sedimenting downward, but possibly also a reduction of their number density occurs due Ž
. to instabilities splintering and aggregation . Such time histories of size spectra have
Ž .
been simulated by Zhang et al. 1992 with a rather simple model considering terminal velocities, heating and cooling by radiation and sublimationrevaporation, respectively,
and also the effects of large scale lifting of cirrus layers. The upper parts of the cirrus cloud layer was horizontally quite inhomogeneous as
shown in the example in Fig. 2, where the effective radii, as derived from the measurements of the OAP-2D2-C probe alone, are plotted. In the lower layers with
relatively higher amounts of diffuse radiation, much smoother radiation fields could be observed.
Below the cirrus cloud layer, some cumuliform clouds were observed, which may also have contained ice particles but will be assumed in this study as water clouds.
3. Radiative transfer model
3.1. Single scattering by non-spherical ice particles Particles in cirrus are generally non-spherical with a dominant hexagonal base
structure. A thorough analysis of holographic images taken in mid-latitude cirrus Ž
. Krupp, 1993 show that single column shaped particles dominate the cold upper layers,
Ž .
whereas the structure becomes more complicated in lower warmer layers. The single scattering properties of ice crystals have been calculated using a ray-trac-
Ž .
ing technique and assuming hexagonal column shapes, as developed by Macke 1993 Ž
. and with the parameterization given by Mitchell and Arnott 1994 and Takano and Liou
Ž .
1989 . No preferred orientation of the particles is assumed. We consider here columns in only 14 size classes between 7 and 49 mm and in 13 size classes from 49 and 1720
mm length of the crystal, where the length has been related to the diameter by the
b
Ž .
relation d s a L with a s 0.7 and b s 1.0 for the smaller crystals L - 100 mm , and a s 0.07 and b s 0.5 for the larger ones.
Application of the ray-tracing technique complemented by Fraunhofer diffraction Ž
. theory, as described by Macke
1993 then provided the scattering functions and
coefficients for each size class. The optical thickness of cirrus cloud is derived from the measured number density and projected particle area by averaging along each leg. It
should be noted that the optical thickness for scattering of the model cirrus cloud is independent of the wavelength.
3.2. Multiple scattering calculations for a homogeneous layer Our first analysis uses arithmetic averages over all quantities measured within a
specific flight leg. We later discuss also the possible effects of horizontal inhomo- geneities within the cirrus.
The multiple scattering calculations have been made with a modified version of a Ž
. scheme, which was used by Masuda et al. 1995 , where the flux density of the radiation
in cirrus cloud is computed by the doubling–adding method for a plane-parallel model
as shown in Fig. 3. The number of discrete directions in the doubling–adding calcula- Ž
. tions is 11. The solar spectral data of the World Radiation Center Iqbal, 1983 has been
Ž .
adopted for the solar irradiance at the top of the atmosphere TOA . The atmospheric column has been divided in the vertical into 17 homogeneous layers, where optical
thickness for molecular scattering and absorbing constituents such as ozone, water Ž
. vapor, and oxygen are obtained from LOWTRAN7 code Kneizys et al., 1988 . From
the surface to 25 km, the input parameters for LOWTRAN7 were based on the nearest radiosonde data, which was released at 12:00 UT from the German research vessel ‘‘MS
Polarstern’’, which has been stationed within the permanent sea ice away from the island, whereas from 25 to 100 km, the standard LOWTRAN7 data for sub-arctic winter
model were used. The vertical profile of water vapor has been modified to obtain saturation within the cloud layer.
The ocean surface is simulated by multiple facets whose slopes have an isotropic Ž
. Gaussian distribution that depends on surface wind speed Cox and Munk, 1953 .
Reflection by the ocean surface is considered for the entire wavelength region, but radiation emerging from below the ocean surface at wavelengths larger than 0.6 mm is
neglected. The effect of whitecaps on the irradiance is accounted for by taking their reflectance to be 0.45 at all wavelengths. Indeed, most of the upward solar radiation
inside of the cirrus cloud field originates from the clouds below it and not from the ocean surface, whose reflectance is about 5 to 8.
For the water clouds below the cirrus, we used the asymmetry factor and single Ž
. scattering albedo of model Sc1 by Stephens 1978, 1979 . The phase function of water
cloud droplets is approximated by the Henyey–Greenstein function for the lower water
Fig. 3. Schematic of the atmosphere–cloud–ocean system as used in these numerical simulations.
cloud. The values of the asymmetry factor g vary with the wavelength. For example, g is 0.84, 0.80, and 0.78 for the wavelengths of 0.55, 1.05, and 1.89 mm, respectively. The
measurements showed that the reflectance, defined as the ratio of the upward flux density to the downward flux density, amounted to 0.38 at the lowest leg. To satisfy this
Ž .
condition, the optical thickness t of the Sc1 layer is chosen to be 3.8.
sc
Computations are carried out for 85 unequally spaced wavelength bands between 0.3 to 3.0 mm. The transmittance functions for absorption by water vapor in each band are
obtained by an exponential sum fitting procedure. The absorption by ozone, oxygen and carbon dioxide has been calculated according to the procedure by Braslau and Dave
Ž .
1973 . The computational accuracy has been compared with the results of a line-by-line Ž
. model Masuda et al., 1995 . The relatively small effect of the aerosol scattering and
absorption inside the clouds is neglected. The optical thickness of the cirrus layer was determined from the number density and
extinction coefficients of particles. The boundaries of layers and optical thickness of cirrus are shown in Table 1. The total optical thickness for scattering amounts to only
5.58 independent of the wavelength up to 3 mm. In this study, the optical properties of ice crystals have been calculated for five
wavelengths of 0.55, 0.77, 1.05, 1.29, and 1.89 mm. These ranges are assumed to be representative for the wavelength regions from 0.3 to 0.7 mm, 0.7 to 0.84 mm, 0.84 to
1.26 mm, 1.26 to 1.32 mm, and 1.32 to 3.00 mm, respectively. Ž
During the time of measurements in this cloud layer, the sun was very low zenith .
angle: 80.58 . The scattering and absorption in the stratosphere reduce the downward radiation by about 25 Wm
y2
, thus a total influx of about 195 Wm
y2
should be
Table 1 Boundaries of layers and optical thickness for radiative transfer calculation
Ž . Altitude m
Optical thickness of cirrus
stratocumulus 100,000
25,000 13,000
10,000 8518
0.16 8212
0.37 7602
0.43 6994
0.40 6385
0.62 5776
0.65 5167
0.74 4557
0.79 3948
0.90 3340
0.51 3035
2000 3.8
1000
observable at the uppermost flight level at 8520 m altitude, while the observed flux density at this level was only 127 Wm
y2
. We exclude in our considerations calibration errors, since both pyranometers were calibrated before and after the campaign, but we
cannot entirely guarantee their exact performance at high altitudes and high speeds Ž
y1
. about 150 to 180 ms
during their operation. The measurements of up and downward solar radiation showed a strong variability of about 10 Wm
y2
after correction for small aircraft attitude variations.
To explain further these differences, we assume an additional layer above the highest flight level, since cirrus particles were indeed sampled even at this highest leg, although
the experimenters observed by eye mostly cloud free air, and the extinction coefficient at Ž
. 8500 m was estimated as 0.6 Koch, 1996 . It is reasonable to add such an additional
cirrus cloud layer above that altitude, where we considered cirrus cloud particles up to an altitude of 10,000 m with the same size distribution as at the second level at 7910 m.
4. Comparison of measured and calculated flux densities
