Large-eddy simulation in an anelastic framework with closed water and entropy balances

Journal of Advances in Modeling Earth Systems

Volumn 7, Issue 3, 2015, Pages 1425-1456

  Pressel, Kyle G ^a   Kaul, Colleen M ^a   Schneider, Tapio ^a,b   Tan, Zhihong ^a,b   Mishra, Siddhartha ^a  

^a ETH ZURICH   (Switzerland)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

clouds; entropy; large eddy simulation; moist thermodynamics; numerical methods

Indexed keywords

BOUNDARY LAYERS; CLOUDS; ENTROPY; EQUATIONS OF MOTION; NUMERICAL METHODS; RUNGE KUTTA METHODS; THERMODYNAMICS;

CENTERED SCHEMES; COARSER RESOLUTION; DISCRETIZATION SCHEME; NUMERICAL TESTS; PROGNOSTIC VARIABLES; REVERSIBLE PROCESS; STRONG STABILITY PRESERVING; WEIGHTED ESSENTIALLY NONOSCILLATORY;

LARGE EDDY SIMULATION;

ADIABATIC PROCESS; ATMOSPHERIC MODELING; BOUNDARY LAYER; CLOUD; ENTROPY; LARGE EDDY SIMULATION; NUMERICAL METHOD; PHASE TRANSITION; STABILITY ANALYSIS; THERMODYNAMICS;

EID: 84944885980     PISSN: None     EISSN: 19422466     Source Type: Journal    
DOI: 10.1002/2015MS000496     Document Type: Article

References (100)

1. Arakawa, A., and, V. R. Lamb, (1977), Computational design of the basic dynamical processes of the UCLA general circulation model, Methods Comput. Phys., 17, 173-265, doi: 10.1016/B978-0-12-460817-7.50009-4.
2. Balsara, D. S., and, C.-W. Shu, (2000), Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405-452, doi: 10.1006/jcph.2000.6443.
3. Bannon, P. R., (1996), On the anelastic approximation for a compressible atmosphere, J. Atmos. Sci., 53, 3618-3628, doi: 10.1175/1520-0469(1996)053<3618:OTAAFA>2.0.CO;2.
4. Batchelor, G. K., (1953), The conditions for dynamical similarity of motions of a frictionless perfect-gas atmosphere, Q. J. R. Meteorol. Soc., 79, 224-235, doi: 10.1002/qj.49707934004.
5. Beare, R. J., et al., (2006), An intercomparison of large-eddy simulations of the stable boundary layer, Boundary Layer Meteorol., 118, 247-272, doi: 10.1007/s10546-004-2820-6.
6. Behnel, S., R. Bradshaw, C. Citro, L. Dalcin, D. S. Seljebotn, and, K. Smith, (2011), Cython: The best of both worlds, Comput. Sci. Eng., 13, 31-39, doi: 10.1109/MCSE.2010.118.
7. Betts, A. K., (1973), Non-precipitating cumulus convection and its parameterization, Q. J. R. Meteorol. Soc., 99, 178-196, doi: 10.1002/qj.49709941915.
8. Blossey, P. N., et al., (2013), Marine low cloud sensitivity to an idealized climate change: The CGILS LES intercomparison, J. Adv. Model. Earth Syst., 5, 234-258, doi: 10.1002/jame.20025.
9. Bony, S., and, J.-L. Dufresne, (2005), Marine boundary layer clouds at the heart of tropical cloud feedback uncertainties in climate models, Geophys. Res. Lett., 32, L20806, doi: 10.1029/2005GL023851.
10. Bretherton, C. S., P. N. Blossey, and, C. R. Jones, (2013), Mechanisms of marine low cloud sensitivity to idealized climate perturbations: A single-LES exploration extending the CGILS cases, J. Adv. Model. Earth Syst., 5, 316-337, doi: 10.1002/jame.20019.
11. Bryan, G. H., and, J. M. Fritsch, (2002), A benchmark simulation for moist nonhydrostatic numerical models, Mon. Weather Rev., 130 (12), 2917-2928, doi: 10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.
12. Byun, D. W., (1990), On the analytical solutions of flux-profile relationships for the atmospheric surface layer, J. Appl. Meteorol., 29, 652-657, doi: 10.1175/1520-0450(1990)029<0652:OTASOF>2.0.CO;2.
13. Cess, R. D., et al., (1990), Intercomparison and interpretation of climate feedback processes in 19 atmospheric general circulation models, J. Geophys. Res., 95, 16,601-16,615, doi: 10.1029/JD095iD10p16601.
14. Cess, R. D., et al., (1996), Cloud feedback in atmospheric general circulation models: An update, J. Geophys. Res., 101, 12,791-12,794, doi: 10.1029/96JD00822.
15. Chase, M. W., Jr,. (1998), NIST-JANAF thermochemical tables, fourth edition, J. Phys. Chem. Ref. Data, Monogr., 9, 1-1951.
16. Chorin, A. C., (1968), Numerical solution of the Navier-Stokes equations, Math. Comput., 22, 745-762.
17. Chow, F. K., and, P. Moin, (2003), A further study of numerical errors in large-eddy simulations, J. Comput. Phys., 184, 366-380, doi: 10.1016/S0021-9991(02)00020-7.
18. Chung, D., G. Matheou, and, J. Teixeira, (2012), Steady-state large-eddy simulations to study the stratocumulus to shallow cumulus cloud transition, J. Atmos. Sci., 69, 3264-3276, doi: 10.1175/JAS-D-11-0256.1.
19. Corrsin, S., (1974), Limitations of gradient transport models in random walks and in turbulence, in Advances in Geophysics, vol. 18A, pp. 25-60, edited by F. N. Frenkiel and R. E. Munn, Academic, doi: 10.1016/S0065-2687(08)60451-3.
20. Cotton, W. R., G. Bryan, and, S. van den Heever, (2011), Storm and Cloud Dynamics, 2nd ed., Academic, Burlington, Mass.
21. Deardorff, J. W., (1974a), Three-dimensional numerical study of the height and mean structure of a heated planetary boundary layer, Boundary Layer Meteorol., 7 (1), 81-106, doi: 10.1007/BF00224974.
22. Deardorff, J. W., (1974b), Three-dimensional numerical study of turbulence in an entraining mixed layer, Boundary Layer Meteorol., 7 (2), 199-226, doi: 10.1007/BF00227913.
23. Deardorff, J. W., (1976), Usefulness of liquid-water potential temperature in a shallow-cloud model, J. Appl. Meteorol., 15, 98-102, doi: 10.1175/1520-0450(1976)015<0098:UOLWPT>2.0.CO;2.
24. Deardorff, J. W., (1980), Stratocumulus-capped mixed layers derived from a three-dimensional model, Boundary Layer Meteorol., 18 (4), 495-527, doi: 10.1007/BF00119502.
25. 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. Weather Rev., 142, 4269-4283, doi: 10.1175/MWR-D-13-00368.1.
26. Dufresne, J.-L., and, S. Bony, (2008), An assessment of the primary sources of spread of global warming estimates from coupled atmosphere-ocean models, J. Clim., 21, 5135-5144, doi: 10.1175/2008JCLI2239.1.
27. Durran, D. R., and, J. B. Klemp, (1982), On the effects of moisture on the Brunt-Väisälä frequency, J. Atmos. Sci., 39, 2152-2158, doi: 10.1175/1520-0469(1982)039<2152:OTEOMO>2.0.CO;2.
28. 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.
29. Emanuel, K. A., (1994), Atmospheric Convection, 580 pp., Oxford Univ. Press, N. Y.
30. Ghosal, S., (1996), An analysis of numerical errors in large-eddy simulations of turbulence, J. Comput. Phys., 125, 187-206, doi: 10.1006/jcph.1996.0088.
31. Ghosh, D., and, J. D. Baeder, (2012), High-order accurate incompressible Navier-Stokes algorithm for vortex-ring interactions with solid wall, AIAA J., 50 (11), 2408-2422, doi: 10.2514/1.J051537.
32. Gottlieb, S., (2005), On high order strong stability preserving Runge-Kutta and multi step time discretizations, J. Sci. Comput., 25 (1), 105-128, doi: 10.1007/BF02728985.
33. Grabowski, W. W., (1998), Toward cloud resolving modeling of large-scale tropical circulations: A simple cloud microphysics parameterization, J. Atmos. Sci., 55 (21), 3283-3298, doi: 10.1175/1520-0469(1998)055<3283:TCRMOL>2.0.CO;2.
34. Harlow, F. H., and, J. E. Welch, (1965), Numerical calculation of timedependent viscous incompressible flow of fluid with free surface, Phys. Fluids, 8 (12), 2182-2189, doi: 10.1063/1.1761178.
35. 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.
36. Heus, T., et al., (2010), Formulation of the Dutch Atmospheric Large-Eddy Simulation (DALES) and overview of its applications, Geosci. Model Dev., 3, 415-444, doi: 10.5194/gmd-3-415-2010.
37. Jiang, G.-S., and, C.-W. Shu, (1996), Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1), 202-228, doi: 10.1006/jcph.1996.0130.
38. Jiang, H., and, W. R. Cotton, (2000), Large eddy simulation of shallow cumulus convection during BOMEX: Sensitivity to microphysics and radiation, J. Atmos. Sci., 57 (4), 582-594, doi: 10.1175/1520-0469(2000)057<0582:LESOSC>2.0.CO;2.
39. Khairoutdinov, M., and, D. Randall, (2006), High-resolution simulation of shallow-to-deep convection transition over land, J. Atmos. Sci., 63 (12), 3421-3436, doi: 10.1175/JAS3810.1.
40. Khairoutdinov, M. F., and, D. A. Randall, (2003), Cloud resolving modeling of the ARM summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities, J. Atmos. Sci., 60 (4), 607-625, doi: 10.1175/1520-0469(2003)060<0607:CRMOTA>2.0.CO;2.
41. Khairoutdinov, M. F., S. K. Krueger, C.-H. Moeng, P. A. Bogenschutz, and, D. A. Randall, (2009), Large-eddy simulation of maritime deep tropical convection, J. Adv. Model. Earth Syst., 1, doi: 10.3894/JAMES.2009.1.15.
42. Kim, J., and, P. Moin, (1985), Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (2), 308-323, doi: 10.1016/0021-9991(85)90148-2.
43. Kravchenko, A., and, P. Moin, (1997), On the effect of numerical errors in large eddy simulations of turbulent flows, J. Comput. Phys., 131 (2), 310-322, doi: 10.1006/jcph.1996.5597.
44. Kurowski, M. J., W. W. Grabowski, and, P. K. Smolarkiewicz, (2014), Anelastic and compressible simulation of moist deep convection, J. Atmos. Sci., 71 (10), 3767-3787, doi: 10.1175/JAS-D-14-0017.1.
45. Larson, V. E., R. Wood, P. R. Field, J.-C. Golaz, T. H. Vonder Haar, and, W. R. Cotton, (2001), Small-scale and mesoscale variability of scalars in cloudy boundary layers: One-dimensional probability density functions, J. Atmos. Sci., 58, 1978-1994, doi: 10.1175/1520-0469(2001)058<1978:SSAMVO>2.0.CO;2.
46. Lemmon, E. W., R. T. Jacobsen, S. G. Penoncello, and, D. G. Friend, (2000), Thermodynamic properties of air and mixtures of nitrogen, argon, and oxygen from 60 to 2000 K at pressures to 2000 MPa, J. Phys. Chem. Ref. Data, 29, 331-385.
47. Lilly, D. K., (1962), On the numerical simulation of buoyant convection, Tellus, 14, 148-172, doi: 10.1111/j.2153-3490.1962.tb00128.x.
48. Lin, J. W.-B., (2012), Why Python is the next wave in earth sciences computing, Bull. Am. Meteorol. Soc., 93, 1823-1824, doi: 10.1175/BAMS-D-12-00148.1.
49. Lipps, F. B., and, R. S. Hemler, (1982), A scale analysis of deep moist convection and some related numerical-calculations, J. Atmos. Sci., 39 (10), 2192-2210, doi: 10.1175/1520-0469(1982)039<2192:ASAODM>2.0.CO;2.
50. Liu, X.-D., S. Osher, and, T. Chan, (1994), Weighted eessentially nonoscillatory schemes, J. Comput. Phys., 115 (1), 200-212, doi: 10.1006/jcph.1994.1187.
51. Lord, S. J., H. E. Willoughby, and, J. M. Piotrowicz, (1984), Role of a parameterized ice-phase microphysics in an axisymmetric, nonhydrostatic tropical cyclone model, J. Atmos. Sci., 41 (19), 2836-2848, doi: 10.1175/1520-0469(1984)041<2836:ROAPIP>2.0.CO;2.
52. Marquet, P., (1993), Exergy in meteorology: Definition and properties of moist available enthalpy, Q. J. R. Meteorol. Soc., 119, 567-590, doi: 10.1002/qj.49711951112.
53. Marquet, P., (2011), Definition of a moist entropy potential temperature: Application to FIRE-I data flights, Q. J. R. Meteorol. Soc., 137, 768-791, doi: 10.1002/qj.787.
54. Matheou, G., D. Chung, L. Nuijens, B. Stevens, and, J. Teixeira, (2011), On the fidelity of large-eddy simulation of shallow precipitating cumulus convection, Mon. Weather Rev., 139 (9), 2918-2939, doi: 10.1175/2011MWR3599.1.
55. Moeng, C.-H., and, J. C. Wyngaard, (1988), Spectral analysis of large-eddy simulations of the convective boundary layer, J. Atmos. Sci., 45 (23), 3573-3587, doi: 10.1175/1520-0469(1988)045<3573:SAOLES>2.0.CO;2.
56. Moin, P., (2010), Fundamentals of Engineering Numerical Analysis, Cambridge Univ. Press, N. Y.
57. Morinishi, Y., T. S. Lund, O. V. Vasilyev, and, P. Moin, (1998), Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys., 143 (1), 90-124, doi: 10.1006/jcph.1998.5962.
58. Nance, L. B., and, D. R. Durran, (1994), A comparison of the accuracy of three anelastic systems and the pseudo-incompressible system, J. Atmos. Sci., 51, 3549-3565, doi: 10.1175/1520-0469(1994)051<3549:ACOTAO>2.0.CO;2.
59. 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.
60. Ooyama, K. V., (1990), A thermodynamic foundation for modeling the moist atmosphere, J. Atmos. Sci., 47 (21), 2580-2593, doi: 10.1175/1520-0469(1990)047<2580:ATFFMT>2.0.CO;2.
61. Ooyama, K. V., (2001), A dynamic and thermodynamic foundation for modeling the moist atmosphere with parameterized microphysics, J. Atmos. Sci., 58 (15), 2073-2102, doi: 10.1175/1520-0469(2001)058<2073:ADATFF>2.0.CO;2.
62. Pauluis, O., (2008), Thermodynamic consistency of the anelastic approximation for a moist atmosphere, J. Atmos. Sci., 65 (8), 2719-2729, doi: 10.1175/2007JAS2475.1.
63. Pauluis, O., and, I. M. Held, (2002a), Entropy budget of an atmosphere in radiative-convective equilibrium. Part I: Maximum work and frictional dissipation, J. Atmos. Sci., 59, 125-139, doi: 10.1175/1520-0469(2002)059<0125:EBOAAI>2.0.CO;2.
64. Pauluis, O., and, I. M. Held, (2002b), Entropy budget of an atmosphere in radiative-convective equilibrium. Part II: Latent heat transport and moist processes, J. Atmos. Sci., 59, 140-149, doi: 10.1175/1520-0469(2002)059<0125:EBOAAI>2.0.CO;2.
65. Pauluis, O., V. Balaji, and, I. M. Held, (2000), Frictional dissipation in a precipitating atmosphere, J. Atmos. Sci., 57, 989-994, doi: 10.1175/1520-0469(2000)057<0989:FDIAPA>2.0.CO;2.
66. Pope, S. B., (2004), Ten questions concerning the large-eddy simulation of turbulent flows, New J. Phys., 6 (1), 35, doi: 10.1088/1367-2630/6/1/035.
67. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and, B. P. Flannery, (1992), Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., Cambridge Univ. Press, N. Y.
68. Raasch, S., and, M. Schröter, (2001), PALM-A large-eddy simulation model performing on massively parallel computers, Meteorol. Z., 10 (5), 363-372, doi: 10.1127/0941-2948/2001/0010-0363.
69. Raymond, D., and, X. Zeng, (2005), Modelling tropical atmospheric convection in the context of the weak temperature gradient approximation, Q. J. R. Meteorol. Soc., 131, 1301-1320, doi: 10.1256/qj.03.97.
70. Raymond, D. J., (2013), Sources and sinks of entropy in the atmosphere, J. Adv. Model. Earth Syst., 5, 755-763, doi: 10.1002/jame.20050.
71. Romps, D. M., and, Z. Kuang, (2010), Do undiluted convective plumes exist in the upper tropical troposphere?, J. Atmos. Sci., 67, 468-484, doi: 10.1175/2009JAS3184.1.
72. Rossum, G., (1995), Python reference manual, technical report, Cent. for Math. and Comput. Sci., Amsterdam. Available at: http://oai.cwi.nl/oai/asset/5008/05008D.pdf.
73. Ruuth, S. J., (2005), Global optimization of explicit strong-stability-preserving Runge-Kutta methods, Math. Comput., 75 (253), 183-207.
74. Satoh, M., T. Matsuno, H. Tomita, H. Miura, T. Nasuno, and, S. Iga, (2008), Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations, J. Comput. Phys., 227, 3486-3514, doi: 10.1016/j.jcp.2007.02.006.
75. Savic-Jovcic, V., and, B. Stevens, (2008), The structure and mesoscale organization of precipitating stratocumulus, J. Atmos. Sci., 65 (5), 1587-1605, doi: 10.1175/2007JAS2456.1.
76. Savre, J., A. M. L. Ekman, and, G. Svensson, (2014), Technical note: Introduction to MIMICA, a large-eddy simulation solver for cloudy planetary boundary layers, J. Adv. Model. Earth Syst., 6, 630-649, doi: 10.1002/2013MS000292.
77. Schmidt, H., and, U. Schumann, (1989), Coherent structure of the convective boundary-layer derived from large-eddy simulations, J. Fluid Mech., 200, 511-562, doi: 10.1017/S0022112089000753.
78. Schroeder, G., K. H. Schlünzen, and, F. Schimmel, (2006), Use of (weighted) essentially non-oscillatory advection schemes in a mesoscale model, Q. J. R. Meteorol. Soc., 132, 1509-1526, doi: 10.1256/qj.04.191.
79. Shu, C.-W., (1998), Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes Math., vol. 1697, edited by, A. Quarteroni, pp. 325-432, Springer, Berlin, doi: 10.1007/BFb0096355.
80. Shu, C.-W., and, S. Osher, (1988), Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (2), 439-471, doi: 10.1016/0021-9991(88)90177-5.
81. Siebesma, A. P., et al., (2003), A large eddy simulation intercomparison study of shallow cumulus convection, J. Atmos. Sci., 60 (10), 1201-1219, doi: 10.1175/1520-0469(2003)60<201:ALESIS>2.0.CO;2.
82. Skamarock, W. C., and, J. B. Klemp, (1993), Adaptive grid refinement for two-dimensional and three-dimensional nonhydrostatic atmospheric flow, Mon. Weather Rev., 121 (3), 788-804, doi: 10.1175/1520-0493(1993)121<0788:AGRFTD>2.0.CO;2.
83. Skamarock, W. C., and, J. B. Klemp, (2008), A time-split nonhydrostatic atmospheric model for weather research and forecasting applications, J. Comput. Phys., 227 (7), 3465-3485, doi: 10.1016/j.jcp.2007.01.037.
84. Smagorinsky, J., (1958), On the numerical integration of the primitive equations of motion for baroclinic flow in a closed region, Mon. Weather Rev., 86 (12), 457-466, doi: 10.1175/1520-0493(1958)086<0457:OTNIOT>2.0.CO;2.
85. Smagorinsky, J., (1963), General circulation experiments with the primitive equations, Mon. Weather Rev., 91 (3), 99-164, doi: 10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.
86. Sommeria, G., and, J. W. Deardorff, (1977), Subgrid-scale condensation in models of nonprecipitating clouds, J. Atmos. Sci., 34 (2), 344-355, doi: 10.1175/1520-0469(1977)034<0344:SSCIMO>2.0.CO;2.
87. Squires, P., and, J. S. Turner, (1962), An entraining jet model for cumulo-nimbus updraughts, Tellus, 14 (4), 422-434, doi: 10.1111/j.2153-3490.1962.tb01355.x.
88. Stevens, B., et al., (2005), Evaluation of large-eddy simulations via observations of nocturnal marine stratocumulus, Mon. Weather Rev., 133, 1443-1462, doi: 10.1175/MWR2930.1.
89. Straka, J. M., R. B. Wilhelmson, L. J. Wicker, J. R. Anderson, and, K. K. Droegemeier, (1993), Numerical solutions of a non-linear density current: A benchmark solution and comparisons, Int. J. Numer. Methods Fluids, 17 (1), 1-22, doi: 10.1002/fld.1650170103.
90. Sullivan, P. P., and, E. G. Patton, (2011), The effect of mesh resolution on convective boundary layer statistics and structures generated by large-eddy simulation, J. Atmos. Sci., 68 (10), 2395-2415, doi: 10.1175/JAS-D-10-05010.1.
91. Tao, W.-K., J. Simpson, and, M. McCumber, (1989), An ice-water saturation adjustment, Mon. Weather Rev., 117 (1), 231-235, doi: 10.1175/1520-0493(1989)117<0231:AIWSA>2.0.CO;2.
92. Tompkins, A. M., (2002), A prognostic parameterization for the subgrid-scale variability of water vapor and clouds in large-scale models and its use to diagnose cloud cover, J. Atmos. Sci., 59, 1917-1942, doi: 10.1175/1520-0469(2002)059<1917:APPFTS>2.0.CO;2.
93. Turk, M. J., B. D. Smith, J. S. Oishi, S. Skory, S. W. Skillman, T. Abel, and, M. L. Norman,. (2011), yt: A Multi-code analysis toolkit for astrophysical simulation data, Astrophys. J. Suppl. Ser., 192 (1), 1-16, doi: 10.1088/0067-0049/192/1/9.
94. 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. Weather Rev., 109 (5), 1094-1102, doi: 10.1175/1520-0493(1981)109<1094:TUOLLW>2.0.CO;2.
95. Vial, J., J.-L. Dufresne, and, S. Bony, (2013), On the interpretation of inter-model spread in CMIP5 climate sensitivity estimates, Clim. Dyn., 41 (11), 3339-3362, doi: 10.1007/s00382-013-1725-9.
96. Webb, M. J., et al., (2006), On the contribution of local feedback mechanisms to the range of climate sensitivity in two GCM ensembles, Clim. Dyn., 27 (1), 17-38, doi: 10.1007/s00382-006-0111-2.
97. Wicker, L. J., and, W. C. Skamarock, (2002), Time-splitting methods for elastic models using forward time schemes, Mon. Weather Rev., 130 (8), 2088-2097, doi: 10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2.
98. Wilbers, I. M., H. P. Langtangen, and, Å. Ødegård, (2009), Using Cython to speed up numerical Python programs, Proc. MekIT 09, 495-512.
99. Zeng, X., W.-K. Tao, and, J. Simpson, (2005), An equation for moist entropy in a precipitating and icy atmosphere, J. Atmos. Sci., 62 (12), 4293-4309, doi: 10.1175/JAS3570.1.
100. Zeng, X., W.-K. Tao, and, J. Simpson, (2008), A set of prognostic variables for long-term cloud- resolving model simulations, J. Meteorol. Soc. Jpn., 86, 839-856.