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Two-way interactions between equatorially-trapped waves and the barotropic flow

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Lateral energy exchange between the tropics and the midlatitudes is a topic of great importance for understanding Earth’s climate system. In this paper, the authors address this issue in an idealized set up through simple shallow water models for the interactions between equatorially trapped waves and the barotropic mode, which supports Rossby waves that propagate poleward and can excite midlatitude teleconnection patterns. It is found here that the interactions between a Kelvin wave and a fixed meridional shear (mimicking the jet stream) generates a non-trivial meridional velocity and meridional convergence in phase with the upward motion that can attain a maximum of about 50%, which oscillates on frequencies ranging from one day to 10 days. When, on the other hand, the barotropic flow is forced by slowly propagating Kelvin waves a complex flow pattern emerges; it consists of a phase-locked barotropic response that is equatorially trapped and that propagates eastward with the forcing Kelvin wave and a certain number of planetary Rossby waves that propagate westward and toward the poles as seen in nature. It is suggested here that the poleward propagating waves are to some sort of multi-way resonant interaction with the phase locked response. Moreover, it is shown here that a numerical scheme with dispersion properties that depend on the direction perpendicular to the direction of propagation, namely the 2D central scheme of Nessyahu and Tadmor, can artificially alter significantly the topology of the wave fields and thus should be avoided in climate models.

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  1. [1]

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

  2. [2]

    Audusse, E, Bouchut, F., Bristeau, M.-O., et al, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25(6), 2004, 2050–2065.

  3. [3]

    Bale, D. S., LeVeque, R. J., Mitran, S., et al, A wave propagation method for conservation laws and balance laws with spatially varying flux functions, SIAM J. Sci. Comput., 24(3), 2002, 955–978.

  4. [4]

    Biello, J. A. and Majda, A. J., Boundary layer dissipation and the nonlinear interaction of equatorial baroclinic barotropic Rossby waves, Geophys. Astrophys. Fluid Dyn., 98(2), 2004, 85–127.

  5. [5]

    Biello, J. A. and Majda, A. J., The effect of meridional and vertical shear on the interaction of equatorial baroclinic and barotropic Rossby waves, Stud. Appl. Math., 112(4), 2003, 341–390.

  6. [6]

    Bond, N. A. and Vecchi, G. A., The influence of the Madden-Julian oscillation on precipitation in Oregon and Washington, Wea. Forecasting, 18(4), 2003, 600–613.

  7. [7]

    Bouchut, F., Sommer, J. L. and Zeitlin, V., Frontal geostrophic adjustment and nonlinear wave phenomena in one dim rotating shallow water, Part 2: high resolution numerical simulations, J. Fluid Mech., 514, 2004, 35–63.

  8. [8]

    Dunkerton, T. J. and Crum, F. X., Eastward propagating ∼2- to 15-day equatorial convection and its relation to the tropical intraseasonal oscillation, J. Geophys. Res., 100(12), 1995, 25781–25790.

  9. [9]

    Durran, D. R., Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Springer-Verlag, New York, 1999.

  10. [10]

    Emanuel, K. A, Atmospheric Convection, Oxford University Press, Oxford, 1994.

  11. [11]

    Emanuel, K. A., An air-sea interaction model of intraseasonal oscillations in the tropics, J. Atmos. Sci., 44(6), 1987, 2324–3240.

  12. [12]

    Ferguson, J., A numerical solution for the barotropic vorticity equation forced by an equatorially trapped wave, Master’s thesis, University of Victoria, 2008.

  13. [13]

    Frierson, D. M. W., Majda, A. J. and Pauluis, O. M., Dynamics of precipitation fronts in the tropical atmosphere, Commun. Math. Sci., 2(4), 2004, 591–626.

  14. [14]

    Gill, A. E., Atmosphere-Ocean Dynamics, Academic Press, New York, 1982.

  15. [15]

    Hendon, H. and Liebmann, B., Organization of convection within the Madden-Julian oscillation, J. Geophys. Res., 99(4), 1994, 8073–8083.

  16. [16]

    Houze, R. A., Jr., Stratiform precipitation in regions of convection: A meteorological Paradox? Bull. Amer. Meteor. Soc., 78(10), 1997, 2179–2196.

  17. [17]

    Johnson, R. H., Rickenbach, T. M., Rutledge, S. A., et al, Trimodal characteristics of tropical convection, J. Climate., 12(8), 1999, 2397–2418.

  18. [18]

    Jones, C., Waliser, D. E., Lau, K. M., et al, The Madden-Julian oscillation and its impact on Northern hemisphere weather predictability, Mon. Wea. Rev., 132(6), 2004, 1462–1471.

  19. [19]

    Hoskins, B. J. and Jin, F.-F., The initial value problem for tropical perturbations to a baroclinic atmosphere, Quart. J. Roy. Meteor. Soc., 117(498), 1991, 299–317.

  20. [20]

    Kasahara, A. and Dias, P. S., Response of planetary waves to stationary tropical heating in a global atmosphere with meridional and vertical shear, J. Atmos. Sci., 43(18), 1986, 1893–1912.

  21. [21]

    Khouider, B. and Majda, A. J., A simple multicloud parametrization for convectively coupled tropical waves, Part I: linear analysis, J. Atmos. Sci., 63(4), 2006, 1308–1323.

  22. [22]

    Khouider, B. and Majda, A. J., Model multicloud parametrizations with crude vertical structure, Theor. Comput. Fluid Dyn., 20(5–6), 2006, 351–375.

  23. [23]

    Khouider, B. and Majda, A. J., A simple multicloud parametrization for convectively coupled tropical waves, Part II: nonlinear simulations, J. Atmos. Sci., 64(2), 2007, 381–400.

  24. [24]

    Khouider, B. and Majda, A. J., Multicloud models for tropical convection: enhanced congestus heating, J. Atmos. Sci., 65(3), 2008, 895–914.

  25. [25]

    Khouider, B. and Majda, A. J., Equatorial convectively coupled waves in a simple multicloud model, J. Atmos. Sci., 65(11), 2008, 3376–3397.

  26. [26]

    Khouider, B. and Majda, A. J., A non-oscillatory balanced scheme for an idealized tropical climate model, Part I: algorithm and validation, Theor. Comput. Fluid Dyn., 19(5), 2005, 331–354.

  27. [27]

    Khouider, B. and Majda, A. J., A non-oscillatory balanced scheme for an idealized tropical climate model, Part II: nonlinear coupling and moisture effects, Theor. Comput. Fluid Dyn., 19(5), 2005, 355–375.

  28. [28]

    Kiladis, G. N., Straub, K. H. and Haertel, P. T., Zonal and vertical structure of the Madden-Julian oscillation, J. Atmos. Sci., 62(8), 2005, 2790–2809.

  29. [29]

    Kiladis, G. N., Wheeler, M. C., Haertel, P. T., et al, Convectively coupled equatorial waves, Rev. Geophys., 47, 2009, RG2003. DOI:10.1029/2008RG000266

  30. [30]

    Lau, W. K. M. and Waliser, D. E., Intraseasonal Variability in the Atmosphere-Ocean Climate System, Praxis, Chichester, 2005.

  31. [31]

    LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.

  32. [32]

    Lin, J.-L., Kiladis, G. N., Mapes, B. E., et al, Tropical intraseasonal variability in 14 IPCC AR4 climate models, Part 1: convective signals, J. Climate, 19(12), 2006, 2665–2690.

  33. [33]

    Lin, J.-L., Lee, M.-I., Kim, D., et al, The impacts of convective parameterization and moisture triggering on AGCM-simulated convectively coupled equatorial waves, J. Climate, 21(5), 2008, 883–909.

  34. [34]

    Lin, X. and Johnson, R. H., Kinematic and thermodynamic characteristics of the flow over the Western Pacific Warm Pool during TOGA COARE, J. Atmos. Sci., 53(5), 1996, 695–715.

  35. [35]

    Liotta, S. F., Romano, V. and Russo, G., Central schemes for systems of balance laws, Hyperbolic Problems: Theory, Numerics, Applications, Seventh International Conference in Zürich, February 1998, M. Fey and R. Jeltsch (eds.), International Series on Numerical Mathematics, 130, Birkhäuser, Boston, 1999, 651–660.

  36. [36]

    Madden, R. A. and Julian, P. R., Description of global scale circulation cells in tropics with a 40–50 day period, J. Atmos. Sci., 29(6), 1972, 1109–1123.

  37. [37]

    Madden, R. A. and Julian, P. R., Observations of the 40-50 day tropical oscillation — A review, Mon. Wea. Rev., 122(5), 1994, 814–837.

  38. [38]

    Majda, A. J., Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes, 9, AMS, 2003.

  39. [39]

    Majda, A. J. and Biello, J. A., The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmos. Sci., 60(15), 2003, 1809–1821.

  40. [40]

    Majda, A. J., Khouider, B., Kiladis, G. N., et al, A model for convectively coupled tropical waves: Nonlinearity, rotation, and comparison with observations, J. Atmos. Sci., 61(17), 2004, 2188–2205.

  41. [41]

    Majda, A. J. and Shefter, M. G., Waves and instabilities for model tropical convective parameterizations, J. Atmos. Sci., 58(8), 2001, 896–914.

  42. [42]

    Mapes, B. E., Convective inhibition, subgridscale triggering energy, and stratiform instability in a toy tropical wave model, J. Atmos. Sci., 57(10), 2000, 1515–1535.

  43. [43]

    Matsuno, T., Quasi-geostrophic motions in the equatorial area, J. Meteor. Soc. Japan, 44(1), 1966, 25–43.

  44. [44]

    Moncrieff, M. W. and Klinker, E., Organized convective systems in the tropical western Pacific as a process in general circulation models: a TOGA-COARE case study, Quart. J. Roy. Meteor. Soc., 123(540), 1997, 805–827.

  45. [45]

    Nakazawa, T., Tropical super clusters within intraseasonal variations over the western Pacific, J. Meteor. Soc. Japan, 66(6), 1988, 823–839.

  46. [46]

    Neelin, J. D., and Yu, J.-Y., Modes of tropical variability under convective adjustment and Madden-Julian oscillation, Part I: analytical theory, J. Atmos. Sci., 51(13), 1994, 1876–1894.

  47. [47]

    Neelin, J. D. and Zeng, N., A quasi-equilibrium tropical circulation model-formulation, J. Atmos. Sci., 57(11), 2000, 1741–1766.

  48. [48]

    Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87(2), 1990, 408–463.

  49. [49]

    Raymond, D. J., A new model for the Madden-Julian oscillation, J. Atmos. Sci., 58(18), 2001, 2807–2819.

  50. [50]

    Pedlosky, J., Geophysical Fluid Dynamics, Springer-Verlag, New York, 1979.

  51. [51]

    Roundy, P., Analysis of convectively coupled Kelvin Waves in the Indian Ocean MJO, J. Atmos. Sci., 65(4), 2008, 1342–1359.

  52. [52]

    Slingo, J. M, Sperber, K. R., Boyle, J. S., et al, Intraseasonal oscillation in 15 atmospheric general circulation models: results from an AMIP diagnostic subproject, Climate Dyn., 12(5), 1996, 325–357.

  53. [53]

    Smith, R. K., The Physics and Parameterization of Moist Atmospheric Convection, Kluwer, Dordresht, 1997.

  54. [54]

    Scinocca, J. F. and McFarlane, N. A., The variability of modeled tropical precipitation, J. Atmos. Sci., 61(16), 2004, 1993–2015.

  55. [55]

    Straub, K. H. and Kiladis, G. N., The observed structure of convectively coupled Kelvin waves: Comparison with simple models of coupled wave instability, J. Atmos. Sci., 60(14), 2003, 1655–1668.

  56. [56]

    Stechmann, S., Majda, A. and Khouider, B., Nonlinear dynamics of hydrostatic internal gravity waves, Theor. Comput. Fluid Dyn., 22(6), 2008, 407–432.

  57. [57]

    Takayabu, Y. N. and Murakami, M., The structure of super cloud clusters observed in 1-20 June 1986 and their relationship to easterly waves, J. Meteor. Soc. Japan, 69(1), 1991, 105–125.

  58. [58]

    Takayabu, Y. N., Large-scale cloud disturbances associated with equatorial waves, Part I: spectral features of the cloud disturbances, J. Meteor. Soc. Japan, 72(3), 1994, 433–448.

  59. [59]

    Yano, J.-I., Moncrieff, M. W. and McWilliams, J. C., Linear stability and single-column analyses of several cumulus parametrization categories in a shallow-water model, Quart. J. Roy. Meteor. Soc., 124(547), 1998, 983–1005.

  60. [60]

    Wang, B. and Xie, X. S., Low-frequency equatorial waves in vertically sheared zonal flow, Part I: stable waves, J. Atmos. Sci., 53(3), 1996, 449–467.

  61. [61]

    Webster, P. J., Response of the tropical atmosphere to local steady forcing, Mon. Wea. Rev., 100(7), 1972, 518–541.

  62. [62]

    Webster, P. J. and Chang, H.-R., Energy accumulation and emanation regions at low latitudes: Impacts of a zonally varying basic state, J. Atmos. Sci., 45(5), 1988, 803–829.

  63. [63]

    Webster, P. J., Seasonality of atmospheric response to sea-surface temperature anomalies, J. Atmos. Sci., 39(1), 1982, 41–52.

  64. [64]

    Webster, P. J. and Holton, J. R., Cross-equatorial response to middle-latitude forcing in a zonally varying basic state, J. Atmos. Sci., 39(4), 1982, 722–733.

  65. [65]

    Wheeler, M. and Kiladis, G. N., Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber-frequency domain, J. Atmos. Sci., 56(3), 1999, 374–399.

  66. [66]

    Wheeler, M., Kiladis, G. N. and Webster, P. J., Large-scale dynamical fields associated with convectively coupled equatorial waves, J. Atmos. Sci., 57(5), 2000, 613–640.

  67. [67]

    Zhang, C. D. and Webster, P. J., Laterally forced equatorial waves in mean zonal flows, Part I: stationary transient forcing, J. Atmos. Sci., 49(7), 1992, 585–607.

  68. [68]

    Zhang, C. D., Madden-Julian oscillation, Rev. Geophys., 43, 2005, RG2003. DOI:10.1029/2004RG000158.

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Correspondence to Boualem Khouider.

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Dedicated to Professor Andrew Majda on the Occasion of his 60th Birthday

Project supported in part by the Natural Sciences and Engineering Research Council of Canada (No. 288339-2004) and the Canadian Foundation for Climate and Atmospheric Sciences (No. GR-7021).

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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).

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  • Equatorially trapped waves
  • Wave interactions
  • Jet stream
  • Rossby waves
  • Phase locked
  • Dispersion relation
  • Central scheme
  • Climate modeling

2000 MR Subject Classification

  • 65M06
  • 65M12
  • 76B15
  • 76B55
  • 76B60
  • 86A10