Editorial Type:
Article
Article Type:
Research Article

Low-Frequency Climate Response and Fluctuation–Dissipation Theorems: Theory and Practice

Andrew J. Majda Department of Mathematics, and Center for Atmosphere–Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

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Boris Gershgorin Department of Mathematics, and Center for Atmosphere–Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

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Yuan Yuan Department of Mathematics, and Center for Atmosphere–Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, New York

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Print Publication:
01 Apr 2010
DOI:
https://doi.org/10.1175/2009JAS3264.1
Page(s):
1186–1201
Restricted access

Abstract

The low-frequency response to changes in external forcing or other parameters for various components of the climate system is a central problem of contemporary climate change science. The fluctuation–dissipation theorem (FDT) is an attractive way to assess climate change by utilizing statistics of the present climate; with systematic approximations, it has been shown recently to have high skill for suitable regimes of an atmospheric general circulation model (GCM). Further applications of FDT to low-frequency climate response require improved approximations for FDT on a reduced subspace of resolved variables. Here, systematic mathematical principles are utilized to develop new FDT approximations on reduced subspaces and to assess the small yet significant departures from Gaussianity in low-frequency variables on the FDT response. Simplified test models mimicking crucial features in GCMs are utilized here to elucidate these issues and various FDT approximations in an unambiguous fashion. Also, the shortcomings of alternative ad hoc procedures for FDT in the recent literature are discussed here. In particular, it is shown that linear regression stochastic models for the FDT response always have no skill for a general nonlinear system for the variance response and can have poor or moderate skill for the mean response depending on the regime of the Lorenz 40-model and the details of the regression strategy. New nonlinear stochastic FDT approximations for a reduced set of variables are introduced here with significant skill in capturing the effect of subtle departures from Gaussianity in the low-frequency response for a reduced set of variables.

Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center for Atmosphere–Ocean Science, Courant Institute, New York University, 251 Mercer St., New York, NY 10012. Email: jonjon@cims.nyu.edu

Abstract

The low-frequency response to changes in external forcing or other parameters for various components of the climate system is a central problem of contemporary climate change science. The fluctuation–dissipation theorem (FDT) is an attractive way to assess climate change by utilizing statistics of the present climate; with systematic approximations, it has been shown recently to have high skill for suitable regimes of an atmospheric general circulation model (GCM). Further applications of FDT to low-frequency climate response require improved approximations for FDT on a reduced subspace of resolved variables. Here, systematic mathematical principles are utilized to develop new FDT approximations on reduced subspaces and to assess the small yet significant departures from Gaussianity in low-frequency variables on the FDT response. Simplified test models mimicking crucial features in GCMs are utilized here to elucidate these issues and various FDT approximations in an unambiguous fashion. Also, the shortcomings of alternative ad hoc procedures for FDT in the recent literature are discussed here. In particular, it is shown that linear regression stochastic models for the FDT response always have no skill for a general nonlinear system for the variance response and can have poor or moderate skill for the mean response depending on the regime of the Lorenz 40-model and the details of the regression strategy. New nonlinear stochastic FDT approximations for a reduced set of variables are introduced here with significant skill in capturing the effect of subtle departures from Gaussianity in the low-frequency response for a reduced set of variables.

Corresponding author address: Andrew J. Majda, Department of Mathematics, and Center for Atmosphere–Ocean Science, Courant Institute, New York University, 251 Mercer St., New York, NY 10012. Email: jonjon@cims.nyu.edu

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  • Abramov, R., 2009: Short-time linear response with reduced-rank tangent map. Chin. Ann. Math., 30 (5) 447–462.

  • Abramov, R., and A. Majda, 2007: Blended response algorithms for linear fluctuation–dissipation for complex nonlinear dynamical systems. Nonlinearity, 20 , 2793–2821.

    • Search Google Scholar
    • Export Citation

  • Abramov, R., and A. Majda, 2008: New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems. J. Nonlinear Sci., 18 , 303–341.

    • Search Google Scholar
    • Export Citation

  • Abramov, R., and A. Majda, 2009: A new algorithm for low-frequency climate response. J. Atmos. Sci., 66 , 286–309.

  • Bell, T., 1980: Climate sensitivity from fluctuation dissipation: Some simple model tests. J. Atmos. Sci., 37 , 1700–1707.

  • Berner, J., and G. Branstator, 2007: Linear and nonlinear signatures in the planetary wave dynamics of an AGCM: Probability density functions. J. Atmos. Sci., 64 , 117–136.

    • Search Google Scholar
    • Export Citation

  • Branstator, G., and J. Berner, 2005: Linear and nonlinear signatures in the planetary wave dynamics of an AGCM: Phase space tendencies. J. Atmos. Sci., 62 , 1792–1811.

    • Search Google Scholar
    • Export Citation

  • Carnevale, G., M. Falcioni, S. Isola, R. Purini, and A. Vulpiani, 1991: Fluctuation-response relations in systems with chaotic behavior. Phys. Fluids, 3A , 2247–2254.

    • Search Google Scholar
    • Export Citation

  • Cohen, B., and G. Craig, 2004: The response time of convective cloud ensemble to a change in forcing. Quart. J. Roy. Meteor. Soc., 130 , 933–944.

    • Search Google Scholar
    • Export Citation

  • Deker, U., and F. Haake, 1975: Fluctuation–dissipation theorems for classical processes. Phys. Rev., 11A , 2043–2056.

  • Delsole, T., 2000: A fundamental limitation of Markov models. J. Atmos. Sci., 57 , 2158–2168.

  • Delsole, T., 2004: Stochastic models of quasigeostrophic turbulence. Surv. Geophys., 25 , 107–149.

  • Franzke, C., and A. Majda, 2006: Low-order stochastic mode reduction for a prototype atmospheric GCM. J. Atmos. Sci., 63 , 457–479.

  • Franzke, C., A. Majda, and G. Branstator, 2007: The origin of nonlinear signatures of planetary wave dynamics: Mean phase space tendencies and contributions from non-Gaussianity. J. Atmos. Sci., 64 , 3987–4003.

    • Search Google Scholar
    • Export Citation

  • Gardiner, C., 2004: Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. 3rd ed. Springer-Verlag, 415 pp.

  • Gerber, E., L. Polvani, and D. Ancukiewicz, 2008a: Annular mode time scales in the Intergovernmental Panel on Climate Change Fourth Assessment Report models. Geophys. Res. Lett., 35 , L22707. doi:10.1029/2008GL035712.

    • Search Google Scholar
    • Export Citation

  • Gerber, E., S. Voronin, and L. Polvani, 2008b: Testing the annular mode autocorrelation time scale in simple atmospheric general circulation models. Mon. Wea. Rev., 136 , 1523–1536.

    • Search Google Scholar
    • Export Citation

  • Goody, R., J. Anderson, and G. North, 1998: Testing climate models: An approach. Bull. Amer. Meteor. Soc., 79 , 2541–2549.

  • Gritsun, A., 2001: Fluctuation–dissipation theorem on attractors of atmospheric models. Russ. J. Numer. Anal. Math. Model., 16 , 115–133.

    • Search Google Scholar
    • Export Citation

  • Gritsun, A., and V. Dymnikov, 1999: Barotropic atmosphere response to small external actions: Theory and numerical experiments. Izv. Atmos. Ocean. Phys., 35 , 511–525.

    • Search Google Scholar
    • Export Citation

  • Gritsun, A., and G. Branstator, 2007: Climate response using a three-dimensional operator based on the fluctuation–dissipation theorem. J. Atmos. Sci., 64 , 2558–2575.

    • Search Google Scholar
    • Export Citation

  • Gritsun, A., G. Branstator, and V. Dymnikov, 2002: Construction of the linear response operator of an atmospheric general circulation model to small external forcing. Russ. J. Numer. Anal. Math. Model., 17 , 399–416.

    • Search Google Scholar
    • Export Citation

  • Gritsun, A., G. Branstator, and A. Majda, 2008: Climate response of linear and quadratic functionals using the fluctuation–dissipation theorem. J. Atmos. Sci., 65 , 2824–2841.

    • Search Google Scholar
    • Export Citation

  • Kirk-Davidoff, D., 2008: On the diagnosis of climate sensitivity using observations of fluctuations. Atmos. Chem. Phys. Discuss., 8 , 12409–12434.

    • Search Google Scholar
    • Export Citation

  • Kubo, R., M. Toda, and N. Hashitsume, 1985: Statistical Physics II: Nonequilibrium Statistical Mechanics. Springer-Verlag, 279 pp.

  • Langen, P., and V. Alexeev, 2005: Estimating 2 × CO2 warming in an aquaplanet GCM using the fluctuation–dissipation theorem. Geophys. Res. Lett., 32 , L23708. doi:10.1029/2005GL024136.

    • Search Google Scholar
    • Export Citation

  • Leith, C., 1975: Climate response and fluctuation dissipation. J. Atmos. Sci., 32 , 2022–2026.

  • Lorenz, E., 1995: Predictability: A problem partly solved. Proc. Seminar on Predictability, Reading, United Kingdom, ECMWF, 1–18.

  • Majda, A., and X. Wang, 2006: Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows. Cambridge University Press, 564 pp.

    • Search Google Scholar
    • Export Citation

  • Majda, A., and X. Wang, 2010: Linear response theory for statistical ensembles in complex systems with time-periodic forcing. Comm. Math. Sci., 8 (1) 145–172.

    • Search Google Scholar
    • Export Citation

  • Majda, A., I. Timofeyev, and E. Vanden-Eijnden, 1999: Models for stochastic climate prediction. Proc. Natl. Acad. Sci. USA, 96 , 14687–14691.

    • Search Google Scholar
    • Export Citation

  • Majda, A., I. Timofeyev, and E. Vanden-Eijnden, 2003: Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci., 60 , 1705–1722.

    • Search Google Scholar
    • Export Citation

  • Majda, A., R. Abramov, and M. Grote, 2005: Information Theory and Stochastics for Multiscale Nonlinear Systems. CRM Monogr. Series, Vol. 25, American Mathematical Society, 133 pp.

    • Search Google Scholar
    • Export Citation

  • Majda, A., C. Franzke, A. Fischer, and D. Crommelin, 2006: Distinct metastable atmospheric regimes despite nearly Gaussian statistics: A paradigm model. Proc. Natl. Acad. Sci. USA, 103 , 8309–8314.

    • Search Google Scholar
    • Export Citation

  • Majda, A., C. Franzke, and B. Khouider, 2008: An applied mathematics perspective on stochastic modelling for climate. Philos. Trans. Roy. Soc., 366A , 2429–2453.

    • Search Google Scholar
    • Export Citation

  • Majda, A., C. Franzke, and D. Crommelin, 2009: Normal forms for reduced stochastic climate models. Proc. Natl. Acad. Sci. USA, 106 , 3649–3653.

    • Search Google Scholar
    • Export Citation

  • North, G., R. Bell, and J. Hardin, 1993: Fluctuation dissipation in a general circulation model. Climate Dyn., 8 , 259–264.

  • Palmer, T., 2001: A nonlinear dynamical perspective on model error: A proposal for non-local stochastic–dynamic parametrizations in weather and climate prediction models. Quart. J. Roy. Meteor. Soc., 127 , 279–304.

    • Search Google Scholar
    • Export Citation

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

  • Ring, M., and R. Plumb, 2008: The response of a simplified GCM to axisymmetric forcings: Applicability of the fluctuation–dissipation theorem. J. Atmos. Sci., 65 , 3880–3898.

    • Search Google Scholar
    • Export Citation

  • Risken, F., 1989: The Fokker–Planck Equation. 2nd ed. Springer-Verlag, 472 pp.

  • Sardeshmukh, P., and P. Sura, 2009: Reconciling non-Gaussian climate statistics with linear dynamics. J. Climate, 22 , 1193–1207.

  • von Storch, J., 2004: On statistical dissipation in GCM climate. Climate Dyn., 23 , 1–15.

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