Available Energy: Net Radiation and Soil Heat Flux - Transport in the Atmosphere-Vegetation-Soil Continuum - page 20

where ρ is the density of the air and s is the location along the path (the beam starts
at ininity and ends at the surface, s = 0). Note that the concept of optical mass is also
used for applications where the domain is not the entire atmosphere, but only a inite
layer (e.g., between the top of the atmosphere and an elevated location on a moun-
tain). Then the integration in Eq. ( 2.5 ) is performed from a height larger than zero and
the optical mass will have a smaller magnitude.
The path length through the atmosphere (i.e., where density is non-zero) depends
on the solar zenith angle. To characterize the optical mass of the atmosphere, without
reference to a certain geometry of the radiation (i.e., a certain solar zenith angle), the
vertical optical mass ( m v ) is introduced (note the change of direction of integration
due to the replacement of s by z , see Figure 2.5 ):

∞
∫ ρ
0

m = d z (2.6)

v

Then, if we deine the ratio of the true optical mass ( m a ) and the optical mass along
the local vertical ( m v ) as the relative optical mass 1 ( m r ), the true optical mass can be
decomposed into a component that depends on the atmosphere ( m v ), and a component
that depends on the direction of the radiation ( m r ):

mmm
a

= (2.7)

v r

If we neglect the curvature of Earth's surface and assume that refraction of light by
the atmosphere is absent, then it is easy to see that in the case of slantwise radiation
we have d z = -cos( θ z ) d s (see Figure 2.5 ), yielding

(2.8)

m r ≈ [ cos) ( θ 1

−

z

This approximation is accurate within 1% for solar zenith angles up to 75 degrees
(based on the expressions in Kasten and Young, 1989 ). In principle, deviations from
the approximation are different for different atmospheric constituents, owing to dif-
ferences in their vertical distribution (Iqbal, 1983 ).
According to Beer's law, the reduction of the radiation along a beam, due to a sub-
stance i , can be described as:

dI =− 0, Ikqs

ii λ λ ρ

d (2.9)

λ

where I λ is the spectral lux density through a plane perpendicular to the beam, I λ 0 is
the spectral lux density entering the medium, k λ , i is the monochromatic extinction

1 Note that in some literature the relative optical mass as introduced here is called just optical mass. Often, for ele-
vated locations, the height-correction is applied to the relative optical mass through multiplication with ( p/p 0 ),
where p 0 is a reference pressure at sea level.