Geophysical Fluid Dynamics, Nonautonomous Dynamical Systems, and the Climate Sciences

  • Michael GhilEmail author
  • Eric Simonnet
Part of the Springer INdAM Series book series (SINDAMS, volume 38)


This contribution introduces the dynamics of shallow and rotating flows that characterizes large-scale motions of the atmosphere and oceans. It then focuses on an important aspect of climate dynamics on interannual and interdecadal scales, namely the wind-driven ocean circulation. Studying the variability of this circulation and slow changes therein is treated as an application of the theory of nonautonomous dynamical systems. The contribution concludes by discussing the relevance of these mathematical concepts and methods for the highly topical issues of climate change and climate sensitivity.


Bifurcations Climate sensitivity Double-gyre problem Low-frequency variability Pullback and random attractors Wasserstein distance Wind-driven ocean circulation 



It is a pleasure to thank M. D. Chekroun, D. Kondrashov, H. Liu, J. C. McWilliams, J. D. Neelin and I. Zaliapin for many useful discussions and their continuing interest in the questions studied here. The research reported in Sect. 3.4 was carried out jointly with M. D. Chekroun, L. De Cruz, J. Demayer, S. Pierini and S. Vannitsem [125, 168]. Figure 23 is due to M. D. Chekroun, a steadfast companion on the road to understanding NDS and RDS theory, along with their applications to climate dynamics in general. H. Liu helped with finalizing Fig. 22. It is a pleasure to thank the organizers of the INdAM Workshop on Mathematical Approaches to Climate Change Impacts—P. Cannarsa, D. Mansutti, and A. Provenzale—for the invitation to deliver a lecture and to contribute to this volume. The writing of this research-and-review paper was supported by grants N00014-12-1-0911 and N00014-16-1-2073 from the Multidisciplinary University Research Initiative (MURI) of the Office of Naval Research and by the US National Science Foundation grant OCE 1243175. This paper is TiPES contribution #4; this project has received funding from the European Union Horizon 2020 research and innovation programme under grant agreement No. 820970.


  1. 1.
    Allen, M.R.: Do-it-yourself climate prediction. Nature 401, 627 (1999)CrossRefGoogle Scholar
  2. 2.
    Andronov, A.A., Pontryagin, L.S.: Systèmes grossiers. Dokl. Akad. Nauk. SSSR 14(5), 247–250 (1937)zbMATHGoogle Scholar
  3. 3.
    Araujo, V., Pacifico, M., Pujal, R., Viana, M.: Singular-hyperbolic attractors are chaotic. Trans. Amer. Math. Soc. 361, 2431–2485 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Arnold, V.I.: Geometrical Methods in the Theory of Differential Equations, 334 pp. Springer, Berlin (1983)CrossRefGoogle Scholar
  5. 5.
    Arnold, L.: Random Dynamical Systems, 616 pp. Springer, Berlin (1998)CrossRefGoogle Scholar
  6. 6.
    Bell, D.R.: Degenerate Stochastic Differential Equations and Hypoellipticity. Longman, Harlow (1995)zbMATHGoogle Scholar
  7. 7.
    Berger, A., Siegmund, S.: On the gap between random dynamical systems and continuous skew products. J. Dyn. Diff. Equ. 15, 237–279 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Berloff, P., Hogg, A., Dewar, W.: The turbulent oscillator: a mechanism of low-frequency variability of the wind-driven ocean gyres. J. Phys. Oceanogr. 37, 2363–2386 (2007)CrossRefGoogle Scholar
  9. 9.
    Bhattacharya, K., Ghil, M., Vulis, I.L.: Internal variability of an energy-balance model with delayed albedo effects. J. Atmos. Sci. 39, 1747–1773 (1982). CrossRefGoogle Scholar
  10. 10.
    Bódai, T., Károlyi, G., Tél, T.: A chaotically driven model climate: extreme events and snapshot attractors. Nonlin. Processes Geophys. 18, 573–580 (2011)CrossRefGoogle Scholar
  11. 11.
    Bódai, T., Lucarini, V., Lunkeit, F., Boschi, R.: Global instability in the Ghil-Sellers model. Clim. Dyn. 44, 3361–3381 (2015)CrossRefGoogle Scholar
  12. 12.
    Bogenschütz, T., Kowalski, Z.S.: A condition for mixing of skew products. Aequationes Math. 59, 222–234 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Bouchet, F., Sommeria, J.: Emergence of intense jets and Jupiter’s great red spot as maximum entropy structures. J. Fluid. Mech. 464, 165–207 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Bracco, A., Neelin, J.D., Luo, H., McWilliams, J.C., Meyerson, J.E.: High-dimensional decision dilemmas in climate models. Geosci. Model Dev. 6, 1673–1687 (2013). CrossRefGoogle Scholar
  15. 15.
    Carvalho, A.N., Langa, J.A., Robinson, J.C.: Lower semicontinuity of attractors for non-autonomous dynamical systems. Ergod. Theory Dyn. Syst. 29, 765–780 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Carvalho, A., Langa, J.A., Robinson, J.C.: Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, 391 pp. Springer, Berlin (2012)Google Scholar
  17. 17.
    Cessi, P., Ierley, G.R.: Symmetry-breaking multiple equilibria in quasigeostrophic wind-driven flows. J. Phys. Oceanogr. 25, 1196–1205 (1995)CrossRefGoogle Scholar
  18. 18.
    Chang, P., Ji, L., Li, H., Flugel, M.: Chaotic dynamics versus stochastic processes in El Niño-Southern Oscillation in coupled ocean-atmosphere models. Physica D 98, 301–320 (1996)CrossRefzbMATHGoogle Scholar
  19. 19.
    Chang, K.I., Ide, K., Ghil, M., Lai, C.C.A.: Transition to aperiodic variability in a wind-driven double-gyre circulation model. J. Phys. Oceanogr. 31, 1260–1286 (2001)CrossRefGoogle Scholar
  20. 20.
    Chang, C.P., Ghil, M., Latif, M., Wallace, J.M.: Climate Change: Multidecadal and Beyond, vol. 6, 388 pp. World Scientific Publishing Co./Imperial College Press (2015)Google Scholar
  21. 21.
    Chao, Y., Ghil, M., McWilliams, J.C.: Pacific interdecadal variability in this century’s sea surface temperatures. Geophys. Res. Lett. 27, 2261–2264 (2000)CrossRefGoogle Scholar
  22. 22.
    Chavez, M., Ghil, M., Urrutia Fucugauchi, J.: Extreme Events: Observations, Modeling and Economics, vol. 214, 438 pp. American Geophysical Union/Wiley, Washington/Hoboken (2015)Google Scholar
  23. 23.
    Chekroun, M.D., Simonnet, E., Ghil, M.: Stochastic climate dynamics: random attractors and time-dependent invariant measures. Physica D 240, 1685–1700 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Chekroun, M.D., Kondrashov, D., Ghil, M.: Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation. Proc. Natl. Acad. Sci. USA 108, 11766–11771 (2011)CrossRefGoogle Scholar
  25. 25.
    Chekroun, M.D., Neelin, J.D., Kondrashov, D., McWilliams, J.C., Ghil, M.: Rough parameter dependence in climate models: the role of Ruelle-Pollicott resonances. Proc. Natl. Acad. Sci. USA 111, 1684–1690 (2014). CrossRefGoogle Scholar
  26. 26.
    Chekroun, M.D., Liu, H., Wang, S.: Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I. Springer Briefs in Mathematics. Springer, Berlin (2015)Google Scholar
  27. 27.
    Chekroun, M.D., Ghil, M., Liu, H., Wang, S.: Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discr. Cont. Dyn. Syst. 36(8), 4133–4177 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Collet, P., Tresser, C.: Ergodic theory and continuity of the Bowen-Ruelle measure for geometrical flows. Fyzika 20, 33–48 (1988)Google Scholar
  29. 29.
    Cong, N.D.: Topological Dynamics of Random Dynamical Systems. Oxford Mathematical Monographs, 212 pp. Clarendon Press, Oxford (1997)Google Scholar
  30. 30.
    Crauel, H.: White noise eliminates instability. Arch. Math. 75, 472–480 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Crauel, H., Random probability measures on Polish spaces, vol. 11. Stochastic Monographs. Taylor & Francis, Didcot (2002)Google Scholar
  32. 32.
    Crauel, H.: A uniformly exponential attractor which is not a pullback attractor. Arch. Math. 78, 329–336 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Technical Report 148, cuola Normale Superiore Pisa (1992)Google Scholar
  34. 34.
    Crauel, H., Flandoli, F.: Additive noise destroys a pitchfork bifurcation. J. Dyn. Diff. Equ. 10, 259–274 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Cushman-Roisin, B., Beckers, J.-M.: Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, 2nd edn., 875 pp. Academic, Cambridge (2011)Google Scholar
  36. 36.
    Da Costa, E.D., Colin de Verdière, A.C.: The 7.7 year North Atlantic oscillation. Q. J. R. Meteorol. Soc. 128, 797–818 (2004)Google Scholar
  37. 37.
    Dijkstra, H.A.: Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño, 2nd edn., 532 pp. Springer, Berlin (2005)Google Scholar
  38. 38.
    Dijkstra, H.A.: Nonlinear Climate Dynamics, 367 pp. Cambridge University Press, Cambridge (2013)CrossRefGoogle Scholar
  39. 39.
    Dijkstra, H.A., Ghil, M.: Low-frequency variability of the large-scale ocean circulation: a dynamical systems approach. Rev. Geophys. 43 (2005).
  40. 40.
    Dijkstra, H.A., Katsman, C.A.: Temporal variability of the wind-driven quasi-geostrophic double gyre ocean circulation: basic bifurcation diagrams. Geophys. Astrophys. Fluid Dyn. 85, 195–232 (1997)CrossRefMathSciNetGoogle Scholar
  41. 41.
    Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theor. Prob. Appl. 15, 458–486 (1979)CrossRefzbMATHGoogle Scholar
  42. 42.
    Dorfle, M., Graham, R.: Probability density of the Lorenz model. Phys. Rev. A 27, 1096–1105 (1983)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Drótos, G., Bódai, T., Tél, T.: Probabilistic concepts in a changing climate: a snapshot attractor picture. J. Clim. 28, 3275–3288 (2015)CrossRefGoogle Scholar
  44. 44.
    Dubar, M.: Approche climatique de la période romaine dans l’est du Var: recherche et analyse des composantes périodiques sur un concrétionnement centennal (Ier-IIe siècle apr. J.-C.) de l’aqueduc de Fréjus. Archeoscience 30, 163–171 (2006)CrossRefGoogle Scholar
  45. 45.
    Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Farrel, B.F., Ioannou, P.J.: Structural stability of turbulent jets. J. Atmos. Sci. 60, 2101–2118 (2003)CrossRefGoogle Scholar
  47. 47.
    Feliks, Y., Ghil, M., Simonnet, E.: Low-frequency variability in the mid-latitude atmosphere induced by an oceanic thermal front. J. Atmos. Sci. 61, 961–981 (2004)CrossRefGoogle Scholar
  48. 48.
    Feliks, Y., Ghil, M., Simonnet, E.: Low-frequency variability in the mid-latitude baroclinic atmosphere induced by an oceanic thermal front. J. Atmos. Sci. 64, 97–116 (2007)CrossRefGoogle Scholar
  49. 49.
    Feliks, Y., Ghil, M., Robertson, A.W.: Oscillatory climate modes in the eastern Mediterranean and their synchronization with the North Atlantic Oscillation. J. Clim. 23, 4060–4079 (2010). CrossRefGoogle Scholar
  50. 50.
    Feliks, Y., Ghil, M., Robertson, A.W.: The atmospheric circulation over the North Atlantic as induced by the SST field. J. Clim. 24, 522–542 (2011). CrossRefGoogle Scholar
  51. 51.
    Galanti, E., Tziperman, E.: ENSO’s phase locking to the seasonal cycle in the fast-SST, fast-wave, and mixed-mode regimes. J. Atmos. Sci. 57, 2936–2950 (2000)CrossRefGoogle Scholar
  52. 52.
    Ghil, M.: Cryothermodynamics: the chaotic dynamics of paleoclimate. Physica D 77, 130–159 (1994)CrossRefGoogle Scholar
  53. 53.
    Ghil, M.: Hilbert problems for the geosciences in the 21st century. Nonlinear Process. Geophys. 8, 211–222 (2001)CrossRefGoogle Scholar
  54. 54.
    Ghil, M.: A mathematical theory of climate sensitivity or, How to deal with both anthropogenic forcing and natural variability?. In: Chang, C.P., Ghil, M., Latif, M., Wallace, J.M. (Eds.) Climate Change: Multidecadal and Beyond, pp. 31–51. World Scientific Publishing Co./Imperial College Press, Singapore/London (2015)CrossRefGoogle Scholar
  55. 55.
    Ghil, M.: The wind-driven ocean circulation: applying dynamical systems theory to a climate problem. Discr. Cont. Dyn. Syst. – A 37, 189–228 (2017). CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    Ghil, M., Childress, S.: Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, 512 pp. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  57. 57.
    Ghil, M., Jiang, N.: Recent forecast skill for the El Niño/Southern Oscillation. Geophys. Res. Lett. 25, 171–174 (1998)CrossRefGoogle Scholar
  58. 58.
    Ghil, M., Lucarini, V.: The physics of climate variability and climate change. Rev. Mod. Phys., 1–84 (2019). Submitted. arXiv:1910.00583 Google Scholar
  59. 59.
    Ghil, M., Robertson, A.W.: Solving problems with GCMs: general circulation models and their role in the climate modeling hierarchy. In: Randall, D. (ed.) General Circulation Model Development: Past, Present and Future, pp. 285–325. Academic, San Diego (2000)Google Scholar
  60. 60.
    Ghil, M., Robertson, A.W.: ‘Waves” vs “particles” in the atmosphere’s phase space: a pathway to long-range forecasting? Proc. Natl. Acad. Sci. USA 99, 2493–2500 (2002)CrossRefGoogle Scholar
  61. 61.
    Ghil, M., Vautard, R.: Interdecadal oscillations and the warming trend in global temperature time series. Nature 350, 324–327 (1991)CrossRefGoogle Scholar
  62. 62.
    Ghil, M., Zaliapin, I.: Understanding ENSO variability and its extrema: a delay differential equation approach, vol. 214, ch. 6, pp. 63–78. In: Chavez, M., Ghil, M., Urrutia-Fucugauchi, J. (eds.) Extreme Events: Observations, Modeling and Economics. American Geophysical Union/Wiley, Washington/Hoboken (2015)Google Scholar
  63. 63.
    Ghil, M., Allen, M.R., Dettinger, M.D., Ide, K., Kondrashov, D., Mann, M.E., Robertson, A.W., Saunders, A., Tian, Y., Varadi, F., Yiou, P.: Advanced spectral methods for climatic time series. Rev. Geophys. 40(1), 3.1–3.41 (2002)CrossRefGoogle Scholar
  64. 64.
    Ghil, M., Chekroun, M.D., Simonnet, E.: Climate dynamics and fluid mechanics: natural variability and related uncertainties. Physica D 237, 2111–2126 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  65. 65.
    Ghil, M., Zaliapin, I., Thompson, S.: A delay differential model of ENSO variability: parametric instability and the distribution of extremes. Nonlin. Processes Geophys. 15, 417–433 (2008)CrossRefGoogle Scholar
  66. 66.
    Ghil, M., Yiou, P., Hallegatte, S., Malamud, B.D., Naveau, P., Soloviev, A., Friederichs, P., Keilis-Borok, V., Kondrashov, D., Kossobokov, V., Mestre, O., Nicolis, C., Rust, H., Shebalin, P., Vrac, M., Witt, A., Zaliapin, I.: Extreme events: dynamics, statistics and prediction. Nonlin. Processes Geophys. 18, 295–350 (2011). CrossRefGoogle Scholar
  67. 67.
    Gill, A.E.: Atmosphere-ocean dynamics, 662 pp. Academic, Cambridge (1982)Google Scholar
  68. 68.
    Guckenheimer, J., Williams, R.F.: Structural stability of Lorenz attractors. Publ. Math. I.H.E.S. 50, 59–72 (1979)Google Scholar
  69. 69.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, 2nd edn., 453 pp. Springer, Berlin (1991)Google Scholar
  70. 70.
    Holton, J., Hakim, G.J.: An Introduction to Dynamic Meteorology, 5th edn., 552 pp. Academic, Cambridge (2012)Google Scholar
  71. 71.
    IPCC: Climate change. In: Houghton, J.T., Jenkins, G.J., Ephraums, J.J. (eds.) The IPCC Scientific Assessment, 365 pp. Cambridge University Press, Cambridge (1991)Google Scholar
  72. 72.
    IPCC: Climate change 2001: the scientific basis. In: Houghton, J.T., Ding, Y., Griggs, D.J., Noguer, M., van der Linden, P.J., Dai, X., Maskell, K., Johnson, C.A. (eds.) Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change (IPCC), 944 pp. Cambridge University Press, Cambridge (2001)Google Scholar
  73. 73.
    IPCC: Climate change 2007: the physical science basis. In: Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K.B., Tignor, M., Miller, H.L. (eds.) Contribution of Working Group I to the Fourth Assessment Report of the IPCC. Cambridge University Press, Cambridge (2007)Google Scholar
  74. 74.
    IPCC: Climate change 2013. In: Stocker, T.F., Qin, D., Plattner, G.K., Tignor, M., Allen, S.K., Boschung, J., Nauels, A., Xia, Y., Bex, B., Midgley, B.M. (eds.) The Physical Science Basis: Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge (2013)Google Scholar
  75. 75.
    Jiang, S., Jin, F.-F., Ghil, M.: The nonlinear behavior of western boundary currents in a wind-driven, double-gyre, shallow-water model, pp. 64–67. In: Ninth Conference Atmospheric & Oceanic Waves and Stability, San Antonio. American Meteorological Society, Boston (1993)Google Scholar
  76. 76.
    Jiang, S., Jin, F.-F., Ghil, M.: Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model. J. Phys. Oceanogr. 25, 764–786 (1995)CrossRefGoogle Scholar
  77. 77.
    Jin, F.-F., Neelin, J.D., Ghil, M.: El Niño on the Devil’s staircase: annual subharmonic steps to chaos. Science 264, 70–72 (1994)CrossRefGoogle Scholar
  78. 78.
    Jin, F.-F., Neelin, J.D., Ghil, M.: El Niño/Southern Oscillation and the annual cycle: subharmonic frequency locking and aperiodicity. Physica D 98, 442–465 (1996)CrossRefzbMATHGoogle Scholar
  79. 79.
    Kalnay, E., Atmospheric Modeling, Data Assimilation and Predictability, 341 pp. Cambridge University Press, Cambridge (2003)Google Scholar
  80. 80.
    Katok, A., Haselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, vol. 54, 822 pp. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1995)Google Scholar
  81. 81.
    Katsman, C.A., Dijkstra, H.A., Drijfhout, S.S.: The rectification of the wind-driven ocean circulation due to its instabilities. J. Mar. Res. 56, 559–587 (1998)CrossRefGoogle Scholar
  82. 82.
    Kifer, Y.: Ergodic Theory of Random Perturbations. Birkhäuser, Basel (1988)Google Scholar
  83. 83.
    Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems, vol. 176. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2011)Google Scholar
  84. 84.
    Kohn, J.J.: Pseudo-differential operators and hypoellipticity. Proc. Amer. Math. Soc. Symp. Pure Math. 23, 61–69 (1973)CrossRefMathSciNetzbMATHGoogle Scholar
  85. 85.
    Kondrashov, D., Feliks, Y., Ghil, M.: Oscillatory modes of extended Nile River records (A.D. 622–1922). Geophys. Res. Lett. 32, L10702 (2005).
  86. 86.
    Kondrashov, K., Chekroun, M.D., Robertson, A.W., Ghil, M.: Low-order stochastic model and “past-noise forecasting” of the Madden-Julian oscillation. Geophys. Res. Lett. 40, 5303–5310 (2013)CrossRefGoogle Scholar
  87. 87.
    Kondrashov, D., Chekroun, M.D., Ghil, M.: Data-driven non-Markovian closure models. Physica D 297, 33–55 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  88. 88.
    Kravtsov, S., Berloff, P., Dewar, W.K., Ghil, M., McWilliams, J.C.: Dynamical origin of low-frequency variability in a highly nonlinear mid-latitude coupled model. J. Climate 19, 6391–6408 (2007)CrossRefGoogle Scholar
  89. 89.
    Langa, J.A., Robinson, J.C., Suarez, A.: Stability, instability, and bifurcation phenomena in non-autonomous differential equations. Nonlinearity 15, 887–903 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  90. 90.
    Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, vol. 97. Applied Mathematical Sciences. Springer, Berlin (1994)Google Scholar
  91. 91.
    Ledrappier, F., Young, L.-S.: Entropy formula for random transformations. Probab. Theory Relat. Fields 80, 217–240 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  92. 92.
    Lin, J.W.B., Neelin, J.D.: Influence of a stochastic moist convective parameterization on tropical climate variability. Geophys. Res. Lett. 27, 3691–3694 (2000)CrossRefGoogle Scholar
  93. 93.
    Lin, J.W.B., Neelin, J.D.: Considerations for stochastic convective parameterization. J. Atmos. Sci. 59, 959–975 (2002)CrossRefGoogle Scholar
  94. 94.
    Lin, J.W.B., Neelin, J.D.: Toward stochastic deep convective parameterization in general circulation models. Geophys. Res. Lett. 30, 1162 (2003). Google Scholar
  95. 95.
    Loikith, P.C., Neelin, J.D.: Short-tailed temperature distributions over North America and implications for future changes in extremes. Geophys. Res. Lett. 42, 8577–8585 (2015). CrossRefGoogle Scholar
  96. 96.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefMathSciNetzbMATHGoogle Scholar
  97. 97.
    Lorenz, E.N.: The Essence of Chaos, 240 pp. University of Washington Press, Seattle (1995)Google Scholar
  98. 98.
    Lucarini, V., Sarno, S.: A statistical mechanical approach for the computation of the climatic response to general forcings. Nonlin. Processes Geophys. 18, 7–28 (2011)CrossRefGoogle Scholar
  99. 99.
    Lucarini, V., Blender, R., Herbert, C., Ragone, F., Pascale, S., Wouters, J.: Mathematical and physical ideas for climate science. Rev. Geophys. 52, 809–859 (2014)CrossRefGoogle Scholar
  100. 100.
    Lucarini, V., Ragone, F., Lunkeit, F.: Predicting climate change using response theory: global averages and spatial patterns. J. Stat. Phys. 166, 1036–1064 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  101. 101.
    Madden, R.A., Julian, P.R.: Observations of the 40–50-day tropical oscillations – a review. Mon. Weather Rev. 122, 814–837 (1994)CrossRefGoogle Scholar
  102. 102.
    Majda, A., Wang, X.: Nonlinear dynamics and statistical theories for basic geophysical flows, 551 pp. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  103. 103.
    Mañé, R.: A proof of the C 1-stability conjecture. Publ. Math I.H.E.S. 66, 161–210 (1987)Google Scholar
  104. 104.
    Mantua, N.J., Hare, S., Zhang, Y., Wallace, J.M., Francis, R.C.: A Pacific interdecadal climate oscillation with impacts on salmon production. Bull. Am. Meteorol. Soc. 78, 1069–1079 (1997)CrossRefGoogle Scholar
  105. 105.
    McWilliams, J.C.: Fundamentals of Geophysical Fluid Dynamics, 2nd edn., 272 pp. Cambridge University Press, Cambridge (2011)Google Scholar
  106. 106.
    Meacham, S.P.: Low-frequency variability in the wind-driven circulation. J. Phys. Oceanogr. 30, 269–293 (2000)CrossRefGoogle Scholar
  107. 107.
    Meehl, G.A.: Decadal climate variability and the early-2000s hiatus, vol. 13(3), pp. 1–6. In: Menemenlis, D., Sprintall, J. (eds.) US CLIVAR Variations in Understanding the Earth’s Climate Warming Hiatus: Putting the Pieces Together (2015)Google Scholar
  108. 108.
    Minobe, S., Kuwano-Yoshida, A., Komori, N., Xie, S.-P., Small, R.J.: Influence of the Gulf Stream on the troposphere. Nature 452, 206–209 (2008)CrossRefGoogle Scholar
  109. 109.
    Mittal, A.K., Dwivedi, S., Yadav, R.S.: Probability distribution for the number of cycles between successive regime transitions for the Lorenz model. Physica D 233, 14–20 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  110. 110.
    Moron, V., Vautard, R., Ghil, M.: Interannual oscillations in global sea-surface temperatures. Clim. Dyn. 14, 545–569 (1998)CrossRefGoogle Scholar
  111. 111.
    Nadiga, N.T., Luce, B.P.: Global bifurcation of Shilnikov type in a double-gyre ocean model. J. Phys. Oceanogr. 31, 2669–2690 (2001)CrossRefMathSciNetGoogle Scholar
  112. 112.
    Neelin, J.D., Battisti, D.S., Hirst, A.C., Jin, F.-F., Wakata, Y., Yamagata, T., Zebiak, S.: ENSO Theory. J. Geophys. Res. 104(C7), 14261–14290 (1998)CrossRefGoogle Scholar
  113. 113.
    Newhouse, S.: The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms. Publ. Math. I.H.E.S. 50, 101–150 (1979)Google Scholar
  114. 114.
    NRC: Carbon Dioxide and Climate: A Scientific Assessment, Charney, J.G. et al., (eds.) National Academies Press, Washington (1979)Google Scholar
  115. 115.
    NRC: Natural Climate Variability on Decade-to-Century Time Scales, 630 pp. Martinson, D.G., Bryan, K., Ghil, M., et al., (eds.) National Academies Press, Washington (1995)Google Scholar
  116. 116.
    Ohtomo, N., et al.: Exponential characteristics of power spectral densities caused by chaotic phenomena. J. Phys. Soc. Jpn. 64, 1104–1113 (1995)CrossRefGoogle Scholar
  117. 117.
    Palis, J.: A global perspective for non-conservative dynamics. Ann. I.H. Poincaré 22, 485–507 (2005)MathSciNetzbMATHGoogle Scholar
  118. 118.
    Palmer, T.N.: The prediction of uncertainty in weather and climate forecasting. Rep. Prog. Phys. 63, 71–116 (2000)CrossRefGoogle Scholar
  119. 119.
    Palmer, T.N., Jung, T., Shutts, G.J.: Influence of a stochastic parameterization on the frequency of occurrence of North Pacific weather regimes in the ECMWF model. Geophys. Res. Lett. 32 (2005), Art. No. L23811Google Scholar
  120. 120.
    Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn., 710 pp. Springer, Berlin (1987)Google Scholar
  121. 121.
    Pedlosky, J.: Ocean Circulation Theory. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  122. 122.
    Peixoto, M.: Structural stability on two-dimensional manifolds. Topology 1, 101–110 (1962)CrossRefMathSciNetzbMATHGoogle Scholar
  123. 123.
    Pierini, S.: Low-frequency variability, coherence resonance, and phase selection in a low-order model of the wind-driven ocean circulation. J. Phys. Oceanogr. 41, 1585–1604 (2011)CrossRefGoogle Scholar
  124. 124.
    Pierini, S.: Ensemble simulations and pullback attractors of a periodically forced double-gyre system. J. Phys. Oceanogr. 44, 3245–3254 (2014)CrossRefGoogle Scholar
  125. 125.
    Pierini, S., Ghil, M., Chekroun, M.D.: Exploring the pullback attractors of a low-order quasigeostrophic ocean model: the deterministic case. J. Clim. 29, 4185–4202 (2016). CrossRefGoogle Scholar
  126. 126.
    Plaut, G., Ghil, M., Vautard, R.: Interannual and interdecadal variability in 335 Years of Central England temperatures. Science 268, 710–713 (1995)CrossRefGoogle Scholar
  127. 127.
    Poincaré, H.: Sur les équations de la dynamique et le problème des trois corps. Acta Math. 13, 1–270 (1890)MathSciNetzbMATHGoogle Scholar
  128. 128.
    Pope, V.D., Gallani, M., Rowntree, P.R., Stratton, R.A.: The impact of new physical parameterisations in the Hadley Centre climate model HadAM3. Clim. Dyn. 16, 123–146 (2000)CrossRefGoogle Scholar
  129. 129.
    Rasmussen, M.: Attractivity and bifurcation for nonautonomous dynamical systems. Springer, Berlin (2007)zbMATHGoogle Scholar
  130. 130.
    Robbin, J.: A structural stability theorem. Ann. Math. 94, 447–449 (1971)CrossRefMathSciNetzbMATHGoogle Scholar
  131. 131.
    Robert, R., Sommeria, J.: Statistical equilibrium states for two-dimensional flows. J. Fluid. Mech. 229, 291–310 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  132. 132.
    Robinson, C.: Structural stability of C 1diffeomorphisms. J. Differ. Equ. 22, 28–73 (1976)CrossRefGoogle Scholar
  133. 133.
    Ruff, T.W., Neelin, J.D.: Long tails in regional surface temperature probability distributions with implications for extremes under global warming. Geophys. Res. Lett. 39 (2012).
  134. 134.
    Salmon, R.: Lectures on Geophysical Fluid Dynamics, 378 pp. Oxford University Press, Oxford (1998)Google Scholar
  135. 135.
    Saunders, A., Ghil, M.: A Boolean delay equation model of ENSO variability. Physica D 160, 54–78 (2001). CrossRefMathSciNetzbMATHGoogle Scholar
  136. 136.
    Schneider, S.H., Dickinson, R.E.: Climate modeling. Rev. Geophys. Space Phys. 12, 447–493 (1974)CrossRefGoogle Scholar
  137. 137.
    Sell, G.: Non-autonomous differential equations and dynamical systems. Trans. Am. Math. Soc. 127, 241–283 (1967)Google Scholar
  138. 138.
    Sheremet, V.A., Ierley, G.R., Kamenkovitch, V.M.: Eigenanalysis of the two-dimensional wind-driven ocean circulation problem. J. Mar. Res. 55, 57–92 (1997)CrossRefGoogle Scholar
  139. 139.
    Simonnet, E., Dijkstra, H.A.: Spontaneous generation of low-frequency modes of variability in the wind-driven ocean circulation. J. Phys. Oceanogr. 32, 1747–1762 (2002)CrossRefGoogle Scholar
  140. 140.
    Simonnet, E., Temam, R., Wang, S., Ghil, M., Ide, K.: Successive bifurcations in a shallow-water ocean model, vol. 515, pp. 225–230. Lecture Notes in Physics, Sixteenth International Conference on Numerical Methods in Fluid Dynamics. Springer, Berlin (1995)Google Scholar
  141. 141.
    Simonnet, E., Ghil, M., Ide, K., Temam, R., Wang, S.: Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solutions. J. Phys. Oceanogr. 33, 712–728 (2003)Google Scholar
  142. 142.
    Simonnet, E., Ghil, M., Ide, K., Temam, R., Wang, S.: Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part II: Time-dependent solutions. J. Phys. Oceanogr. 33, 729–752 (2003)Google Scholar
  143. 143.
    Simonnet, E., Ghil, M., Dijkstra, H.A.: Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation. J. Mar. Res. 63, 931–956 (2005)CrossRefGoogle Scholar
  144. 144.
    Simonnet, E., Dijkstra, H.A., Ghil, M.: Bifurcation analysis of ocean, atmosphere and climate models, vol. 14, pp. 187–229. Temam, R., Tribbia, J.J. (eds.) North-Holland, Amsterdam (2009)Google Scholar
  145. 145.
    Sinai, Y.: Gibbs measures in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  146. 146.
    Smale, S.: Structurally stable systems are not dense. American J. Math. 88(2), 491–496 (1966)CrossRefMathSciNetzbMATHGoogle Scholar
  147. 147.
    Smale, S.: Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 199–206 (1967)CrossRefMathSciNetzbMATHGoogle Scholar
  148. 148.
    Small, R.J., DeSzoeke, S.P., Xie, S.P., O’Neill, L., Seo, H., Song, Q., Cornillon, P.: Air–sea interaction over ocean fronts and eddies. Dyn. Atmos. Oceans 45, 274–319 (2008)CrossRefGoogle Scholar
  149. 149.
    Soize, C.: The Fokker-Planck equation for stochastic dynamical systems and its explicit steady state solutions. World Scientific Publishing Co., Singapore (1994)CrossRefzbMATHGoogle Scholar
  150. 150.
    Speich, S., Dijkstra, H.A., Ghil, M.: Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation. Nonlinear Process. Geophys. 2, 241–268 (1995)CrossRefGoogle Scholar
  151. 151.
    Stainforth, D.A., et al.: Uncertainty in predictions of the climate response to rising levels of greenhouse gases. Nature 433, 403–406 (2005)CrossRefGoogle Scholar
  152. 152.
    Stevens, B., Zhang, Y., Ghil, M.: Stochastic effects in the representation of stratocumulus-topped mixed layers, pp. 79–90. Proceedings of ECMWF Workshop on Representation of Sub-grid Processes Using Stochastic-Dynamic Models. Shinfield Park, Reading (2005)Google Scholar
  153. 153.
    Stommel, H.: Thermohaline convection with two stable regimes of flow. Tellus 13, 224–230 (1961)CrossRefGoogle Scholar
  154. 154.
    Stommel, H.: The Gulf Stream: A Physical and Dynamical Description, 2nd edn., 248 pp. Cambridge University Press, London (1965)Google Scholar
  155. 155.
    Sushama, L., Ghil, M., Ide, K.: Spatio-temporal variability in a mid-latitude ocean basin subject to periodic wind forcing. Atmosphere-Ocean 45, 227–250 (2007). CrossRefGoogle Scholar
  156. 156.
    Sverdrup, H.U.: Wind-driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proc. Natl. Acad. Sci. USA 33, 318–326 (1947)CrossRefMathSciNetGoogle Scholar
  157. 157.
    Sverdrup, H.U., Johnson, M.W., Fleming, R.H.: The Oceans: Their Physics, Chemistry and General Biology. Prentice-Hall, New York (1942). Available at
  158. 158.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn., 648 pp. Springer, New York (1997)Google Scholar
  159. 159.
    Thompson, P.D.: Numerical Weather Analysis and Prediction, 170 pp. Macmillan, New York (1961)Google Scholar
  160. 160.
    Trefethen, L.N., Trefethen, A., Reddy, S.C., Driscoll, T.A.: Hydrodynamic stability without eigenvalues. Science 261, 578–584 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  161. 161.
    Tucker, W.: Lorenz attractor exists. C. R. Acad. Sci. Paris 328(12), 1197–1202 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  162. 162.
    Tziperman, E., Stone, L., Cane, M., Jarosh, H.: El Niño chaos: overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator. Science 264, 72–74 (1994)CrossRefGoogle Scholar
  163. 163.
    Tziperman, E., Cane, M.A., Zebiak, S.E.: Irregularity and locking to the seasonal cycle in an ENSO prediction model as explained by the quasi-periodicity route to chaos. J. Atmos. Sci. 50, 293–306 (1995)CrossRefGoogle Scholar
  164. 164.
    Vallis, G.: Atmospheric and Oceanic Fluid Dynamics, 745 pp. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  165. 165.
    Vannitsem, S.: Dynamics and predictability of a low-order wind-driven ocean–atmosphere coupled model. Clim. Dyn. 42, 1981–1998 (2014)CrossRefzbMATHGoogle Scholar
  166. 166.
    Vannitsem, S., Ghil, M.: Evidence of coupling in ocean–atmosphere dynamics over the North Atlantic. Geophys. Res. Lett. 44, 2016–2026 (2017). Google Scholar
  167. 167.
    Vannitsem, S., Demaeyer, J., De Cruz, L., Ghil, M.: A 24-variable low-order coupled ocean-atmosphere model: OA-QG-WS v2. Geosci. Model Dev. 7, 649–662 (2014)CrossRefGoogle Scholar
  168. 168.
    Vannitsem, S., Demaeyer, J., De Cruz, L., Ghil, M.: Low-frequency variability and heat transport in a low-order nonlinear coupled ocean-atmosphere model. Physica D 309, 71–85 (2015). CrossRefMathSciNetzbMATHGoogle Scholar
  169. 169.
    Weeks, E.R., Tian, Y., Urbach, J.S., Ide, K., Swinney, H.L., Ghil, M.: Transitions between blocked and zonal flows in a rotating annulus with topography. Science 278, 1598–1601 (1997)CrossRefGoogle Scholar
  170. 170.
    Williams, R.F.: The structure of Lorenz attractors. Publ. Math. I.H.E.S. 50, 73–99 (1979)Google Scholar
  171. 171.
    Wunsch, C.: The interpretation of short climate records, with comments on the North Atlantic and Southern Oscillations. Bull. Am. Meteorol. Soc. 80, 245–255 (1999)CrossRefGoogle Scholar
  172. 172.
    Young, L.S.: What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108, 733–754 (2002)CrossRefMathSciNetzbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Ecole Normale Supérieure and PSL UniversityParisFrance
  2. 2.University of CaliforniaLos AngelesUSA
  3. 3.Institut de Physique de NiceCNRS & Université Côte d’AzurNiceFrance

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