(palaeo-)climate sensitivity: ideas and deﬁnitions from the …...michel cruciﬁx – ringberg...

Michel Crucifix – Ringberg Grand Challenge Workshop: Earth's Climate Sensitivities 25 Mars 2015

(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)

Friday 27 March 15


static def. dynamical

assume time scale sep.

FD theorem

attractor measure

assume long-memory processes

Lovejoy approach

Friday 27 March 15


E = F �R

E = F � �T (� omitted)

if equilibrium

0 = F �R

T = �F/�

E : energy imbalanceF : Radiative forcingR : Radiative responseT : Global average of surface air temperature

Friday 27 March 15


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)

Friday 27 March 15




FD theorem

attractor measure


Lovejoy approach question basic model

Bayesian model selection / calibration

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Dynamic extension

CpdT

dt= F � �T

T =F

�

⇣1� e

� �Cp

t⌘

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Multivariate model (e.g.: box model)

CdT

dt= F� ⇤T

⇠i eigenvalues of C�1⇤

T / 1�X

aie�⇠it

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

Friday 27 March 15


Non-linear extension

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

CdT

dt= F� ⇤(T )T (1)

Friday 27 March 15




FD theorem

attractor measure




Friday 27 March 15

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”)

CdT = (F� ⇤(T )T)dt +B(T )d!

Going stochastic

Friday 27 March 15




FD theorem

attractor measure




Friday 27 March 15


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”)

PT (t)

hT (t1)2ihT (t1)3i· · ·hT (t1), T (t2)i

Probability density

Moments of distribution at time t

Covariances

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⇤ controls both mean response and variability

Very simple expression if linear and gaussian:

Friday 27 March 15


if LINEAR model and GAUSSIAN noise (Wiener process):

dX = �adt+ �d!

hx2i = �

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.

Friday 27 March 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, 472–486 2015

• 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 l inear fluctuation-dissipation theorem: the role of causality, Journal of Statistical Mechanics: Theory and Experiment, 2012, P05013 2012

• Andrew J. Majda, Boris Gershgorin, and Yuan Yuan, Low-Frequency Climate Response and Fluctuation–Dissipation Theorems: Theory and Practice, J. Atmos. Sci., 67, 1186–1201 2010

• Bernd Schalge et al., Fluctuation Theorem in an Atmospheric Circulation Model, 201, http://arxiv.org/abs/1211.1181

• Fenwick C. Cooper and Peter H. Haynes, Climate Sensitivity via a Nonparametric Fluctuation–Dissipation Theorem, J. Atmos. Sci., 68, 937–953 2011

(thanks Retto Knutti and Steve Sherwood)

Friday 27 March 15

http://arxiv.org/abs/1211.1181

http://arxiv.org/abs/1211.1181




FD theorem

attractor measure




Friday 27 March 15


M. Ghil proposes a measure of climate sensitivity based on the ‘deformation’ of the random attractor under forcing change, measured by the Wallerstein distance...

Forced climate characterized, at a time ‘t’, by its random attractor...

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

Friday 27 March 15




FD theorem

attractor measure




Friday 27 March 15


Linear stochastic model

dX = �ATdt + ⌃d!E(!)

!

!1

!2

!3

!i associated with the (negative) eigenvalues of A

� = �2slope

Friday 27 March 15


Mitchell 1976 “artist view” corresponds to this model

I I

I I. MO 1DAY 3.HR

0.1 1O-2 10-3 10-4 PHtIOD IN YEARS

FIG. 1. Estimate of relative variance of climate over all periods (wavelengths) of variation, from those comparable to the age of the Earth to about one hour. Stippled area represents total variance on all spatial scales of variation. Dashed curves in lower part of the figure indicate the con- tributions to the total variance from processes characterized by spatial scales less than those indicated (in kilometers). Strictly periodic components of variation are represented by spikes of arbitrary width. Solid triangles indicate scaling relationship between the spikes and the amplitude of other

& W

features of the spectrum (see text).

FIGURE FROM : J. M. M. Mitchell, An overview of climatic variability and its causal mechanisms, Quat. Res., 6, 481–494 1976

Friday 27 March 15


Spectrum of the linear stochastic model

dX = �ATdt + ⌃d!E(!)

!

!1

!2

!3

!i associated with the (negative) eigenvalues of A

the ‘plateaus’ are where the statistical momentscan be consistently defined (quasi-stationary process)

Friday 27 March 15


Lovejoy (2013) (book and articles)

S. Lovejoy et al.: Do GCMs predict the climate . . . or macroweather? 441

Climate Weather

m

Macroweather

Fig. 2. A composite temperature spectrum: the GRIP (Summit) ice core �18O, a temperature proxy, low resolution (left, brown) along withthe first 91 kyr at high resolution (left, green), with the spectrum of the (mean) 75� N 20th century reanalysis (20CR, Compo et al., 2011)temperature spectrum, at 6 h resolution, from 1871 to 2008, at 700mb (right). The overlap (from 10–138 yr scales) is used for calibrating theformer (moving them vertically on the log–log plot). All spectra are averaged over logarithmically spaced bins, ten per order of magnitude infrequency. Three regimes are shown corresponding to the weather regime with �w = 2 (the diurnal variation and harmonic at 12 h are visibleat the extreme right). The central low frequency weather “plateau” is shown along with the theoretically predicted �mw = 0.2–0.4 regime.Finally, at longer timescales (left), a new scaling climate regime with exponent �c⇡ 1.4 continues to about 100 kyr. Note that a recent revisedchronology may modify the very lowest frequencies. Reproduced from Lovejoy and Schertzer (2012b). The black lines are reference lineswith the (absolute) slopes indicated.

!��%$��

��

�� $�� &��

��

��#��#�

��

�$� �#�

�!� ��!�

��

�� '�!�( �!��

)!��

)"��

��

��

��

��

Fig. 3. Empirical RMS temperature fluctuations (S(1t)): on the left top we show grid-point-scale (2� ⇥ 2�) daily scale fluctuations for both75� N and globally averaged along with reference slope ⇠ (2)/2 =�0.4⇡ H (20CR at 700mb). On the lower left, we see at daily resolution,the corresponding globally averaged structure function. Also shown (bottom) are the average of the three in situ surface series as well as threemultiproxy structure functions described in Lovejoy and Schertzer (2012b) (the ensemble average of the RMS fluctuations of the Huang,2004, Moberg et al., 2005, and Ljundqvist, 2010, multiproxies). The surface curve is the mean of three surface series (NASA GISS, NOAACDC and HADCRUT3, all 1881–2008). At the right we show the Vostok palaeotemperature series. Also shown is the interglacial “window”.This is a simplification of a figure in Lovejoy and Schertzer (2012b).

www.earth-syst-dynam.net/4/439/2013/ Earth Syst. Dynam., 4, 439–454, 2013

FIGURE FROM : S. Lovejoy, D. Schertzer, and D. Varon, Do GCMs predict the climate ... or macroweather?, Earth System Dynamics, 4, 439—454 2013

Friday 27 March 15

S. Lovejoy, Climate Dynamics, 201410.1007/s00382-014-2128-2

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

Michel Crucifix – Ringberg Grand Challenge Workshop: Earth's Climate Sensitivities 25 Mars 2015Friday 27 March 15

(palaeo-)climate sensitivity: ideas and deﬁnitions from the …...michel cruciﬁx – ringberg...

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