(Palaeo-)climate sensitivity: ideas and definitions from the NPG literature

Transcription

1 (Palaeo-)climate sensitivity: ideas and definitions from the NPG literature Michel Crucifix Université catholique de Louvain & Belgian National Fund of Scientific Research Ringberg Grand Challenge Workshop: Earth's Climate Sensitivities 25 Mars 2015 (thanks to WCRP and BELSPO for funding my participation)

2 FD theorem assume time scale sep. attractor measure static def. dynamical assume long-memory processes Lovejoy approach

3 E = F R E = F T ( omitted) if equilibrium 0=F R T = F/ E : energy imbalance F : Radiative forcing R : Radiative response T : Global average of surface air temperature

4 E = F T is a diagnostic model supposes a consistent link between T, E, F and observable quantities different models if T = radiative temperature, or global average of temperature, or related to classical state variables like enthalpy (accounting for snow and sea-ice molt)

5 FD theorem assume time scale sep. attractor measure static def. dynamical assume long-memory processes question basic model Bayesian model selection / calibration Lovejoy approach

6 Dynamic extension C p dt dt = F T T = F 1 e C p t

7 Multivariate model (e.g.: box model) C dt dt = F T T / 1 X ai e it i eigenvalues of C 1 classical hypothesis : one time scale will dominate, slow responses considered constant and fast responses integrated in the definition of the forcing go back to 1-D case

8 Non-linear extension C dt dt = F (T )T (1) may exhibit all sorts of exotic behaviors (chaos, limit cycles, etc.) that WILL be relevant for climate system dynamics, at least on some time scales most naturally defined in a (quasi-)linear setting again, the hope is that some sort of time scale separation will apply

9 FD theorem assume time scale sep. attractor measure static def. dynamical assume long-memory processes question basic model Bayesian model selection / calibration Lovejoy approach

10 Going stochastic CdT =(F (T )T)dt + B(T )d! going stochastic may be justified either using a formal argument of separation of time scales (requires averaging over characteristic period + time-mixing hypothesis) or simply heuristically (i.e., by saying it works )

11 FD theorem assume time scale sep. attractor measure static def. dynamical assume long-memory processes question basic model Bayesian model selection / calibration Lovejoy approach

12 Whether we adopt a stochastic approach, OR, consider chaotic dynamics (with ergodic theory), (or both...) variables are described in terms of density distribution ( space measures ) P T (t) Probability density ht (t 1 ) 2 i Moments of distribution at time t ht (t 1 ) 3 i ht (t 1 ),T(t 2 )i Covariances

13 controls both mean response and variability Very simple expression if linear and gaussian:

14 if LINEAR model and GAUSSIAN noise (Wiener process): dx = adt + d! ) hx 2 i = 2 2a c(t) =e a t... GCMs urges caution (non-gaussian) see Cooper and Haynes for attempts at tackling non- Gaussianity with a non-parametric approach. Not trivial.

15 some references on this: (thanks Retto Knutti and Steve Sherwood) David Fuchs, Steven Sherwood, and Daniel Hernandez, An Exploration of Multivariate Fluctuation Dissipation Operators and Their Response to Sea Surface Temperature Perturbations, J. Atmos. Sci., 72, Peter L. Langen and Vladimir A. Alexeev, Estimating 2 CO 2 warming in an aquaplanet GCM using the fluctuation-dissipation theorem, Geophys. Res. Lett., 32, 2005 Valerio Lucarini and Matteo Colangeli, Beyond the linear fluctuation-dissipation theorem: the role of causality, Journal of Statistical Mechanics: Theory and Experiment, 2012, P Andrew J. Majda, Boris Gershgorin, and Yuan Yuan, Low-Frequency Climate Response and Fluctuation Dissipation Theorems: Theory and Practice, J. Atmos. Sci., 67, Bernd Schalge et al., Fluctuation Theorem in an Atmospheric Circulation Model, 201, Fenwick C. Cooper and Peter H. Haynes, Climate Sensitivity via a Nonparametric Fluctuation Dissipation Theorem, J. Atmos. Sci., 68, (thanks Retto Knutti and Steve Sherwood)

16 FD theorem assume time scale sep. attractor measure static def. dynamical assume long-memory processes question basic model Bayesian model selection / calibration Lovejoy approach

17 Forced climate characterized, at a time t, by its random attractor... M. Ghil proposes a measure of climate sensitivity based on the deformation of the random attractor under forcing change, measured by the Wallerstein distance... M. Ghil, A Mathematical Theory of Climate Sensitivity or, How to Deal With Both Anthropogenic Forcing and Natural Variability? in Climate Change: Multidecadal and Beyond, 2014

18 FD theorem assume time scale sep. attractor measure static def. dynamical assume long-memory processes question basic model Bayesian model selection / calibration Lovejoy approach

19 Linear stochastic model E(!) dx = ATdt + d!! 3! 2! 1 slope = 2! i associated with the (negative) eigenvalues of A!

20 Mitchell 1976 artist view corresponds to this model I I I I. MO 1DAY 3.HR PHtIOD IN YEARS 0.1 1O FIGURE FROM : J. M. M. Mitchell, An overview of climatic variability and its causal mechanisms, Quat. Res., 6,

21 Spectrum of the linear stochastic model E(!) dx = ATdt + d! the plateaus are where the statistical moments can be consistently defined (quasi-stationary process)! 3! 2! 1! i associated with the (negative) eigenvalues of A!

22 Lovejoy (2013) (book and articles) m Climate Macroweather Weather FIGURE FROM : S. Lovejoy, D. Schertzer, and D. Varon, Do GCMs predict the climate... or macroweather?, Earth System Dynamics, 4,

23 S. Lovejoy, Climate Dynamics, /s Various natural forcings (volcanos, solar, etc.) participate to the overall multifractality Simple linear regression between temperature and CO2 Take the linear regression off the signal (subtraction) Verify that the residual is statistically similar to what occurs during the last millennium Concludes that it is plausible. he finds: effective (transient?) climate sensitivity : 2.33 K