Numerical simulations of the humid
atmosphere above a mountain

• Authors: Arthur Bousquet , Mickaël D. Chekroun , Youngjoon Hong , Roger M. Temam , Joseph Tribbia
• Languages of publication: EN

Abstracts

EN

New avenues are explored for the numerical study of the two dimensional inviscid hydrostatic primitive
equations of the atmosphere with humidity and saturation, in presence of topography and subject to
physically plausible boundary conditions for the system of equations. Flows above a mountain are classically
treated by the so-called method of terrain following coordinate system. We avoid this discretization
method which induces errors in the discretization of tangential derivatives near the topography. Instead we
implement a first order finite volume method for the spatial discretization using the initial coordinates x
and p. A compatibility condition similar to that related to the condition of incompressibility for the Navier-
Stokes equations, is introduced. In that respect, a version of the projection method is considered to enforce
the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. For
the spatial discretization, a modified Godunov type method that exploits the discrete finite-volume derivatives
by using the so-called Taylor Series Expansion Scheme (TSES), is then designed to solve the equations.
We report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale
forcing on the velocity equation is numerically investigated.

Keywords

EN

primitive equations; humidity; finite volume; phase change; projection method; stochastic parameterizations

• Publisher: De Gruyter Open
• Journal: Mathematics of Climate and Weather Forecasting
• Year: 2015
• Volume: 1
• Issue: 1

Dates

received

23 - 7 - 2015

accepted

30 - 11 - 2015

online

31 - 12 - 2015

Contributors

author

Arthur Bousquet

• Department of Mathematics, Pennsylvania State University, PA, USA

author

Mickaël D. Chekroun

• Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA, USA

author

Youngjoon Hong

• Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, IL, USA

author

Roger M. Temam

• Temam: Institute for Scientific Computing and Applied Mathematics, Indiana University,
  Bloomington, Indiana, USA

author

Joseph Tribbia

• Climate Dynamics and Predictability (CDP) section in the Division of Climate and Global Dynamics (CGD) at
  the National Center for Atmospheric Research (NCAR)

References

• [1] Arthur Bousquet, Michele Coti Zelati, and Roger Temam. Phase transition models in atmospheric dynamics. Milan Journalof Mathematics, pages 1–30, 2014.
• [2] Arthur Bousquet, Gung-Min Gie, Youngjoon Hong, and Jacques Laminie. A higher order finite volume resolution method fora system related to the inviscid primitive equations in a complex domain. Numerische Mathematik, 128(3):431–461, 2014.
• [3] Chongsheng Cao and Edriss S. Titi. Global well-posedness of the three-dimensional viscous primitive equations of largescale ocean and atmosphere dynamics. Ann. of Math. (2), 166(1):245–267, 2007.
• [4] M. D. Chekroun, J. D. Neelin, D. Kondrashov, J. C. McWilliams, and M. Ghil. Rough parameter dependence in climate modelsand the role of ruelle-pollicott resonances. Proceedings of the National Academy of Sciences, 111(5):1684–1690, 2014.
• [5] M.D. Chekroun, D. Kondrashov, and M. Ghil. Predicting stochastic systems by noise sampling, and application to the elniño-southern oscillation. Proceedings of the National Academy of Sciences, 108(29):11766–11771, 2011.
• [6] Q. S. Chen, J. Laminie, A. Rousseau, R. Temam, and J. Tribbia. A 2.5D model for the equations of the ocean and the atmosphere.Anal. Appl. (Singap.), 5(3):199–229, 2007.[Crossref]
• [7] Qingshan Chen, Ming-Cheng Shiue, and Roger Temam. The barotropic mode for the primitive equations. J. Sci. Comput.,45(1-3):167–199, 2010.[Crossref]
• [8] Qingshan Chen, Ming-Cheng Shiue, Roger Temam, and Joseph Tribbia. Numerical approximation of the inviscid 3D primitiveequations in a limited domain. ESAIM Math. Model. Numer. Anal., 46(3):619–646, 2012.[Crossref]
• [9] Qingshan Chen, Roger Temam, and Joseph J. Tribbia. Simulations of the 2.5D inviscid primitive equations in a limited domain.J. Comput. Phys., 227(23):9865–9884, 2008.
• [10] Alexandre Joel Chorin. A numerical method for solving incompressible viscous flow problems [J. Comput. Phys. 2 (1967),no. 1, 12–36]. J. Comput. Phys., 135(2):115–125, 1997. With an introduction by Gerry Puckett, Commemoration of the 30thanniversary {of J. Comput. Phys.}.
• [11] Michele Coti Zelati, Michel Frémond, Roger Temam, and Joseph Tribbia. The equations of the atmosphere with humidity andsaturation: uniqueness and physical bounds. Phys. D, 264:49–65, 2013.
• [12] Michele Coti Zelati and Roger Temam. The atmospheric equation of water vapor with saturation. Boll. Unione Mat. Ital. (9),5(2):309–336, 2012.
• [13] J. Dudhia. A nonhydrostatic version of the penn state-ncar mesoscale model: Validation tests and simulation of an atlanticcyclone and cold front. Monthly Weather Review, 121(5):1493–1513, 1993.
• [14] Dale R Durran and Joseph B Klemp. A compressible model for the simulation of moist mountain waves. Monthly WeatherReview, 111(12):2341–2361, 1983.
• [15] D.R. Durran and J.B. Klemp. A compressible model for the simulation of moist mountain waves. Monthly Weather Review,111(12):2341–2361, 1983.
• [16] D.L. Dwoyer, J.A. Sanders, M. Ghil, F. Verhulst, M.Y. Hussaini, S. Childress, and R.G. Voigt. Topics in Geophysical Fluid Dynamics:Atmospheric Dynamics, DynamoTheory, and Climate Dynamics. Number v. 58-60 in AppliedMathematical Sciences.Springer-Verlag, 1985.
• [17] J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Reviews of modern physics, 57(3):617–656,1985.
• [18] B. F. Farrell. Optimal excitation of perturbations in viscous shear flow. Physics of Fluids, 31(8):2093, 1988.[Crossref]
• [19] B.F. Farrell and P.J. Ioannou. Generalized stability theory. part i: Autonomous operators. Journal of the atmospheric sciences,53(14):2025–2040, 1996.
• [20] Matthew F Garvert, Bradley Smull, and Cliff Mass. Multiscale mountain waves influencing a major orographic precipitationevent. Journal of the atmospheric sciences, 64(3):711–737, 2007.
• [21] M. Ghil, M.R. Allen, M.D. Dettinger, K. Ide, D. Kondrashov, M.E. Mann, A.W. Robertson, A. Saunders, Y. Tian, F. Varadi, et al.Advanced spectral methods for climatic time series. Reviews of Geophysics, 40(1), 2002.[Crossref]
• [22] Gung-Min Gie and Roger Temam. Cell centered finite volume discretization method for a general domain in R2 using convexquadrilateral meshes.
• [23] Gung-Min Gie and Roger Temam. Cell centered finite volume methods using Taylor series expansion scheme without fictitiousdomains. Int. J. Numer. Anal. Model., 7(1):1–29, 2010.
• [24] Adrian E Gill. Atmosphere-ocean dynamics. New York : Academic Press, 1982.
• [25] George J. Haltiner. Numerical weather prediction. Wiley New York, 1971.
• [26] George J. Haltiner and Roger T. Williams. Numerical Prediction and Dynamic Meteorology. Wiley, 2 edition, 5 1980.
• [27] Robert L Haney. On the pressure gradient force over steep topography in sigma coordinate ocean models. Journal of PhysicalOceanography, 21(4):610–619, 1991.[Crossref]
• [28] Robert A Houze Jr and Socorro Medina. Turbulence as a mechanism for orographic precipitation enhancement. Journal ofthe atmospheric sciences, 62(10):3599–3623, 2005.
• [29] A. Kasahara. Various vertical coordinate systems used for numerical weather prediction. Mon. Wea. Rev., 102(540):953–981, 1997.
• [30] Daniel J Kirshbaum and Dale R Durran. Factors governing cellular convection in orographic precipitation. Journal of theatmospheric Sciences, 61(6):682–698, 2004.
• [31] Joseph B Klemp, William C Skamarock, and Oliver Fuhrer. Numerical consistency of metric terms in terrain-following coordinates.Monthly weather review, 131(7):1229–1239, 2003.
• [32] Georgij M. Kobelkov. Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations. C. R. Math.Acad. Sci. Paris, 343(4):283–286, 2006.
• [33] Georgy M. Kobelkov. Existence of a solution “in the large” for ocean dynamics equations. J. Math. Fluid Mech., 9(4):588–610, 2007.[Crossref]
• [34] Randall J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. CambridgeUniversity Press, Cambridge, 2002.
• [35] Shian-Jiann Lin. A finite-volume integration method for computing pressure gradient force in general vertical coordinates.Quarterly Journal of the Royal Meteorological Society, 123(542):1749–1762, 1997.
• [36] Jacques-Louis Lions, Roger Temam, and Shou Hong Wang. New formulations of the primitive equations of atmosphere andapplications. Nonlinearity, 5(2):237–288, 1992.[Crossref]
• [37] Simone Marras, Margarida Moragues, Mariano Vázquez, Oriol Jorba, and Guillaume Houzeaux. Simulations of moist convectionby a variational multiscale stabilized finite element method. Journal of Computational Physics, 252:195–218, 2013.
• [38] A.M. Moore and R. Kleeman. The singular vectors of a coupled ocean-atmosphere model of enso. i: Thermodynamics, energeticsand error growth. Quarterly Journal of the Royal Meteorological Society, 123(7):509–522, 1974.
• [39] Joseph Oliger and Arne Sundström. Theoretical and practical aspects of some initial boundary value problems in fluiddynamics. SIAM J. Appl. Math., 35(3):419–446, 1978.
• [40] Joseph Pedlosky. Geophysical Fluid Dynamics. Springer, 2nd edition, 4 1992.
• [41] C. Penland and P.D. Sardeshmukh. The optimal growth of tropical sea surface temperature anomalies. Journal of climate,8(8):1999–2024, 1995.[Crossref]
• [42] Madalina Petcu, Roger M. Temam, and Mohammed Ziane. Some mathematical problems in geophysical fluid dynamics.In Handbook of numerical analysis. Vol. XIV. Special volume: computational methods for the atmosphere and the oceans,volume 14 of Handb. Numer. Anal., pages 577–750. Elsevier/North-Holland, Amsterdam, 2009.
• [43] S. C. Reddy, P.J. Schmid, and D.S. Henningson. Pseudospectra of the orr-sommerfeld operator. SIAM Journal on AppliedMathematics, 53(1):15–47, 1993.[Crossref]
• [44] Roger F Reinking, Jack B Snider, and Janice L Coen. Influences of storm-embedded orographic gravity waves on cloud liquidwater and precipitation. Journal of Applied Meteorology, 39(6):733–759, 2000.[Crossref]
• [45] R. R. Rogers and M.K. Yau. A short course in cloud physics. Pergamon Press Oxford ; New York, 3rd edition, 1989.
• [46] Richard Rotunno and Robert A Houze. Lessons on orographic precipitation from the mesoscale alpine programme. QuarterlyJournal of the Royal Meteorological Society, 133(625):811–830, 2007.
• [47] A. Rousseau, R. Temam, and J. Tribbia. Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity.Discrete Contin. Dyn. Syst., 13(5):1257–1276, 2005.
• [48] A. Rousseau, R. Temam, and J. Tribbia. Numerical simulations of the inviscid primitive equations in a limited domain. InAnalysis and simulation of fluid dynamics, Adv. Math. Fluid Mech., pages 163–181. Birkhäuser, Basel, 2007.
• [49] Laurent Schwartz. Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg, No.IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée. Hermann, Paris, 1966.
• [50] Ronald B Smith. The influence of mountains on the atmosphere. Advances in geophysics., 21:87–230, 1979.
• [51] Ronald B Smith and Yuh-Lang Lin. The addition of heat to a stratified airstreamwith application to the dynamics of orographicrain. Quarterly Journal of the Royal Meteorological Society, 108(456):353–378, 1982.
• [52] R. Témam. Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. I. Arch.Rational Mech. Anal., 32:135–153, 1969.
• [53] R. Témam. Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II.Arch. Rational Mech. Anal., 33:377–385, 1969.
• [54] Roger Temam and Joseph Tribbia. Open boundary conditions for the primitive and Boussinesq equations. J. AtmosphericSci., 60(21):2647–2660, 2003.
• [55] L.N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. PrincetonUniversity Press, 2005.
• [56] L.N. Trefethen, A. Trefethen, S.C. Reddy, T. Driscoll, et al. Hydrodynamic stability without eigenvalues. Science,261(5121):578–584, 1993.
• [57] M. Xue, K.K. Droegemeier, and V. Wong. The Advanced Regional Prediction System (ARPS)–A multi-scale nonhydrostatic atmosphericsimulation and prediction model. Part I: Model dynamics and verification. Meteorology and atmospheric physics,75(3-4):161–193, 2000.
• [58] Yinglong J Zhang, Eli Ateljevich, Hao-Cheng Yu, Chin H Wu, and CS Jason. A new vertical coordinate system for a 3dunstructured-grid model. Ocean Modelling, 85:16–31, 2015.

Identifiers

DOI

10.1515/mcwf-2015-0005