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Numerical aspects of applying the fluctuation dissipation theorem to study climate system sensitivity to external forcings. (English) Zbl 1382.86004
Summary: The fluctuation dissipation theorem (FDT), a classical result coming from statistical mechanics, suggests that, under certain conditions, the system response to external forcing can be obtained using the statistics of natural fluctuation of the system. The application of the FDT to the most sophisticated climate models and the real climate system represents a difficult problem due to the huge dimensionality of these systems and the lack of the data available for proper sampling of the system natural variability. As a consequence, one has to use some regularization procedures constraining the form of permitted perturbations. Naturally, the skill of the FDT depends on the type and parameters of the regularization procedure. In the present paper we apply FDT to predict the response of a recent version of the NCAR climate system model (CCSM4) to salinity and temperature forcing anomalies in the North Atlantic. We study the sensitivity of our results to the amount of available data and to key parameters used in our numerical algorithm.

##### MSC:
 86-08 Computational methods for problems pertaining to geophysics 86A10 Meteorology and atmospheric physics 60H30 Applications of stochastic analysis (to PDEs, etc.) 62P12 Applications of statistics to environmental and related topics
POP
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##### References:
 [1] R. V. Abramov and A. J. Majda, Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems. Nonlinearity 20 (2007), No. 12, 2793-2821. · Zbl 1134.37365 [2] R. V. Abramov and A. J. Majda, A new algorithm for low frequency climate response. J.Atmosph. Sci. 66 (2009), No. 2, 2863. [3] T. L. Bell, Climate sensitivity from fluctuation dissipation: Some simple model tests. J. Atmosph. Sci. 37 (1980), No. 8, 1700-1707. [4] H. B. Callen and T. A. Welton, Irreversibility and generalized noise. Phys. Review83 (1951), 34-41. · Zbl 0044.41201 [5] I. Cionni, G. Visconti, and F. Sassi, Fluctuation dissipation theorem in a general circulation model. Geophys. Res. Letters31 (2004) , No. 9, L09206. [6] M. Colangeli and V. Lucarini, Elements of a unified framework for response formulae. J. Stat. Mech. (2014) P01002. doi: 10.10 88/1742-5 4 68/2014/01/P01002 [7] F. C. Cooper and P. H. Haynes, Climate sensitivity via a nonparametric fluctuation-dissipation theorem. J. Atmos. Sci. 68 (2011), 937-953. [8] V. P. Dymnikov and A. S. Gritsun, Current problems in the mathematical theory of climate. Izv. Atmos. Ocean Phys. 41 (2005) , No. 3, 263-284. [9] U. Deker and F. Haake, Fluctuation-dissipation theorems for classical processes. Phys. Rev. A (1975), No. 11, 2043-2056. [10] D. Fuchs, S. Sherwood, and D. Hernandez, An exploration of multivariate fluctuation dissipation operators and their response to sea surface temperature perturbations. J. Atmosph. Sci. 72 (2014), No. 1, 472-486. [11] G. Gallavotti, Chaotic hypotesis: Onsanger reciprocity and fluctuation-dissipation theorem. J. Stat. Phys. 84 (1996), 899-926. [12] G. Gallavotti, Chaotic dynamics, fluctuations, nonequilibrium ensembles. Chaos8 (1998), 384-392. · Zbl 0965.82011 [13] P. R. Gent, G. Danabasoglu, L. J. Donner, M. Holland, E. C. Hunke, S. R. Jayne, D. M. Lawrence, R. B. Neale, P. J. Rasch, M. Vertenstein, P. H. Worley, Z.-L.Yang, and M. Zhang, The community climate system model version 4. J. Climate24 (2011), No. 19, 4973-4991. [14] G. A. Gottwald, J. P. Wormell, and J. Wouters, On spurious detection of linear response and misuse of the fluctuation-dissipation theorem in finite time series. PhysicaD331 (2016), 89-101. · Zbl 1364.37164 [15] A. S. Gritsun and V. P. Dymnikov, Barotropic atmosphere response to small external actions: Theory and numerical experiments. Izv. Atmos. Ocean Phys. 35 (1999), No. 4, 511-525. [16] A. Gritsun, G. Branstator, and V. P. Dymnikov, Construction of the linear response operator of an atmospheric general circulation model to small external forcing. Russ. J. Numer. Anal. Math. Modelling17 (2002), No. 5, 399-416. · Zbl 1007.86003 [17] A. S. Gritsun and G. Branstator, Climate response using a three-dimensional operator based on the fluctuation-dissipation theorem. J. Atmosph. Sci. 64 (2007), No. 7, 2558-2575. [18] A. S. Gritsun, G. Branstator, and A. J. Majda, Climate response of linear and quadratic functional using the fluctuation-dissipation theorem. J. Atmosph. Sci. 65 (2008), No. 9, 2824-2841. [19] A. Gritsun, Construction of the response operators onto small external forcings for the general circulation atmospheric models with time-periodic right hand sides. Izv. Atmos. OceanPhys. 46 (2010), No. 6, 748-756. [20] M. Hairer and J. C. Mattingly, Ergodic properties of highly degenerate 2D stochastic Navier-Stokes equations. Comptes Rendus Mathematique. Academie des Sciences339 (2004), 879-882. · Zbl 1059.60073 [21] E. C. Hunke and W. H. Lipscomb, CICE: The Los Alamos sea ice model user’s manual, version 4. Los Alamos National Laboratory Tech. Report LA-CC-06-012, 2008, 76 pp. [22] IPCC Fifth Assessment Report: Climate Change (AR5), 2013, [23] R. Kraichnan, Classical fluctuation-relaxation theorem. Phys. Rev. 113 (1959), 1181-1182. [24] R. Kubo, Statistical-mechanical theory of irreversible processes, I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Japan12 (1957), 570-586. [25] D. M. Lawrence, K. W. Oleson, M. G. Flanner, C. G. Fletcher, P. J. Lawrence, S. Levis, S. C. Swenson, and G. B. Bonan, The CCSM4 land simulation, 1850-2005: Assessment of surface climate and new capabilities. J. Climate25 (2012), 2240-2260. [26] C. E. Leith, Climate response and fluctuation dissipation. J. Atmosph. Sci. 32 (1975), No. 10, 2022-2025. [27] V. Lucarini and S. Sarno, A statistical mechanical approach for the computation of the climatic response to general forcings. Nonlin. Processes Geophys. 18 (2011), 7-28. [28] V. Lucarini, Stochastic perturbations to dynamical systems: a response theory approach. J. Stat. Phys. 146 (2012), 774-786. · Zbl 1245.82037 [29] G. Magnusdottir, C. Deser, and R. Saravanan, The effects of North Atlantic SST and sea-ice anomalies on the winter circulation in CCM3. J. Climate17 (2004), 857-876. [30] A. J. Majda, R. V. Abramov, and M. J. Grote, Information theory and stochastics for multiscale nonlinear systems. CRM Monograph Series25. American Mathematical Society, Providence, R.I. 2005. · Zbl 1082.60002 [31] A. J. Majda, B. Gershgorin, and Y. Yuan, Low-frequency climate response and fluctuation dissipation theorems: theory and practice. J. Atmosph. Sci. 67 (2010), No. 4,1186-1201. [32] A. J. Majda and X. Wang, Linear response theory for statistical ensembles in complex systems with time-periodic forcing. Commun. Math. Sci. 8 (2010), No. 1,145-172. · Zbl 1201.37109 [33] A. Majda, Challenges in climate science and contemporary applied mathematics. Commun. Pure Applied Math. 65 (2012), No. 7, 920-948. · Zbl 1242.86009 [34] U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Fluctuation-dissipation: response theory in statistical physics. Phys. Rep. 461 (2008), No. 4-6,111-195. [35] R. B. Neale, J. H. Richter, and M. Jochum, The impact of convection on ENSO: From a delayed oscillator to a series of events. J. Climate21 (2008), 5904-5924. [36] H. Nyquist, Thermal agitation of electric charge in conductors. Phys. Review32 (1928), 110-113. [37] T. N. Palmer, A nonlinear dynamical perspective on model error: Aproposal for non-local stochastic-dynamic parameterization in weather and climate prediction models. Quqrt. J. Roy. Meteor. Soc. 127 (2001), 279-304. [38] F. Ragone, V. Lucarini, and F. Lunkeit, A new framework for climate sensitivity and prediction: a modelling perspective. Climate Dynamics (2015), 1-13. doi: 10.1007/s00382-015-2657-3. [39] M. J. Ring and R. A. Plumb, The response of a simplified GCMto axisymmetric forcings:Applicability of the fluctuation-dissipation theorem. J. Atmos. Sci. 65 (2008), 3880-3898. [40] H. Risken, The Fokker-PlanckEquation. Springer, Berlin, 1989. [41] D. Ruelle, General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. Phys. Lett. A245 (1998), 220. · Zbl 0940.82035 [42] D. Ruelle, Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95 (1999), 393. · Zbl 0934.37010 [43] C. A. Shields, D. A. Bailey, G. Danabasoglu, M. Jochum, J. T. Kiehl, S. Levis, and S. Park, The low-resolution CCSM4. J. Cli-mate25 (2012), 3993-4014. [44] R. Smith et al., The parallel ocean program (POP) reference manual, ocean component of the Community Climate System Model (CCSM). LANL Tech. ReportLAUR-10-01853. 2010,140 pp. [45] G. J. Shutts, A Kinetic energy backscatter algorithm for use in ensemble prediction systems. Quart. J. Roy. Meteor. Soc. 131 (2005), 3079-3102.
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