Climate Science, Waves and PDEs for the Tropics

  • Andrew J. Majda
Part of the Abel Symposia book series (ABEL, volume 7)


A reader’s guide to recent applied mathematics development in multi-scale modeling in the tropics is provided here including the mathematical theory of precipitation fronts as well as singular limits with variable coefficients in the fast variables.


Partial Differential Equation Tropical Convection Planetary Scale Suitable Weak Solution Geophysical Flow 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Biello, J.A., 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), 341–390 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Biello, J.A., Majda, A.J.: A new multiscale model for the Madden–Julian oscillation. J. Atmos. Sci. 62, 694–1721 (2005) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Biello, J.A., Majda, A.J.: Modulating synoptic scale convective activity and boundary layer dissipation in the IPESD models of the Madden–Julian oscillation. Dyn. Atmos. Ocean. 42, 152–215 (2006) CrossRefGoogle Scholar
  4. 4.
    Biello, J.A., Majda, A.J.: Intraseasonal multi-scale moist dynamics of the tropical troposphere. Commun. Math. Sci. 8(2), 519–540 (2010) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Biello, J.A., Majda, A.J., Moncrieff, M.W.: Meridional momentum flux and superrotation in the multi-scale IPESD MJO model. J. Atmos. Sci. 64, 1636–1651 (2007) CrossRefGoogle Scholar
  6. 6.
    Bourgeois, A.J., Beale, J.T.: Validity of the quasigeostrophic model for large-scale flow in the atmosphere and ocean. SIAM J. Math. Anal. 25(4), 1023–1068 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chao, W.C.: On the origin of the tropical intraseasonal oscillation. J. Atmos. Sci. 44, 1940–1949 (1987) CrossRefGoogle Scholar
  8. 8.
    Emanuel, K.A.: An air-sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci. 44, 2324–2340 (1987) CrossRefGoogle Scholar
  9. 9.
    Embid, P., Majda, A.J.: Averaging over fast gravity waves for geophysical flows with arbitrary potential vorticity. Commun. Partial Differ. Equ. 21(3–4), 619–658 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Embid, P.F., Majda, A.J.: Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers. Geophys. Astrophys. Fluid Dyn. 87(1–2), 1–50 (1998) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Frierson, D.M.W., Majda, A.J., Pauluis, O.M.: Large scale dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation limit. Commun. Math. Sci. 19, 591–626 (2004) MathSciNetGoogle Scholar
  12. 12.
    Gill, A.E.: Atmosphere-Ocean Dynamics. Academic Press, New York (1982) Google Scholar
  13. 13.
    Grabowski, W.W.: Coupling cloud processes with the large-scale dynamics using the cloud-resolving convection parameterization (CRCP). J. Atmos. Sci. 58, 978–997 (2001) CrossRefGoogle Scholar
  14. 14.
    Grabowski, W.W.: MJO-like coherent structures: sensitivity simulations using the cloud-resolving convection parameterization (CRCP). J. Atmos. Sci. 60, 847–864 (2003) CrossRefGoogle Scholar
  15. 15.
    Grabowski, W.W., Moncrieff, M.W.: Moisture-convection feedback in the tropics. Q. J. R. Meteorol. Soc. 130, 3081–3104 (2004) CrossRefGoogle Scholar
  16. 16.
    Hendon, H.H., Salby, M.L.: The life cycle of the Madden–Julian oscillation. J. Atmos. Sci. 51, 2225–2237 (1994) CrossRefGoogle Scholar
  17. 17.
    Hendon, H.H., Liebmann, B.: Organization of convection within the Madden–Julian oscillation. J. Geophys. Res. 99, 8073–8084 (1994) CrossRefGoogle Scholar
  18. 18.
    Houze, R.A.: Mesoscale convective systems. Rev. Geophys. 42, RG4003 (2004) CrossRefGoogle Scholar
  19. 19.
    Johnson, R.H., Hamilton, P.J.: The relationship of surface pressure features to the precipitation and airflow structure of an intense midlatitude squall line. Mon. Weather Rev. 116, 1444–1473 (1988) CrossRefGoogle Scholar
  20. 20.
    Khouider, B., Majda, A.J.: A non-oscillatory balanced scheme for an idealized tropical climate model. Part I: Algorithm and validation. Theor. Comput. Fluid Dyn. 19, 331–354 (2005) zbMATHCrossRefGoogle Scholar
  21. 21.
    Khouider, B., 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, 355–375 (2005) zbMATHCrossRefGoogle Scholar
  22. 22.
    Khouider, B., Majda, A.J.: A simple multi-cloud model for convectively coupled tropical waves. Part I: Linear analysis. J. Atmos. Sci. 63, 1308–1323 (2006) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Khouider, B., Majda, A.J.: A simple multicloud parameterization for convectively coupled tropical waves. Part II. Nonlinear simulations. J. Atmos. Sci. 64, 381–400 (2007) CrossRefGoogle Scholar
  24. 24.
    Klainerman, S., Majda, A.J.: Compressible and incompressible fluids. Commun. Pure Appl. Math. 35(5), 629–651 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Klein, R.: An applied mathematical view of meteorological modeling. In: Hill, J.M., Moore, R. (eds.) Applied Mathematics Entering the 21st Century: Invited Talks from the ICIAM 2003 Congress. Proceedings in Appl. Math., vol. 116, 227–289. SIAM, Philadelphia (2004) Google Scholar
  26. 26.
    Klein, R., Angew, Z.: Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. Math. Mech. 80, 765–777 (2000) zbMATHGoogle Scholar
  27. 27.
    Klein, R., Majda, A.J.: Systematic multiscale models for deep convection on mesoscales. Theor. Comput. Fluid Dyn. 20, 525–551 (2006) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lau, W.K.M., Waliser, D.E. (eds.): Intraseasonal Variability in the Atmosphere Ocean Climate System. Springer, Berlin (2005) Google Scholar
  29. 29.
    Lin, J.L., et al.: Tropical intraseasonal variability in 14 IPCC AR4 climate models. Part I. Convective signals. J. Climate 19, 2665–2690 (2006) CrossRefGoogle Scholar
  30. 30.
    Madden, R., Julian, P.: Detection of a 40–50 day oscillation in the zonal wind in the tropical Pacific. J. Atmos. Sci. 28, 702–708 (1971) CrossRefGoogle Scholar
  31. 31.
    Madden, R.A., Julian, P.R.: Observations of the 40–50-day tropical oscillation–a review. Mon. Weather Rev. 122, 814–837 (1994) CrossRefGoogle Scholar
  32. 32.
    Majda, A.J.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Applied Mathematical Sciences, vol. 53. Springer, New York (1984) zbMATHCrossRefGoogle Scholar
  33. 33.
    Majda, A.J.: Real world turbulence and modern applied mathematics. In: Mathematics: Frontiers and Perspectives, pp. 137–151. Am. Math. Soc., Providence (2000) Google Scholar
  34. 34.
    Majda, A.J.: Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, vol. 9. AMS/CIMS, New York (2003) zbMATHGoogle Scholar
  35. 35.
    Majda, A.J.: Multiscale models with moisture and systematic strategies for superparameterization. J. Atmos. Sci. 64, 2726–2734 (2007) CrossRefGoogle Scholar
  36. 36.
    Majda, A.J.: New multiscale models and self-similarity in tropical convection. J. Atmos. Sci. 64, 1393–1404 (2007) CrossRefGoogle Scholar
  37. 37.
    Majda, A.J., Biello, J.A.: The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves. J. Atmos. Sci. 60(15), 1809–1821 (2003) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Majda, A.J., Biello, J.A.: A multiscale model for the intraseasonal oscillation. Proc. Natl. Acad. Sci. USA 101, 4736–4741 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Majda, A.J., Embid, P.: Averaging over fast gravity waves for geophysical flow with unbalanced initial data. Theor. Comput. Fluid Dyn. 11, 155–169 (1998) zbMATHCrossRefGoogle Scholar
  40. 40.
    Majda, A.J., Dutrifoy, A.: The dynamics of equatorial long waves: a singular limit with fast variable coefficients. Commun. Math. Sci. 4(2), 375–397 (2006) MathSciNetzbMATHGoogle Scholar
  41. 41.
    Majda, A.J., Dutrifoy, A.: Fast wave averaging for the equatorial shallow water equations. Commun. Partial Differ. Equ. 32(10), 1617–1642 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Majda, A.J., Klein, R.: Systematic multiscale models for the tropics. J. Atmos. Sci. 60, 393–408 (2003) CrossRefGoogle Scholar
  43. 43.
    Majda, A.J., Souganidis, P.: The existence and uniqueness of weak solutions for precipitation fronts: a novel hyperbolic free boundary problem in several space variables. Commun. Pure Appl. Math. 63(10), 1351–1361 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Majda, A.J., Stechmann, S.N.: The skeleton of tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. USA 106(21), 8417–8422 (2009) CrossRefGoogle Scholar
  45. 45.
    Majda, A.J., Stechmann, S.N.: A simple dynamical model with features of convective momentum transport. J. Atmos. Sci. 66, 373–392 (2009) CrossRefGoogle Scholar
  46. 46.
    Majda, A.J., Rosales, R.R., Tabak, E.G., Turner, C.V.: Interaction of large-scale equatorial waves and dispersion of Kelvin waves through topographic resonances. J. Atmos. Sci. 56(24), 4118–4133 (1999) MathSciNetCrossRefGoogle Scholar
  47. 47.
    Majda, A.J., Stechmann, S.N., Khouider, B.: Madden–Julian oscillation analog and intraseasonal variability in a multicloud model above the equator. Proc. Natl. Acad. Sci. USA 104, 9919–9924 (2007) zbMATHCrossRefGoogle Scholar
  48. 48.
    Majda, A.J., Dutrifoy, A., Schochet, S.: A simple justification of the singular limit for equatorial shallow-water dynamics. Commun. Pure Appl. Math. 62, 322–333 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Maloney, E.D., Hartman, D.L.: Frictional moisture convergence in a composite life cycle of the Madden–Julian oscillation. J. Climate 11, 2387–2403 (1998) CrossRefGoogle Scholar
  50. 50.
    Mapes, B.E., Tulich, S., Lin, J.L., Zuidema, P.: The mesoscales convection life cycle: building block or prototype for large-scale tropical waves? Dyn. Atmos. Ocean. 42, 3–29 (2006) CrossRefGoogle Scholar
  51. 51.
    Moncrieff, M.W.: Analytic representation of the large-scale organization of tropical convection. J. Atmos. Sci. 61, 1521–1538 (2004) MathSciNetCrossRefGoogle Scholar
  52. 52.
    Moncrieff, M.W., Shapiro, M., Slingo, J., Molteni, F.: Collaborative research at the intersection of weather and climate. WMO Bull. 56, 204–211 (2007) Google Scholar
  53. 53.
    Neelin, J.D., Held, I.M., Cook, K.H.: Evaporation-wind feedback and low-frequency variability in the tropical atmosphere. J. Atmos. Sci. 44, 2341–2348 (1987) CrossRefGoogle Scholar
  54. 54.
    Nakazawa, T.: Tropical super clusters within intraseasonal variations over the western Pacific. J. Meteorol. Soc. Jpn. 66, 823–839 (1988) Google Scholar
  55. 55.
    Pauluis, O., Frierson, D.M.W., Majda, A.J.: Precipitation fronts and the reflection and transmissions of tropical disturbances. Q. J. R. Meteorol. Soc. 134, 913–930 (2008) CrossRefGoogle Scholar
  56. 56.
    Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1979) zbMATHCrossRefGoogle Scholar
  57. 57.
    Philander, S.G.: El Nino, La Nina, and The Southern Oscillation. Academic Press, San Diego (1990) Google Scholar
  58. 58.
    Raymond, D.J.: A new model of the Madden–Julian oscillation. J. Atmos. Sci. 58, 2807–2819 (2001) CrossRefGoogle Scholar
  59. 59.
    Roundy, P., Frank, W.: A climatology of waves in the equatorial region. J. Atmos. Sci. 61, 2105–2132 (2004) MathSciNetCrossRefGoogle Scholar
  60. 60.
    Salby, M., Hendon, H.: Intraseasonal behavior of clouds, temperature, and motion in the tropics. J. Atmos. Sci. 51, 2207–2224 (1994) CrossRefGoogle Scholar
  61. 61.
    Salby, M.L., Garcia, R.R., Hendon, H.H.: Planetary-scale circulations in the presence of climatological and wave-induced heating. J. Atmos. Sci. 51, 2344–2367 (1994) CrossRefGoogle Scholar
  62. 62.
    Schochet, S.: Resonant nonlinear geometric optics for weak solutions of conservation laws. J. Differ. Equ. 113(2), 473–504 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Schochet, S.: Fast singular limits of hyperbolic PDEs. J. Differ. Equ. 114(2), 476–512 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Smith, R.K. (ed.): The Physics and Parametrization of Moist Atmospheric Convection. NATO Advanced Study Institute Series C. Mathematical and Physical Sciences, vol. 505. Kluwer Academic, Norwell (1997) Google Scholar
  65. 65.
    Stechmann, S.N., Majda, A.J.: The structure of precipitation fronts for finite relaxation time. Theor. Comput. Fluid Dyn. 20, 377–404 (2006) CrossRefGoogle Scholar
  66. 66.
    Wang, B., Rui, H.: Dynamics of the coupled moist Kelvin–Rossby wave on an equatorial beta-plane. J. Atmos. Sci. 47, 397–413 (1990) CrossRefGoogle Scholar
  67. 67.
    Wheeler, M., Kiladis, G.N.: Convectively coupled equatorial waves: analysis of clouds and temperature in the wave number-frequency domain. J. Atmos. Sci. 56, 374–399 (1999) CrossRefGoogle Scholar
  68. 68.
    Zhang, C.: Madden–Julian oscillation. Rev. Geophys. 43, G2003 (2005) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Atmosphere/Ocean Science (CAOS), Courant Institute of Mathematical Sciences (CIMS)New York UniversityNew YorkUSA

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