T. Allmendinger
DOI:
10.4236/acs.2018.82014 216 Atmospheric and Climate Sciences
steady state where a constant limiting temperature of the black body is reached:
( )
44
blackbody,lim atm
TT
σ
Φ= −
(4a)
or rearranged
44
blackbody,lim atm
TT
σσ
⋅ =Φ+ ⋅
(4b)
Hence, the Stefan-Boltzmann relation was applied to the whole atmos-
phere—or rather to its lowest layer—, which obviously does not exhibit any sur-
rounding walls, as it was the case for the measurements of Dulong and Petit. In-
stead, it is held together by the gravity of the Earth. However, this implies a con-
siderably inhomogeneity—even when cloud and dust effects are neglected—, not
only because of the vertical pressure and temperature gradients, but also because
of horizontal gradients caused by the ball shape of the Earth, which leads to lati-
tude-dependant solar irradiation intensities. In addition, differences in altitude,
due to mountains, as well as diurnal and seasonal fluctuations contribute to
permanent but not consistent motions which hardly admit steady-state condi-
tions.
When, instead of a blackbody, a coloured
solid opaque body
(abbrev.
SOB
) is
used, these equations can be modified to the Equations ((5a) and (5b)) since the
thermal emission power of a coloured solid opaque body turned out to be equal
to the thermal emission power of a blackbody [11]:
( )
44
SOB,lim atm
s
TT
βσ
Φ⋅ = −
(5a)
or rearranged
44
SOB,lim atms
TT
σ βσ
⋅ =Φ⋅ + ⋅
(5b)
yielding formula (6) for calculating the limiting temperature of an irradiated co-
loured SOB:
4
4
SOB,lim atm
s
TT
β
σ
Φ⋅
= +
(6)
Therein,
β
s
indicates the
solar reflection coefficient
, expressing the portion of
absorbed irradiation. But normally, instead of this solar absorption coefficient,
the complementary
solar reflection coefficient α
s
(or the “albedo”) is used which
entails its
indirect
determination according to the relation
β
s
= 1 −
α
s
. Since this
may lead to uncertainties due to scattering of the reflected radiation, a method
for the
direct
determination
of the solar reflection coefficient
β
s
was developed
by the author, measuring the warming-up of coloured solid opaque plates [11].
Thus, it spares you the trouble of determining the albedo and will briefly be de-
scribed in the next chapter, in particular since it is suitable to empirically assess
the Stefan-Boltzmann relation.
In any case it should be realized that the incident solar intensity Φ and the
ambient surface air temperature
T
atm
are not independent of one another, apart
from the fact that the latter one is ill-defined. However, this interdependency is
not distinctly computable because of the variety of relevant further parameters
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