Radiative Transfer • Shortwave Absorption: – Clouds, H20, O3, some CO2 • Shortwave Reflection: – Clouds, surface, atmosphere • Longwave Absorption: – Clouds, H20, CO2, CH4, N2O Planck’s Law • Based on assumption of local thermodynamic equilibrium – (Not valid at very high altitudes in atmosphere) Bυ (T ) = 2 υ 3 υ  c e kT − 1   k = Boltzmann ' s constant = Planck ' s constant 2 υ = frequency c = speed of light Stefan-Boltzmann Law is the integral of the Planck function over all frequencies and all angles in a hemisphere: ∞ π ∫ Bυ (T )dυ = σ T 0 2π k σ= 2 3 15c 5 4 4 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: 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. 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 Principal Atmospheric Absorbers • H2O: Bent triatomic, with permanent dipole moment and pure rotational bands as well as rotation-vibration transitions • O3: Like water, but also involved in photodissociation • CO2: No permanent dipole moment, so no pure rotational transitions, but temporary dipole during vibrational transitions • Other gases: N2O, CH4 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 Extended Layer Models TOA : σ T2 4 = σ Te 4 → T2 = Te Middle Layer : 2σ T = σ T2 + σ Ts = σ Te + σ Ts 4 1 4 Surface : σ Ts 4 = σ Te 4 + σ T14 1 → Ts = 3 4 Te 1 T1 = 2 4 Te 4 4 4 Effects of emissivity<1 Surface : 2ε Aσ TA 4 = ε Aσ T14 + ε Aσ Ts 4 ( 2) → TA = 5 1 4 Te 321K < Ts Stratosphere : 2ε tσ Tt 4 = ε Aσ T2 4 ( 2) → Tt = 1 1 4 Te 214K < Te Full calculation of radiative equilibrium: 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) 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 When is a fluid unstable to convection? • Pressure and hydrostatic equilibrium • Buoyancy • Stability Hydrostatic equilibrium: Weight : − g ρδ xδ yδ z Pressure : pδ xδ y − ( p + δ p ) δ xδ y dw F = MA : ρδ xδ yδ z = − g ρδ xδ yδ z − δ pδ xδ y dt dw ∂p = −g − α , α = 1 ρ = specific volume ∂z dt Pressure distribution in atmosphere at rest: R* RT Ideal gas : α = , R≡ p m 1 ∂p g =− Hydrostatic : p ∂z RT Isothermal case : p = p0 e −z H , RT H≡ = "scale height " g Earth: H~ 8 Km Buoyancy: Weight : − g ρbδ xδ yδ z Pressure : pδ xδ y − ( p + δ p ) δ xδ y dw F = MA : ρbδ xδ yδ z = − g ρbδ xδ yδ z − δ pδ xδ y dt ∂p ∂p dw = −g − α b but =−g αe dt ∂z ∂z αb − αe dw → =g ≡B αe dt Buoyancy and Entropy Specific Volume: α = 1ρ Specific Entropy: s α = α ( p, s  ∂T  ∂α  =  δ s =   ∂s  p  ∂p B=g (δα ) p (δα ) p g  ∂T  α =  α  ∂p   δ s s  ∂T   δ s = −   δ s ≡ Γδ s  ∂z  s s The adiabatic lapse rate: First Law of Thermodynamics : dsrev dT dα Q =T = cv +p dt dt dt dp dT d (α p ) = cv + −α dt dt dt dT dp −α = ( cv + R ) dt dt dp dT = cp −α dt dt Adiabatic : c p dT − α dp = 0 Hydrostatic : c p dT + gdz = 0 → ( dT dz ) =− g s cp ≡ −Γ d g Γ= c p 1K Γ = Earth’s atmosphere: 100 m Model Aircraft Measurements (Renno and Williams, 1995) Radiative equilibrium is unstable in the troposphere Re-calculate equilibrium assuming that tropospheric stability is rendered neutral by convection: Radiative-Convective Equilibrium Better, but still too hot at surface, too cold at tropopause