Available Energy: Net Radiation and Soil Heat Flux - Transport in the Atmosphere-Vegetation-Soil Continuum - page 21

Figure 2.5 Deinition of coordinates for radiation travelling through the atmosphere.

coeficient (in m 2 kg -1 ) for a given substance i and q i is the speciic concentration
(mass fraction) of substance i in the air. With the relationship between d s and d z (see
earlier and Figure 2.5 ) and integrating over z we obtain:

∞
∫

(2.10)

I =⋅ − I exp( mkqdzI

ii

ρ ) = τ

λ λ 0 r λ , λ λθ 0 , i

0

where τ λθ , i is the monochromatic transmissivity for substance i along a path with
zenith angle θ z (dimensionless). The integral in Eq. ( 2.10 ) is often referred to as the
optical thickness or normal optical depth δ λ , i (Wallace and Hobbs, 2006 ):

≡ ∞

∫ d

δ kq ρ z (2.11)

λ , i λ , i i

0

The transmissivity τ λθ , i will have a value between zero and one: it is the fraction of
radiation that is transmitted.
In general, the extinction coeficient depends on height: in the case of extinction
by absorption the exact spectral location, strength and width of the absorption lines
depend on local pressure and temperature and hence on height. But if we would
assume the extinction coeficient to be constant and allow only the concentration q i
to vary with height, the optical thickness can be written as k

∞
∫ , where the
integral represents the total amount of the substance under consideration (e.g., total
ozone column, or amount of precipitable water). If we would assume both the extinc-
tion coeficient k λ , i and the mass fraction q i to be constant with height, the optical
thickness could be replaced by k λ ∙q i ∙m v . This expression can also be used for variable
concentrations if q i is interpreted as a weighted (with ρ i ) average concentration.
The monochromatic transmissivity τ λθ , i can be decomposed into a vertical compo-
nent and an angle-dependent part as follows:

qz ρ d

λ , i i

0

= ( ) = ( )

m m

− δ m − δ

r

τ = e e τ

r

λ , i r λ , i

(2.12)

λ θ , i λ v, i