Jökull - 01.12.1968, Page 30

where G0 is the insulation with a cloudless sky,
N the cloud cover and a, b and m numerical
constants which depend on the latitude. A
somewhat better approximation may be achiev-
ed with the use oí' relative duration o£ sun-
shine instead o£ cloud cover (Einarsson 1966).
The part of the solar radiation absorbed in
the water is G (1 — a) where a is the albedo.
The albedo for the direct radiation depends
on the sun’s altitude and the roughness of the
water surface. When the sun is high it is only
a few per cent. The albedo for diffuse radia-
tion is not so variable, for the sea surface it is
5—10% (Cox and Munk 1955). Monthly values
of the albedo for total insolation are given by
Budyko (1956). Some uncertainity in the esti-
mation of the albedo is not likely to lead to
serious errors as it is only a few per cent when
the insolation is substantial. For rivers in
canyons or narrow valleys topographical effects
on the sunshine must be taken into account
as Devik (1931) has done for the river Glomma
in Norway.
Terrestrial radiation
If the net radiation from the water surface
is not observed directly the heat loss by ter-
restrial radiation is calculated as the difference
between the radiation emitted by the water
surface and the back radiation from the atmos-
phere absorbed in the water. The flux of radia-
tion from the water surface is toTw4, where t,
is the emissivity, a the Stefan-Boltzmann con-
stant and Tw the water temperature in °K.
The flux of back radiation from the atmos-
phere is the diffcrence between the flux of
radiation from the ground ancl the net flux
of terrestrial radiation. If the temperature o£
the ground is assumed the same as the air
temperature the flux of radiation from the
ground is very nearly: oTa4, where Ta is the
air temperature in °K, as the ground acts al-
most like a black body. There are several em-
pirical formulas for the net flux of terrestrial
radiation with cloudless sky, of which the best
known are these by Ángström and Brunt.
Brunt’s formula is: oTa4 (1 — a — b \/e), where
a and b are numerical constants and e is the
vapour pressure of the water in the air near
the surface. The absorptivity of the water sur-
364 JÖKULL 18. ÁR
face is the same as the emissivity. With Brunt’s
formula for the net flux of terrestrial radia-
tion we thus obtain the following expression
for the rate of heat loss from a water surface
by terrestrial radiation with cloudless skv:
s = toTw4 — t (oTa4 — oTa4 (1 — a — b Ve))
= toTa4 (1 - a - b Ve) + %a (Tw4 - Ta4).
Under cloudy sky the back radiation from
the atmosphere increases and the heat loss is
reduced. The factor to (Tw4 — Ta4) is not
affected by the cloudiness as can be seen by
considering the extreme case that the net flux
from the ground were zero (or the flux from
the atmosphere were equal to the flux from
the ground), in that case the rate of heat loss
from the water surface would just be: X,a (Tw4
— Ta4); (Devik 1931).
Combining solar and terrestrial radiation the
rate of heat loss by radiation from a water
surface is:
si = Tw4 (1 — a — b Ve) ci
+ (Tw4 Ta4) — G0 (1 — a) c2. (20)
The cloudiness reductions, ci and c«, mav
eventually be the same for insolation and ter-
restrial radiation.
The coefficients a and b in Brunt’s formula
are somewhat different in the various sources
and possibly they differ from place to place.
When e is expressed in millibars the values are
for instance:
a = 0.61 and b = 0.05
according to Budyko (1956) and
a = 0.44 and b = 0.08
according to Haltiner and Martin (1957).
The absorptivity of water is very strong.
Devik (1931) used t = 0.95 and Dingman,
Weeks and Yen (1968) use t = 0.97 according
to newer observations.
It must be emphasized that the empirical
formulas for radiation are found in a statistical
way and represent mean values and can not be
expected to give accurate results for shorter
periods.