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Climate Science, Waves and PDEs for the Tropics

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Nonlinear Partial Differential Equations

Part of the book series: Abel Symposia ((ABEL,volume 7))

Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Biello, J.A., Majda, A.J.: A new multiscale model for the Madden–Julian oscillation. J. Atmos. Sci. 62, 694–1721 (2005)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  4. Biello, J.A., Majda, A.J.: Intraseasonal multi-scale moist dynamics of the tropical troposphere. Commun. Math. Sci. 8(2), 519–540 (2010)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chao, W.C.: On the origin of the tropical intraseasonal oscillation. J. Atmos. Sci. 44, 1940–1949 (1987)

    Article  Google Scholar 

  8. Emanuel, K.A.: An air-sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci. 44, 2324–2340 (1987)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  12. Gill, A.E.: Atmosphere-Ocean Dynamics. Academic Press, New York (1982)

    Google Scholar 

  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)

    Article  Google Scholar 

  14. Grabowski, W.W.: MJO-like coherent structures: sensitivity simulations using the cloud-resolving convection parameterization (CRCP). J. Atmos. Sci. 60, 847–864 (2003)

    Article  Google Scholar 

  15. Grabowski, W.W., Moncrieff, M.W.: Moisture-convection feedback in the tropics. Q. J. R. Meteorol. Soc. 130, 3081–3104 (2004)

    Article  Google Scholar 

  16. Hendon, H.H., Salby, M.L.: The life cycle of the Madden–Julian oscillation. J. Atmos. Sci. 51, 2225–2237 (1994)

    Article  Google Scholar 

  17. Hendon, H.H., Liebmann, B.: Organization of convection within the Madden–Julian oscillation. J. Geophys. Res. 99, 8073–8084 (1994)

    Article  Google Scholar 

  18. Houze, R.A.: Mesoscale convective systems. Rev. Geophys. 42, RG4003 (2004)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  24. Klainerman, S., Majda, A.J.: Compressible and incompressible fluids. Commun. Pure Appl. Math. 35(5), 629–651 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  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. Klein, R., Angew, Z.: Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. Math. Mech. 80, 765–777 (2000)

    MATH  Google Scholar 

  27. Klein, R., Majda, A.J.: Systematic multiscale models for deep convection on mesoscales. Theor. Comput. Fluid Dyn. 20, 525–551 (2006)

    Article  MathSciNet  Google Scholar 

  28. Lau, W.K.M., Waliser, D.E. (eds.): Intraseasonal Variability in the Atmosphere Ocean Climate System. Springer, Berlin (2005)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  31. Madden, R.A., Julian, P.R.: Observations of the 40–50-day tropical oscillation–a review. Mon. Weather Rev. 122, 814–837 (1994)

    Article  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  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. 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)

    MATH  Google Scholar 

  35. Majda, A.J.: Multiscale models with moisture and systematic strategies for superparameterization. J. Atmos. Sci. 64, 2726–2734 (2007)

    Article  Google Scholar 

  36. Majda, A.J.: New multiscale models and self-similarity in tropical convection. J. Atmos. Sci. 64, 1393–1404 (2007)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  38. Majda, A.J., Biello, J.A.: A multiscale model for the intraseasonal oscillation. Proc. Natl. Acad. Sci. USA 101, 4736–4741 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  41. Majda, A.J., Dutrifoy, A.: Fast wave averaging for the equatorial shallow water equations. Commun. Partial Differ. Equ. 32(10), 1617–1642 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Majda, A.J., Klein, R.: Systematic multiscale models for the tropics. J. Atmos. Sci. 60, 393–408 (2003)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Majda, A.J., Stechmann, S.N.: The skeleton of tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. USA 106(21), 8417–8422 (2009)

    Article  Google Scholar 

  45. Majda, A.J., Stechmann, S.N.: A simple dynamical model with features of convective momentum transport. J. Atmos. Sci. 66, 373–392 (2009)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  51. Moncrieff, M.W.: Analytic representation of the large-scale organization of tropical convection. J. Atmos. Sci. 61, 1521–1538 (2004)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  54. Nakazawa, T.: Tropical super clusters within intraseasonal variations over the western Pacific. J. Meteorol. Soc. Jpn. 66, 823–839 (1988)

    Google Scholar 

  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)

    Article  Google Scholar 

  56. Pedlosky, J.: Geophysical Fluid Dynamics. Springer, New York (1979)

    Book  MATH  Google Scholar 

  57. Philander, S.G.: El Nino, La Nina, and The Southern Oscillation. Academic Press, San Diego (1990)

    Google Scholar 

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

    Article  Google Scholar 

  59. Roundy, P., Frank, W.: A climatology of waves in the equatorial region. J. Atmos. Sci. 61, 2105–2132 (2004)

    Article  MathSciNet  Google Scholar 

  60. Salby, M., Hendon, H.: Intraseasonal behavior of clouds, temperature, and motion in the tropics. J. Atmos. Sci. 51, 2207–2224 (1994)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  62. Schochet, S.: Resonant nonlinear geometric optics for weak solutions of conservation laws. J. Differ. Equ. 113(2), 473–504 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  63. Schochet, S.: Fast singular limits of hyperbolic PDEs. J. Differ. Equ. 114(2), 476–512 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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. Stechmann, S.N., Majda, A.J.: The structure of precipitation fronts for finite relaxation time. Theor. Comput. Fluid Dyn. 20, 377–404 (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  68. Zhang, C.: Madden–Julian oscillation. Rev. Geophys. 43, G2003 (2005)

    Google Scholar 

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Majda, A.J. (2012). Climate Science, Waves and PDEs for the Tropics. In: Holden, H., Karlsen, K. (eds) Nonlinear Partial Differential Equations. Abel Symposia, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25361-4_12

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