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
4.1. Effect of an additional cirrus layer aboÕe the highest flight leÕel Fig. 4a compares the leg-averaged values of measured and computed upward and
downward solar broadband flux densities of the measurements on 12 March 1993. Calculations have been made by changing the optical thickness of the additional layer,
denoted by t above the highest leg. In Fig. 4b, the root mean square difference between
a
calculated and measured layer-average values are shown as a function of t .
a
The optical thickness of the additional layer should be determined to fit the measured radiation values at the highest flight level. We found a best match for t s 0.6. We
a
therefore select the model with this value of t to investigate the effect of the optical
a
properties of cirrus clouds on the flux density in the following subsections. 4.2. Effect of the background atmosphere
The optical thickness for molecular scattering and absorption has been calculated with the LOWTRAN 7 code based on the radiosonde data at 12:00 UT. No significant
difference was obtained when we used the radiosonde data at 06:00 UT instead. The optical thickness of stratospheric aerosol in the visible region of the spectrum is about
0.004 while after the eruption of Mount Pinatubo in 1991, it reached values of up to
Ž .
0.2–0.3 at the Mauna Loa Observatory Dutton et al., 1994 . It dropped to values of Ž
. about 0.05 21 months after the eruption Dutton et al., 1994 . We included into our
calculation stratospheric aerosols composed of spherical droplets of 75 sulfuric acid in water solution. Values of single scattering albedo and asymmetry factor of these
Ž .
particles has been taken from the compilation in WMO 1986 . In this sensitivity study, the value of t was decreased to 0.55 to obtain a total optical
a
thickness of cirrus cloud and the aerosol above the highest leg of 0.6. This means that part of the additional cirrus layer in the reference model is replaced by the stratospheric
aerosols. This inclusion of stratospheric aerosols showed little effect on the upward and downward flux densities. For example, the difference from the reference model is less
Ž . Ž
. Ž
. Ž
Fig. 4. a Comparison of upward dashed lines and downward solid lines flux density 12 March 1993, .
10:25–11:50, Location : 778N, 68E for each of the 10 measurement legs. Closed circles: measurements, error bars show the standard deviations. Open circles, triangles, and crosses, respectively, denote the calculation for
Ž . Ž .
the optical thickness of the additional layer t of 0.0, 0.6, and 1.4. b Root mean square difference of
a
calculation with the measurements as a function of the optical thickness of the additional layer.
than 0.8 and 0.2 Wm
y2
at 8500 and 3000 m, respectively. Thus, it is not very important to include the stratospheric aerosols explicitly in a radiative transfer model, when
particular attention is paid only to the radiation inside a cloud layer in the troposphere. 4.3. Effect of optical properties of cirrus cloud
We also examined the effects of errors in the measured cloud properties on calculated radiation fields.
Ž . The optical thickness of cirrus cloud t
was determined from the number density
c
and extinction coefficients of particles. We investigated the effect of t on the flux
c
density by modifying t values as shown in Fig. 5a, where original values of t are
c c
Ž . Fig. 5. a The effect of the optical thickness of cirrus cloud on the upward and downward flux density. Open
Ž . circles: optical thickness of cirrus t
is multiplied by 0.5, triangles: reference model, crosses: t is multiplied
c c
Ž . Ž .
Ž .
by 1.5. b The same as a , but for the effect of the single scattering albedo v . Open circles: single Ž
. scattering co-albedo 1y v
is multiplied by 0.5, triangles: reference model, crosses: single scattering Ž .
Ž . co-albedo is multiplied by 1.5. c The same as a , but for the effect of the phase function. Open circles:
reference model, triangles: phase functions are modified so that asymmetry factors of particles decrease by 5–11 depending on the size and wavelength.
Ž .
Ž .
multiplied by 0.5 open circles and by 1.5 crosses . The downward flux density is influenced by 10–15 Wm
y2
with this 50 change of t . The effect of t on the upward
c c
flux density is smaller than on the downward flux. It is obvious that the reflectance at each level increases with increasing values of t .
c
The single scattering properties of cirrus particles have been calculated only for the five wavelength intervals given in Section 3. The absorption by the cloud may be
underestimated or overestimated due to this simple treatment. To estimate the effect of Ž
. the single scattering albedo v
on the uprdownward flux density in the cirrus cloud, Ž
. the single scattering co-albedo 1 y v
of cirrus particles, has artificially been de- Ž
. Ž
. creased less absorption or increased more absorption by 50 of its value. The results
of this sensitivity study are shown in Fig. 5b by open circles and by crosses, respectively, for smaller and higher absorption. It can be seen that the effect of the
change of the single scattering co-albedo by 50 on the uprdownward flux density ranges only between values of 2 and 3 Wm
y2
. The shape of the phase function is one of the candidates to alter the flux density in
Ž .
Ž the cirrus cloud. Here, the phase functions are artificially modified, by P Q - y 1 y c
Ž ..
Ž .
P cos Q
P P Q
where the constant c is assumed to be 0.35. By this modification, Ž
. Ž
. P Q in the forward direction Q - 90 decreases while that in the backward direction
Ž .
Q 90 increases, resulting in a smaller asymmetry factor. The magnitude of the decrease of asymmetry factor depends on the shape of the single scattering phase
Ž .
function, P Q . It decreased by 5–11 for the single scattering phase functions used in our calculation. This effect of phase function shapes is shown in Fig. 5c. Difference
from the reference model is larger for the downward flux density than for the upward flux density. As expected, the reflectance at each leg increases by this modification.
4.4. Effect of temporal changes of the cloud field structure In the previous subsections, the cirrus cloud was assumed to be stable during the time
of measurements, which extended over a period of about 2 h. A sensitivity study was made for measurement and modeling error for that case. In this section, the flux density
Ž .
is compared separately in the upper part 6000–8500 m
and in the lower part Ž
. 3000–5500 m of the cirrus cloud layer, allowing the change of cloud properties in the
other part. Fig. 5a suggests, that a decrease of t shifts the flux density in the lower part
c
closer to the measurements. Fig. 6a shows the flux densities in the lower part. Here, values of t in the upper part
c
as well as of t are decreased by 50. Almost all calculated values are now within the
a
error bar. The root mean square difference from the measurements are shown in Fig. 6b Ž
. as a function of a multiplier for t
upper and t . If the optical thickness in the upper
c a
part decreases by 50, the RMS error in the lower part becomes about 3 Wm
y2
. Ž
. However, it is not clear that the decrease in t
upper is true or is it an effective
c
decrease due to the inhomogeneity of cirrus cloud discussed in the next subsection. Similar calculations were done by increasing t in the lower part as well as t
by 50.
c sc
The reflectance at 6000 m was 0.51 for the reference model while it was 0.57 for Ž
. measurement. By increasing t
lower and t by 50, it becomes 0.58. Furthermore,
c sc
Ž . Fig. 6. a The same as Fig.5a, but for the lower part of cirrus layer. Open circles: reference model, triangles:
Ž . t at the upper part and t are multiplied by 0.5. b Root mean square difference of calculation with the
c a
Ž .
measurements at the lower part as a function of multiplier for t upper and t .
c a
the calculated flux density is within the error bars except for the downward flux at 8500 m.
5. Monte Carlo simulations for spatially structured clouds
