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The interaction of equatorial waves with a barotropic shear: a potential test case for climate model dynamical cores

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Abstract

We compare the performances of two different numerical methods to solve the equatorial shallow water equations in a background meridional shear: A coarse-resolution Galerkin-truncation-based method and a finite-volume method, for the case of an equatorial Rossby wave. In the presence of the barotropic shear, the Rossby wave quickly loses its energy through shear interaction and excites other equatorially trapped waves despite the fact that the PDE system is linear. The two methods handle this sudden energy exchange across scales very differently. While the finite volume converges statistically to a coherent large-scale solution, the Galerkin truncation method results in large-scale and slow oscillations where energy is bumped back and forth within the resolved-scale spectrum, involving higher-order equatorial wave modes heavily depending on numerical resolution, that is, the number of Galerkin basis functions. The addition of an artificial viscosity for the coarse-resolution Galerkin method heavily damps the energy oscillations without significantly changing its transient dynamics. This provides an example of interaction across scales where the resolution of scales that are apparently not representative of the statistical solution are important for the transient dynamics. Therefore, non-linear interactions of equatorially trapped waves may provide an interesting test bed for the validation of climate model dynamical cores.

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References

  1. Arakawa A.: Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part 1. J. Comput. Phys. 1, 119–143 (1966)

    Article  MATH  Google Scholar 

  2. Bale D., LeVeque R., Mitran S., Rossmanith J.: A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24, 955–978 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Durran, D.: Numerical methods for fluid dynamics with applications to geophysics. Springer Series: Texts in Applied Mathematics, vol. 32, 2nd edn (2010)

  4. Ferguson J., Khouider B., Namazi M.: Two-way interactions between equatorially-trapped waves and the barotropic flow. Chin. Ann. Math. Ser. B 30, 539–568 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ham F.E., Lien F.S., Strong A.B.: A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids. J. Comput. Phys. 177, 117–133 (2002)

    Article  MATH  Google Scholar 

  6. Held I.M., Suarez M.J.: A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models. Bull. Am. Meteor. Soc. 75, 1825–1830 (1994)

    Article  Google Scholar 

  7. Hickel S., Adams N.A., Domaradzki J.A.: An adaptive local deconvolution method for implicit LES. J. Comput. Phys. 213, 413–436 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jablonowski C., Williamson D.L.: A baroclinic instability test case for atmospheric model dynamical cores. Q. J. R. Meteorol. Soc. 132, 2943–2975 (2006)

    Article  Google Scholar 

  9. Khouider B., Majda A.: A simple multicloud parametrization for convectively coupled tropical waves. Part I: linear analysis. J. Atmos. Sci. 63, 1308–1323 (2006)

    Article  MathSciNet  Google Scholar 

  10. Khouider B., Majda A.: Equatorial convectively coupled waves in a simple multicloud model. J. Atmos. Sci. 65, 3376–3397 (2008)

    Article  Google Scholar 

  11. Khouider B., Majda A.: A non-oscillatory balanced scheme for an idealized tropical climate model. Part I: algorithm and validation. Theor. Comput. Fluid Dyn. 19, 331–354 (2005)

    Article  MATH  Google Scholar 

  12. Khouider B., Majda A.: A non-oscillatory balanced scheme for an idealized tropical climate model. Part II: nonlinear interactions and effects of moisture. Theor. Comput. Fluid Dyn. 19, 355–375 (2005)

    Article  MATH  Google Scholar 

  13. Khouider B., St Cyr A., Majda A., Tribbia J.: MJO and convectively coupled waves in a coarse resolution GCM using a simple multicloud parameterization. J. Atmos. Sci 68, 240–264 (2011)

    Article  Google Scholar 

  14. Kiladis G.N., Straub K., Haertel P.: Zonal and vertical structure of the Madden-Julian oscillation. J. Atmos. Sci. 62, 2790–2809 (2005)

    Article  Google Scholar 

  15. Leveque R.: Balancing source terms and flux gradients in high-order Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Levy M.N., Nair R.D., Tufo H.M.: High-order Galerkin methods for scalable global atmospheric models. Comput. Geosci. 33, 1022–1035 (2007)

    Article  Google Scholar 

  17. Liotta S.F., Romano V., Russo G.: Central scheme for balance laws of relaxation. SIAM J. Numer. Anal. 38, 1337–1356 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Majda A., Biello J.: The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves. J. Atmos. Sci. 60, 1809–1821 (2003)

    Article  MathSciNet  Google Scholar 

  19. Majda A., Khouider B.: A Numerical strategy for efficient modeling of equatorial waves. Proc. Nat. Acad. Sci. 98, 1341–1346 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Majda A.: Introduction to PDEs and Waves for the Atmosphere and Ocean. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  21. Nair R.D., Jablonowski C.: Moving vortices on the sphere: a test case for horizontal advection problems. Mon. Weather Rev. 136, 699–711 (2008)

    Article  Google Scholar 

  22. Nessyahu H., Tadmor E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ralston A., Rabinowitz P.: A First Course in Numerical Analysis. McGraw Hill, New York (1978)

    MATH  Google Scholar 

  24. Schumacher C., Houze R.A., Kraucunas I.: The tropical dynamical response to latent heating estimates derived from the TRMM precipitation radar. J. Atmos. Sci. 61, 1341–1358 (2004)

    Article  Google Scholar 

  25. Smagorinsky J.: General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev. 91, 99–164 (1963)

    Article  Google Scholar 

  26. Smolarkiewicz P.K., Prusa J.M.: VLES modeling of geophysical fluids with nonoscillatory forward-in-time schemes. Int. J. Numer. Methods Fluids 39, 799–819 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. St-Cyr A., Jablonowski C., Dennis J.M., Tufo H.M., Thomas S.J.: A comparison of two shallow water models with non-conforming adaptive grids. Mon. Weather Rev. 136, 1898–1922 (2008)

    Article  Google Scholar 

  28. Taylor M., Tribbia J., Iskandarani M.: The spectral element method for the shallow water equations on the sphere. J. Comput. Phys. 130, 92–108 (1997)

    Article  MATH  Google Scholar 

  29. 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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wheeler M., Kiladis G.: Convectively Coupled Equatorial Waves: Analysis of Clouds and Temperature in the Wavenumber–Frequency Domain. J. Atmos. Sci 56, 374–399 (1999)

    Article  Google Scholar 

  31. You D., Moin P.: A dynamic global-coefficient subgrid-scale eddy-viscosity model for large-eddy simulation in complex geometries. Phys. Fluids 19, 065110 (2007)

    Article  Google Scholar 

  32. Zhang C., Webster P.: Effect of zonal flow on equatorially trapped waves. J. Atmos. Sci. 46, 3632–3652 (1989)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, PO BOX 3045, STN CSC, Victoria, BC, V8W 3P4, Canada

    Maryam Namazi & Boualem Khouider

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  1. Maryam Namazi
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  2. Boualem Khouider
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Correspondence to Boualem Khouider.

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Communicated by Klein.

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Namazi, M., Khouider, B. The interaction of equatorial waves with a barotropic shear: a potential test case for climate model dynamical cores. Theor. Comput. Fluid Dyn. 27, 149–176 (2013). https://doi.org/10.1007/s00162-012-0257-y

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