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© Crown copyright Met Office
Radiation scheme for Earth’s atmosphere…and what might not work for exoplanets
James Manners 6/12/11
© Crown copyright Met Office
Table of Contents
• Two-stream equations
• Solar and thermal spectrum
• Gaseous absorption: lines / continuum
• Rayleigh scattering
• Aerosols – Mie scattering and absorption
• (Clouds)
© Crown copyright Met Office
Some fundamentals….
dI / ds = − kaI + kaB
I (n)
ds
Kirchhoff’s Law: absorption proportional to emission
Requires Local Thermodynamic Equilibrium (LTE)(valid up to ~65km for Earth)
Consider a single frequency:
© Crown copyright Met Office
Some fundamentals….
dI / ds = − kaI + kaB − ksI + ksS
I (n)
ds
I (n' )
Conservative scattering (no change in frequency)
© Crown copyright Met Office
Some fundamentals….
dI / ds = − kaI + kaB − ksI +
I (n)
ds
I (n' )
Phase function:
P(n', n) : probability of scattering into direction n from n‘
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Two-stream approximation
F+
F–
I (n)Height (z)
Integrate over up and down directions:
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Two-stream equations
Linear simultaneous equations can be solved to give F± on levels.
All we need to know:
• ka : absorption coefficient• ks : scattering coefficient• g : asymmetry of scattering (1st moment of phase function)• Q± : up and down source terms
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Source terms Q±
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Split between solar (short-wave) and thermal (long-wave) radiation
Treat scattering ofdirect solar radiation
Treat thermal emission from atmosphere
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Split between solar (short-wave) and thermal (long-wave) radiation
Q−
Q+
SW LW
Q+
Q−
Source term =scattering from direct beam
Source term =Plankian emission at temperature of layer
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“Single scattering” properties(ka, ks, g) for each process
• Gaseous absorption (ka = f(ν), ks = 0)
(including continuum absorption)
• Rayleigh scattering (ka = 0, ks = f(ν), g = 0)
• Aerosol particles (ka = f(ν), ks = f(ν), g = f(ν) )
• Clouds: water droplets and ice crystals
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Gaseous absorption
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Gaseous absorption
• SW: H2O, CO2, O3, O2
• LW: H2O, CO2, O3, N2O, CH4, CFCs, HFCs
• Absorption line data from HITRAN (HIgh-resolution TRANsmission molecular absorption database).
• HITRAN data designed for Earth’s atmosphere:
• Reference Temperature 296K
• Line broadening coefficients for Earth P/T
• Isotope abundance for Earth atmosphere
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Gaseous absorption
Doppler (temperature) broadening: Gaussian line shape
Collision (pressure) broadening: Lorentz line shape
Absorption spectrum needs to be adjusted for P/T broadening of lines.
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k-distribution
ka ka
g0.0 0.5 1.0
ka
weight
k-terms
Order wavenumber bins by absorption at reference P/T:
Number of monochromatic calculations reduced from 50,000 to 6(in this example).
© Crown copyright Met Office
Correlated-k method
ka
0.0 0.5 1.0g
• Use same ordering (mapping to “g-space”) for all P/T.
P / T P' / T'
ka
0.0 0.5 1.0g
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LW bands:overlapping gaseous absorption
Relative abundances at 10km(mid-latitude summer, ~ tropopause)
1 2 3 5 7 8 9
4 6
H2O
CO2
O3 CH4
N2O
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SW bands:
123456
line data
cross sections
HITRAN 2008
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Random overlap of absorption lines
H2O
N2O
CH4
k-terms
Requires 2*2*6 = 24 monochromatic calculations
LW band 7
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Equivalent extinction
H2O
N2O
CH4
k-terms
Calculate single “equivalent extinction” coefficient using clear-sky atmosphere with minor gas.
Requires 2*1*1 = 2 monochromatic calculations
LW band 7Major gas
Minor gas
Minor gas
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“Grey” processesSlowly changing with wavelength (considered constant over band).
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Continuum absorption
• Absorption poorly modelled far from line centres.
• Empirical fit to continuum.
• Add absorption due to:• Self-broadened H2O continuum• Foreign-broadened H2O continuum
• Other gases continua may be important for exoplanets.
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Rayleigh scattering
• Scattering from particles of size << wavelength of light
ks λ-4 , ka = 0
• Depends linearly on number density
• Scattering efficiency depends on molecule – will be different for exoplanets.
• Symmetric phase function (g = 0)
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Aerosol absorption and scattering
• Particle size >> wavelength of light
• ka, ks, g calculated using Mie-Debye theory (equivalent to geometric optics for large spherical particles)
• Assume single size distribution for each aerosol species (+ humidity dependence)
• Strong forward scattering peak (g > 0)
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δ-rescaling
• 2-stream approximation loses accuracy for strongly asymmetric scattering
ks, g
ks' = ks(1 – f )
g' = (g – f )/(1 – f )
Forward scattering fraction f = g2
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Cloud droplets and ice crystals
• Cloud droplets similar to aerosols except:• Effective radius depends on number of CCN• Sub-grid structure (cloud fraction etc.)
• Cloud ice based on ice-aggregates:• Optical properties parametrised based on external
database
• δ-rescaling required
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New configuration for exoplanets…
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External “spectral files”:
• Spectral bands• Solar spectrum• Gases considered• k-terms for each gas• Coefficients for pressure / temperature scaling• Single scattering properties for:
• Continuum absorption• Rayleigh scattering• Aerosols & clouds
Tools available• Corr_k.f90 : generates k-terms from line database• Scatter.f90 : generates properties for aerosol
particle size distributions
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Model code and parameter changes:
• Orbital parameters & solar constant
• Gas and aerosol species (+ mixing ratios)
• Surface albedo and emissivity
• …bound to be others I’ve forgotten
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Questions and answers
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Some fundamentals…
Change in radiance (I) for beam direction n
Absorption from beam
Scattering from beam
Scattering into beam n from all other beams n'
Emission into beam