Thermodynamics and
Statistical Mechanics of Climate
Valerio Lucarini
Institute of Meteorology, U. Hamburg
Dept of Mathematics and Statistics, U. Reading
100° Congresso Nazionale
Societa’ Italiana di Fisica
Pisa, 22/09/2014
Introduction
2
Climate Science and Physics“A solved problem, just some well-known
equations and a lot of integrations”
“who cares about the
mathematical/physical consistency of
models: better computers, better
simulations, that’s it!
“I regret to inform the author that geophysical
problems related to climate are of little
interest for the physical community…”
“Who cares of energy and entropy? We are
interested in T, P, precipitation”
3
Scales of Motions
(Stommel/Smagorinsky)
Scales of Motions
- different models for different scales -
5
ff
Atmospheric Motions
Three contrasting approaches:
Those who like maps, look for features/particles
Those who like regularity, look for waves
Those who like irreversibility, look for
turbulence
Let’s see schematically these 3 visions of
the world
6
Atmospheric (macro) turbulence
7
Energy, enstrophy cascades, 2D vs 3D
Note:
NOTHING
is really 2D
in the
atmosphere
Features/ParticlesFocus is on specific (self)organised structures
Hurricane physics/track
8
Waves in the atmosphere
9
Large and small scale patterns
“Waves” in the atmosphere?
10
Hayashi-Fraedrich decomposition
“Waves” in
GCMs
11
GCMs differ in
representation of
large scale
atmospheric
processes
Just Kinematics?
What we see are
only unstable
waves and their
life cycle
Non-equilibriumA Non-equilibrium Statistical Mechanical
System is in contact with several reservoirs
12
Gallavotti, 2014
Non-equilibrium in the Earth system
(Kleidon, 2011)
climate
Multiscale
Local evolution in
the phase space
NWP
vs.
Statistical
properties on the
attractor
Climate Modeling
G
O
A
L
S
O
F
M
O
D
E
L
L
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G
PR
ED
ICT
AB
ILIT
Y
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Climate Models uncertainties
Uncertainties of the 1st kind
Are our initial conditions correct? Not so relevant for
CM, crucial for NWP
Uncertainties of the 2nd kind
Are we representing all the most relevant processes for
the scales of our interest? Are we representing them
well? (structural uncertainty)
Are our heuristic parameters appropriate? (parametric
uncertainty)
Uncertainty on the metrics:
Are we comparing properly and in a meaningful way
our outputs with the observational data?
Toy
Zen and the Art of … Climate Modelling
Thermodynamics of Climate
17
Energy & Climate Response
Perfect Model
NESS→Transient → NESS
Forcing
τ
Total warming
18
L. and Ragone, 2011
Energy and Climate Response
Actual GCMs
Not only bias: bias control ≠ bias final state
Bias depends on climate state! Dissipation
Forcing τ
19
L. and Ragone, 2011
20
Energy Budget
Total energy of the climatic system:
ρ is the local density
e is the total energy per unit mass
u, and k indicate the internal, potential
and kinetic energy components
Energy budget
KPkudVedVEkinetic
potentialstaticmoist
KPE
21
Detailed Balances
Kinetic energy budget
Potential Energy budget
Total Energy Budget
WDKPCdVK
),(2
WQdVP
HQ
21
HndSHdVE
ˆ
),( KPCW
WORK
DISSIPATION
FLUXES
22
Johnson’s idea (2000)
Partitioning the Domain
Better than it
seems!
QdVQdVWP
0Q 0Q
Q+ Q-
23
Carnot Efficiency
Lorenz (1955) Energy Cycle
We have
Hot Cold reservoirs
Work:
“Carnot Efficiency”:
0
W =F+ +F-
F+F+ =
Q+ -Q-
Q+F+
0
)( ),( )(
KDndissipatio
KACconversion
AGheatingaldifferenti
DW
24
Entropy Production
Contributions of dissipation plus heat
transport:
We can quantify the “excess” of entropy
production, degree of irreversibility with α:
EP:
Sin W( ) = dV-Ñ×H
TW
ò + dVe2
TW
ò » dV-Ñ×H
TW
ò + Smin W( )
a = dV-Ñ×H
TW
ò Smin W( ) = Be-1
11minSSin
A very imperfect engine
Something interesting to get out of this picture
Work
Entropy Production25
T1 T2
QinQout
dissipation
Q1 Q2
W
Irreversible Heat
transport
Transport, Mixing, Adjustment
Vertical Transport of Energy
Convective adjustment
Irreversible mixing
Horizontal Transports
Baroclinic adjustment
Irreversible mixing
26
W
C
W C
2-box model(s)
Results on IPCC GCMs
Hor vs Vert EP in
IPCC models
Warmer climate:
Hor↓ Vert↑
Venus, Mars, Titan27
vertinS
horinS
TE
>
TE
<
TE
<
L., Ragone, Fraedrich, 2011
28
Snowball HysteresisSwing of S* by ±10% starting from present climate
hysteresis experiment with full climate model
Global average surface temperature TS
Wide (~ 10%) range of S* bistable regime -TS ~ 50 K
d TS/d S* >0 everywhere, almost linear
SB
W
L., Lunkeit, Fraedrich, 2010
29
Thermodynamic Efficiencyd η /d S* >0 in SB regime
Large T gradient due to large albedo gradient
d η /d S* <0 in W regime
System thermalized by efficient LH fluxes
η decreases at transitions System more stable
Similar behaviour for total Dissipation
η=0.04
Δθ=10K
A 3D picture
30
Parametric Analysis of Climate Change
Structural Properties (Boschi et al. 2013)
Is there a common framework?
Going from a 1D to a 2D parameter
exploration we gain completeness, we lose
focus
Necessarily so?
Can find an overall equivalence between the
atmospheric opacity and incoming radiation
perturbations
Concept of radiative forcing…
If so, we gain some sort of universality
31
Parametrizations
32
EP vs Emission Temperature
Parametrizations
33
Dissipation vs Emission Temperature
Parametrizations
34
Efficiency vs Emission Temperature
Now we reduce the length of
the year
35
36
Phase Transition
37
Width bistability vs length year (L. et al. 2013)
Fast orbiting planets cannot be in Snowball Earth
Climate Change as a problem in
Non-equilibrium Statistical
Mechanics
38
IPCC Scenarios
39
Models’ Response
40
Climate Response
IPCC scenario 1% increase p.y.
41
42
Response theoryThe 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!
43
Perturbed chaotic (Axiom A) flow:
Change in expectation value of Φ:
nth order perturbation:
Ruelle (’98) Kubo-like Response
Theory
44
This is a perturbative theory…with a causal Green function:
Expectation value of an operator evaluated over the
unperturbed invariant measure ρSRB(dx)
where: and
Linear term:
Linear Green:
Linear suscept:
Fe
1( )(t) = dsò GF
(1) s( )e t -s( )
GF
(1) t( ) = r0 dx( )ò Q t( )LP t( )F
cF
(1) w( ) = dtò exp iwt[ ]GF
(1) t( )
Applicability of FDT
If measure is singular, FDT has a boundary term
Forced and Free fluctuations non equivalent
Recently(Cooper, Alexeev, Branstator ….): FDT is OK
In fact, coarse graining sorts out the problem
Parametrization by Wouters and L. 2012, 2013 has noise45
GF
(1) t( ) = dxr0 x( )ò Q t( ) X x( ) ×ÑF x t( )( ) FDT ¯
GF
(1) t( ) = - dxò r0 x( )Ñ× r0 x( ) X x( )( )
r0 x( )F x t( )( )
OR ¯
GF
(1) t( ) = -C s x( )F x t( )( )éë
ùû
Linear (and nonlinear)
Spectroscopy of L63
46Resonances have to do with UPOs
L. 2009
cz
1( ) w( )
e t( ) = cos wt( )
Stochastic forcing
Therefore, and
We obtain:
The linear correction vanishes; only even
orders of perturbations give a contribution
No time-dependence
Convergence to unperturbed measure
der F( ) =e2 dt1GF
2( ) t1,t1( )ò + o(e 4 ) =
=1 2e2 r0 dx( ) dt1Q t1( )òò Xi¶i X j¶ jF f t1 x( )
47
e t( ) =eh t( ) =edW t( ) dt
h t( )h ¢t( ) =d t - ¢t( )h t( ) = 0
Fourier Transform
We end up with the linear susceptibility...
Let’s rewrite he equation:
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
Correlations Power Spectra
2
122
,
2
1
22
,, AAPAP
48
We choose observable A, forcing f(t)
Let’s perform an ensemble of experiments
Linear response:
Fantastic, we estimate
…and we obtain:
…we can predict
Broadband forcing
Af
1( )t( ) = dsò GA
(1) s( ) f t -s( )
c f
(1) w( ) =A
f
1( )w( )
f w( )
GA
(1) s( )
Ag
1( )t( ) = dsò GA
(1) s( )g t -s( )
Lorenz 96 modelExcellent toy model of the atmosphere
Advection, Dissipation, Forcing
Test Bed for Data assimilation schemes
Popular within statistical physicists
Evolution Equations
Spatially extended, 2 Parameters: N & F
Properties are intensive
Fxxxxx iiiii 211
Nii xxNi ,...,1
50
e= xj
2 2 Nj=1
N
å m= xj Nj=1
N
å
F®F +ee t( )
Broadband forcing G(1)(t)
51
e t( ) = Q t( )
Inverse FT of the susceptibility
Response to any forcing with the same spatial pattern but with general time pattern
Spectroscopy – Im [χ(1)(ω)]
Rigorous extrapolation
LW HF
52
e t( ) = 2cos wt( )L. and Sarno 2011
(Non-)Differentiability of the
measure for the climate system
53CO2 S*
Boschi et al. 2013
Observable: globally averaged TS
Forcing: increase of CO2 concentration
Linear response:
Let’s perform an ensemble of experiments
Concentration at t =0
Fantastic, we estimate
…and we predict:
A Climate Change experiment
T S f
(1)t( ) = dsò GTS
(1) s( ) f t -s( )
d
dtT S f
(1)t( ) =eGTS
(1) t( )
f t( ) =eQ t( )
T S g
(1)t( ) = dsò GTS
(1) s( )g t -s( )
Model Starter
and
Graphic User Interface
Spectral Atmospheremoist primitive equations
on levels
Sea-Icethermodynamic
Terrestrial Surface: five layer soil
plus snow
Vegetations(Simba, V-code,
Koeppen)
Oceans:LSG, mixed layer,or climatol. SST
PlaSim: Planet Simulator
Key features
• portable
• fast
• open source
• parallel
• modular
• easy to use
• documented
• compatible
Step 1
We double instantaneously [CO2]
360 ppm 720 ppm
We look at response of the surface
temperature TS
We average over the N members of the
ensemble
This is how we probe the system
56
What we get – CO2 doubling
57
N = 200
Linear Susceptibility
58
cTS
(1) w( ) =
dtò exp iwt[ ]GTS
(1) t( )
Step 2We increase the [CO2]
360 ppm 720 ppm at 1% per year
[CO2] is doubled after τ≈ 70 years
We keep [CO2] constant after that
Note: radiative forcing is ≈ log[CO2]
Our forcing amounts to a linear increase
gτ(t) is a ramp function reaching 1 at τ
We look at response of TS
We average over the N ensemble members
We predict using
If linear response holds.. 59
T S gt
(1)t( ) = dsò GTS
(1) s( )gt t -s( )
Climate Change Prediction - TS
60ECS= Â cTS
(1) 0( ){ } =2
pdwò Re[ Ts
(1)(w)]
Bibliography Lucarini V., Thermodynamic Efficiency and Entropy Production in the Climate
System, Phys Rev. E 80, 021118 (2009)
Lucarini, V., K. Fraedrich, and F. Lunkeit, Thermodynamic Analysis ofSnowball Earth Hysteresis Experiment: Efficiency, Entropy Production, andIrreversibility. Q. J. R. Meterol. Soc., 136, 2-11 (2010)
Lucarini, V., K. Fraedrich, and F. Ragone, New results on the thermodynamicalproperties of the climate system. J. Atmos. Sci., 68, 2438-2458 (2011)
Lucarini V., S. Sarno, A Statistical Mechanical Approach for the Computation ofthe Climatic Response to General Forcings. Nonlin. Processes Geophys., 18, 7-28 (2011)
Lucarini, V., Stochastic perturbations to dynamical systems: a response theoryapproach. J Stat Phys. 146, 774-786 (2012)
Lucarini V., Modeling Complexity: the case of Climate Science, in “Models,Simulations, and the Reduction of Complexity”, Gähde U, Hartmann S, Wolf J.H., De Gruyter Eds., Hamburg (2013)
Boschi R., S. Pascale, V. Lucarini: Bistability of the climate around the habitablezone: a thermodynamic investigation, Icarus 226, 1724-1742 (2013)
Lucarini V. and S. Pascale, Entropy Production and Coarse Graining of theClimate Fields in a General Circulation Model, Climate Dynamics DOI10.1007/s00382-014-2052-5 (2014))
Lucarini V., R. Blender, C. Herbert, F. Ragone, S. Pascale, J. Wouters,Mathematical and Physical Ideas for Climate Science, Arxiv (2014)
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