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Mathematical Analysis of the Jin-Neelin Model of El Niño-Southern-Oscillation

Abstract

The Jin-Neelin model for the El Niño–Southern Oscillation (ENSO for short) is considered for which the authors establish existence and uniqueness of global solutions in time over an unbounded channel domain. The result is proved for initial data and forcing that are sufficiently small. The smallness conditions involve in particular key physical parameters of the model such as those that control the travel time of the equatorial waves and the strength of feedback due to vertical-shear currents and upwelling; central mechanisms in ENSO dynamics.

From the mathematical view point, the system appears as the coupling of a linear shallow water system and a nonlinear heat equation. Because of the very different nature of the two components of the system, the authors find it convenient to prove the existence of solution by semi-discretization in time and utilization of a fractional step scheme. The main idea consists of handling the coupling between the oceanic and temperature components by dividing the time interval into small sub-intervals of length k and on each sub-interval to solve successively the oceanic component, using the temperature T calculated on the previous sub-interval, to then solve the sea-surface temperature (SST for short) equation on the current sub-interval. The passage to the limit as k tends to zero is ensured via a priori estimates derived under the aforementioned smallness conditions.

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References

  1. [1]

    Adams, R. A. and Fournier, J. J. F., Sobolev Spaces, 140, Academic Press, Amsterdam, 2003.

    MATH  Google Scholar 

  2. [2]

    Aubin, J.–P., Un théoreme de compacité, C. R. Acad. Sci. Paris, 256(24), 1963, 5042–5044.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    Barnston, A. G., Tippett, M. K., Heureux, M. L., et al., Skill of real–time seasonal ENSO model predictions during 2002–2011 — is our capability improving?, Bull. Amer. Meteo. Soc., 93(5), 2012, 631–651.

    Article  Google Scholar 

  4. [4]

    Bjerknes, J., Atmospheric teleconnections from the equatorial Pacific, Monthly Weather Review, 97(3), 1969, 163–172.

    Article  Google Scholar 

  5. [5]

    Brézis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer–Verlag, New York, 2011.

    MATH  Google Scholar 

  6. [6]

    Camargo, S. J. and Sobel, A. H., Western North Pacific tropical cyclone intensity and ENSO, Journal of Climate, 18(15), 2005, 2996–3006.

    Article  Google Scholar 

  7. [7]

    Cane, M. A. and Zebiak, S. E., A theory for El Nieno and the Southern Oscillation, Science, 228, 1985, 1085–1088.

    Article  Google Scholar 

  8. [8]

    Cane, M. A., Experimental forecasts of El Nieno, Nature, 321, 1986, 827–832.

    Article  Google Scholar 

  9. [9]

    Cane, M. A. and Sarachik, E. S., Forced baroclinic ocean motions, II, The linear equatorial bounded case, J. of Marine Research, 35(2), 1977, 395–432.

    Google Scholar 

  10. [10]

    Cao, C. and Titi, E. S., Global well–posedness of the three–dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math. (2), 166(1), 2007, 245–267.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    Cazenave, T. and Haraux, A., An Introduction to Semilinear Evolution Equations, 13, Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford, 1998.

    MATH  Google Scholar 

  12. [12]

    Chang, P., Ji, L., Wang, B. and Li, T., Interactions between the seasonal cycle and El Nieno–Southern Oscillation in an intermediate coupled ocean–atmosphere model, Journal of the Atmospheric Sciences, 52(13), 1995, 2353–2372.

    Article  Google Scholar 

  13. [13]

    Chekroun, M. D., Ghil, M. and Neelin, J. D., Pullback attractor crisis in a delay differential ENSO model, Advances in Nonlinear Geosciences, to appear, A. Tsonis, Ed. Springer–Verlag, 2018, 1–33.

    Book  Google Scholar 

  14. [14]

    Chekroun, M. D., Kondrashov, D. and Ghil, M., Predicting stochastic systems by noise sampling, and application to the El Nieno–Southern Oscillation, Proc. Natl. Acad. Sci USA, 108(29), 2011, 11766–11771.

    Article  Google Scholar 

  15. [15]

    Chekroun, M. D., Neelin, J. D., Kondrashov, D., et al., Rough parameter dependence in climate models: The role of Ruelle–Pollicott resonances, Proc. Natl. Acad. Sci USA, 111(5), 2014, 1684–1690.

    Article  Google Scholar 

  16. [16]

    Chekroun, M. D., Simonnet, E. and Ghil, M., Stochastic climate dynamics: Random attractors and timedependent invariant measures, Physica D., 240(21), 2011, 1685–1700.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    Chen, C., Cane, M. A., Henderson, N., et al., Diversity, nonlinearity, seasonality, and memory effect in ENSO simulation and prediction using empirical model reduction, Journal of Climate, 29(5), 2016, 1809–1830.

    Article  Google Scholar 

  18. [18]

    Chorin, A., Numerical solution of the Navier–Stokes equations, Math. Comput., 22, 1968, 745–762.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    Coti Zelati, M., Huang, A., Kukavica, I., et al., The primitive equations of the atmosphere in presence of vapour saturation, Nonlinearity, 28(3), 2015, 625–668.

    MathSciNet  Article  MATH  Google Scholar 

  20. [20]

    Dijkstra, H. A., Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Nieno, 28, Springer–Verlag Science & Business Media, 2005.

    Google Scholar 

  21. [21]

    Engel, K.–J. and Nagel, R., One–Parameter Semigroups for Linear Evolution Equations, 194, Graduate Texts in Mathematics, Springer–Verlag, New York, 2000.

    MATH  Google Scholar 

  22. [22]

    Guilyardi, E., Wittenberg, A., Fedorov, A., et al., Understanding El Nieno in ocean–atmosphere general circulation models: Progress and challenges, Bulletin of the American Meteorological Society, 90(3), 2009, 325–340.

    Article  Google Scholar 

  23. [23]

    Huang, A. and Temam, R., The linearized 2D inviscid shallow water equations in a rectangle: Boundary conditions and well–posedness, Archive for Rational Mechanics and Analysis, 211(3), 2014, 1027–1063.

    MathSciNet  Article  MATH  Google Scholar 

  24. [24]

    Huang, A. and Temam, R., The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction, Commun. Pure Appl. Anal., 13(5), 2014, 2005–2038.

    MathSciNet  Article  MATH  Google Scholar 

  25. [25]

    Huang, A. and Temam, R., The linear hyperbolic initial and boundary value problems in a domain with corners, Discrete and Continuous Dynamical Systems, Series B, 19(6), 2014, 1627–1665.

    MathSciNet  MATH  Google Scholar 

  26. [26]

    Huang, A. and Temam, R., The 2D nonlinear fully hyperbolic inviscid shallow water equations in a rectangle, J. Dynam. Differential Equations, 27(3–4), 2015, 763–785.

    MathSciNet  Article  MATH  Google Scholar 

  27. [27]

    Jin, F.–F., An equatorial ocean recharge paradigm for ENSO, Part I: Conceptual model, Journal of the Atmospheric Sciences, 54(7), 1997, 811–829.

    Article  Google Scholar 

  28. [28]

    Jin, F.–F. and Neelin, J. D., Modes of interannual tropical ocean–atmosphere interaction–A unified view, Part I: Numerical results, Journal of the Atmospheric Sciences, 50(21), 1993, 3477–3503.

    MathSciNet  Article  Google Scholar 

  29. [29]

    Jin, F.–F. and Neelin, J. D., Modes of interannual tropical ocean–atmosphere interaction–A unified view, Part III: Analytical results in fully coupled cases, Journal of the atmospheric sciences, 50(21), 1993, 3523–3540.

    MathSciNet  Article  Google Scholar 

  30. [30]

    Jin, F.–F., Neelin, J. D. and Ghil, M., El Nieno on the Devil’s staircase: Annual subharmonic steps to chaos, Science, 274, 1994, 70–72.

    Article  Google Scholar 

  31. [31]

    Jin, F.–F., Neelin, J. D. and Ghil, M., El Nieno/Southern Oscillation and the annual cycle: Subharmonic frequency locking and aperiodicity, Physica D, 98, 1996, 442–465.

    Article  MATH  Google Scholar 

  32. [32]

    Kiladis, G. N., Wheeler, M. C., Haertel, P. T, et al., Convectively coupled equatorial waves, Reviews of Geophysics, 47(2), 2009.

    Google Scholar 

  33. [33]

    Kirtman, B. P. and Schopf, P. S., Decadal variability in ENSO predictability and prediction, Journal of Climate, 11(11), 1998, 2804–2822.

    Article  Google Scholar 

  34. [34]

    Kobelkov, G. M., Existence of a solution “in the large” for ocean dynamics equations, J. Math. Fluid Mech., 9(4), 2007, 588–610.

    MathSciNet  Article  MATH  Google Scholar 

  35. [35]

    Kondrashov, D., Kravtsov, S., Robertson, A. W. and Ghil, M., A hierarchy of data–based ENSO models, J. Climate, 18(21), 1995, 4425–4444.

    Article  Google Scholar 

  36. [36]

    Kukavica, I. and Ziane, M., The regularity of solutions of the primitive equations of the ocean in space dimension three, C. R. Math. Acad. Sci. Paris, 345(5), 2007, 257–260.

    MathSciNet  Article  MATH  Google Scholar 

  37. [37]

    Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires, 2nd ed., 68, Gauthier–Villars Dunod, 1969.

    MATH  Google Scholar 

  38. [38]

    Lions, J. L. and Magenes, E., Non–homogeneous Boundary Value Problems and Applications. Vol. I, 2nd ed., 68, Springer–Verlag, New York, 1972.

    Book  MATH  Google Scholar 

  39. [39]

    Lions, J.–L., Temam, R. and Wang, S., New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5(2), 1992, 237–288.

    MathSciNet  Article  MATH  Google Scholar 

  40. [40]

    Lions, J.–L., Temam, R. and Wang, S., On the equations of the large–scale ocean, Nonlinearity, 5(5), 1992, 1007–1053.

    MathSciNet  Article  MATH  Google Scholar 

  41. [41]

    Lyon, B. and Barnston, A. G., ENSO and the spatial extent of interannual precipitation extremes in tropical land areas, Journal of Climate, 18(23), 2005, 5095–5109.

    Article  Google Scholar 

  42. [42]

    Marchuk, G. I., Methods of Numerical Mathematics, 2nd ed., Springer–Verlag, New York, Heidelberg, Berlin, 1982.

    Book  MATH  Google Scholar 

  43. [43]

    Matsuno, T., Quasi–geostrophic motions in the equatorial area, Journal of the Meteorological Society of Japan., Ser. II, 44(1), 1966, 25–43.

    Google Scholar 

  44. [44]

    McCreary Jr, J. P. and Anderson, D. L. T., A simple model of El Nieno and the Southern Oscillation, Monthly Weather Review, 112(5), 1984, 934–946.

    Article  Google Scholar 

  45. [45]

    McCreary Jr, J. P. and Anderson, D. L. T., Simple models of El Nieno and the Southern Oscillation, in Elsevier oceanography series, Nihoul, J. C. J.(ed), 40, Elsevier Amsterdam, 1985, 345–370.

    Google Scholar 

  46. [46]

    McCreary Jr, J. P. and Anderson, D. L. T., An overview of coupled ocean–atmosphere models of El Nieno and the Southern Oscillation, Journal of Geophysical Research: Oceans, 96(S01), 1991, 3125–3150.

    Article  Google Scholar 

  47. [47]

    McPhaden, M. J., Zebiak, S. E. and Glantz, M. H., ENSO as an integrating concept in earth science, science, 314(5806), 2006, 1740–1745.

    Article  Google Scholar 

  48. [48]

    Mechoso, C. R., Neelin, J. D. and Yu, J.–Y., Testing simple models of ENSO, J. Atmos. Sci., 60, 2003, 305–318.

    Article  Google Scholar 

  49. [49]

    Neelin, J. D., The slow sea surface temperature mode and the fast–wave limit: Analytic theory for tropical interannual oscillations and experiments in a hybrid coupled model, J. of the Atmos. Sci., 48(4), 1991, 584–606.

    Article  Google Scholar 

  50. [50]

    Neelin, J. D., Battisti, D. S., Hirst, A. C., et al., ENSO theory, Journal of Geophysical Research: Oceans, 103(C7), 1998, 14261–14290.

    Article  Google Scholar 

  51. [51]

    Neelin, J. D., Dijkstra, H. A., Ocean–atmosphere interaction and the tropical climatology, Part I: The dangers of flux correction, Journal of climate, 8(5), 1995, 1325–1342.

    Article  Google Scholar 

  52. [52]

    Neelin, J. D. and Jin, F.–F., Modes of interannual tropical ocean–atmosphere interaction–a unified view, Part II: Analytical results in the weak–coupling limit, Journal of the atmospheric sciences, 50(21), 1993, 3504–3522.

    MathSciNet  Article  Google Scholar 

  53. [53]

    Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, 44, Applied Mathematical Sciences, Springer–Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  54. [54]

    Penland, C. and Sardeshmukh, P. D., The optimal growth of tropical sea–surface temperature anomalies, J. Climate, 8(8), 1995, 1999–2024.

    Article  Google Scholar 

  55. [55]

    Philander, S. G. H., El Nieno, La Niena, and the Southern Oscillation, Academic Press, San Diego, 1992.

    Google Scholar 

  56. [56]

    Sarachik, E. S. and Cane, M. A., The El Nieno–Southern Oscillation Phenomenon, Cambridge University Press, New York, 2010.

    Book  Google Scholar 

  57. [57]

    Temam, R., Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires (II), Arch. Ration. Mech. Anal., 33, 1969, 377–385.

    Article  MATH  Google Scholar 

  58. [58]

    Temam, R., Infinite–Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., 68, Applied Mathematical Sciences, Springer–Verlag, New York, 1997.

    Book  MATH  Google Scholar 

  59. [59]

    Tziperman, E., Cane, M. A. and Zebiak, S. E., Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi–periodicity route to chaos, Journal of the Atmospheric Sciences, 52(3), 1995, 293–306.

    Article  Google Scholar 

  60. [60]

    Tziperman, E., Stone, L., Cane, M. and Jarosh, H., El Nieno chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean–atmosphere oscillator, Science, 264(5155), 1994, 72–74.

    Article  Google Scholar 

  61. [61]

    Wang, C. and Picaut, J., Understanding ENSO physics—A review, in Earth’s Climate: The Ocean–Atmosphere Interaction, Geophys. Monogr., 147, 2004, 21–48.

    Google Scholar 

  62. [62]

    Wang, C. and Wang, X., Classifying El Nieno Modoki I and II by different impacts on rainfall in southern China and typhoon tracks, Journal of Climate, 26(4), 2013, 1322–1338.

    Article  Google Scholar 

  63. [63]

    Yanenko, N. N., The method of fractional steps: the solution of problems of mathematical physics in several variables, Springer–Verlag, New York–Heidelberg, 1971.

    Book  MATH  Google Scholar 

  64. [64]

    Zebiak, S. E., A simple atmospheric model of relevance to El Nieno, Journal of the Atmospheric Sciences, 39(9), 1982, 2017–2027.

    Article  Google Scholar 

  65. [65]

    Zebiak, S. E., Atmospheric convergence feedback in a simple model for El Nieno, Monthly weather review, 114(7), 1986, 1263–1271.

    Article  Google Scholar 

  66. [66]

    Zebiak, S. E. and Cane, M. A., A model El Nieno–southern oscillation, Monthly Weather Review, 115(10), 1987, 2262–2278.

    Article  Google Scholar 

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Acknowledgements

MDC is grateful to David Neelin for the numerous inspiring discussions about the JN model and ENSO modeling in general, and to Dmitri Kondrashov for the useful discussions regarding the numerical integration of the JN model.

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Correspondence to Yining Cao.

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This work was supported by the Office of Naval Research Multidisciplinary University Research Initiative (No. N00014-16-1-2073), the National Science Foundation (Nos. OCE-1658357, DMS-1616981, DMS- 1206438, DMS-1510249) and the Research Fund of Indiana University.

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Cao, Y., Chekroun, M.D., Huang, A. et al. Mathematical Analysis of the Jin-Neelin Model of El Niño-Southern-Oscillation. Chin. Ann. Math. Ser. B 40, 1–38 (2019). https://doi.org/10.1007/s11401-018-0115-3

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Keywords

  • El Niño–Southern Oscillation
  • Coupled nonlinear hyperbolic-parabolic systems
  • Fractional step method
  • Semigroup theory

2000 MR Subject Classification

  • 35K55
  • 35L50
  • 35M33
  • 47D03
  • 76U05