# A new framework for climate sensitivity and prediction: a modelling perspective

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## Abstract

The sensitivity of climate models to increasing CO_{2} concentration and the climate response at decadal time-scales are still major factors of uncertainty for the assessment of the long and short term effects of anthropogenic climate change. While the relative slow progress on these issues is partly due to the inherent inaccuracies of numerical climate models, this also hints at the need for stronger theoretical foundations to the problem of studying climate sensitivity and performing climate change predictions with numerical models. Here we demonstrate that it is possible to use Ruelle’s response theory to predict the impact of an arbitrary CO_{2} forcing scenario on the global surface temperature of a general circulation model. Response theory puts the concept of climate sensitivity on firm theoretical grounds, and addresses rigorously the problem of predictability at different time-scales. Conceptually, these results show that performing climate change experiments with general circulation models is a well defined problem from a physical and mathematical point of view. Practically, these results show that considering one single CO_{2} forcing scenario is enough to construct operators able to predict the response of climatic observables to any other CO_{2} forcing scenario, without the need to perform additional numerical simulations. We also introduce a general relationship between climate sensitivity and climate response at different time scales, thus providing an explicit definition of the inertia of the system at different time scales. This technique allows also for studying systematically, for a large variety of forcing scenarios, the time horizon at which the climate change signal (in an ensemble sense) becomes statistically significant. While what we report here refers to the linear response, the general theory allows for treating nonlinear effects as well. These results pave the way for redesigning and interpreting climate change experiments from a radically new perspective.

## Keywords

Climate response Climate sensitivity Linear response theory IPCC climate change scenarios GCM ensemble simulations## Notes

### Acknowledgments

The authors wish to thank C. Franzke and G. Gallavotti for commenting on an earlier version of the manuscript. F.R. wish to thank T. Bodai and S. Schubert for useful discussions. F.R. and V.L. acknowledge funding from the Cluster of Excellence for Integrated Climate Science (CLISAP) and from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement No. 257106. The authors acknowledge the Newton Institute for Mathematical Sciences (Cambridge, UK), hosting the 2013 programme “Mathematics for the Fluid Earth” during which part of this work was discussed.

## References

- Abramov RV, Majda A (2008) New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems. J Nonlinear Sci 18:303–341CrossRefGoogle Scholar
- Allen MR, Frame DJ (2007) Call off the quest. Science 318:582–583CrossRefGoogle Scholar
- Bracegirdle TJ, Stephenson BD (2013) On the robustness of emergent constraints used in multimodel climate change projections of arctic warming. J Clim 26:669–678CrossRefGoogle Scholar
- Chekroun MD, Simonnet E, Ghil M (2011a) Stochastic climate dynamics: random attractors and time-dependent invariant measures. Phys D 240:1685–1700CrossRefGoogle Scholar
- Chekroun MD, Kondrashov D, Ghil M (2011b) Predicting stochastic systems by noise sampling, and application to the El Nino-Southern Oscillation. Proc Natl Acad Sci USA 108:11766–11771CrossRefGoogle Scholar
- Colangeli M, Lucarini V (2014) Elements of a unified framework for response formulae. J Stat Mech P01002. doi: 10.1088/1742-5468/2014/01/P01002
- Cooper FC, Haynes PH (2011) Climate sensitivity via a nonparametric fluctuation-dissipation theorem. J Atmos Sci 68:937–953CrossRefGoogle Scholar
- Cooper FC, Esler JG, Haynes PH (2013) Estimation of the local response to a forcing in a high dimensional system using the fluctuation-dissipation theorem. Nonlinear Process Geophys 20:239–248CrossRefGoogle Scholar
- Cox PM, Pearson D, Booth BB, Friedlingstein P, Huntingford C, Jones CD, Luke CM (2013) Sensitivity of tropical carbon to climate change constrained by carbon dioxide variability. Nature 494:341–344CrossRefGoogle Scholar
- Fraedrich K, Kirk E, Lunkeit FP (1998) Portable university model of the atmosphere. Technical Report 16, Deutsches KlimarechenzentrumGoogle Scholar
- Fraedrich K, Jansen H, Luksch U, Lunkeit F (2005) The planet simulator: towards a user friendly model. Meteorol Z 14:299–304CrossRefGoogle Scholar
- Gallavotti G (1996) Chaotic hypotesis: onsanger reciprocity and fluctuation-dissipation theorem. J Stat Phys 84:899–926CrossRefGoogle Scholar
- Galloway J, Heimann J, Le Quéré C, Levitus S, Ramaswamy V (2013) Climate sensitivity in the Anthropocene. Q J R Meteorol Soc 139:1121–1131CrossRefGoogle Scholar
- Gritsun A, Branstator G (2007) Climate response using a three-dimensional operator based on the Fluctuation Dissipation theorem. J Atmos Sci 64:2558–2575CrossRefGoogle Scholar
- Hasselmann K, Sausen R, Maier-Reimer E, Reinhard V (1993) On the cold start problem in transient simulations with coupled atmosphere ocean models. Clim Dyn 9:53–61CrossRefGoogle Scholar
- Held IM, Soden BJ (2006) Robust responses of the hydrological cycle to global warming. J Clim 19:5686–5699CrossRefGoogle Scholar
- IPCC (2007a) In: Parry ML, Canziani OF, Palutikof JP, van der Linden PJ, Hanson CE (eds) Climate Change 2007: Impacts, adaptation and vulnerability. Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, UKGoogle Scholar
- IPCC (2007b) In: Solomon S, Qin D, Manning M, Chen Z, Marquis M, Averyt KB, Tignor M, Miller HL (eds) Climate change 2007: the physical science basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USAGoogle Scholar
- IPCC (2013) In: Stocker TF, Qin D, Plattner G-K, Tignor M, Allen SK, Boschung J, Nauels A, Xia Y, Bex V, Midgley PM (eds) Climate Change 2013: the physical science basis. Contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change stocker. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USAGoogle Scholar
- Kalnay E (2003) Atmospheric modeling, data assimilation and predictability. Cambridge University Press, CambridgeGoogle Scholar
- Knutti R, Hegerl GC (2008) The equilibrium sensitivity of the Earth’s temperature to radiation changes. Nat Geosci 1:735–743CrossRefGoogle Scholar
- Kubo R (1966) The fluctuation-dissipation theorem. Rep Prog Phys 29:255–284CrossRefGoogle Scholar
- Langen PL, Alexeev VA (2005) Estimating 2 × CO
_{2}warming in an aquaplanet GCM using the fluctuation-dissipation theorem. Geophys Res Lett 32(23):L23708CrossRefGoogle Scholar - Lorenz E (1979) Forced and free variations of weather and climate. J Atmos Sci 36:1367–1376CrossRefGoogle Scholar
- Lovejoy S (2014) Scaling fluctuation analysis and statistical hypotesis of anthtropogenic warming. Clim Dyn 42:2339–2351CrossRefGoogle Scholar
- Lucarini V, Saarinen JJ, Peiponen K-E, Vartiainen EM (2005) Kramers-kronig relations in optical materials research. Springer, HeidelbergGoogle Scholar
- Lucarini V (2008) Response theory for equilibrium and non-equilibium statistical mechanics: causality and generalized Kramers–Kronig relations. J Stat Phys 131:543–558CrossRefGoogle Scholar
- Lucarini V (2009) Evidence of dispersion relations for the nonlinear response of Lorenz 63 System. J Stat Phys 134:381–400CrossRefGoogle Scholar
- Lucarini V, Sarno S (2011) A statistical mechanical approach for the computation of the climatic response to general forcings. Nonlin Processes Geophys 18:7–28CrossRefGoogle Scholar
- Lucarini V, Colangeli M (2012) Beyond the linear fluctuation-dissipation theorem: the role of causality. J Stat Mech P05013. doi: 10.1088/1742-5468/2012/05/P05013
- Lucarini V, Blender R, Herbert C, Pascale S, Ragone F, Wouters J (2014) Mathematical and physical ideas for climate science. http://arxiv.org/abs/1311.1190
- Lunt DJ, Haywood AM, Schmidt GA, Salzmann U, Valdes PJ, Dowset HJ (2010) Earth system sensitivity inferred from Pliocene modelling and data. Nat Geosci 3:60–64CrossRefGoogle Scholar
- Nicolson AM (1973) Forming the fast Fourier transform of a step response in time-domain metrology. Electron Lett 9:317–318CrossRefGoogle Scholar
- Otto A et al (2013) Energy budget constraints on climate response. Nat Geosci 6:415–416CrossRefGoogle Scholar
- Penland C (1996) A stochastic model of IndoPacific sea surface temperature anomalies. Phys D 98:534–558CrossRefGoogle Scholar
- Penland C (2003) Noise out of chaos and why it won’t go away. B Am Meteor Soc 84:921–925CrossRefGoogle Scholar
- Previdi M, Liepert BG, Peteet D, Hansen J, Beerling DJ, Broccoli AJ, Frolking S, Galloway JN, Heimann M, Le Quéré C, Levitus S, Ramaswamy V (2013) Climate sensitivity in the Anthropocene. Q J R Meteorol Soc 139:1121–1131CrossRefGoogle Scholar
- Ruelle D (1998a) General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. Phys Lett A 245:220–224CrossRefGoogle Scholar
- Ruelle D (1998b) Nonequilibrium statistical mechanics near equilibrium: computing higher-order terms. Nonlinearity 11:5–18CrossRefGoogle Scholar
- Ruelle D (2009) A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22:855–870CrossRefGoogle Scholar
- Saltzman B (2001) Dynamical paleoclimatology. Academic Press, New YorkGoogle Scholar
- Sardeshmukh P, Compo GP, Penland C (2000) Changes in probability assoicated with El Nino. J Clim 13:4268–4286CrossRefGoogle Scholar
- Sherwood SC, Bony S, Dufresne J-L (2014) Spread in model climate sensitivity traced to atmospheric convective mixing. Nature 505:37–42CrossRefGoogle Scholar
- Shukla J, Hagedorn R, Hoskins B, Kinter J, Marotzke J, Miller M, Palmer T, Slingo J (2009) Revolution in climate prediction is both necessary and possible: a declaration at the world modelling summit for climate prediction. B Am Meteor Soc 90:175–178CrossRefGoogle Scholar
- Stein U, Alpert P (1993) Factor separation in numerical simulations. J Atmos Sci 50:2107–2115CrossRefGoogle Scholar
- Winton M, Takahashi K, Held IM (2010) Importance of ocean heat uptake efficacy to transient climate change. J Clim 23:2333–2344CrossRefGoogle Scholar
- Wouters J, Lucarini V (2013) Multi-level dynamicsl systems: connecting the Ruelle response theory and the Mori-Zwanzig approach. J Stat Phys 151:850–860CrossRefGoogle Scholar