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Lecture 8.
Terrestrial infrared radiative processes.
Part 1:Line-by-line (LBL) method for solving IR radiative transfer.
Objectives:
- Fundamentals of the thermal IR radiative transfer.
- Line-by-line computations of radiative transfer in IR. Required reading: L02: 4.2.1-4.2.
- Fundamentals of the thermal IR radiative transfer. Recall the equation of radiative transfer (Eq.[3.25 a, b], Lecture 3) for upward and downward intensities in the plane-parallel atmosphere
τ
μ τμϕ λ (^) λ↑ λ↑ ↑
= I − J
d
dI [3.25a]
− μ λ^ (τ;τ−μ;ϕ)= λ↓(τ;−μ;ϕ)− λ↓(τ;−μ; ϕ )
↓
dI d I J [3.25b]
and its solutions (Eq.[3.26a,b], Lecture 3):
μ τ μτ τ μ ϕ^ τ
μ
τ μ ϕ τ μ ϕ^ τ^ τ
λ
τ
τ
λ λ
+ − ′− ′^ ′
= −^ −
↑
↑ ↑
∫ J d
I I
1 exp( ) ( ; ; )
( ; ; ) ( ; ; )exp( )
*^ * [3.26]
τ μ ϕ τ μ
τ τ μ
μ
τ μ ϕ μ ϕ^ τ
λ
τ
λ λ
+ − − ′ ′ −^ ′
↓
↓ ↓
∫ J d
I I
1 exp( ) ( ; ; )
( ; ; ) ( 0 ; ; )exp( )
0
[3.26b]
Infrared radiative transfer in the absorbing/emitting atmosphere: For a non-scattering medium in the local thermodynamical equilibrium, the source
function is given by the Plank’s function B λ (T)(see Lecture 4) and
βe ,λ = βa,λ =k λ ρ ,
where βe ,λ and βa ,λ are the volume extinction and absorption coefficients, and k λ is the
mass absorption coefficient.
Assuming that the thermal infrared radiation from the earth’s atmosphere is independent on the azimuthal angle ϕ, the equation of infrared radiative transfer (in the wavenumber domain) for the monochromatic upward and downward intensities can be expressed as:
( ; ) I ( ; ) B(T )
d
dI
ν τ ν τ μ ν μ ↑τ μ = ↑ − [8.1a]
( ; ) I ( ; ) B(T )
d
dI
ν τ ν τ μ ν
− μ ↓τ−μ = ↓ − − [8.1b]
and the solutions as
μ τ μτ τ^ τ
τ μ τ μ^ τ^ τ
ν
τ τ
ν ν
+ − ′− ′^ ′
= −^ −
∫
↑ ↑
B T d
I I
1 exp( ) ( ( ))
( ; ) ( *; )exp( * )
τ τ μ
τ τ μ
μ τ μ μ^ τ
ν
τ
ν ν
∫
↓ ↓
B T d
I I
1 exp( ) ( ( ))
( ; ) ( 0 ; )exp( )
0
[8.2b]
Thus the formal solutions for monochromatic upward and downward intensities given by Eq.[8.3a,b] in terms of transmittance are:
ν ν τ
ν ν ν τ − ′ ′−′^ ′
∫
↑
B dT d d
I B T
( ) ( ;^ )
[8.4a]
τ μ ν τ^ ν τττ μ τ
τ
↓ (^) ∫ d d
I ( ; ) B ( )dT ( ; )
0
[8.4b]
NOTE: Eq.(8.4 a, b) can be also written in terms of the weighing function which is defined as
τ ν ττ μ^ ν τ τ μ d W ( , ', )= dT( − ', ) [8.5]
Let’s re-write the solutions of the radiative transfer equation for upward and downward radiances in the altitude coordinate z.
k d z
z z
τ (^) ν = (^) ∫′ ν ρgas ′′ [8.6]
Thus transmission between z and z’ along the path at μ is
T ( z,z, ) exp(^1 k gasdz )
z z
′ = − ∫ ′ ′ ′
ν μ μ ν^ ρ [8.7]
and
dT ( (^) dz,zz, ) k exp( (^1) k gasdz ) z z
=− gas^ − ′′ ′
′ μ μ^ ∫′^ ρ
μ ρ ν ν ν [8.8]
Thus
k dz B T z k d z
I z I k dz
gas
z gas
z z
z gas
′ ′
+ ^ − ′′
= ^ − ′
∫ ∫
∫
′
↑ ↑
μμμμ μμμμ ρρρρ^ ρρρρ
μμμμ μμμμ μμμμ ρρρρ
νννν νννν νννν
νννν νννν νννν (^1) exp (^1) ( ( ))
( , ) ( 0 , )exp^1
0
(^0) [8.9a]
I z k dz B T z k gasd z
z
z z gas^
− = ^ − ′′
∫ ∫
∞ ′
( , )^1 exp^1 [8.9b]
- Line-by-line (LBL) computations of radiative transfer in IR. LBL method is considered to be an “exact” computation of radiative transfer in the gaseous absorbing/emitting inhomogeneous atmosphere, accounting for all (known) gas absorption lines in the wavenumber range from 0 to about 23,000 cm-1. Strategy to perform LBL calculations (i.e., to solve Eq.[8.3a,b]) for the plane- parallel atmosphere: For a given wavenumber ν:ν:ν:ν: For the j-th atmospheric layer:(homogeneous; temperature Tj, pressure pj, length ∆Zj) For n-th gas: Absorption coefficient k νννν,j,n is
= =
L l
nl j nl j j
L l
k j n k jnl S T f T p
1
, , , 1
where l = 1, ..L in the number of absorbing lines of n-th gas at a selected ν; Sνν νν,n,l and f νννν,n,l are the intensity and profile of the l-th line.
Optical depthττ ττνννν,j,n, of n-th gas of j-th layer
τ ν,j ,n =k ν,n,jun, j
where un,j is the slant path for n-gas in j-th layer (i.e., the amount of n-th gas in j-th layer).
Repeating above calculations for all gases n=1, ..., N, we find optical depth of j-th layer
j n
N ν,j (^) n 1 ν,,
=
Repeating above calculations for all layers j=1, ..., J, we find optical depth of each layer. Using calculated optical depth of each layer, we find the monochromatic upward and downward intensities from Eq.[8.3a,b].
NOTE: Spectral intensify requires the calculations of spectral transmission which requires the calculations of monochromatic optical depth which are done with LBL computations.
LBL spectral resolution:
- Because LBL computes each line of absorbing gases in a non-homogeneous
atmopshere, the adequate selection of an integration step (i.e., interval dνννν )))) is required
to calculate the spectral transmittance in the interval ∆ν∆ν∆ν∆ν (∆ν >∆ν >∆ν >∆ν > dν)ν)ν)ν).. ..
- Because P decreases exponentially with altitude, the line half-width and hence the integration step should be smaller at higher altitudes in the atmosphere.
- Because of these variable resolutions, the absorptions coefficients of two consequent layers must be merged – it is done by interpolating the coarser-resolution of lower layers into the finer-resolution of the higher-level => spectral absorptance for a given slant path is computed with the finest spectral resolution.
Continuum Absorption lines may have long wings (e.g., depending on a line half-width). To simplify calculations, the wings of a line are cut at a given distance from the line center Thus the absorption coefficient of the line may be expressed as
kν = Sfν + k ν^ c [8.13]
where k ννννc^ gives the absorption fraction in the wings (called continuum absorption).
CKD (Clough, Kneizys, and Davies) continuum model: includes continuum absorption due to water vapor, nitrogen, oxygen, carbon dioxide, and ozone. The water vapor continuum is based upon a water vapor monomer line shape formalism applied to all spectral regions from the microwave to the shortwave. The absorption coefficient for the water vapor continuum is the sum of the self- and foreign (air)-broadening components:
[ ( w )] self
k c^ = self pw+ air p−p σ ν σ σ [8.14]
where pw and p denote the water vapor partial pressure and the air (ambient) pressure (in atm), respectively, and σself and σair are the self- and air-broadening coefficients for water vapor. In the 8-12 μm region, the self-broadening coefficient is parameterized as (Roberts et al., 1976):
σ (^) self (ν,Tr)= a+bexp(− βν ) [8.15]
where Tr = 296 K, a=4.18, b=5578, and b=7.87x10-3^ (for σself in cm^2 g-1^ atm-1). And σair /σself = 0.002 at Tr = 296 K. Moreover, σself depends on as:
σself(ν ,T )= σself( ν,Tr)exp(c(Tr/T− 1 ))
where c=6.
NOTE: There are two competing effects on absorption of the water vapor continuum: absorption coefficient increases as the temperature decreases, but colder atmospheric conditions have less water vapor.
LBL numerical codes: LBLRTM and FASTCODE LBLRTM is developed in the ATMOSPHERIC AND ENVIRONMENTAL RESEARCH INC. (available to the scientific community) http://www.rtweb.aer.com/ General info: A radiance algorithm has been used to treat the vertically inhomogeneous atmospheres resulting in substantially improved accuracy, and the model is directly applicable to longwave cooling rate calculations. A layered atmosphere is used with each layer assumed to be in local thermodynamic equilibrium with respect to absorption in the layer. The spectral lines are optimally sampled at each layer using an algorithm that effectively provides optimal sampling over the line. An accelerated approximation to the Voigt line
