Thermodynamics of Climate
– Part 2 –
Efficiency, Irreversibility, Tipping Points
Valerio Lucarini
Meteorological Institute, University of Hamburg
Dept. Mathematics and Statistics, University of Reading
valerio.lucarini@uni-hamburg,de
Cambridge, 12/11/2013
1
Scales of Motions (Smagorinsky)
E
N
E
R
G
Y
F
L
O
W
FEQ®NHP
FEQ®SHP
3
Energy & GW – Perfect GCM
Forcing
τ
L. and Ragone, 2011
Total warming
 NESS→Transient → NESS
 Applies to the whole climate and to to all climatic subdomains
 for atmosphere τ is small, always quasi-equilibrated 4
Energy and GW – Actual GCMs
L. and Ragone, 2011
Forcing
τ
 Not only bias: bias control ≠ bias final state
Bias depends on climate state!  Dissipation
5
Non-equilibrium in the Earth system
climate
Multiscale
(Kleidon, 2011)
Looking for the big picture
Global structural properties (Saltzman 2002).
Deterministic & stochastic dynamical systems
Example: stability of the thermohaline circulation
Stochastic forcing: ad hoc “closure theory” for noise
Stat Mech & Thermodynamic perspective
Planets are non-equilibrium thermodynamical systems
Thermodynamics: large scale properties of the climate
system; definition of robust metrics for GCMs, data
Stat Mech for Climate response to perturbations
EQ
NON EQ7
Thermodynamics of the CS
The CS generates entropy (irreversibility),
produces kinetic energy with efficiency η
(engine), and keeps a steady state by balancing
fluxes with surroundings (Ozawa et al., 2003)
Fluid motions result from mechanical work,
and re-equilibrate the energy balance.
We have a unifying picture connecting the
Energy cycle to the MEPP (L. 2009);
This approach helps for understanding many
processes (L et al., 2010; Boschi et al. 2013):
Understanding mechanisms for climate transitions;
Defining generalised sensitivities
8
Proposing parameterisations
Energy Budget
Total energy of the climatic system:




E     dVe   dV  u
   k   P   K  
 moist static kinetic


 potential

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
E   P   K 
9
Detailed Balances
Kinetic energy budget
K    dV 2  C( P, K )  D  W
WORK
W  C ( P, K )

Potential Energy budget
 
2



Q  1      H 
P   dVQ  W

Total Energy Budget
 

E     dV    H     dSnˆ H


FLUXES
DISSIPATION
10
Johnson’s idea (2000)
Partitioning the Domain
P    W 
   dVQ   
  

dV

Q



Better than it
seems!

Q  0
Q  0



11
Long-Term averages
E   P   K   0
Stationarity:
Work = Dissipation
 K    W  W  D  0
Work = Input-Output
  
  0
P    W  W  
A different view on Lorenz Energy cycle
  
  W





differential heating
G ( A)
conversion
C ( A, K )

D

dissipation
D(K )
0
12
Entropy
Mixing neglected (small on global scale), LTE: Q  sT
Entropy Balance of the system:
S    dV

Q 
T
  dV
Q 

T
  dVs    dVs       


Long Term average:
S         0           0
Note: if the system is stationary, its entropy does
not grow  balance between generation and
boundary fluxes
13
Carnot Efficiency
Mean Value Theorem:
      

We have
      

    0
Hot Cold
Work:
Carnot Efficiency:
reservoirs
  
      

 

W 







  
    








14
Bounds on Entropy Production
Minimal Entropy Production (Landau):
 dVQ 


W




 
Sin   S min   




  dVT     2




Efficiency:
entropy production
entropy fluctuations
Min entropy production is due to dissipation:
2




S min     dV  
T 

and the rest?
15
Entropy Production
Contributions of dissipation plus heat
transport:
2
  1 
  1 


S in     dV H      dV
  dV H     Smin  
T 
T  
T 

We can quantify the “excess” of entropy
production, degree of irreversibility with α:
  1 
   dV H    Smin   Be  1
T 

Heat Transport down the T gradient
increases irreversibility
16
MEPP re-examined
Let’s look again at the Entropy production:
S in    S min  1        1   
If heat transport down the temperature is
strong, η is small
If the transport is weak, α is small.

MEPP: joint optimization of heat transport
and of the production of mechanical work
17
But…
Two ways to compute EP:
é
Ñ × H rad ù
SEarth (W) = ò dV êsmat ú = 0,
T û
ë
W
ß
smat
æ1ö
æ1ö
= - Ñ × H mat × ç ÷ = Qmat × ç ÷
èT ø
èT ø
T
e2
1
1
Sin (W) = ò dVQmat = - ò dVQrad = > 0
T
T
W
W
Direct vs
Material vs
Indirect
Radiative
18
GCMs entropy budget
All in units mW m-2K-1. Hyperdiffusion in
the atmosphere and mixing in the ocean
each contribute about 1 unit.
A 2D formula
1
surf  1
TOA 1

S in     dRnet      dRnet
TE TS  S
TE

S

 
Sinvert  
Sinhor  
EP from 2D radiative fields only
Separation between effect of vertical
(convection)/horizontal processes (large
scale heat transport)
Lower bounds on Lorenz Energy Cycle
High precision, very low computational
20
cost; planetary systems?
4-box model of entropy budget
EP & Co from 2D radiative fields only
High precision, very low res needed
Poleward transport
1
2
3
4
Fluid
Vertical
transport
Surface
2-box × 2-box
21
Results on IPCC GCMs
Sinvert   
TE<
>
E
T
TE<
Sinhor   
L., Ragone, Fraedrich, 2011
Hor vs Vert EP in
IPCC models
Warmer climate:
Hor↓ Vert↑
Venus, Mars, Titan
22
PlaSim: Planet Simulator
Spectral Atmosphere
moist primitive equations
on  levels
Vegetations
(Simba, V-code,
Koeppen)
Sea-Ice
thermodynamic
Oceans:
LSG, mixed layer,
or climatol. SST
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
MoSt – The Model Starter
Snowball Hysteresis
Swing of S* by ±10% starting from present climate
  hysteresis experiment!
Global average surface temperature TS
Wide (~ 10%) range of S* bistable regime -TS ~ 50 K
d TS/d S* >0 everywhere, almost linear
L., Lunkeit, Fraedrich, 2010
W
SB
26
Thermodynamic Efficiency
d 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
27
Entropy Production
d Sin/d S* >0 in SB & W regime
Entropy production is “like” TS… but better than TS!
 Sin is about 400%  benchmark for SB vs W regime
Sin is an excellent state variable
System MUCH more irreversible in W state (Bejan)
28
Generalized Sensitivities
CO2 concentration ranging from 50 to 1750 ppm
  no bistability!
Efficiency
  0.002
Energy
Cycle
W  0.06W m2
L., Lunkeit, Fraedrich, 2010
EP
 S  0.0004Wm-2K -1
in
Irreversibility
  0.7
29
d)
100 ppm CO2
Heating Patterns
1000 ppm CO2
3

KE @ Surface
1000-100 ppm Differences
Temperature
LH Heating
3
Bringing it together…
Parametric Analysis of Climate Change
Structural Properties of the system (Boschi, L.,
Pascale 2012)
η
Upper Manifold
S*
S*
Lower Manifold
CO2
CO2
32
Bringing it together…
Parametric Analysis of Climate Change
Structural Properties of the system (Boschi, L.,
Pascale 2012)
TS
Upper Manifold
S*
S*
Lower Manifold
CO2
CO2
33
Bringing it together…
Parametric Analysis of Climate Change
Structural Properties of the system (Boschi, L.,
Pascale 2012)
Smat
Upper Manifold
S*
S*
Lower Manifold
CO2
CO2
34
A 3D picture
35
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
36
Parametrizations
EP vs Emission Temperature
37
Parametrizations
Dissipation vs Emission Temperature
38
Parametrizations
Efficiency vs Emission Temperature
39
Parametrizations
Heat Transport vs Emission Temperature
40
Now we change the LOD
Will we recover similar relations?
41
Temp vs EP
42
Temp vs Efficiency
43
Temp vs Transport
44
45
Phase Transition
 Width bistability vs length year (L. et al. 2013)
Fast rotating planets cannot be in Snowball Earth46
Planetary Atmospheres
Vast range of planetary atmospheres
Rotation rate - Orbital Phase lock
Atmospheric opacities/ incoming radiation
Thermodynamics 
habitable super-Earths?
Conclusions
Unifying picture connecting Energy cycle to EP;
Snow Ball hysteresis experiment
Mechanisms involved in climate transitions;
Analysis of the impact of [CO2] increase
Generalized set of climate sensitivities
Simplified 2D formula for studying GCMs
Many challenges ahead:
Analysis of GCMs performance
Thermodynamics of celestial bodies
Basic macroscopic thermodynamics
Next:
Entropy Production & Coarse Graining (20/11)
48
Tipping Points & EVT (25/11)
Bibliography
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habitable zone: a thermodynamic investigation, Icarus (2013)
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in General Circulation Model Development: Past, Present and Future,
D.A. Randall Ed. (Academic Press, 2002)
 Kleidon, A., Lorenz, R.D. (Eds.) Non-equilibrium thermodynamics and
the production of entropy: life, Earth, and beyond (Springer, 2005)
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Climate System, Phys Rev. E 80, 021118 (2009)
 Lucarini, V., K. Fraedrich, and F. Ragone, 2011: New results on the
thermodynamical properties of the climate system. J. Atmos. Sci., 68,
2438-2458
 Lucarini V., Blender R., Herbert C., Pascale S., Wouters, J.,
Mathematical and Physical Ideas for Climate Science, arXiv:1311.1190
[physics.ao-ph] (2013)
 Lucarini V. S. Pascale, Entropy Production and Coarse Graining of the
Climate Fields in a General Circulation Model, sub Clim. Dyn. (2013)
49
 Saltzman B., Dynamic Paleoclimatology (Academic Press, 2002)