1. Introduction
Ž .
In the planetary boundary layer PBL the solar energy received by the surface primes mechanisms of heat exchange between the surface and the atmosphere causing the
development of different physical quantity fluxes. Such fluxes mainly have a turbulent character and show irregular patterns characterised by random variations of physical
quantities in space and time.
To facilitate radiation studies at the surface, the incoming and reflected flux are Ž .
Ž . normally divided into a short-wave radiation flux, b solar radiation, confined to the
Ž . spectral range between 0.3 and 2.2 mm, and c long-wave radiation flux, originated by
the sky and the surface and delimited between 6.8 and 100 mm. The 2.2–6.8 mm. interval includes both short- and long-wave, which is, however, less than 5 of the total
Ž .
radiation Geiger, 1966 . The radiation balance at the surface is described by the following relation:
Q s S y S q L y L
n i
o i
o
where Q is the net radiation flux, S and S the short-wave radiation flux coming from
n i
o
the sky and that reflected by the surface, L and L those in the long-wave coming from
i o
the atmosphere and emitted by the surface. The fluxes in the PBL are regulated by the diurnal cycle of the surface energy
balance: Q y G q H q E s 0
n
where G is the vertical heat flux into the surface, while H and E represent the latent and sensible heat flux in the atmosphere.
The data-set analysed in this work was collected at an Antarctic site where the surface is permanently covered by snow. The consequent high albedo values contribute
to make it the coldest continent of the earth. The annual net radiation flux above the Antarctic surface is negative since in winter the visible radiation is practically absent,
while in summer the high albedo reduces the radiation absorbed by the surface. The annual deficit in radiative energy must be balanced by the transfer of sensible and latent
heat between the surface and the PBL.
Different systematic parameterisations have been proposed to model the incoming Ž .
Ž .
short-wave radiation S . Kasten and Czeplak 1980 proposed the following formula
i
obtained for clear-sky conditions: S s a sin C y a
Ž .
i 1
2
where C is the solar elevation angle, a and a are experimental coefficients, which are
1 2
strongly connected with the local conduction parameters, such as the altitude and the air turbidity.
Ž .
Another parameterisation was made by Haurwitz 1945 :
b
S s asin C exp ycrsin C
Ž .
Ž .
i
Again a, b and c are experimental coefficients. In this equation, the exponential term represents the reduction factor due to the local conditions of atmospheric turbulence.
The cloud cover has a strong influence in the long and short-wave radiation fluxes. Both the incoming and reflected visible radiation show a marked dependence on the
cloud cover variation. On the contrary, the long-wave radiation emitted by the surface does not depend on the cloud cover. In Antarctica changes in cloud amount can
determine considerable variations in albedo. In fact, on snow-covered surfaces the reflected radiation strongly depends on the incident direction of the light so that the
presence of clouds can modify the ratio between direct and indirect radiation. The incoming short-wave flux is composed of two terms: a direct flux with a single angular
direction and a component of diffused light due to scattering in the atmosphere which has, on the average, a smaller incident angle with the surface and is thus more reflected.
Both are partially reflected by the surface and therefore depend on several factors such as the kind of soil, the surface humidity and its geometry.
Ž .
Iqbal 1983 proposed a parameterisation of albedo with the elevation angle:
X X
X
a C s a q 1 y a exp y0.1C y 1 y a r2
Ž .
Ž .
Ž .
where a
X
is the albedo at the maximum solar elevation angle and C the solar elevation angle.
Another parameterisation obtained from a set of data recorded at the South Pole was Ž
. Ž
proposed by Carroll and Fitch 1981 . In clear-sky conditions N F 0.25, with N cloud .
cover index and for C F 208 their relation is: a C s 0.98 y 0.0075C
Ž .
so that for high elevation angles the albedo reaches a minimum value of 0.83. Ž
. Ž
. For C F 188 in overcast conditions N G 0.7 , Carroll and Fitch’s 1981 parameteri-
sation changes in: a C s 0.93 y 0.0050C
Ž .
so that for C 188, the albedo value is 0.84. The incoming long-wave radiation is the sum of the radiation emitted and transmitted
by the various layers of the atmosphere. However, it is possible to consider the entire atmosphere as a grey-body with an effective emissivity ´ . Thus, the incoming long-wave
a
Ž .
flux becomes Paltridge and Platt, 1976 : L s ´ s T
4 i
a a
where s is the Stefan–Boltzmann constant and T the air temperature near the surface.
a
Ž .
Kondratyev 1969 was the first to emphasise the connection between ´ and the
a
vertical profile of humidity and temperature from the surface to 1–2 km a.m.s.l. For convenience, relationships were formulated to relate ´ exclusively to the values
a
of humidity and temperature near the surface. Ž
. The Brunt 1932 releationship makes ´ dependent on the square root of the water
a
Ž .
vapour pressure according to the heat conduction theory. Also in Brutsaert 1975 , the effective emissivity is determined only by the water vapour pressure. There are
equations that use both air temperature and water vapour pressure, as is the case of Idso
Ž .
Ž .
1983 , or the dew-point temperature, like Berdhal and Martin 1984 . Another class of formulas utilises only the air temperature T at the surface to obtain the incoming
a
Ž .
long-wave radiation. An example is Swinbank’s 1963 formula: L s 5.31 = 10
y1 4
T
6 i
a
Ž .
The Idso and Jackson 1969 expression is:
2 y4
4
L s 1 y 0.261exp y7.77 = 10 273 y T
s T
Ž .
½ 5
i a
a
Ž .
and the Deacon 1970 formula: L s L
y 0.035 Zr1000 s T
4
Ž .
i Swin
a
Ž .
where L is the relation proposed by Swinbank 1963 and Z the height of the station
Sw in
in meters a.m.s.l. Ž .
These expressions are based on the following facts: 1 a shallow surface layer with a temperature close to T contains sufficient water vapour and CO to provide essentially
a 2
Ž . full radiation in the H O and CO wave-band, and 2 the atmospheric water vapour
2 2
content depends on its temperature. Several studies showed that the diurnal trend of measured L differs from this
i
parameterisation, according to a well-defined distribution: during night-time the Swin- Ž
. bank
1963 formula underestimates the real atmospheric emission by a constant
amount, while overestimates it the rest of the day with a maximum in the early afternoon.
Ž .
Ž .
A similar trend was also found by Arnfield 1979 and Idso 1972 , through the Ž
. analysis of the deviation from Idso and Jackson’s 1969 formula. The deviation always
occurs during two well-defined periods of the day: during the first, which lasts almost 10 h, both relationships underestimate atmospheric emission, while in the second, corre-
sponding to the early afternoon, they overestimate it.
Ž .
Paltridge 1970 realised that the air temperature at the surface was only a convenient approximation of the atmospheric temperature, and that it would have been more correct
to use the emission barycenter temperature located at 200–300 m above the surface. In fact, during the night conditions when a thermal inversion generally take place, the
surface air temperature is lower than the long-wave emission temperature of the sky; during the diurnal hours, when the maximum lapse-rate values are observed, the
opposite occurs.
Different empirical relationships have been developed to relate the daily global irradiance with a cloud cover index.
In 1980 Kasten and Czeplak proposed the relationship: S s S
1 q b N
b 2
Ž .
i ifor
1
where N is the cloud cover index previously introduced, b and b empirical coeffi-
1 2
cients that depend on the climate of the site, and S and S the incoming short-wave
i ifor
radiation and the expected radiation flux for clear sky conditions, respectively. Using
Table 1 Coefficients of cloud type k
i
Cloud type k
i
Cirrus 0.04
Cirrostratus 0.08
Altocumulus 0.16
Altostratus 0.20
Cumulus 0.20
Stratocumulus 0.22
Fog 0.25
data recorded in Hamburg over a 10-year period, they obtained the following mean values:
b s y0.75
1
b s 3.4
2
Ž .
Concerning long-wave radiation, Arnfield 1979 proposed a relationship to correlate the sky radiation flux with the amount and type of cloud cover:
´ s ´ 1 q Ýk c c
Ž .
a i
i
In this formula ´ represents the atmospheric emissivity in clear sky condition, k is
i
the ith coefficient related to the individual cloud type i, c the total cloud cover fraction and c the fractional cloud amount of any cloud type. Table 1 shows the k values used
i i
Ž .
by Arnfield 1979 . Ž
. Paltridge 1970 tried to correlate the increase in infrared radiation coming from the
sky with a cloud cover index by analysing a set of data recorded in Aspendale, Australia. He found that any 1r10 of increase in the cloud cover index corresponds to a
mean increase of 0.6 mW cm
y2
in the long-wave radiation flux. This work aims at verifying the admissibility of the daily mean values of infrared
radiation from the sky at the Antarctic measuring site with respect to the values of the Ž
. Ž
. Ž
. Swinbank 1963 , Deacon 1970 and Idso and Jackson 1969 relationships. It shows
how to correlate the incoming solar radiation with the deviation of L from the expected
i
Ž .
Swinbank’s 1963 values for a clear day. It was possible to relate this deviation to the cloud cover amount according to the cloud height using the formula of Kasten and
Ž .
Czeplak 1980 . Finally, a particular geometric surface is modelled to explain the Ž
. deviation from Iqbal’s 1983 formulation of the daily pattern of the albedo values, with
regard to the solar azimuth and elevation angle.
2. Materials and methods
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