Ž .
Fig. 8. Comparison of the optical depth retrieved from lidar measurements at 532 nm and from the cloud albedo analysis using ARAT flux measurements for leg MA2.
comparable between the two. In the optically denser part of the cloud, values are in good Ž
. agreement about 20 or better . On the other hand, CODs retrieved from passive
measurements are larger in the other parts of the cloud. Most likely, this discrepancy is due to a combination of effects: cloud inhomogeneity and the fact that the lidar and the
pyranometers have different fields of view. It could also be related to the simplified
Ž Ž ..
COD calculation used i.e. Eq. 6 . This point will be further investigated.
5. Retrieval of cloud microphysics
Lidar measurements show the existence of small-scale structures, which appear to be associated with convective updrafts in the cloud layer. In such updrafts, the vertical
distribution of the liquid water content in water vapor-saturated air follows an adiabatic increase, which can be assumed to be linear with altitude over a few hundred meters
Ž .
above the cloud base Brenguier, 1991 . The cloud extinction coefficient a , is defined as the extinction cross-section averaged over the whole droplet size distribution
`
2
a z ,l s p Q
r , l r N r d r , 7
Ž .
Ž .
Ž . Ž .
H
e
where Q is the extinction efficiency. From Mie theory calculations, this factor depends
e
on the size parameter x s 2p rrl, which is a function of droplet radius r but also of wavelength l. Q is maximal for x s 6 at visible wavelengths. At x values larger than
e
10, the extinction efficiency decreases to the asymptotic value Q s 2, with damped
e
Ž .
oscillations see for example Pinnick et al., 1983 . Assuming the droplet distributions in the sampled clouds under study have a modal radius larger than a few microns, we can
consider that at the measurement wavelength l s 532 nm, Q can be approximated by
e
Ž .
an analytical expression Pinnick et al, 1983 , Q
x s 2 1 q x
y2 r3
, 8
Ž . Ž
. Ž .
e
and calculated for the value of x corresponding to the mode radius of the distribution. Q varies between 2.23 and 2.08 for mean droplet radius ranging between 2 and 10 mm.
e
This range encompasses most of the cases of droplet distributions in warm clouds Ž
. Bower et al., 1994 , assuming a factor 0.6 between the mean and effective radius, as
estimated from a gamma droplet size distribution. Ž .
The extinction coefficient a given by Eq. 7 is thus proportional to the second order moment of the droplet size distribution. Neglecting high order fluctuations, assumed to
be of second order at the measurement scale, a can be simply written as the ratio of the Ž
. liquid water content and the effective radius Pinnick et al., 1983 . In the case of an
adiabatic increase of liquid water content with height, the effective radius at cloud top Ž
. can be written as PBB
3 C z
w t
r s
Q .
9
Ž .
e Ž z .
e
t
4r a z
Ž .
t
As the value of Q depends on the average droplet radius, a correction may be applied
e
Ž . Ž
. using Eq. 8 . However, this correction remains small less than 5 and in this paper,
we have used the average value Q s 2.16, close to the value Q s 2.2, obtained by
e e
PBB. Should drizzle occur near cloud top, the extinction coefficient can be written as the sum of two independent contributions: extinction due to cloud droplets and to drizzle
as there is no overlapping between their size distributions. The drizzle liquid water content being smaller than the cloud one, and the effective radius of drizzle being more
than 10 times larger than the droplet one, the extinction coefficient is mainly sensitive to the cloud droplet contribution.
Ž The temperature at cloud base height z
being close to 28C as obtained from
b
. soundings , the coefficient C
defining the adiabatic increase of liquid water is equal to
w y3
y4
Ž .
1.5 10 g m
Brenguier, 1991 . The maximum cloud geometrical depth is z s 520
t
Ž .
y1
m Fig. 3 and the maximum extinction coefficient at the cloud top is about 0.17 m Ž
. Fig. 6 . Thus, we obtain r s 7.3 mm at cloud top in the strongest updraft. The error is
e
estimated to be about 15 from results plotted in Fig. 6, and from the approximation in Q . This value is close to the value r s 7 mm measured in situ by PBB. Assuming
e e
effective radius is weakly varying close to cloud top, extinction fluctuations at cloud top reported in Figs. 5 and 6 can be mostly attributed to variations in liquid water content.
Ž .
Following PBB, the droplet concentration N z at cloud top is
t 3
2 3
K a 1
16 K r
a
t t
N z s
s ,
10
Ž .
Ž .
t 2
2
p Q r
z 9
Q p C z
Ž .
e e
t e
w t
where K is a size parameter depending on the ratio of the volume to effective radius Ž
. see Martin et al., 1994 and PBB for more details taken equal to 0.8 for marine
stratocumulus, and r is the liquid water density taken equal to 10
6
g m
y3
. Using Eq. Ž
. 10 , the number of droplets deduced from lidar measurements in updrafts at cloud top is
Ž .
y3
about N z s 390 cm
, meaning that a fairly large number of nuclei have been
t
activated. This value is also in good agreement with in situ measurements of PBB. Ž
. Ž
. Ž
. According to Eq. 10 , uncertainties are large 30 because N z
is proportional to
t
the third power of the extinction coefficient at the cloud top. The vertical distribution of the cloud droplet concentration can be considered, to the
Ž Ž .
Ž .
. Ž
. first order, as constant with height
N z s N z
, see PBB . However, N z
t
depends on the horizontal location in the cloud, and differences may be observed Ž
. between updrafts and downdrafts. From Eq.
10 and cloud droplet concentration
obtained at cloud top, the vertical profile of the extinction coefficient in an adiabatic updraft is given as a first approximation by the expression
2r3 y3
w x
a z s 2.87 = 10
z ,
11
Ž .
Ž .
Ž .
where z is in meters and a is in inverse meters. The adiabatic limit given by Eq. 11
Ž . Ž
y3
. has been plotted on Fig. 6. Recall that the coefficient in Eq. 11
i.e. 2.87 = 10 is
calculated using the largest lidar-derived extinction value and cloud thickness value Ž
. Ž
. observed i.e. for the strongest updraft observed by lidar . Eq. 11 should provide an
estimate of the maximum extinction coefficient to be expected as a function of altitude. The fact that most of the observations are found systematically to the left of the dotted
Ž curve instead of falling on the curve as expected if the increase in liquid water was
. adiabatic is consistent with the fact that, in Fig. 6, we have not selected data from
updrafts only and is compatible with the uncertainty related to the lidar-derived extinctions.
Ž .
Inversely, the theoretical limit provided by Eq. 11 can be used to define the cloud Ž
. geometrical thickness or cloud base altitude, indifferently , if the cloud top altitude,
Ž droplet concentration and adiabatic liquid water content are known as was done in Fig.
. 6 . This inverse approach can be of interest in the context of space-borne studies. A
scatter plot, such as the one shown in Fig. 6, can be used to retrieve a first guess for the Ž
. Ž .
altitude of the extinction at the top of the most energetic updraft s .
6. Conclusion
