• Batchelor, G. K., 1977: An Introduction to Fluid Dynamics. Cambridge University Press, 615 pp.

  • Berner, J., , G. J. Shutts, , M. Leutbecher, , and T. N. Palmer, 2009: A spectral stochastic kinetic energy backscatter scheme and its impact on flow-dependent predictability in the ECMWF Ensemble Prediction System. J. Atmos. Sci., 66, 603626.

    • Search Google Scholar
    • Export Citation
  • Buckwar, E., , and R. Winkler, 2006: Multistep methods for SDEs and their application to problems with small noise. SIAM J. Numer. Anal., 44, 779803.

    • Search Google Scholar
    • Export Citation
  • Buckwar, E., , A. Rößler, , and R. Winkler, 2010: Stochastic Runge–Kutta methods for Itô SODEs with small noise. SIAM J. Sci. Comput., 32, 17891808.

    • Search Google Scholar
    • Export Citation
  • Caya, A., , R. Laprise, , and P. Zwack, 1998: Consequences of using the splitting method for implementing physical forcings in a semi-implicit semi-Lagrangian model. Mon. Wea. Rev., 126, 17071713.

    • Search Google Scholar
    • Export Citation
  • Cullen, M. J. P., 2001: Alternative implementations of the semi-Lagrangian semi-implicit schemes in the ECMWF model. Quart. J. Roy. Meteor. Soc., 127, 27872802.

    • Search Google Scholar
    • Export Citation
  • Cullen, M. J. P., , and D. J. Salmond, 2003: On the use of a predictor–corrector scheme to couple the dynamics with the physical parameterizations in the ECMWF model. Quart. J. Roy. Meteor. Soc., 129, 12171236.

    • Search Google Scholar
    • Export Citation
  • Diamantakis, M., , T. Davies, , and N. Wood, 2007: An iterative time-stepping scheme for the Met Office's semi-implicit semi-Lagrangian non-hydrostatic model. Quart. J. Roy. Meteor. Soc.,133, 997–1011.

  • Dubal, M., , N. Wood, , and A. Staniforth, 2004: Analysis of parallel versus sequential splittings for time-stepping physical parameterizations. Mon. Wea. Rev., 132, 121132.

    • Search Google Scholar
    • Export Citation
  • Dubal, M., , N. Wood, , and A. Staniforth, 2005: Mixed parallel–sequential-split schemes for time-stepping multiple physical parameterizations. Mon. Wea. Rev., 133, 9891002.

    • Search Google Scholar
    • Export Citation
  • Dubal, M., , N. Wood, , and A. Staniforth, 2006: Some numerical properties of approaches to physics–dynamics coupling for NWP. Quart. J. Roy. Meteor. Soc., 132, 2742.

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer, 465 pp.

  • Durran, D. R., , and J. B. Klemp, 1983: A compressible model for the simulation of moist mountain waves. Mon. Wea. Rev., 111, 23412361.

    • Search Google Scholar
    • Export Citation
  • Eckermann, S. D., 2011: Explicitly stochastic parameterization of nonorographic gravity wave drag. J. Atmos. Sci., 68, 17491765.

  • Egger, J., 2003: Advection equation with oscillating forcing: Numerical analysis. Mon. Wea. Rev., 131, 984989.

  • Ewald, B. D., 2012: Weak versions of stochastic Adams–Bashforth and semi-implicit leapfrog schemes for SDEs. Comput. Methods Appl. Math., 12, 2331.

    • Search Google Scholar
    • Export Citation
  • Ewald, B. D., , and R. Temam, 2003: Analysis of stochastic numerical schemes for the evolution equations of geophysics. Appl. Math. Lett., 16, 12231229.

    • Search Google Scholar
    • Export Citation
  • Ewald, B. D., , and C. Penland, 2009: Numerical generation of stochastic differential equations in climate models. Handbook of Numerical Analysis: Computational Methods for the Atmosphere and the Oceans, R. Temam and J. Tribbia, Eds., Elsevier, 279–306.

  • Ewald, B. D., , C. Penland, , and R. Temam, 2004: Accurate integration of stochastic climate models with application to El Niño. Mon. Wea. Rev., 132, 154164.

    • Search Google Scholar
    • Export Citation
  • Itô, K., 1951: On stochastic differential equations. Mem. Amer. Math. Soc., 4, 151.

  • Jazwinski, A. H., 2007: Stochastic Processes and Filtering Theory. Academic Press, 376 pp.

  • Kloeden, P. E., , and E. Platen, 1999: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, 636 pp.

  • Lauritzen, P. H., Ed., 2011: Numerical Techniques for Global Atmospheric Models. Springer, 382 pp.

  • Leith, C. E., 1990: Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer. Phys. Fluids, 2A, 297299.

  • Majda, A. J., , and B. Khouider, 2002: Stochastic and mesoscopic models for tropical convection. Proc. Natl. Acad. Sci. USA, 99, 11231128.

    • Search Google Scholar
    • Export Citation
  • Orszag, S. A., 1970: Transform method for the calculation of vector-coupled sums: Application to the spectral form of the vorticity equation. J. Atmos. Sci., 27, 890895.

    • Search Google Scholar
    • Export Citation
  • Papanicolaou, G. C., , and W. Kohler, 1974: Asymptotic theory of mixing stochastic ordinary differential equations. Commun. Pure Appl. Math., 27, 641668.

    • Search Google Scholar
    • Export Citation
  • Phillips, N., 1959: An example of non-linear computational instability. The Atmosphere and Sea in Motion, B. Bolin, Ed., Oxford University Press, 501–504.

  • Plant, R. S., , and G. C. Craig, 2008: A stochastic parameterization for deep convection based on equilibrium statistics. J. Atmos. Sci., 65, 87105.

    • Search Google Scholar
    • Export Citation
  • Rümelin, W., 1982: Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal., 19, 604613.

  • Schumann, U., 1995: Stochastic backscatter of turbulence energy and scalar variance by random subgrid-scale fluxes. Proc. Roy. Soc. London, 451A, 293318.

    • Search Google Scholar
    • Export Citation
  • Shutts, G., 2006: Upscale effects in simulations of tropical convection on an equatorial beta-plane. Dyn. Atmos. Oceans, 42, 3058.

  • Staniforth, A., , N. Wood, , and J. Côté, 2002a: A simple comparison of four physics–dynamics coupling schemes. Mon. Wea. Rev., 130, 31293135.

    • Search Google Scholar
    • Export Citation
  • Staniforth, A., , N. Wood, , and J. Côté, 2002b: Analysis of the numerics of physics–dynamics coupling. Quart. J. Roy. Meteor. Soc., 128, 27792799.

    • Search Google Scholar
    • Export Citation
  • Stratonovich, R. L., 1966: A new representation for stochastic integrals and equations. J. SIAM Control, 4, 362371.

  • Teixeira, J., , and C. A. Reynolds, 2008: Stochastic nature of physical parameterizations in ensemble prediction: A stochastic convection approach. Mon. Wea. Rev., 136, 483496.

    • Search Google Scholar
    • Export Citation
  • Termonia, P., , and R. Hamdi, 2007: Stability and accuracy of the physics–dynamics coupling in spectral models. Quart. J. Roy. Meteor. Soc., 133, 15891604, doi:10.1002/qj.119.

    • Search Google Scholar
    • Export Citation
  • van Kampen, N. G., 1981: Itô versus Stratonovich. J. Stat. Phys., 24, 175187.

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The Impact of Noisy Physics on the Stability and Accuracy of Physics–Dynamics Coupling

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  • 1 Naval Research Laboratory, Monterey, California
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Abstract

The coupling of the dynamical core of a numerical weather prediction model to the physical parameterizations is an important component of model design. This coupling between the physics and the dynamics is explored here from the perspective of stochastic differential equations (SDEs). It will be shown that the basic properties of the impact of noisy physics on the stability and accuracy of common numerical methods may be obtained through the application of the basic principles of SDEs. A conceptual model setting is used that allows the study of the impact of noise whose character may be tuned to be either very red (smooth) or white (noisy). The change in the stability and accuracy of common numerical methods as the character of the noise changes is then studied. Distinct differences are found between the ability of multistage (Runge–Kutta) schemes as compared with multistep (Adams–Bashforth/leapfrog) schemes to handle noise of various characters. These differences will be shown to be attributable to the basic philosophy used to design the scheme. Additional experiments using the decentering of the noisy physics will also be shown to lead to strong sensitivity to the quality of the noise. As an example, the authors find the novel result that noise of a diffusive character may lead to instability when the scheme is decentered toward greater implicitness. These results are confirmed in a nonlinear shear layer simulation using a subgrid-scale mixing parameterization. This subgrid-scale mixing parameterization is modified stochastically and shown to reproduce the basic principles found here, including the notion that decentering toward implicitness may lead to instability.

Corresponding author address: Dr. Daniel Hodyss, Naval Research Laboratory, Marine Meteorology Division, 7 Grace Hopper Ave., Stop 2, Monterey, CA 93943. E-mail: daniel.hodyss@nrlmry.navy.mil

Abstract

The coupling of the dynamical core of a numerical weather prediction model to the physical parameterizations is an important component of model design. This coupling between the physics and the dynamics is explored here from the perspective of stochastic differential equations (SDEs). It will be shown that the basic properties of the impact of noisy physics on the stability and accuracy of common numerical methods may be obtained through the application of the basic principles of SDEs. A conceptual model setting is used that allows the study of the impact of noise whose character may be tuned to be either very red (smooth) or white (noisy). The change in the stability and accuracy of common numerical methods as the character of the noise changes is then studied. Distinct differences are found between the ability of multistage (Runge–Kutta) schemes as compared with multistep (Adams–Bashforth/leapfrog) schemes to handle noise of various characters. These differences will be shown to be attributable to the basic philosophy used to design the scheme. Additional experiments using the decentering of the noisy physics will also be shown to lead to strong sensitivity to the quality of the noise. As an example, the authors find the novel result that noise of a diffusive character may lead to instability when the scheme is decentered toward greater implicitness. These results are confirmed in a nonlinear shear layer simulation using a subgrid-scale mixing parameterization. This subgrid-scale mixing parameterization is modified stochastically and shown to reproduce the basic principles found here, including the notion that decentering toward implicitness may lead to instability.

Corresponding author address: Dr. Daniel Hodyss, Naval Research Laboratory, Marine Meteorology Division, 7 Grace Hopper Ave., Stop 2, Monterey, CA 93943. E-mail: daniel.hodyss@nrlmry.navy.mil
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