The longwave radiation emitted by a surface (Fup) is described by the Stefan-
Fup = ε σ T04 ,
where T0 is the surface temperature, ε is the emittance of the surface, and σ is the
The incoming longwave radiation from the atmosphere (Fdn) can be determined
by an appropriate radiative transfer model (Kondratyev 1969, Curry and Ebert 1992).
Using such models, however, requires data on the distribution of air temperature
and humidity up to heights of at least 30 km. Therefore, since the works of Brunt
(1952) and ngstrm (Geiger 1965, Matveev 1969), Fdn has been parameterized from
standard meteorological observations using its empirical dependence on cloud
amount and on the temperature and humidity of the atmospheric surface layer. In
these parameterizations, the incoming longwave radiation is estimated from
Fdn = ε* (n, T , e) σ T 4 ,
where ε* is the effective longwave emittance of the atmosphere, a function of cloud
amount and air temperature (T) and vapor pressure (e) at a height of 2 m.
Many functional expressions for the effective emittance of the atmosphere have
been published. These are generally based on readily available observations and
contain empirical coefficients obtained with a variety of temporal averaging meth-
ods. We consider here some frequently used functions.
In Brunt's (1952) parameterization (e.g., Matveev 1969), ε* for a clear sky
depends only on the water vapor content of the atmosphere and is described by
ε* = aB + bB e 1/2 ,
where e is the vapor pressure in millibars, and aB and bB are empirical coefficients.
On the basis of observations in middle latitudes, Brunt found aB = 0.526 and
bB = 0.065. These coefficients, however, are not universal; Kondratyev (1969) shows
that their values change with the measurement site. Below we will show Marshu-
nova's (1961) confirmation of this site dependence.
Brunt deduced the longwave radiation balance (B = Fup Fdn) by introduc-
ing a cloud multiplier. That is, the longwave radiation balance in the presence
of clouds is
B = B0 (1 - cB n) ,
where n is fractional total cloud amount, B0
Table 6. The coefficients aM and bM
is the longwave radiation balance for a clear
for use in eq 9, derived by Marshu-
sky, and cB is the average weighting coeffi-
nova (1961) from observations at var-
ious Arctic stations.
above 60N, cB = 0.81 (Berliand 1956).
Marshunova's (1961) parameterization is
essentially an application of Brunt's meth-
NP-3, NP-4, 1954-1957
ods to meteorological conditions in the Arc-