283, doi:10.1175/MWR-D-13-00368.1.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1989: Improving the anelastic approximation. J. Atmos. Sci., 46, 1453–1461, doi:10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Dutton, J. A., and G. H. Fichtl, 1969: Approximate equations of motion for gases and liquids. J. Atmos. Sci., 26, 241–254, doi:10.1175/1520-0469(1969)026<0241:AEOMFG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Gough, D. O., 1969: The anelastic approximation for thermal convection. J. Atmos. Sci., 26, 448–456, doi:10.1175/1520-0469(1969)026<0448:TAAFTC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Grabowski, W. W., and P. K. Smolarkiewicz, 1990: Monotone finite-difference approximations to the advection-condensation problem. Mon. Wea. Rev., 118, 2082–2097, doi:10.1175/1520-0493(1990)118<2082:MFDATT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Grabowski, W. W., and T. L. Clark, 1991: Cloud-environment interface instability: Rising thermal calculations in two spatial dimensions. J. Atmos. Sci., 48, 527–546, doi:10.1175/1520-0469(1991)048<0527:CIIRTC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Grabowski, W. W., and P. K. Smolarkiewicz, 2002: A multiscale anelastic model for meteorological research. Mon. Wea. Rev., 130, 939–956, doi:10.1175/1520-0493(2002)130<0939:AMAMFM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Hauf, T., and H. Höller, 1987: Entropy and potential temperature. J. Atmos. Sci., 44, 2887–2901, doi:10.1175/1520-0469(1987)044<2887:EAPT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Klein, R., 2009: Asymptotics, structure, and integration of sound-proof atmospheric flow equations. Theor. Comput. Fluid Dyn., 23, 161–195, doi:10.1007/s00162-009-0104-y.

    • Search Google Scholar
    • Export Citation

  • Klein, R., and O. Pauluis, 2012: Thermodynamic consistency of a pseudoincompressible approximation for general equations of state. J. Atmos. Sci., 69, 961–968, doi:10.1175/JAS-D-11-0110.1.

    • Search Google Scholar
    • Export Citation

  • Klemp, J. B., and R. B. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci., 35, 1070–1096, doi:10.1175/1520-0469(1978)035<1070:TSOTDC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Klemp, J. B., W. C. Skamarock, and J. Dudhia, 2007: Conservative split-explicit time integration methods for the compressible nonhydrostatic equations. Mon. Wea. Rev., 135, 2897–2913, doi:10.1175/MWR3440.1.

    • Search Google Scholar
    • Export Citation

  • Lipps, F., and R. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci., 39, 2192–2210, doi:10.1175/1520-0469(1982)039<2192:ASAODM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nonaka, A., A. S. Almgren, J. B. Bell, M. J. Lijewski, C. Malone, and M. Zingale, 2010: MAESTRO: An adaptive low Mach number hydrodynamics algorithm for stellar flows. Astrophys. J., Suppl. Ser., 188, 358–383, doi:10.1088/0067-0049/188/2/358.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19, 173–179, doi:10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • O’Neill, W. P., and R. Klein, 2014: A moist pseudo-incompressible model. Atmos. Res., 142, 133–141, doi:10.1016/j.atmosres.2013.08.004.

    • Search Google Scholar
    • Export Citation

  • Ooyama, K. V., 1990: A thermodynamic foundation for modeling the moist atmosphere. J. Atmos. Sci., 47, 2580–2593, doi:10.1175/1520-0469(1990)047<2580:ATFFMT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2008: The dry-entropy budget of a moist atmosphere. J. Atmos. Sci., 65, 3779–3799, doi:10.1175/2008JAS2679.1.

  • Satoh, M., 2003: Conservative scheme for a compressible nonhydrostatic model with moist processes. Mon. Wea. Rev., 131, 1033–1050, doi:10.1175/1520-0493(2003)131<1033:CSFACN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Soong, S.-T., and Y. Ogura, 1973: A comparison between axisymmetric and slab-symmetric cumulus cloud models. J. Atmos. Sci., 30, 879–893, doi:10.1175/1520-0469(1973)030<0879:ACBAAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Tapp, M. C., and P. W. White, 1976: A non-hydrostatic mesoscale model. Quart. J. Roy. Meteor. Soc., 102, 277–296, doi:10.1002/qj.49710243202.

    • Search Google Scholar
    • Export Citation

  • Tripoli, G. J., and W. R. Cotton, 1981: The use of ice-liquid water potential temperature as a thermodynamic variable in deep atmospheric models. Mon. Wea. Rev., 109, 1094–1102, doi:10.1175/1520-0493(1981)109<1094:TUOLLW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Vasil, G. M., D. Lecoanet, B. P. Brown, T. S. Wood, and E. G. Zweibel, 2013: Energy conservation and gravity waves in sound-proof treatments of stellar interiors. II. Lagrangian constrained analysis. Astrophys. J., 773, 169, doi:10.1088/0004-637X/773/2/169.

    • Search Google Scholar
    • Export Citation
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