12.842 / 12.301 Past and Present Climate › ... › lecture-notes › part3_lec2.pdf“General...
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MIT OpenCourseWare http://ocw.mit.edu
12.842 / 12.301 Past and Present ClimateFall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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Radiative Transfer Chapter 3, Hartmann
• Shortwave Absorption: – Clouds, H20, O3, some CO2
• Shortwave Reflection: – Clouds, surface, atmosphere
• Longwave Absorption: – Clouds, H20, CO2, CH4, N2O
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Planck’s Law• Based on assumption of local thermodynamic
equilibrium – (Not valid at very high altitudes in atmosphere)
2 υ3
B Tυ ( ) = 2 υ
c e kT −1 k = Boltzmann 's constant = Planck 's constant
υ = frequency
= c speed of light
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∞B T d (
Stefan-Boltzmann Law is the integral of the Planck function over all frequencies and all angles in a
hemisphere:
) υ σT 4π ∫0 υ =
5 4 2π kσ = 2 3 15c
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Absorption and Emission in a Gas:
Photon energy Eυ = υ
Atomic energy levels Eυ = n υ, n = 0,1,2,3 …
An isolated atom can absorb only those photons whose energy is equal to the difference between two atomic energy levels
Molecules have additional energy levels:
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These images are copyrighted by Academic Press, NY, 1999. The images are in a chapter entitled "Quasi-equilibrium thinking" (author is Kerry Emanuel) that appears in the book entitled “General circulation model development: past, present and future”, edited by D. Randall, Ed. ISBN: 0125780109. Used with permission.
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For molecules in a gas:
Etotal = Eatomic + Evibrational + Erotational + Etranslational
Translational energy is the kinetic energy of molecular motions in a gas, proportional to the gas temperature. Not quantized.
Molecules in a gas can absorb more frequencies than isolated atoms.
Collisions between molecules can carry away energy or supply energy to interactions between matter and photons.
Natural, pressure and Doppler broadening
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Principal Atmospheric Absorbers
• H2O: Bent triatomic, with permanent dipolemoment and pure rotational bands as wellas rotation-vibration transitions
• O3: Like water, but also involved in photodissociation
• CO2: No permanent dipole moment, so nopure rotational transitions, but temporarydipole during vibrational transitions
• Other gases: N2O, CH4
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Radiative Equilibrium
• Equilibrium state of atmosphere and surface in the absence of non-radiative enthalpy fluxes
• Radiative heating drives actual state toward state of radiative equilibrium
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Extended Layer Models
TOA : σT24 =σTe
4 → T2 = Te
Middle Layer : 2σT14 =σT2
4 +σTs 4 =σTe
4 +σTs 4
Surface : σTs 4 =σTe
4 +σT14
1 1→ Ts = 3 4Te T1 = 2 4Te
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Effects of emissivity<1
Surface : 2ε σT 4 = ε σ T 4 +ε σ T 4A A A 1 A s
5 4→ TA = ( 2) 1
Te 321K < Ts
Stratosphere : 2ε σT 4 = ε σ T 4t t A 2
4→ T = ( 12) 1
T 214K < Tt e e
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Full calculation of radiative equilibrium:
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Problems with radiative equilibrium solution:
• Too hot at and near surface • Too cold at a near tropopause • Lapse rate of temperature too large in the
troposphere • (But stratosphere temperature close to
observed)
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Missing ingredient: Convection
• As important as radiation in transporting enthalpy in the vertical
• Also controls distribution of water vapor and clouds, the two most important constituents in radiative transfer
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When is a fluid unstable to convection?
• Pressure and hydrostatic equilibrium • Buoyancy • Stability
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Hydrostatic equilibrium:
Weight : − gρ x y zδ δ δ
Pressure : δ δx y − ( p + p) xp δ δ δ y
dwF = MA : ρ x y z = − gρ x y z −δ δ δ x yδ δ δ δ δ δ pdt
dw ∂p 1 dt
= −g −α∂z
, α = ρ = specific volume
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Pressure distribution in atmosphere at rest:
Ideal gas : α = RT , R ≡
R *
p m
Hydrostatic : 1 ∂p = −
g
p z RT ∂
Isothermal case : p = p e − z
H , H ≡RT
= "scale height " 0 g
Earth: H~ 8 Km
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Buoyancy:
δ δ δ
Pressure : p x y − ( p + p) Weight : − gρb x y z
δ δ δ δ δ x y
dwF = MA : bδ δ δ = − gρb x y z −δ δ δ x yρ x y z δ δ δ pdt
dw ∂p ∂p g dt
= −g −αb ∂z but
∂z = − αe
→dw
= g αb −αe ≡ B
dt αe
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1ρ
Buoyancy and Entropy
Specific Volume: α =
Specific Entropy: s
α =α ( ,p s
(δα ) p = ∂α
δ s =
∂T δ s
∂s p ∂p s
(δα ) p g T∂ ∂T B = g α =α
∂p s
δ s = − ∂z
s
δ s ≡Γ δ s
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The adiabatic lapse rate:First Law of Thermodynamics :
ds dT dα rev Q T= = cv + pdt dt dt
dT d (α p) dp= c + −αv dt dt dt
= (c + R) dT −α dp
v dt dt
= c dT
−α dp p dt dt
Adiabatic : c dT −αdp = 0p
Hydrostatic : c dT + gdz = 0p
→ (dT dz )s
= − g cp
≡ −Γ d
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Γ = g cp
Earth’s atmosphere: Γ =1 K 100 m
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Model Aircraft Measurements(Renno and Williams, 1995)
Image courtesy of AMS.
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Radiative equilibrium is unstable in the troposphere
Re-calculate equilibrium assuming that tropospheric stability is rendered neutral by
convection:
Radiative-Convective Equilibrium
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Better, but still too hot at surface, too cold at tropopause