Ensemble Simulations and Pullback Attractors of a Periodically Forced Double-Gyre System

Stefano Pierini Dipartimento di Scienze e Tecnologie, Università di Napoli Parthenope, Napoli, Italy

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Abstract

A primitive equation ocean model has recently reproduced with reasonable realism the synchronization between the North Pacific Oscillation and the last two Kuroshio Extension decadal cycles observed from altimetry. However, the timing of the cycles is imperfect: could a different model initialization improve this fundamental aspect of the phenomenon? Ensemble simulations stemming from many initial conditions should be carried out to answer this question, but doing that with a primitive equation model is highly computationally expensive. A preliminary analysis is therefore performed here with a nonlinear low-order ocean model, which identifies a significant paradigm of intrinsic oceanic double-gyre low-frequency variability. The chaotic pullback attractors of the periodically forced model are first recognized to be periodic and cycloergodic. Two parameters are then introduced to analyze the topological structure of the pullback attractors as a function of the forcing period; their joint use allows one to identify four forms of sensitivity to initialization corresponding to different system behaviors. The model response under periodic forcing turns out to be, in most cases, very sensitive to initialization. Implications concerning the primitive equation model are finally discussed.

Corresponding author address: Stefano Pierini, Dipartimento di Scienze e Tecnologie, Università di Napoli Parthenope, Centro Direzionale, Isola C4, 80143 Napoli, Italy. E-mail: stefano.pierini@uniparthenope.it

Abstract

A primitive equation ocean model has recently reproduced with reasonable realism the synchronization between the North Pacific Oscillation and the last two Kuroshio Extension decadal cycles observed from altimetry. However, the timing of the cycles is imperfect: could a different model initialization improve this fundamental aspect of the phenomenon? Ensemble simulations stemming from many initial conditions should be carried out to answer this question, but doing that with a primitive equation model is highly computationally expensive. A preliminary analysis is therefore performed here with a nonlinear low-order ocean model, which identifies a significant paradigm of intrinsic oceanic double-gyre low-frequency variability. The chaotic pullback attractors of the periodically forced model are first recognized to be periodic and cycloergodic. Two parameters are then introduced to analyze the topological structure of the pullback attractors as a function of the forcing period; their joint use allows one to identify four forms of sensitivity to initialization corresponding to different system behaviors. The model response under periodic forcing turns out to be, in most cases, very sensitive to initialization. Implications concerning the primitive equation model are finally discussed.

Corresponding author address: Stefano Pierini, Dipartimento di Scienze e Tecnologie, Università di Napoli Parthenope, Centro Direzionale, Isola C4, 80143 Napoli, Italy. E-mail: stefano.pierini@uniparthenope.it
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  • Arnold, V. I., 1983: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, 334 pp.

  • Bódai, T., and T. Tél, 2012: Annual variability in a conceptual climate model: Snapshot attractors, hysteresis in extreme events, and climate sensitivity. Chaos, 22, 023110, doi:10.1063/1.3697984.

    • Search Google Scholar
    • Export Citation
  • Bódai, T., G. Károlyi, and T. Tél, 2011: A chaotically driven model climate: Extreme events and snapshot attractors. Nonlinear Processes Geophys., 18, 573580, doi:10.5194/npg-18-573-2011.

    • Search Google Scholar
    • Export Citation
  • Bódai, T., G. Károlyi, and T. Tél, 2013: Driving a conceptual model climate by different processes: Snapshot attractors and extreme events. Phys. Rev., 87E, 022822, doi:10.1103/PhysRevE.87.022822.

    • Search Google Scholar
    • Export Citation
  • Boyles, R., and W. A. Gardner, 1983: Cycloergodic properties of discrete-parameter nonstationary stochastic processes. IEEE Trans. Inf. Theory, 29, 105114, doi:10.1109/TIT.1983.1056613.

    • Search Google Scholar
    • Export Citation
  • Chekroun, M. D., E. Simonnet, and M. Ghil, 2011: Stochastic climate dynamics: Random attractors and time-dependent invariant measures. Physica D, 240, 16851700, doi:10.1016/j.physd.2011.06.005.

    • Search Google Scholar
    • Export Citation
  • Crucifix, M., 2012: Oscillators and relaxation phenomena in Pleistocene climate theory. Philos. Trans. Roy. Soc. London, A370, 11401165, doi:10.1098/rsta.2011.0315.

    • Search Google Scholar
    • Export Citation
  • Crucifix, M., 2013: Why could ice ages be unpredictable? Climate Past, 9, 22532267, doi:10.5194/cp-9-2253-2013.

  • De Saedeleer, B., M. Crucifix, and S. Wieczorek, 2013: Is the astronomical forcing a reliable and unique pacemaker for climate? A conceptual model study. Climate Dyn., 40, 273294, doi:10.1007/s00382-012-1316-1.

    • Search Google Scholar
    • Export Citation
  • Eckmann, J. P., and D. Ruelle, 1985: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys., 57, 617656, doi:10.1103/RevModPhys.57.617.

    • Search Google Scholar
    • Export Citation
  • Gardner, W. A., A. Napolitano, and L. Paura, 2006: Cyclostationarity: Half a century of research. Signal Process., 86, 639697, doi:10.1016/j.sigpro.2005.06.016.

    • Search Google Scholar
    • Export Citation
  • Ghil, M., M. D. Chekroun, and E. Simonnet, 2008: Climate dynamics and fluid mechanics: Natural variability and related uncertainties. Physica D, 237, 21112126, doi:10.1016/j.physd.2008.03.036.

    • Search Google Scholar
    • Export Citation
  • Grassberger, P., and I. Procaccia, 1983: Measuring the strangeness of strange attractors. Physica D, 9, 189208, doi:10.1016/0167-2789(83)90298-1.

    • Search Google Scholar
    • Export Citation
  • Jiang, S., F. Jin, and M. Ghil, 1995: Multiple equilibria, periodic and aperiodic solutions in a wind-driven, double-gyre, shallow water model. J. Phys. Oceanogr., 25, 764786, doi:10.1175/1520-0485(1995)025<0764:MEPAAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Kloeden, P. E., and M. Rasmussen, 2011: Nonautonomous Dynamical Systems. American Mathematical Society, 264 pp.

  • Kramer, W., H. A. Dijkstra, S. Pierini, and P. J. van Leeuwen, 2012: Measuring the impact of observations on the predictability of the Kuroshio Extension in a shallow-water model. J. Phys. Oceanogr., 42, 317, doi:10.1175/JPO-D-11-014.1.

    • Search Google Scholar
    • Export Citation
  • Leutbecher, M., and T. N. Palmer, 2008: Ensemble forecasting. J. Comput. Phys., 227, 35153539, doi:10.1016/j.jcp.2007.02.014.

  • Lewis, J. M., 2005: Roots of ensemble forecasting. Mon. Wea. Rev., 133, 18651885, doi:10.1175/MWR2949.1.

  • Lorenz, E. N., 1982: Low-order models of atmospheric circulations. J. Meteor. Soc. Japan, 60, 255267.

  • Lorenz, E. N., 1984: Irregularity: A fundamental property of the atmosphere. Tellus, 36A, 98110, doi:10.1111/j.1600-0870.1984.tb00230.x.

    • Search Google Scholar
    • Export Citation
  • Mertins, A., 1999: Signal Analysis: Wavelets, Filter Banks, Time-Frequency Transforms, and Applications. John Wiley and Sons, 317 pp.

  • Molteni, F., R. Buizza, T. N. Palmer, and T. Petroliagis, 1996: The ECMWF Ensemble Prediction System: Methodology and validation. Quart. J. Roy. Meteor. Soc., 122, 73119, doi:10.1002/qj.49712252905.

    • Search Google Scholar
    • Export Citation
  • Olbers, D., 2001: A gallery of simple models from climate physics. Stochastic Climate Models, P. Imkeller and J. von Storch, Eds., Progress in Probability, Vol. 49, Birkhäuser Verlag, 3–63.

  • Ott, E., 2002: Chaos in Dynamical Systems. Cambridge University Press, 478 pp.

  • Parker, W. S., 2010: Predicting weather and climate: Uncertainty, ensembles and probability. Stud. Hist. Philos. Sci., 41B, 263272, doi:10.1016/j.shpsb.2010.07.006.

    • Search Google Scholar
    • Export Citation
  • Pierini, S., 2006: A Kuroshio Extension System model study: Decadal chaotic self-sustained oscillations. J. Phys. Oceanogr., 36, 16051625, doi:10.1175/JPO2931.1.

    • Search Google Scholar
    • Export Citation
  • Pierini, S., 2010: Coherence resonance in a double-gyre model of the Kuroshio Extension. J. Phys. Oceanogr., 40, 238248, doi:10.1175/2009JPO4229.1.

    • Search Google Scholar
    • Export Citation
  • Pierini, S., 2011: Low-frequency variability, coherence resonance and phase selection in a low-order model of the wind-driven ocean circulation. J. Phys. Oceanogr., 41, 15851604, doi:10.1175/JPO-D-10-05018.1.

    • Search Google Scholar
    • Export Citation
  • Pierini, S., 2012: Stochastic tipping points in climate dynamics. Phys. Rev., 85E, 027101, doi:10.1103/PhysRevE.85.027101.

  • Pierini, S., 2014: Kuroshio Extension bimodality and the North Pacific Oscillation: A case of intrinsic variability paced by external forcing. J. Climate, 27, 448454, doi:10.1175/JCLI-D-13-00306.1.

    • Search Google Scholar
    • Export Citation
  • Pierini, S., H. A. Dijkstra, and A. Riccio, 2009: A nonlinear theory of the Kuroshio Extension bimodality. J. Phys. Oceanogr., 39, 22122229, doi:10.1175/2009JPO4181.1.

    • Search Google Scholar
    • Export Citation
  • Pikovsky, A. S., and J. Kurths, 1997: Coherence resonance in noise-driven excitable systems. Phys. Rev. Lett., 78, 775778, doi:10.1103/PhysRevLett.78.775.

    • Search Google Scholar
    • Export Citation
  • Qiu, B., and S. Chen, 2010: Eddy-mean flow interaction in the decadally-modulating Kuroshio Extension system. Deep-Sea Res. II, 57, 10971110, doi:10.1016/j.dsr2.2008.11.036.

    • Search Google Scholar
    • Export Citation
  • Rasmussen, M., 2007: Attractivity and Bifurcation for Nonautonomous Dynamical Systems. Springer, 212 pp.

  • Romeiras, F., C. Grebogi, and E. Ott, 1990: Multifractal properties of snapshot attractors of random maps. Phys. Rev., 41A, 784799, doi:10.1103/PhysRevA.41.784.

    • Search Google Scholar
    • Export Citation
  • Stocker T. F., and Coauthors, Eds., 2014: Climate Change 2013: The Physical Science Basis. Cambridge University Press, 1535 pp.

  • Tebaldi, C., and R. Knutti, 2007: The use of the multi-model ensemble in probabilistic climate projections. Philos. Trans. Roy. Soc. London, A365, 20532075, doi:10.1098/rsta.2007.2076.

    • Search Google Scholar
    • Export Citation
  • Toth, Z., and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev., 125, 32973319, doi:10.1175/1520-0493(1997)125<3297:EFANAT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Van der Pol, B., 1926: On relaxation oscillations. Philos. Mag., 2, 978992.

  • Vannitsem, S., 2014a: Dynamics and predictability of a low-order wind-driven ocean–atmosphere coupled model. Climate Dyn., 42, 19811998, doi:10.1007/s00382-013-1815-8.

    • Search Google Scholar
    • Export Citation
  • Vannitsem, S., 2014b: Stochastic modelling and predictability: Analysis of a low-order coupled ocean–atmosphere model. Philos. Trans. Roy. Soc. London, A372, 20130282, doi:10.1098/rsta.2013.0282.

    • Search Google Scholar
    • Export Citation
  • Vannitsem, S., and L. De Cruz, 2014: A 24-variable low-order coupled ocean–atmosphere model: OA-QG-WS v2. Geosci. Model Dev., 7, 649662, doi:10.5194/gmd-7-649-2014.

    • Search Google Scholar
    • Export Citation
  • Veronis, G., 1963: An analysis of wind-driven ocean circulation with a limited number of Fourier components. J. Atmos. Sci., 20, 577593, doi:10.1175/1520-0469(1963)020<0577:AAOWDO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
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