coarse-graining and entropy production in a climate model - part 3- valerio lucarini klimacampus,...
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Coarse-graining and Entropy Production in a Climate Model
- Part 3-
Valerio LucariniKlimacampus, Meteorological Institute, University of Hamburg
Department of Mathematics & Statistics, University of Reading
Email: [email protected]
Cambridge, 20/11/2013
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TodayInterplay between order and chaos in the
climate systemFrom space, mid-latitude cyclones look like
Von Karman’s vorticesBut atmosphere is NOT 2D at all
Dynamical/thermodynamical processesWe’ll focus on entropy production and
coarse graining: irreversibility at various scales! How to account for them?
EP dissipation parametrizations
Scales of Motions (Smagorinsky)
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Update
ff
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Atmospheric MotionsThree contrasting approaches:
Those who like maps, look for features/particlesThose who like regularity, look for wavesThose who like irreversibility, look for
turbulence
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CommentsNone of the 3 approaches is fully satisfactory
The atmosphere has indeed features, but organised motions are just part of the story
Particles have a life-cycle: what generates them, what dissipates them?
The waves we observe result from unstable processes – energy budgets are crucial
Nonlinearities turbulent mixing and dissipationScaling properties are observed at larger scales:
turbulence is macro-turbulenceBaroclinic (3D) / barotropic (2D) interplay Lorenz Energy Cycle
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Other peculiarities of the atmosphereNegative viscosity phenomena:
Turbulent momentum fluxes can be directed towards the region of high momentum (Jets)
Turbulent: fast, non-zonally symmetric (but far from microscopic, infinitesimal!)
Discovered by V. StarrThe atmosphere is not a “simple”, Onsager-
like dissipative system – eddy viscosity?
Such phenomena are due to long-range correlations (travelling waves!)
… but how can this be sustained?
Disequilibrium in the Earth system
(Kleidon, 2011)
climate
Disequilibrium
Work
Irreversibility
Multiscale
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Main Formulas for Climate Thermodynamics
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Entropy ProductionContributions of dissipation plus heat
transport:
Note:If heat transport along T is strong, η is smallIf the transport along T is weak, α is small
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But…Two ways to compute EP:
Direct vs Indirect
Material vs Radiative
T R A N S P O R T
E N E R G Y
E N E R G Y
T R A N S P O R T
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Transport, Mixing
Vertical Transport of Energy Convective adjustmentIrreversible mixing
Horizontal TransportsBaroclinic adjustmentIrreversible mixing
W
C
W C
2-box model(s)
4-box model of entropy budget
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3 4
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Poleward transport
Vertical transport
Fluid
Surface
2×2 box model
Results on IPCC GCMs
Hor vs Vert EP in IPCC models
Warmer climate:Hor↓ Vert↑
Venus, Mars, Titan15
vertinS
horinS
L., Ragone, Fraedrich, 2011
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A step backwards
Let’s consider a quasi-equilibrium system where Onsager relations apply
with A positive definite quadratic formEP is locally positive!Let’s use extensively the Parseval
Theorem for
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We obtain:
Where is the entropy produced at space scale kl and t scale ωm
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Coarse GrainingWe perform a coarse graining on our data
cut-off for |ωm| > |ωMAX| and |kl |> |kMAX|
Estimate of EP:
And: the coarser the graining (lower values for the cut-off), the lower the value of the estimate of the EP
Each scale contributes with a positive termDifferent scales tell about different processes
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Local MadnessIn the stratosphere
eddy (high frequency/wavenumber) heat fluxes are locally against the T gradient
NEGATIVE EPHow it possible?Work on the systemLocal vs Global?
Scales?
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Entropy Production in the CS
It is NOT a diffusive system, as we have seen
It is a non-equilibrium, multiphase system
What are our expectations on the properties of ?
Climate is VERY heterogeneous (horizontal vs vertical), and seasonally and daily forced
Feedbacks, re-equilibration processes
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Computing entropy production
Direct Formula
Indirect Formula
We use high resolution fields
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Coarse graining the fieldsWe perform coarse graining with a given
temporal kernel τ and spatial kernel v
Coarse grained estimates
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Temporal Coarse Graining
Time averaging kernel
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Time and Space Coarse GrainingAveraging in time and along the pressure levels
Direct Method Indirect Method
1 D COLUMN 1 D COLUMN
CL
IMA
TO
LO
GY
CL
IMA
TO
LO
GY
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EP – 1 d resolution – Indirect MethodWe use the indirect formula
Ver Av
Hor
Av
1 D COLUMN
2 D
HO
R F
LO
W
0 D
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EP - 1d resolution – Direct MethodSpatial coarse graining does not kill EP!
Ver Av
Hor
Av
1 D COLUMN
2 D
HO
R F
LO
W
0 D
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Transport, Mixing – Indirect FormulaHorizontal Transports
Baroclinic adjustmentIrreversible mixing 6 mWK-1m-2
W C
2-box model
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Transport, Mixing – Direct FormulaHorizontal Transports
Hydrological cycleDissipation of KE18 mWK-1m-2
W C
2-box model
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CommentsFull Coarse graining
Indirect formula gives 0: the system is reduced to a 0D body, which absorbs and emits radiation at a given temperature
Direct formula gives min EP = SKE
Concluding we get: 13 mWK-1m-2 SKE
6 mWK-1m-2 Shor
33 mWK-1m-2 Sver
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Comment The coarser the graining, the lower the
estimated value of entropy productionWe lose information about transfer of energy
across domains at different temperatures
From indirect method: can separate vertical/ horizontal only transfer processesLosing info on the vertical structure is very badremoving all resolution gives 0
Direct Method: we retain the effect of KE dissipation also at lowest resolution
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EP 1d res: Direct-Indirect MethodDifference depends on effective resolutionDifference >0, 0 with full resolution
NVER
Hor
Av
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InterestingLet’s compare the effect of coarse graining
Note that
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InterpretationSystem is forced, fluctuates and
dissipatesThe correlation btw radiative heating
rates & temperature >0 at all scales Energy input temperature increase
The correlation btw material heating rates & temperature <0 at all scales T fluctuation relaxation
System driven by radiative forcing, T fluctuations dissipated by material fluxes
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What we have learnedPositive contribution to EP comes from
all time and space scalesNot so locally! Waves allow for this.13 mWK-1m-2 SKE
6 mWK-1m-2 Shor
33 mWK-1m-2 Sver
This looks like being a general resultIt may be a way to generalize Onsager-like
properties to far from equilibrium systems
Should be used in other, simpler systems as well (Aquaplanet, Rayleigh-Benard)
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Bibliography Lucarini V. and S. Pascale, Entropy Production and Coarse Graining of the
Climate Fields in a General Circulation Model, submitted to Clim. Dyn. (2013)
Boschi R., S. Pascale, V. Lucarini: Bistability of the climate around the habitable zone: a thermodynamic investigation, Icarus (2013)
Johnson D.R., Entropy, the Lorenz Energy Cycle and Climate, 659-720 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)
Lucarini V., Thermodynamic Efficiency and Entropy Production in the Climate System, Phys Rev. E 80, 021118 (2009)
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)
Lucarini, V., K. Fraedrich, and F. Ragone, 2011: New results on the thermodynamical properties of the climate system. J. Atmos. Sci., 68, 2438-2458
Saltzman B., Dynamic Paleoclimatology (Academic Press, 2002)