mit opencourseware ://ocw.mit.edu/.../lecture-notes/part3_lec6.pdf · n) by chain rule, f ... •...
TRANSCRIPT
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MIT OpenCourseWarehttp://ocw.mit.edu
12.842 / 12.301 Past and Present Climate�� Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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What controls the temperature gradient in middle and high
latitudes?
Annual mean temperature, northern hemisphere, about 4 km altitude
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Issues
• Temperature gradient controlled by eddies of horizontal dimensions ~ 3000 km
• Familiar highs and lows on weather maps• Eddy physics not simple • Concept of criticality does not
apply...critical T gradient = 0 (notobserved)
• While eddies try to wipe out T gradient, they do not succeed
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Example of surface pressure distribution
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Concept of Available Potential Energy
• Difference between potential energy integrated over atmosphere and the minimum value that could be obtained by an adiabatic redistribution of mass
1 2
APE = PE1 – PE2 > 0
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Complicated by rotation:
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Is there a simple principle that governs the middle and high
latitude temperature gradient, or do we have to deal with the
eddies in all their complexity?
No generally agreed upon answer to this question
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One possibility: Atmosphere arranges itself to maximize the rate
of entropy production
s� = ∫VQ�
Trad
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s� = ∫VQ�
Trad
Consider two extreme possibilities:
1. Radiative equilibrium in each column:
Q� rad = 0 → s� = 0
2. Eddies succeed in wiping out T gradient:
1 s� = T ∫V Q� rad = 0
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Maximum entropy production somewhere between∴ radiative equilibrium and zero T gradient states
Absorption and emission of radiation thermodynamically reversible processes:
s�reversible = ∫ Q� rad
V T But entropy is a state variable, so no net change in long-term average:
s�reversible + s�irreversible = 0
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→ s�irreversible = − ∫ Q�
Trad
V
Q� shortwave Q� longwave � − −
Tsurface Temission
But Q� longwave = −Q� shortwave
⎛ 1 1 ⎞ → s�irreversible =Q� shortwave ⎜⎜ − ⎟⎟ > 0
⎝Temission Tsurface ⎠
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Irreversible entropy production by microphysical processes:
• Mixing of cloudy and clear air • Fall of rain and snow • Frictional dissipation of wind energy
Nevertheless, sum total of all of these constrained by
⎛ 1 1 ⎞ s�irreversible = Q� shortwave ⎜⎜ − ⎟⎟
⎝ Temission Tsurface ⎠
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Convective generation of entropy augmented if there are also
horizontal temperature gradients:
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Simple Two-Box Model:
∂T2 = =0 T2eq
−T2 −
V (T2 −T1 )∂T1 = 0 =
T1eq −T1
+V (T2 −T1 )∂t τ rad L ∂t τ rad L
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Definitions: 1T ≡ 2 (T1 +T2 ) ,eq eq
Teq T2eq −T1eq
,Δ ≡
T T2 T1Δ ≡ −
τ radVχ ≡ L
Solution: 1 + 2 T1 + 2 ,T T = T eq eq
ΔTT eqΔ =
1 2χ+
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Entropy Production:
−s� = T2 − T2
+ T1 − T1eq eq
irreversible T2τ rad T1τ rad
1 ⎛ ΔTeq ⎞2
χ = ⎜ ⎟ 2τ rad ⎝ T ⎠ ( + χ )2 −
1 ⎜⎛ ΔTeq
⎟⎞
1 2 4 ⎝ T ⎠
1 1 ⎛ ΔT ⎞2
s maximum when � χ = 1− ⎜ eq⎟2 4 ⎝ T ⎠
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1 1Δ =T ΔT ≅ Δ Teq 2 eq
1 ⎛ ΔTeq ⎞ 2
+ − ⎜4 ⎝ T ⎠ 1 1 ⎟
Can this be generalized to include more boxes, processes?
Remember that real eddies affect distributions of clouds, water vapor
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Global Climate Modeling
• General philosophy: – Simulate large-scale motions of atmosphere,
oceans, ice – Solve approximations to full radiative transfer
equations – Parameterize processes too small to resolve – Some models also try to simulate
biogeochemical processes
– First GCMs developed in 1960s
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Model Partial Differential Equations
• Conservation of momentum • Conservation of mass • Conservation of water • First law of Thermodynamics • Equation of state • Radiative transfer equations
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Alternative Grids:
Classical spherical coordinates Conformal mapping of cube onto sphere
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A spherical grid based on the Fibonacci sequence. The grid is highly uniform and isotropic.
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Some Fundamental Numerical Constraints
Courant-Friedrichs-Lewy (CFL) condition:
c tΔ < 1,
Δx
where c is the phase speed of the fastest wave in the system, Δt is the time step used by the model, and Δx is a characteristic spacing between grid points.
Typical size of model: 20 levels, grid points spaced ~120 km apart, 10-15 variables to defines state of atmosphere or ocean at each grid point: ~1,000,000-5,000,000 variables. Typical time step: 20 minutes. Thus 70,000,000 -350,000,000 variables calculated per simulated day.
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Unresolved physical processes must be handled parametrically
• Convection • Thin and/or broken clouds • Cloud microphysics • Aerosols and chemistry (e.g.
photochemical processes, ozone
• Turbulence, including surface fluxes • Sea ice • Land ice • Land surface processes
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ForcingsForcings and Feedbacks in Climateand ModelsFeedbacks in Climate Models
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Forcings and Feedbacks Consider the total flux of radiation through the top of the atmosphere:
FTOA = Fsolar − FIR
Each term on the right can be regarded as function of the surface temperature, Ts, and many other variables xi :
FTOA = FTOA (Ts , x1, x2 ,..... xN ) By chain rule,
FTOA 0 ∂FTOA δTs +∑
N ∂FTOA xiδ = = δ ∂Ts i=1 ∂xi
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Now let’s call the Nth process a “forcing”, Q:
∂F N −1 ∂FδFTOA 0 TOA δTs +∑ TOA xi = = δ δ + Q
∂Ts i=1 ∂xi
∂FTOA N −1 ∂FTOA ∂xi δ δ Q= δTs +∑ Ts +
∂ ∂ T∂Ts i=1 xi s
Then
∂T 1s ≡ λR = − N −1∂Q ∂FTOA ∂FTOA ∂xi+∑ ∂ ∂ T∂Ts i=1 xi s
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−1⎛ ∂FTOA ⎞ Climate sensitivityLet S ≡ −⎜ ⎟ ⎝ ∂Ts ⎠ without feedbacks
∂T Ss ≡ λR = N −1 ∂FTOA ∂xi∂Q 1− S∑i=1 ∂xi ∂Ts
Climate sensitivity Feedback factors; can be of either sign
Note that feedback factors do NOT add linearly in their collective effects on climate sensitivity
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Examples of Forcing:
• Changing solar constant • Changing concentrations of non-
interactive greenhouse gases
• Volcanic aerosols • Manmade aerosols • Land use changes
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Solar Sunspot Cycle
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Examples of Feedbacks:
• Water vapor • Ice-albedo • Clouds • Surface evaporation • Biogeochemical feedbacks
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Estimates of Climate Sensitivity∂T Ss ≡ λ = R N −1 ∂F ∂x∂Q 1− S∑ TOA i
i=1 ∂xi ∂Ts
−1⎛ ∂F ⎞
S ≡ − TOA ⎜ ⎟ ⎝ ∂Ts ⎠
Suppose that Ts = Te + constant and that shortwave radiation is insensitive to Ts:
∂F ∂FTOA = −σTe 4 , TOA = − σTe
4 = −4σTe 3 = −3.8 Wm −2 K −1
∂T ∂Ts s
= 0.26 ( −2 )−1S K Wm
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Examples of Forcing Magnitudes:
• A 1.6% change in the solar constant, equivalent to 4 Wm-2, would produce about 1oC change in surface temperature
• Doubling CO2, equivalent to 4 Wm-2, would produce about 1oC change in surface temperature
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Contributions to net radiative forcing change, 1750Contributions to net radiative forcing change, 1750‐‐2004:2004:
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Examples of feedback magnitudes:
• Experiments with one-dimensional radiative-convective models suggest that holding the relative humidity fixed,
⎛ ∂FTOA ⎞⎛ ∂q ⎞ −2 −1⎜ ⎟⎜ ⎟ ≅ 2 Wm K , ⎝ ∂q ⎠⎝ ∂Ts ⎠RH
S ⎛⎜∂FTOA ⎞
⎟⎛⎜ ∂q ⎞
⎟ ≅ 0.5 ⎝ ∂q ⎠⎝ ∂Ts ⎠RH
This, by itself, doubles climate sensitivity; with other positive feedbacks, effect on sensitivity is even larger
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Free Natural Variability of the Climate System
Deterministic versus chaotic dynamics
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Chaotic DynamicsChaotic Dynamics
Predictability time, τ
Note : lim(τ ) =τ ≠ 0ε→0
pre
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Climate chaosClimate chaos
• Atmosphere known to be chaotic on time scales at least as large as several months
• Ocean known to be chaotic on time scales of at least 6 months and perhaps as long as hundreds of years
• Coupled atmosphere-ocean system may be chaotic on time scales as long as several thousand years
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Global mean temperature (black) andsimulations using many differentglobal models (colors) including allforcings
To quantify natural, chaoticTo quantify natural, chaotic variability, necessary tovariability, necessary to
run large ensemblesrun large ensembles
Same as above, but models run with only natural forcings
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How Do We Know If We Have ItHow Do We Know If We Have It Right?Right?
• Very few tests of model as whole: annual and diurnal cycles, 20th century climate, weather forecasts, response to orbital variations
• Fundamentally ill-posed: Far more free parameters than tests
• Alternative: Rigorous, off-line tests of model subcomponents. Arduous, unpopular: Necessary but not sufficient for model robustness: Model as whole may not work even though subcomponents are robust
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Global mean temperature (black) andsimulations using many differentglobal models (colors) including allforcings
To some extent,To some extent, ““successsuccess””of 20of 20thth century simulationscentury simulations is a result of model curveis a result of model curve
fittingfitting
Same as above, but models runwith only natural forcings
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RootRoot--meanmean--squaresquare error in zonally anderror in zonally and annually averagedannually averaged SW radiation (top)SW radiation (top) and LW radiationand LW radiation (bottom) for individual(bottom) for individual AR4 models (colors)AR4 models (colors) and for ensembleand for ensemble mean (black dashed)mean (black dashed)
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Observed time mean, zonally averaged oceanObserved time mean, zonally averaged ocean temperature (black contours), and modeltemperature (black contours), and model--mean minusmean minus
observed temperature (colors) for the period 1957observed temperature (colors) for the period 1957--19901990