Advertisement

Chinese Annals of Mathematics, Series B

, Volume 40, Issue 1, pp 1–38 | Cite as

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

  • Yining CaoEmail author
  • Mickaël D. Chekroun
  • Aimin Huang
  • Roger Temam
Article
  • 20 Downloads

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.

Keywords

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

2000 MR Subject Classification

35K55 35L50 35M33 47D03 76U05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

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.

References

  1. [1]
    Adams, R. A. and Fournier, J. J. F., Sobolev Spaces, 140, Academic Press, Amsterdam, 2003.zbMATHGoogle Scholar
  2. [2]
    Aubin, J.–P., Un théoreme de compacité, C. R. Acad. Sci. Paris, 256(24), 1963, 5042–5044.MathSciNetzbMATHGoogle 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.CrossRefGoogle Scholar
  4. [4]
    Bjerknes, J., Atmospheric teleconnections from the equatorial Pacific, Monthly Weather Review, 97(3), 1969, 163–172.CrossRefGoogle Scholar
  5. [5]
    Brézis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer–Verlag, New York, 2011.zbMATHGoogle 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.CrossRefGoogle Scholar
  7. [7]
    Cane, M. A. and Zebiak, S. E., A theory for El Nieno and the Southern Oscillation, Science, 228, 1985, 1085–1088.CrossRefGoogle Scholar
  8. [8]
    Cane, M. A., Experimental forecasts of El Nieno, Nature, 321, 1986, 827–832.CrossRefGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.zbMATHGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.CrossRefGoogle Scholar
  18. [18]
    Chorin, A., Numerical solution of the Navier–Stokes equations, Math. Comput., 22, 1968, 745–762.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.zbMATHGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefGoogle 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.MathSciNetCrossRefGoogle 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.CrossRefGoogle 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.CrossRefzbMATHGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires, 2nd ed., 68, Gauthier–Villars Dunod, 1969.zbMATHGoogle 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.CrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.CrossRefGoogle Scholar
  42. [42]
    Marchuk, G. I., Methods of Numerical Mathematics, 2nd ed., Springer–Verlag, New York, Heidelberg, Berlin, 1982.CrossRefzbMATHGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefGoogle Scholar
  53. [53]
    Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, 44, Applied Mathematical Sciences, Springer–Verlag, New York, 1983.CrossRefzbMATHGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefzbMATHGoogle Scholar
  58. [58]
    Temam, R., Infinite–Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., 68, Applied Mathematical Sciences, Springer–Verlag, New York, 1997.CrossRefzbMATHGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefzbMATHGoogle 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.CrossRefGoogle Scholar
  65. [65]
    Zebiak, S. E., Atmospheric convergence feedback in a simple model for El Nieno, Monthly weather review, 114(7), 1986, 1263–1271.CrossRefGoogle Scholar
  66. [66]
    Zebiak, S. E. and Cane, M. A., A model El Nieno–southern oscillation, Monthly Weather Review, 115(10), 1987, 2262–2278.CrossRefGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yining Cao
    • 1
    Email author
  • Mickaël D. Chekroun
    • 2
  • Aimin Huang
    • 1
  • Roger Temam
    • 1
  1. 1.Department of Mathematics and the Institute for Scientific Computing and Applied MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Department of Atmospheric and Oceanic SciencesUniversity of CaliforniaLos AngelesUSA

Personalised recommendations