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Some conservation issues for the dynamical cores of NWP and climate models. (English) Zbl 1132.86314
Summary: The rationale for designing atmospheric numerical model dynamical cores with certain conservation properties is reviewed. The conceptual difficulties associated with the multiscale nature of realistic atmospheric flow, and its lack of time-reversibility, are highlighted. A distinction is made between robust invariants, which are conserved or nearly conserved in the adiabatic and frictionless limit, and non-robust invariants, which are not conserved in the limit even though they are conserved by exactly adiabatic frictionless flow. For non-robust invariants, a further distinction is made between processes that directly transfer some quantity from large to small scales, and processes involving a cascade through a continuous range of scales; such cascades may either be explicitly parameterized, or handled implicitly by the dynamical core numerics, accepting the implied non-conservation. An attempt is made to estimate the relative importance of different conservation laws. It is argued that satisfactory model performance requires spurious sources of a conservable quantity to be much smaller than any true physical sources; for several conservable quantities the magnitudes of the physical sources are estimated in order to provide benchmarks against which any spurious sources may be measured.

86A10 Meteorology and atmospheric physics
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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[1] Andrews, D.G.; McIntyre, M.E., On wave action and its relatives, J. fluid mech., 89, 647-664, (1978) · Zbl 0431.76011
[2] Arakawa, A., A personal perspective on the early years of general circulation modeling at UCLA, (), 1-65
[3] Arakawa, A.; Lamb, V.R., A potential entrophy and energy conserving scheme for the shallow water equations, Mon. weather rev., 109, 18-36, (1981)
[4] Bannon, P.R., Hydrostatic adjustment: lamb’s problem, J. atmos. sci., 52, 1743-1752, (1995)
[5] Barry, L.; Craig, G.C.; Thuburn, J., A GCM investigation into the nature of baroclinic adjustment, J. atmos. sci., 57, 1141-1155, (2000)
[6] Bates, J.R., A dynamical stabilizer in the climate system: a mechanism suggested by a simple model, Tellus, 51A, 349-372, (1999)
[7] Bates, J.R.; Li, Y.; Brandt, A.; McCormick, S.F.; Ruge, J., A global shallow-water numerical model based on the semi-Lagrangian advection of potential vorticity, Quart. J. roy. meteorol. soc., 121, 1981-2005, (1995)
[8] Bridges, T.J.; Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve simplecticity, Phys. lett. A., 284, 184-193, (2001) · Zbl 0984.37104
[9] Bryan, G.H.; Fritsch, J.M., A reevaluation of ice – liquid water potential temperature, Mon. weather rev., 132, 2421-2431, (2004)
[10] D.M. Burridge, J. Haseler, A model for medium range weather forecasting—Adiabatic formulation. ECMWF Tech. Rep. 4, Reading, United Kingdom, 1977, 46pp.
[11] Chapman, D.; Browning, K.A., Measurements of dissipation rate in frontal zones, Quart. J. roy. meteorol. soc., 127, 1939-1959, (2001)
[12] Cho, J.-Y.; Lindborg, E., Horizontal velocity structure functions in the upper troposphere and lower stratosphere 1. observations, J. geophys. res., 106, 10,223-10,232, (2001)
[13] Cho, J.-Y., Horizontal wavenumber spectra of winds, temperature and trace gases during the Pacific exploratory missions. part I: climatology, J. geophys. res., 104, 5697-5716, (1999)
[14] Cho, J.-Y.; Newell, R.E.; Barrick, J.D., Horizontal wavenumber spectra of winds, temperature and trace gases during the Pacific exploratory missions. part II: gravity waves, quasi-two-dimensional turbulence, and vortical modes, J. geophys. res., 104, 16,297-16,308, (1999)
[15] M.J.P. Cullen, Modelling atmospheric and oceanic flows. Acta Numerica, in preparation. · Zbl 1120.76069
[16] Cullen, M.J.P.; Douglas, R.J., Large-amplitude nonlinear stability results for atmospheric circulations, Quart. J. roy. meteorol. soc., 129, 1969-1988, (2003)
[17] M.J.P. Cullen, D. Salmond, P.K. Smolarkiewicz, Key numerical issues for the future development of the ECMWF model. in: Proceedings of the ECMWF Workshop on Developments in Numerical Methods for Very High Resolution Global Models, June, 2000, pp. 183-206.
[18] Curry, J.A.; Webster, P.J., Thermodynamics of atmospheres and oceans, (1999), Academic Press, 471pp
[19] Dewan, E.M., Stratospheric wave spectra resembling turbulence, Science, 204, 832-835, (1979)
[20] Dritschel, D.G.; Ambaum, M.H.P., A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields, Quart. J. roy. meteorol. soc., 123, 1097-1130, (1997)
[21] Durran, D.R., Numerical methods for wave equations in geophysical fluid dynamics, (1999), Springer, 465pp
[22] Egger, J., Numerical generation of entropies, Mon. weather rev., 127, 2211-2216, (1999)
[23] Frank, J.; Reich, S., Conservation properties of smoothed particle hydrodynamics applied to the shallow water equations, Bit, 43, 40-54, (2003)
[24] Frank, J.; Reich, S., The Hamiltonian particle-mesh method for the spherical shallow water equations, Atmos. sci. lett., 5, 89-95, (2004)
[25] Goody, R., Sources and sinks of climate entropy, Quart. J. roy. meteorol. soc., 126, 1953-1970, (2000)
[26] Grabowski, W.W.; Smolarkiewicz, P.K., Monotone finite-difference approximations to the advection – condensation problem, Mon. weather rev., 118, 2082-2097, (1990)
[27] A.R. Gregory, Numerical simulations of winter stratosphere dynamics, Ph.D. Thesis, University of Reading, 1999.
[28] Gregory, A.R.; West, V., The sensitivity of a model’s stratospheric tape recorder to the choice of advection scheme, Quart. J. roy. meteorol. soc., 128, 1827-1846, (2002)
[29] Haynes, P.H., Forced, dissipative generalizations of finite-amplitude wave activity conservation relations for zonal and non-zonal basic flows, J. atmos. sci., 45, 2352-2362, (1988)
[30] Haynes, P.H.; Anglade, J., The vertical-scale cascade in atmospheric tracers due to large-scale differential advection, J. atmos. sci., 54, 1121-1136, (1997)
[31] Hoskins, B.J.; McIntyre, M.E.; Robinson, A.W., On the use and significance of isentropic potential vorticity maps, Quart. J. roy. meteorol. soc., 111, 877-946, (1985)
[32] Johnson, D.R., “general coldness of climate models” and the second law: implications for modeling the Earth system, J. clim., 10, 2826-2846, (1997)
[33] Johnson, D.R.; Lenzen, A.J.; Zapotocny, T.H.; Schaak, T.K., Numerical uncertainties in the simulation of reversible isentropic processes and entropy conservation, J. clim., 13, 3860-3884, (2000)
[34] Kennedy, P.J.; Shapiro, M.A., The energy budget in a clear air turbulence zone as observed by aircraft, Mon. weather rev., 103, 650-654, (1975)
[35] Koshyk, J.N.; Boer, G.J., Parametrization of dynamical subgrid-scale processes in a spectral GCM, J. atmos. sci., 52, 965-976, (1995)
[36] Koshyk, J.N.; Hamilton, K.; Mahlman, J.D., Simulation of the k−5/3 mesoscale spectral regime in the GFDL SKYHI general circulation model, Geophys. res. lett., 26, 843-846, (1999)
[37] Laursen, L.; Eliasen, E., On the effects of the damping mechanisms in an atmospheric general circulation model, Tellus, 41A, 385-400, (1989)
[38] LeVeque, R.J., Numerical methods for conservation laws, (1992), Birkhauser, 214pp · Zbl 0847.65053
[39] Li, Y.; Ruge, J.; Bates, J.R.; Brandt, A., A proposed adiabatic formulation of 3-dimensional global atmospheric models based on potential vorticity, Tellus, 52, 129-139, (2000)
[40] Lilly, D.K., Stratified turbulence and the mesoscale variability of the atmosphere, J. atmos. sci., 40, 749-761, (1983)
[41] Lin, S.-J.; Rood, R.B., An explicit flux-form semi-Lagrangian shallow-water model on the sphere, Quart. J. roy. meteorol. soc., 123, 2477-2498, (1997)
[42] Lindborg, E.; Cho, J.-Y., Horizontal velocity structure functions in the upper troposphere and lower stratosphere. 2. theoretical considerations, J. geophys. res., 106, 10,233-10,241, (2001)
[43] Mahowald, N.M.; Plumb, R.A.; Rasch, P.J.; del Corral, J.; Sassi, F.; Heres, W., Stratospheric transport in a three-dimensional isentropic coordinate model, J. geophys. res., 107, (2002), Art. no. 4254
[44] Mason, P.J.; Brown, A.R., On subgrid models and filter operations in large eddy simulations, J. atmos. sci., 56, 2101-2114, (1999)
[45] McCalpin, J.D., A quantitative analysis of the dissipation inherent in semi-Lagrangian advection, Mon. weather rev., 116, 2330-2336, (1988)
[46] McIntyre, M.E.; Norton, W.A., Potential vorticity inversion on a hemisphere, J. atmos. sci., 57, 1214-1235, (2000)
[47] McIntyre, M.E.; Shepherd, T.G., An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on arnold’s stability theorems, J. fluid mech., 181, 527-565, (1987) · Zbl 0646.76030
[48] Mohebalhojeh, A.R.; Dritschel, D.G., The contour-advective semi-Lagrangian algorithms for many-layer primitive-equation models, Quart. J. roy. meteorol. soc., 130, 347-364, (2004)
[49] Nastrom, G.D.; Gage, K.S., A climatology of atmospheric wavenumber spectra observed by commercial aircraft, J. atmos. sci., 42, 950-960, (1985)
[50] Peixoto, J.P.; Oort, A.H.; de Almeida, M.; Tomé, A., Entropy budget of the atmosphere, J. geophys. res., 96, 10,981-10,988, (1991)
[51] Peixoto, J.P.; Oort, A.H., Physics of climate, (1992), American Institute of Physics
[52] Pointin, Y., Wet equivalent potential temperature and enthalpy as prognostic variables in cloud modeling, J. atmos. sci., 41, 651-660, (1984)
[53] Reich, S., Backward error analysis for numerical integrators, SIAM J. numer. anal., 36, 1549-1570, (1999) · Zbl 0935.65142
[54] Sadourny, R., The dynamics of finite-difference models of the shallow-water equations, J. atmos. sci., 32, 680-689, (1974)
[55] Sadourny, R.; Basdevant, C., Parameterization of subgrid scale barotropic and baroclinic eddies in quasi-geostrophic models: anticipated potential vorticity method, J. atmos. sci., 42, 1353-1363, (1985)
[56] Salmon, R., Lectures on geophysical fluid dynamics, (1998), Oxford University Press, 378pp
[57] Salmon, R., Poisson-bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations, J. atmos. sci., 61, 2016-2036, (2004)
[58] Shutts, G., A kinetic energy backscatter algorithm for use in ensemble prediction systems, Quart. J. roy. meteorol. soc., 131, 3079-3102, (2005)
[59] Simmons, A.J.; Burridge, D.M., An energy and angular-momentum conserving vertical finite-difference scheme and hybrid vertical coordinates, Mon. weather rev., 109, 758-766, (1981)
[60] Smith, K.S., Comments on “the k−3 and k−5/3 energy spectrum of atmospheric turbulence: quasigeostrophic two-level model simulation”, J. atmos. sci., 61, 937-942, (2004)
[61] Straus, D.M.; Ditlevsen, P., Two-dimensional turbulence properties of the ECMWF reanalyses, Tellus, 51A, 749-772, (1999)
[62] Thuburn, J., Dissipation and cascades to small scales in numerical models using a shape-preserving advection scheme, Mon. weather rev., 123, 1888-1903, (1995)
[63] Thuburn, J.; Lagneau, V., Eulerian Mean, contour integral, and finite-amplitude wave activity diagnostics applied to a single layer model of the winter stratosphere, J. atmos. sci., 56, 689-710, (1999)
[64] Thuburn, J.; McIntyre, M.E., Numerical advection schemes, cross-isentropic random walks, and correlations between chemical species, J. geophys. res., 102, 6775-6797, (1997)
[65] Thuburn, J.; Tan, D.G.-H., A parameterization of mixdown time for atmospheric chemicals, J. geophys. res., 102, 13,037-13,049, (1997)
[66] Tripoli, G.J.; Cotton, W.R., The use of ice – liquid water potential temperature as a thermodynamic variable in deep atmosphere models, Mon. weather rev., 109, 1094-1102, (1981)
[67] Tung, K.K., Reply to comments on “the k−3 and k−5/3 energy spectrum of atmospheric turbulence: quasigeostrophic two-level model simulation”, J. atmos. sci., 61, 943-948, (2004)
[68] Tung, K.K.; Welch Orlando, W.T., The k−3 and k−5/3 energy spectrum of atmospheric turbulence: quasigeostrophic two-level model simulation, J. atmos. sci., 60, 824-835, (2003)
[69] WGNE, WMO Atmospheric Research and Environment Programme, CAS/JSC Working Group on Numerical Experimentation, Report No. 18, 2003.
[70] White, A.A., A view of the equations of meteorological dynamics and various approximations, (), 1-100 · Zbl 1036.86001
[71] Williamson, D.L.; Drake, J.B.; Hack, J.J.; Jakob, R.; Swarztrauber, P.N., A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. comput. phys., 102, 211-224, (1992) · Zbl 0756.76060
[72] T.J. Woollings, Entropy and potential vorticity in atmospheric dynamical core models, PhD Thesis, University of Reading, 2004.
[73] Woollings, T.J.; Thuburn, J., Entropy sources in a dynamical core atmospheric model, Quart. J. roy. meteorol. soc., 132, 43-59, (2006)
[74] Zapotocny, T.H.; Lenzen, A.J.; Johnson, D.R.; Reames, F.M.; Schaak, T.K., A comparison of inert trace constituent transport between the university of wisconsin isentropic-sigma model and the NCAR community climate model, Mon. weather rev., 125, 120-142, (1997)
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