Usual approximations to the equations of atmospheric motion: A variational perspective

Journal of the Atmospheric Sciences

Volumn 71, Issue 7, 2014, Pages 2452-2466

  Tort, Marine ^a   Dubos, Thomas ^a  

^a ECOLE POLYTECHNIQUE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EULER EQUATIONS; KINETICS;

ASYMPTOTIC CONSISTENCIES; COMPRESSIBLE EULER EQUATIONS; CONSERVATION OF ENERGY; EULER-LAGRANGE EQUATIONS; GENERAL CURVILINEAR COORDINATES; ORDER-OF-MAGNITUDE ANALYSIS; VARIATIONAL APPROACHES; VARIATIONAL FRAMEWORK;

LAGRANGE MULTIPLIERS;

ATMOSPHERIC MODELING; ATMOSPHERIC MOTION; CORIOLIS FORCE; EULERIAN ANALYSIS; GEOPOTENTIAL; GRAVITY FIELD; KINETIC ENERGY; LAGRANGIAN ANALYSIS; MOMENTUM; POTENTIAL VORTICITY; PRESSURE GRADIENT;

EID: 84903902085     PISSN: 00224928     EISSN: 15200469     Source Type: Journal    
DOI: 10.1175/JAS-D-13-0339.1     Document Type: Article

References (34)

1. Bénard, P., 2014: An oblate-spheroid geopotential approximation for global meteorology. Quart. J. Roy. Meteor. Soc., 140, 170- 184, doi:10.1002/qj.2141.
2. Cho, J. Y. K., and L. M. Polvani, 1996: The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets. Science, 273, 335-337, doi:10.1126/science.273.5273.335.
3. Cho, J, K. Menou, B. M. S. Hansen, and S. Seager, 2003: The changing face of the extrasolar giant planet HD 209458b. Astrophys. J. Lett., 587, L117, doi:10.1086/375016.
4. Cotter, C. J., and D. Holm, 2013: Variational formulations of sound-proof models. Quart. J. Roy. Meteor. Soc., doi:10.1002/ qj.2260, in press.
5. Dellar, P. J., 2011: Variations on a beta-plane: Derivation of non-traditional beta-plane equations from Hamilton's principle on a sphere. J. FluidMech., 674, 174-195, doi:10.1017/ S0022112010006464.
6. 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
7. Durran, D. R., 2008: A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow. J. Fluid Mech., 601, 365-379, doi:10.1017/S0022112008000608.
8. Gates, W. L., 2004: Derivation of the equations of atmospheric motion in oblate spheroidal coordinates. J. Atmos. Sci., 61, 2478-2487, doi:10.1175/1520-0469(2004)061,2478: DOTEOA.2.0.CO;2.
9. Gerkema, T., and V. I. Shrira, 2005: Near-inertial waves in the ocean: Beyond the traditional approximation. J. Fluid Mech., 529, 195-219, doi:10.1017/S0022112005003411.
10. Gerkema, T., J. T. F. Zimmerman, L. R. M. Maas, and H. van Haren, 2008: Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev.Geophys., 46, 1-33, doi:10.1029/ 2006RG000220.
11. Holm, D. D., J. E. Marsden, and T. S. Ratiu, 2002: The Euler- Poincar_e equations in geophysical fluid dynamics. Large-Scale Atmosphere-Ocean Dynamics, J. Norbury and I. Roulstone, Eds., Cambridge University Press, 251-300.
12. Hua, B. L., D. W. Moore, and S. Le Gentil, 1997: Inertial nonlinear equilibration of equatorial flows. J. Fluid Mech., 331, 345-371, doi:10.1017/S0022112096004016.
13. Klein, R., and O. Pauluis, 2012: Thermodynamic consistency of a pseudo-incompressible approximation for general equations of state. J. Atmos. Sci., 69, 961-968, doi:10.1175/ JAS-D-11-0110.1.
14. Lipps, F. B., and R. S. 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.
15. Mayne, N. J., and Coauthors, 2014: The unified model, a fully compressible, non-hydrostatic, deep atmosphere global circulation model, applied to hot jupiters. Astron. Astrophys., 561, A1, doi:10.1051/0004-6361/201322174.
16. Morrison, P. J., 1998:Hamiltonian description of the ideal fluid. Rev. Mod. Phys., 70, 467-521, doi:10.1103/RevModPhys.70.467.
17. Müller, P., 1995: Ertel's potential vorticity theorem in physical oceanography. Rev. Geophys., 33, 67-87, doi:10.1029/ 94RG03215.
18. Müller, R., 1989: A note on the relation between the traditional approximation and the metric of the primitive equations. Tellus, 41A, 175-178, doi:10.1111/ j.1600-0870.1989.tb00374.x.
19. Newcomb, W. A., 1967: Exchange invariance in fluid systems in magneto-fluid and plasma dynamics. Proc. Symp. Appl. Math., 18, 152-161.
20. Ogura, Y., andN. 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.
21. Padhye, N., and P. J. Morrison, 1996: Fluid element relabeling symmetry. Phys. Lett., 219A, 287-292, doi:10.1016/ 0375-9601(96)00472-0.
22. Phillips, N. A., 1966: The equations of motion for a shallow rotating atmosphere and the traditional approximation. J. Atmos. Sci., 23, 626-628, doi:10.1175/1520-0469(1966)023,0626: TEOMFA.2.0.CO;2.
23. Roulstone, I., and S. J. Brice, 1995: On the Hamiltonian formulation of the quasi-hydrostatic equations. Quart. J. Roy. Meteor. Soc., 121, 927-936, doi:10.1002/qj.49712152410.
24. Salmon, R., 1988: Hamiltonian fluid dynamics. Annu. Rev. Fluid Mech., 20, 225-256, doi:10.1146/annurev.fl.20.010188.001301.
25. Showman, A. P., and L. M. Polvani, 2011: Equatorial superrotation on tidally locked exoplanets. Astrophys. J., 738, 71, doi:10.1088/ 0004-637X/738/1/71.
26. Staniforth, A., 2014a: Spheroidal and spherical geopotential approximations. Quart. J. Roy. Meteor. Soc., doi:10.1002/ qj.2324, in press.
27. Staniforth, A., 2014b: Deriving consistent approximate models of the global atmosphere using Hamilton's principle. Quart. J. Roy. Meteor. Soc., doi:10.1002/qj.2273, in press.
28. Tort, M., and T. Dubos, 2013: Dynamically consistent shallowatmosphere equations with a complete Coriolis force. Quart. J. Roy. Meteor. Soc., doi:10.1002/qj.2274, in press.
29. Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press, 745 pp.
30. White, A. A., and R. A. Bromley, 1995: Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Quart. J. Roy. Meteor. Soc., 121, 399-418, doi:10.1002/qj.49712152208.
31. White, A. A., and G. W. Inverarity, 2012: A quasi-spheroidal system for modelling global atmospheres: Geodetic coordinates. Quart. J. Roy. Meteor. Soc., 138, 27-33, doi:10.1002/qj.885.
32. White, A. A., and N. Wood, 2012: Consistent approximate models of the global atmosphere in non-spherical geopotential coordinates. Quart. J. Roy. Meteor. Soc., 138, 980-988, doi:10.1002/qj.972.
33. White, A. A., B. J. Hoskins, I. Roulstone, and A. Staniforth, 2005: Consistent approximate models of the global atmosphere: Shallow, deep, hydrostatic, quasi-hydrostatic and non-hydrostatic. Quart. J. Roy. Meteor. Soc., 131, 2081-2107, doi:10.1256/ qj.04.49.
34. White, A. A., A. Staniforth, and N. Wood, 2008: Spheroidal coordinate systems for modelling global atmospheres. Quart. J. Roy. Meteor. Soc., 134, 261-270, doi:10.1002/ qj.208.