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
What makes it so difficult to model the
geophysical fluids?
› Some gross mistakes in our models
› Some conceptual/epistemological issues

What is a response?
› Examples and open problems

Recent results of the perturbation theory for
non-equilibrium statistical mechanics
› Deterministic & Stochastic Perturbations
› Spectroscopy/Noise/Broadband analysis

Applications on systems of GFD interest
› Climate Change prediction
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
The response theory is a Gedankenexperiment:
› a system, a measuring device, a clock, turnable
knobs.
Changes of the statistical properties of a system
in terms of the unperturbed system
 Divergence in the response  tipping points
 Suitable environment for a climate change
theory

› “Blind” use of several CM experiments
› We struggle with climate sensitivity and climate
response

Deriving parametrizations!
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
Axiom A dynamical systems are very special
› hyperbolic on the attractor
› SRB invariant measure
 Smooth on unstable (and neutral) manifold
 Singular on stable directions (contraction!)

When we perform numerical simulations, we
implicitly set ourselves in these hypotheses
› Chaotic hypothesis by Gallavotti & Cohen (1995, 1996):
systems with many d.o.f. can be treated as if Axiom A
› These are, in some sense, good physical models!!!

Response theory is expected to apply in more
general dynamical systems AT LEAST FOR SOME
observables
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
Perturbed chaotic flow as:

Change in expectation value of Φ:
 nth
order perturbation:
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
with a causal Green function:
› Expectation value of an operator evaluated over
the unperturbed invariant measure ρSRB(dx)
where:
and

Linear term:
F

Linear Green:

Linear suscept:

(1)
(t) =
ò ds G (s ) e(t - s )
(1)
F
( t ) = ò r0 ( dx)Q (t ) LP (t ) F
(1)
(1)
c F (w ) = ò dt exp [iw t ] GF ( t )
(1)
F
G
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GF(1) ( t ) =
ò dxr ( x)Q (t ) X ( x) ×× F ( x (t ))
0
FDT ¯
(1)
F
G

( t ) = - ò dxr0 ( x)
Ñ Ñ( r0 ( x) X ( x))
r0 ( x)
F ( x ( t ))
If measure is singular, FDT has a boundary term
› Forced and Free fluctuations non equivalent
Recent studies (Cooper, Alexeev, Branstator ….):
FDT approximately works
 In fact, coarse graining sorts out the problem

› Parametrization by Wouters and L. 2012 has noise
› The choice of the observable is crucial
› Gaussian approximation may be dangerous
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GF(1) ( t ) = - ò dxr0 ( x) s ( x) F ( x ( t ))
X ( x) = x̂ j
ß
GF(1) ( t ) = - ò dxr0 ( x) ¶ j log éër0 ( x)ùûF ( x ( t ))
F ( x) = xk
ß
GF(1) ( t ) = - ò dxr0 ( x) ¶ j log éër0 ( x)ùû xk ( t )
r0 ( x) = exp éë-b xi xi ùû ß
GF(1) ( t ) = b ò dxr0 ( x) xi xk ( t ) = b Ci, j ( t )

Various degrees of approximation
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
FDT or not, in-phase and out-of-phase responses
are connected by Kramers-Kronig relations:
› Measurements of the real (imaginary) part of the
susceptibility  K-K  imaginary (real) part
Every causal linear model obeys these
constraints
 K-K exist also for nonlinear susceptibilities

with
 (1) ( )  [  (1) ( )]*
Kramers, 1926; Kronig, 1927
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(1)
c z (w )
L. 2009

Resonances have to do with UPOs
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e( t ) = eh ( t ) = e dW,( t ) dt
 t  t   t  t 
Therefore,  t   0 and
 We obtain:

ò dt G (t , t ) + o(e ) =
=1 2 e ò r ( dx) ò dt Q (t )X ¶ X ¶ F ( f t x) + o(e
de r (F) = e
( 2)
2
1
F
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1
1
2
1
0
1
1
i i
j
j
The linear correction vanishes; only even
orders of perturbations give a contribution
 No time-dependence
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 Convergence to unperturbed measure e

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4
)

Fourier Transform
  ,   
2
     
2
1
2
We end up with the linear susceptibility...
 Let’s rewrite he equation:

P ,  A  P  A    ,   A

2
  1  
2
2
So: difference between the power spectra
› → square modulus of linear susceptibility
› Stoch forcing enhances the Power Spectrum
Can be extended to general (very) noise
 KK  linear susceptibility  Green function

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
Excellent toy model of the atmosphere
› Advection, Dissipation, Forcing
Test Bed for Data assimilation schemes
 Popular within statistical physicists
 Evolution Equations

xi  xi 1 xi 1  xi 2   xi  F i  1,..., N xi  xi  N

Spatially extended, 2 Parameters: N & F
N
e= å x 2 N
2
j
m= å xj N
j=1

N
Properties are intensive
j=1
F ® F + ee(t )
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LW
HF
e(t ) = 2cos (wt )
L. and Sarno 2011
Rigorous extrapolation
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(1)

 Squared modulus of e  
Blue: Using stoch pert; Black: deter forcing
 ... And many many many less integrations

L. 2012
dW
e( t ) =
dt
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 We
choose observable A, forcing e
 Let’s perform an ensemble of experiments
T
(1)
(1)
 Linear response:
r ( e)t = ò ds Ge (s ) T (t - s )
 Fantastic,
 …and
 …we
c
we estimate
we obtain:
can predict r
(1)
e
(w ) =
(1)
e
G
(1)
(e)t =
T
r
( A)w
T (w )
(1)
T
(s )
(1)
d
s
G
ò e (s ) T (t - s )
e( t ) = Q ( t )


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Inverse FT of the susceptibility
Response to any forcing with the same spatial
pattern but with general time pattern
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Noise due to finite length L of integrations and
of number of ensemble members N
(1)
-g
-u
 We assume
c w »w ; T w »w

A

( )
We can make predictions for timescales:
tcrit ³ cN

( )
-
1
1
2(g +u )
2(g +u )
L
Or for frequencies:
wcrit £ cN
1
2(g +u )
1
2(g +u )
L
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Boschi et al. 2013
CO2
S*
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 Observable:
globally averaged TS
 Forcing: increase of CO2 concentration
T
(1)
(1)
 Linear response: r (T S )t = ò ds GT (s ) T ( t - s )
 Let’s perform an ensemble of experiments
T ( t ) = eQ ( t )
› Concentration  at t=0
T
 Fantastic, we estimate d (1)
r T = G(1) t
S
( S)t
dt
 …and
we predict: r
(1)
(TS)t
R
=
TS
()
ò ds G (s ) R(t - s )
(1)
TS
PlaSim: Planet Simulator
Spectral Atmosphere
moist primitive equations
on  levels
Sea-Ice
thermodynamic
Oceans:
LSG, mixed layer,
or climatol. SST
Vegetations
(Simba, V-code,
Koeppen)
Terrestrial Surface:
five layer soil
plus snow
Model Starter
and
Graphic User Interface
Key features
• portable
• fast
• open source
• parallel
• modular
• easy to use
• documented
• compatible
N = 200
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CLIMATE SENSITIVITY
DTS = × { c
(1)
TS
( 0)} =
2
p
(1)
d
w
r
(TS)w
×
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Impact of deterministic and stochastic forcings to nonequilibrium statistical mechanical systems
 Frequency-dependent response obeys strong constraints
› We can reconstruct the Green function –
Spectroscopy/Broadband
 Δ expectation of observable ≈ variance of the noise
› SRB measure is robust with respect to noise
 Δ power spectral density ≈ l linear susceptibility |2
› More general case: Δ power spectral density >0
 We can predict climate change given the scenario of
forcing and some baseline experiments
› Limits to prediction
› Decadal time scales
› Now working on IPCC/Climateprediction.net
data
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D. Ruelle, Phys. Lett. 245, 220 (1997)
D. Ruelle, Nonlinearity 11, 5-18 (1998)
C. H. Reich, Phys. Rev. E 66, 036103 (2002)
R. Abramov and A. Majda, Nonlinearity 20, 2793 (2007)
U. Marini Bettolo Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Phys.
Rep. 461, 111 (2008)
D. Ruelle, Nonlinearity 22 855 (2009)
V. Lucarini, J.J. Saarinen, K.-E. Peiponen, E. Vartiainen: Kramers-Kronig
Relations in Optical Materials Research, Springer, Heidelberg, 2005
V. Lucarini, J. Stat. Phys. 131, 543-558 (2008)
V. Lucarini, J. Stat. Phys. 134, 381-400 (2009)
V. Lucarini and S. Sarno, Nonlin. Proc. Geophys. 18, 7-27 (2011)
V. Lucarini, J. Stat. Phys. 146, 774 (2012)
J. Wouters and V. Lucarini, J. Stat. Mech. (2012)
J. Wouters and V. Lucarini, J Stat Phys. 2013 (2013)
V. Lucarini, R. Blender, C. Herbert, S. Pascale, J. Wouters, Mathematical
Ideas for Climate Science, in preparation (2013)
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