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Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation

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Abstract

The response of a low-frequency mode of climate variability, El Niño–Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane–Zebiak model (Zebiak and Cane 1987), from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle–Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution (Chekroun et al. 2019) to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by Gaspard (2002) and made explicit in the second part of this contribution (Tantet et al. 2019). Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in Chekroun et al. (2019), complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems.

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Notes

  1. If peaks in the power spectra are continuous, the decay of correlations may be slower at first but still be exponentially fast for infinite times (as formalized by the Paley–Wiener theorem). In general, however, peaks may be discontinuous and prevent the exponential decay of correlations. Such behavior is not visible in this study.

  2. The matrix C(t) is in fact the covariance matrix of a periodic Ornstein-Uhlenbeck process with a drift given by the Jacobian matrix A(t) generating the fundamental matrix M(t) and with the diffusion matrix D(t), both evaluated along the limit cycle.

References

  1. Avram, F., Leonenko, N.N., Suvak, N.: On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions. Markov Process. Relat. Fields 19, 249–298 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Bagheri, S.: Effects of weak noise on oscillating flows: linking quality factor, Floquet modes, and Koopman spectrum. Phys. Fluids 26(9), 094104 (2014)

    ADS  Google Scholar 

  3. Bittracher, A., Koltai, P., Junge, O.: Pseudogenerators of spatial transfer operators. SIAM J. Appl. Dyn. Syst. 14(3), 1478–1517 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1(2), 301–322 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Cao, Y., Chekroun, M.D., Huang, A., Temam, R.: Mathematical analysis of the jin-neelin model of el niño-southern oscillation. Chin. Ann. Math. B 40(1), 1–38 (2019)

    MathSciNet  MATH  Google Scholar 

  6. Cerrai, S.: Second-order PDE’s in finite and infinite dimension: a probabilistic approach, vol. 1762. Springer, New York (2001)

    MATH  Google Scholar 

  7. 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. 111(5), 1684–1690 (2014)

    ADS  Google Scholar 

  8. Chekroun, M.D., Tantet, A., Neelin, J.D., Dijkstra, H.A.: Ruelle-Pollicott resonances of stochastic systems in reduced state space. Part I: Theory. J. Stat. Phys. (Submitted) (2019)

  9. Chen, C., Cane, M.A., Henderson, N., Lee, D.E., Chapman, D., Kondrashov, D., Chekroun, M.D.: Diversity, nonlinearity, seasonality, and memory effect in ENSO simulation and prediction using empirical model reduction. J. Clim. 29(5), 1809–1830 (2016)

    ADS  Google Scholar 

  10. Crommelin, D., Vanden-Eijnden, E.: Reconstruction of diffusions using spectral data from time series. Commun. Math. Sci. 4(3), 651–668 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Crommelin, D.T., Vanden-Eijnden, E.: Fitting time series by continuous-time markov chains: a quadratic programming approach. J. Comput. Phys. 217(2), 782–805 (2006)

    ADS  MathSciNet  MATH  Google Scholar 

  12. Dellnitz, M., Junge, O.: Almost invariant sets in Chua’s circuit. Int. J. Bifurc. Chaos 7(11), 2475–2485 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Dellnitz, M., Junge, O.: On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36(2), 491–515 (1999)

    MathSciNet  MATH  Google Scholar 

  14. Dellnitz, M., Froyland, G., Horenkamp, C., Padberg-Gehle, K., Gupta, A.S.: Seasonal variability of the subpolar gyres in the Southern Ocean: a numerical investigation based on transfer operators. Nonlinear Process. Geophys. 16(6), 655–663 (2009)

    ADS  Google Scholar 

  15. Deser, C., Alexander, M., Xie, S.-P., Phillips, A.S., Variability, S.S.T.: Patterns and mechanisms. Annu. Rev. Mar. Sci. 2(1), 115–143 (2010)

    ADS  Google Scholar 

  16. Deuflhard, P., Dellnitz, M., Junge, O., Schütte, C.: Computation of essential molecular dynamics by subdivision techniques. In: Deuflhard, P., Hermans, J., Leimkuhler, B., Mark, A.E., Reich, S., Skeel, R.D. (eds.) Comput. Mol. Dyn. Chall. Methods Ideas, vol. 45, pp. 98–115. Springer, Berlin (1999)

    Google Scholar 

  17. Dijkstra, H.A.: Nonlinear Climate Dynamics. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  18. Dyatlov, S., Zworski, M.: Stochastic stability of Pollicott-Ruelle resonances. Nonlinearity 28(10), 3511 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  19. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2001)

    MATH  Google Scholar 

  20. Flandoli, F., Gubinelli, M., Priola, E.: Flow of diffeomorphisms for SDEs with unbounded Holder continuous drift. Bull. Sci. Math. 134(4), 405–422 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Froyland, G.: Computer-assisted bounds for the rate of decay of correlations. Commun. Math. Phys. 189(1), 237–257 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

  22. Froyland, G., Padberg-Gehle, K.: Almost-invariant sets and invariant manifolds—connecting probabilistic and geometric descriptions of coherent structures in flows. Physica D 238(16), 1507–1523 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  23. Froyland, G., Padberg-Gehle, K., England, M., Treguier, A.: Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. 98(22), 224503 (2007)

    ADS  Google Scholar 

  24. Froyland, G., Gottwald, G.A., Hammerlindl, A.: A computational method to extract macroscopic variables and their dynamics in multiscale systems. SIAM J. Appl. Dyn. Syst. 13(4), 1816–1846 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Froyland, G., Stuart, R.M., van Sebille, E.: How well-connected is the surface of the global ocean? Chaos 24(3), 033126 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  26. Gaspard, P.: Trace formula for noisy flows. J. Stat. Phys. 106(1–2), 57–96 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Gaspard, P., Nicolis, G., Provata, A., Tasaki, S.: Spectral signature of the pitchfork bifurcation: Liouville equation approach. Phys. Rev. E 51(1), 74–94 (1995)

    ADS  MathSciNet  Google Scholar 

  28. Goldenberg, S.B., O’Brien, J.J.: Time and space variability of tropical Pacific wind stress. Mon. Weather Rev. 109, 1190–1207 (1981)

    ADS  Google Scholar 

  29. Jin, F.-F.: Tropical ocean-atmosphere interaction, the Pacific cold tongue, and the El Niño-Southern oscillation. Science 274, 76 (1996)

    ADS  Google Scholar 

  30. Jin, F.-F., Neelin, J.D.: Modes of interannual tropical ocean-atmosphere interaction—a unified view. Part I: Numerical results. J. Atmos. Sci. 50(21), 3477–3503 (1993)

    ADS  Google Scholar 

  31. Jin, F.-F., Neelin, J.D.: Modes of interannual tropical ocean-atmosphere interaction—a unified view. Part III: Analytical results in fully coupled cases. J. Atmos. Sci. 50(21), 3523–3540 (1993)

    ADS  Google Scholar 

  32. Klus, S., Koltai, P., Schütte, C.: On the numerical approximation of the Perron–Frobenius and Koopman operator, arXiv (2015), pp. 1–19

  33. Koltai, Péter: Efficient approximation methods for the global long-term behavior of dynamical systems - Theory, algorithms and examples, Ph.D. thesis, Technische Universität at München, p. 162 (2010)

  34. Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK users’ guide: solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods, pp. xv + 137 (1997)

  35. Mauroy, A., Mezić, I.: Global stability analysis using the eigenfunctions of the Koopman operator. IEEE Trans. Autom. Control 61(11), 3356–3369 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Mauroy, A., Mezić, I., Moehlis, J.: Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics. Physica D 261, 19–30 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  39. Neelin, D.S., Battisti, J.D., Hirst, A.C., Jin, F.-F., Wakata, Y., Yamagata, T., Zebiak, S.E.: ENSO theory. J. Geophys. Res. 103(C7), 14261–14290 (1998)

    ADS  Google Scholar 

  40. Pavliotis, G.A.: Stochastic Processes and Applications. Springer, New York (2014)

    MATH  Google Scholar 

  41. Pollicott, M.: Meromorphic extensions of generalised zeta functions. Invent. Math. 85(1), 147–164 (1986)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Roulston, M.S., Neelin, J.D.: The response of an ENSO model to climate noise, weather noise and intraseasonal forcing. Geophys. Res. Lett. 27(22), 3723–3726 (2000)

    ADS  Google Scholar 

  43. Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  44. Ruelle, D.: Locating resonances for axiom a dynamical systems. J. Stat. Phys. 44(3–4), 281–292 (1986)

    ADS  MathSciNet  MATH  Google Scholar 

  45. Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  46. Schütte, C., Fischer, A., Huisinga, W., Deuflhard, P.: A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys. 151(1), 146–168 (1999)

    ADS  MathSciNet  MATH  Google Scholar 

  47. Schütte, C., Fischer, A., Huisinga, W., Deuflhard, P.: A direct approach to conformational dynamics based on hybrid Monte Carlo. J. Comput. Phys. 151(1), 146–168 (1999)

    ADS  MathSciNet  MATH  Google Scholar 

  48. Smith, T.M., Reynolds, R.W.: Reconstruction of historical sea surface temperatures using empirical orthogonal functions. J. Clim. 9, 1403–1420 (1996)

    ADS  Google Scholar 

  49. Tantet, A., van der Burgt, F.R., Dijkstra, H.A.: An early warning indicator for atmospheric blocking events using transfer operators. Chaos 25(3), 036406 (2015)

    ADS  Google Scholar 

  50. Tantet, A., Lucarini, V., Dijkstra, H.A.: Resonances in a chaotic attractor crisis of the Lorenz flow. J. Stat. Phys. 170(3), 584–616 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  51. Tantet, A., Lucarini, V., Lunkeit, F., Dijkstra, H.A.: Crisis of the chaotic attractor of a climate model: a transfer operator approach. Nonlinearity 31(5), 2221 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  52. Tantet, A., Chekroun, M.D., Neelin, J.D., Dijkstra, H.A.: Ruelle-Pollicott resonances of stochastic systems in reduced state space. Part II: Stochastic Hopf Bifurcation. J. Stat. Phys. (submitted) (2019)

  53. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1, 391–421 (2014)

    MathSciNet  MATH  Google Scholar 

  54. Ulam, S.M.: Problems in Modern Mathematics, science edn. Wiley, New York (1964)

    MATH  Google Scholar 

  55. Vaidya, U., Mehta, P.G.: Lyapunov measure for almost everywhere stability. IEEE Trans. Autom. Control 53(1), 307–323 (2008)

    MathSciNet  MATH  Google Scholar 

  56. van Sebille, E., England, M.H., Froyland, G., Van Sebille, E.: Origin, dynamics and evolution of ocean garbage patches from observed surface drifters. Environ. Res. Lett. 7(4), 044040 (2012)

    ADS  Google Scholar 

  57. van der Vaart, P., Dijkstra, H.A., Jin, F.-F.: The Pacific cold tongue and the ENSO mode: a unified theory within the Zebiak–Cane model. J. Atmos. Sci. 57, 967–988 (2000)

    ADS  Google Scholar 

  58. von Storch, H., Zwiers, F.: Stastistical Analysis in Climate Research. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  59. Wiesenfeld, K.: Noisy precursors of nonlinear instabilities. J. Stat. Phys. 38(5), 1071–1097 (1985). (en)

    ADS  MathSciNet  Google Scholar 

  60. Wiesenfeld, K.A., Knobloch, E.: Effect of noise on the dynamics of a nonlinear oscillator. Phys. Rev. A 26(5), 2946–2953 (1982)

    ADS  MathSciNet  Google Scholar 

  61. Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307–1346 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  62. Zebiak, S.E., Cane, M.A.: A model of El Nino-Southern Oscillation. Mon. Weather Rev. 115(31), 2262–2278 (1987)

    ADS  Google Scholar 

  63. Zworski, M.: Mathematical study of scattering resonances. Bull. Math. Sci. 7(1), 1–85 (2017)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work has been partially supported by the Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) Grant N00014-12-1-0911 and N00014-16-1-2073 (MDC), by the National Science Foundation Grant DMS-1616981(MDC) and AGS-1540518 (JDN), by the LINC Project (No. 289447) funded by EC’s Marie-Curie ITN (FP7-PEOPLE-2011-ITN) Program (AT and HD) and by the Utrecht University Center for Water, Climate and Ecosystems (AT).

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Tantet, A., Chekroun, M.D., Neelin, J.D. et al. Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation. J Stat Phys 179, 1449–1474 (2020). https://doi.org/10.1007/s10955-019-02444-8

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