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
As can be seen in Fig. 2, the observed cirrus cloud is horizontally very inhomoge- neous along the flight path at 8520 m. Similar inhomogeneities have also been found at
all other altitudes. To investigate the effect of such inhomogeneities on the radiative flux densities,
preliminary analyses were made using a Monte Carlo code, with a version of the model Ž
. described in Kobayashi 1988, 1989 , which includes all cloud–cloud interactions and
also the mutual shadowing. In this method, the distance a photon travels until it interacts with a cloud particle is
determined by the volume scattering coefficient. The photon–particle interaction follows Ž
. the description of Cashwell and Everett
1959 , i.e., the photon path lengths are calculated at the boundaries of the clouds. At the boundary, however, photons continue
to travel in the same direction until they enter a new cloud, etc. The path length inside the new cloud is again determined by the volume scattering coefficient and the fraction
of the path length remaining from the cloud that the photon has previously exited. The photon path, therefore, can be traced accurately even if the photon enters the edge of a
cloud. The redirection of the photon takes place at the end of its trajectory and is determined by the phase function in the usual way. Radiative parameters are prescribed
and kept constant within each cloud.
The purpose of the Monte Carlo calculation is not to fit the calculated flux density to the measured one, but to investigate the effect of inhomogeneity of cirrus on the flux
density. The atmosphere and surface model, therefore, are much simpler. We use here a one-layer cirrus cloud model, where phase functions are vertically averaged considering
the optical thickness of each layer. The top and bottom of the cirrus layer are located at 8518 and 3035 m, respectively. Both additional cirrus and stratocumulus layers intro-
duced in the previous section are omitted. The atmospheric molecules are also neglected. As mentioned in Section 3, the reflectance at the lowest leg was 0.38. In the broken
cloud-surface model, the angular surface reflection is assumed to be Lambertian with an albedo of 0.38.
Two kinds of cloud geometry are considered. The first is a plane-parallel model with Ž .
an optical thickness t of 5.58, analogous to the studies above. The other contains
c
clouds of two different optical thicknesses: the thicker cuboidal clouds are arranged on a Ž
. regular grid surrounded by the thinner clouds Fig. 7 . The fraction of the thicker area is
w Ž
. Ž
.x assumed to be 0.5. The optical thickness of thick and thin areas t
thick and t thin
c c
are 11.16 and 0.0 for the type A and 8.37 and 2.79 for the type B, respectively. Average optical thicknesses for both types are the same as for the plane-parallel model. Solar
Ž .
azimuth angles w measured from x axis are 08 and 458 whereas solar zenith angle is
fixed to be 80.58. Fig. 8a shows the upward, downward, and absorbed flux densities calculated by the
Monte Carlo method. Except for the type A with w s 08, the flux densities for regular array models are very similar to those for the plane-parallel model. For this type A with
w s 08, parts of the incident photons reach directly the bottom of the cloud through the open area, which results in an enhancement of the downward flux density at the bottom
of the cloud. For the other cases, the incident photons hit the top or the side of cloud due Ž
. to large solar zenith angle even if t
thin is zero. It should be noted that the results in
c
Fig. 8a are only for the direct solar radiation on the cloud layer with the incident zenith angle of 80.58.
Fig. 7. Cloud field model for Monte Carlo calculation. Thick and thin cloud areas are shown by meshes and hatches, respectively.
Ž . Fig. 8. a Upward, downward, and absorbed flux density for the cirrus cloud model with Lambertian surface
calculated by the Monte Carlo method. P–P and Reg denote the plane parallel and regular array models, respectively. For plane-parallel model, t is 5.58. Fraction of the thick cloud area for regular array model is
c
Ž .
Ž .
0.5. t thick and t
thin are 11.16 and 0.0 for type A and 8.37 and 2.79 for type B, respectively. Solar
c c
Ž . zenith angle is 80.58. Solar azimuth angles are measured from x-axis are 08 and 458. b Upward and
Ž .
downward flux density at the top and bottom of the cirrus cloud layer for P–P and Reg A:0 models.
In an actual situation, the diffused photons are also included in the downward radiation at the top of the cloud. By considering the effect of the diffuse radiation, the
above results may be changed to some extent. Fig. 8b shows the upward and downward flux density at the top and bottom of the plane-parallel model and the regular array
Ž .
model type A with w s 08. It is seen that the slopes for the regular array model Ž
. Ž
. dashed lines are steeper than those for the plane-parallel model solid lines . This trend
Ž .
is similar to the case where t is decreased by 50 Fig. 5a where the slopes for the
c
Ž .
decreased optical thickness model open circles in Fig. 5a are steeper than those for the Ž
. reference model open triangles in Fig. 5a . Thus, the upward and downward flux
density for the broken cloud model with the same average optical thickness of plane-parallel model shows similar trend to those for the plane-parallel model with
smaller optical thickness.
Radiative transfer calculation using a more realistic multi-layered broken cloud model is necessary to further examine the effect of cloud inhomogeneity on the radiation field.
6. Summary
