Numerical simulations of the humid atmosphere above a mountain

Arthur Bousquet; Mickaël D. Chekroun; Youngjoon Hong; Roger M. Temam; Joseph Tribbia

Mathematics of Climate and Weather Forecasting (2015)

  • Volume: 1, Issue: 1
  • ISSN: 2353-6438

Abstract

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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.

How to cite

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Arthur Bousquet, et al. "Numerical simulations of the humid atmosphere above a mountain." Mathematics of Climate and Weather Forecasting 1.1 (2015): null. <http://eudml.org/doc/276585>.

@article{ArthurBousquet2015,
abstract = {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.},
author = {Arthur Bousquet, Mickaël D. Chekroun, Youngjoon Hong, Roger M. Temam, Joseph Tribbia},
journal = {Mathematics of Climate and Weather Forecasting},
keywords = {primitive equations; humidity; finite volume; phase change; projection method; stochastic parameterizations},
language = {eng},
number = {1},
pages = {null},
title = {Numerical simulations of the humid atmosphere above a mountain},
url = {http://eudml.org/doc/276585},
volume = {1},
year = {2015},
}

TY - JOUR
AU - Arthur Bousquet
AU - Mickaël D. Chekroun
AU - Youngjoon Hong
AU - Roger M. Temam
AU - Joseph Tribbia
TI - Numerical simulations of the humid atmosphere above a mountain
JO - Mathematics of Climate and Weather Forecasting
PY - 2015
VL - 1
IS - 1
SP - null
AB - 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.
LA - eng
KW - primitive equations; humidity; finite volume; phase change; projection method; stochastic parameterizations
UR - http://eudml.org/doc/276585
ER -

References

top
  1. [1] Arthur Bousquet, Michele Coti Zelati, and Roger Temam. Phase transition models in atmospheric dynamics. Milan Journal of Mathematics, pages 1–30, 2014.  Zbl1319.35181
  2. [2] Arthur Bousquet, Gung-Min Gie, Youngjoon Hong, and Jacques Laminie. A higher order finite volume resolution method for a system related to the inviscid primitive equations in a complex domain. Numerische Mathematik, 128(3):431–461, 2014.  Zbl1303.65073
  3. [3] Chongsheng Cao and Edriss S. Titi. Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. of Math. (2), 166(1):245–267, 2007.  Zbl1151.35074
  4. [4] M. D. Chekroun, J. D. Neelin, D. Kondrashov, J. C. McWilliams, and M. Ghil. Rough parameter dependence in climate models and the role of ruelle-pollicott resonances. Proceedings of the National Academy of Sciences, 111(5):1684–1690, 2014.  
  5. [5] M.D. Chekroun, D. Kondrashov, and M. Ghil. Predicting stochastic systems by noise sampling, and application to the el niño-southern oscillation. Proceedings of the National Academy of Sciences, 108(29):11766–11771, 2011.  
  6. [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] Zbl1117.35319
  7. [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] Zbl1203.86009
  8. [8] Qingshan Chen, Ming-Cheng Shiue, Roger Temam, and Joseph Tribbia. Numerical approximation of the inviscid 3D primitive equations in a limited domain. ESAIM Math. Model. Numer. Anal., 46(3):619–646, 2012. [Crossref] Zbl1308.76197
  9. [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.  Zbl1317.86007
  10. [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 30th anniversary {of J. Comput. Phys.}.  
  11. [11] Michele Coti Zelati, Michel Frémond, Roger Temam, and Joseph Tribbia. The equations of the atmosphere with humidity and saturation: uniqueness and physical bounds. Phys. D, 264:49–65, 2013.  Zbl1286.86013
  12. [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.  Zbl1256.35174
  13. [13] J. Dudhia. A nonhydrostatic version of the penn state-ncar mesoscale model: Validation tests and simulation of an atlantic cyclone and cold front. Monthly Weather Review, 121(5):1493–1513, 1993.  
  14. [14] Dale R Durran and Joseph B Klemp. A compressible model for the simulation of moist mountain waves. Monthly Weather Review, 111(12):2341–2361, 1983.  
  15. [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. [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. [17] J.-P. Eckmann and D. Ruelle. Ergodic theory of chaos and strange attractors. Reviews of modern physics, 57(3):617–656, 1985.  Zbl0989.37516
  18. [18] B. F. Farrell. Optimal excitation of perturbations in viscous shear flow. Physics of Fluids, 31(8):2093, 1988. [Crossref] 
  19. [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. [20] Matthew F Garvert, Bradley Smull, and Cliff Mass. Multiscale mountain waves influencing a major orographic precipitation event. Journal of the atmospheric sciences, 64(3):711–737, 2007.  
  21. [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. [22] Gung-Min Gie and Roger Temam. Cell centered finite volume discretization method for a general domain in R2 using convex quadrilateral meshes.  
  23. [23] Gung-Min Gie and Roger Temam. Cell centered finite volume methods using Taylor series expansion scheme without fictitious domains. Int. J. Numer. Anal. Model., 7(1):1–29, 2010.  
  24. [24] Adrian E Gill. Atmosphere-ocean dynamics. New York : Academic Press, 1982.  
  25. [25] George J. Haltiner. Numerical weather prediction. Wiley New York, 1971.  
  26. [26] George J. Haltiner and Roger T. Williams. Numerical Prediction and Dynamic Meteorology. Wiley, 2 edition, 5 1980.  
  27. [27] Robert L Haney. On the pressure gradient force over steep topography in sigma coordinate ocean models. Journal of Physical Oceanography, 21(4):610–619, 1991. [Crossref] 
  28. [28] Robert A Houze Jr and Socorro Medina. Turbulence as a mechanism for orographic precipitation enhancement. Journal of the atmospheric sciences, 62(10):3599–3623, 2005.  
  29. [29] A. Kasahara. Various vertical coordinate systems used for numerical weather prediction. Mon. Wea. Rev., 102(540):953– 981, 1997.  
  30. [30] Daniel J Kirshbaum and Dale R Durran. Factors governing cellular convection in orographic precipitation. Journal of the atmospheric Sciences, 61(6):682–698, 2004.  
  31. [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. [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.  Zbl1102.35003
  33. [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] Zbl1132.35443
  34. [34] Randall J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.  Zbl1010.65040
  35. [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. [36] Jacques-Louis Lions, Roger Temam, and Shou Hong Wang. New formulations of the primitive equations of atmosphere and applications. Nonlinearity, 5(2):237–288, 1992. [Crossref] Zbl0746.76019
  37. [37] Simone Marras, Margarida Moragues, Mariano Vázquez, Oriol Jorba, and Guillaume Houzeaux. Simulations of moist convection by a variational multiscale stabilized finite element method. Journal of Computational Physics, 252:195–218, 2013.  Zbl1286.65126
  38. [38] A.M. Moore and R. Kleeman. The singular vectors of a coupled ocean-atmosphere model of enso. i: Thermodynamics, energetics and error growth. Quarterly Journal of the Royal Meteorological Society, 123(7):509–522, 1974.  
  39. [39] Joseph Oliger and Arne Sundström. Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math., 35(3):419–446, 1978.  Zbl0397.35067
  40. [40] Joseph Pedlosky. Geophysical Fluid Dynamics. Springer, 2nd edition, 4 1992.  Zbl0429.76001
  41. [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. [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. [43] S. C. Reddy, P.J. Schmid, and D.S. Henningson. Pseudospectra of the orr-sommerfeld operator. SIAM Journal on Applied Mathematics, 53(1):15–47, 1993. [Crossref] Zbl0778.34060
  44. [44] Roger F Reinking, Jack B Snider, and Janice L Coen. Influences of storm-embedded orographic gravity waves on cloud liquid water and precipitation. Journal of Applied Meteorology, 39(6):733–759, 2000. [Crossref] 
  45. [45] R. R. Rogers and M.K. Yau. A short course in cloud physics. Pergamon Press Oxford ; New York, 3rd edition, 1989.  
  46. [46] Richard Rotunno and Robert A Houze. Lessons on orographic precipitation from the mesoscale alpine programme. Quarterly Journal of the Royal Meteorological Society, 133(625):811–830, 2007.  
  47. [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.  Zbl1092.35089
  48. [48] A. Rousseau, R. Temam, and J. Tribbia. Numerical simulations of the inviscid primitive equations in a limited domain. In Analysis and simulation of fluid dynamics, Adv. Math. Fluid Mech., pages 163–181. Birkhäuser, Basel, 2007.  Zbl1291.86008
  49. [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. [50] Ronald B Smith. The influence of mountains on the atmosphere. Advances in geophysics., 21:87–230, 1979.  
  51. [51] Ronald B Smith and Yuh-Lang Lin. The addition of heat to a stratified airstreamwith application to the dynamics of orographic rain. Quarterly Journal of the Royal Meteorological Society, 108(456):353–378, 1982.  
  52. [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.  Zbl0195.46001
  53. [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.  Zbl0207.16904
  54. [54] Roger Temam and Joseph Tribbia. Open boundary conditions for the primitive and Boussinesq equations. J. Atmospheric Sci., 60(21):2647–2660, 2003.  
  55. [55] L.N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, 2005.  Zbl1085.15009
  56. [56] L.N. Trefethen, A. Trefethen, S.C. Reddy, T. Driscoll, et al. Hydrodynamic stability without eigenvalues. Science, 261(5121):578–584, 1993.  Zbl1226.76013
  57. [57] M. Xue, K.K. Droegemeier, and V. Wong. The Advanced Regional Prediction System (ARPS)–A multi-scale nonhydrostatic atmospheric simulation and prediction model. Part I: Model dynamics and verification. Meteorology and atmospheric physics, 75(3-4):161–193, 2000.  
  58. [58] Yinglong J Zhang, Eli Ateljevich, Hao-Cheng Yu, Chin H Wu, and CS Jason. A new vertical coordinate system for a 3d unstructured-grid model. Ocean Modelling, 85:16–31, 2015.  

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