growth. For comparison, the cloud base conditions are taken the same as that used in Ž
. Ž
. Twomey 1959, 1977 ; p s 800 hPa, T s 108C, and RH s 100. This leads to the
b b
b
starting conditions of haze growth process below the cloud base: p s 809.21 hPa, Ž
. T s 10.938C, and RH s 95.
Then for prescribed size groups of dry solute and properties of the corresponding solution droplets considering van’t Hoff factor as a function of molality, the critical
radius and the critical supersaturation are evaluated to obtain supersaturation change and haze size evolution below the cloud, which will provide the initial condition for
nucleation and growth of droplets above the base. The nucleation of droplets takes place and droplet size distribution forms as
n y n
jq 1 jy1
f r s
. 36
Ž . Ž .
j
2 r y r
Ž .
j jy1
The subsequent estimation of droplet growth process involves estimation of temperature, supersaturation and mixing ratios of water vapor and liquid water which leads to the
maximum supersaturation. For the Maxwellian droplet growth system, our model results for cloud supersatura-
Ž .
tion are checked with that of Rogers and Yau 1989 . In the DK droplet growth system, Ž
. the predictions of our model are compared to those of the model by Young 1993 which
uses the droplet growth equation of Fuchs. The output of the present model agrees with that of others when compared under the same condition.
3. Results and discussion
3.1. Analytical formula of Twomey Ž
. Twomey 1959 derived an analytical formula to predict the maximum supersatura-
tion in dew point elevation in 8C, s at a fixed cloud base condition, p s 800 hPa,
ma x b
T s 108C,
b 1
y3 3r2
1.63 = 10 w
kq 2 u
s s
, 37
Ž .
ma x
CkB kr2, 3r2
Ž .
Ž .
where B kr2, 3r2 the complete beta function which can be expressed by gamma function G
G 3r2 P G kr2
Ž .
Ž .
B kr2, 3r2 s .
Ž .
G 3r2 q kr2
Ž .
Ž .
where C and k are parameters of CCN activity spectrum in Eq. 15 . Due to the Ž
. Ž
. Ž
. different nature of s
, S-1 in Eq. 15 cannot be replaced with Eq. 37 .
ma x
Ž .
Since Eq. 37 is obtained at the specific cloud base conditions, it does not include Ž
. the variation. In order to overcome this shortcoming, Twomey 1977 revised it, so that
1 3r2
3r2
2 A w
kq 2 u
S y 1 s
. 38
Ž .
Ž .
max , TW 1r2
BG CkB kr2, 3r2
Ž .
1
Ž .
Eq. 38
includes the temperature and pressure variation at cloud base through parameters A, B, and G;
gL g
c
A s y
,
2
R T R T c
a b
v b
p
1 ´ L
2 c
B s 4pr G q
,
L
ž
p c p T
s p
b b
2
and 1
1 G s
, L
L 1
r
c c
L
y 1 q
ž
R T KT
r D
v b
b s`
where g is the acceleration of gravity. Ž
. In the following, we will use Eq. 37 to numerically evaluate the accuracy of the
work. 3.2. Analytical formula of Fukuta–Xu
Ž .
Fukuta and Xu 1996
employed a different treatment within the condition of Ž
. Maxwellian droplet growth, and obtained an analytical formula prognostic for cloud
maximum supersaturation:
1 3
3 X
1
A
Ž .
y 2 kq2
Ž .
2 kq2 Ž
. 2 kq2
kq 2
S y 1 s
2 k q 3 F C
w ,
39
Ž .
Ž .
Ž .
max , FX X
u
G where
´ L g 1
T
c b
X
A s y
,
2
c ´ L
R T
p c
a b
´ L
2
p
c b
X
B s q
,
2
´ e c R T
bs p
a b
4pr B
X L
X
F s ,
3r
a
4p B
X X
G s .
2
L 1
c
r q
a 2
ž
r D KR T
s` v `
1
For p s800 hPa, T s108C, 2 A
3r 2
B
y1
G
1r 2
s6.9=10
y2
is used erroneously for percentage supersatu-
b b
Ž .
y6
ration in Pruppacher and Klett 1978 . It has to be replaced with 7.8=10 for fractional supersaturation.
2
In the original derivation, the bracketed term in B is missing, and values for B and G in Table 4.4 Ž
. Twomey, 1977 are in error.
Ž .
Ž .
Ž .
Then, from Eqs. 15 and 39 , the maximum activated CCN or droplet number concentration in cloud is
k 3
1 3 k
X
A
Ž .
2 kq2 Ž
. 2 kq2
Ž .
kq2 X
2 kq2
N s 2 k q 3 F
C w
. 40
Ž .
Ž .
X u
G There is no need to mention the advantage of analytical solutions for their clarity in
the relationship among the variables and readiness for use. However, they are restricted here within the Maxwellian condition of droplet growth. In addition, the solutions do not
incorporate the Kelvin effect of droplet curvature and the Raoult effect of solute and start with zero size after activation. The output of the present cloud model can now be
used to compare with those of analytical solutions and their limits of error can be assessed or correction factor can be obtained for their practical use.
3.3. Cloud supersaturation predicted by present model In the present study of microphysical cloud modeling, we take p s 809.21 hPa,
Ž .
T s 10.98C, and RH s 95. This selection leads to the cloud base forming at Ž
. p s 800 hPa, T s 108C, and RH s 100, so that we can compare the model output
b b
b
with analytical solutions under the same cloud base conditions. In the activation and growth phase, haze particles in 98 groups begin to grow after
Ž .
the cloud base the smallest two groups are too small to activate , and the radius of
y3
Ž .
solution droplets of the 98 groups ranges from 2.9 = 10 to 5.6 mm at S-1 s 0. As
the parcel continues to rise, cloud supersaturation increases so that more and more droplets become activated. In the meantime, droplets become larger and larger through
condensation growth. The resultant moisture removal and latent heat addition cause increases of cloud supersaturation to be slower. The interaction of these two opposing
processes results in the evolution of supersaturation to finally reach a point of their balance, and the maximum supersaturation is achieved. Corresponding to the supersatu-
ration, the maximum number of CCN is activated. Afterwards, as the supersaturation decreases due to its depletion by condensation, but the droplet number concentration
remains nearly constant without further activation of CCN.
Fig. 1a and b show the supersaturation change in clouds predicted by the present cloud model with a given constant updraft, w s 2 mrs. Fig. 1a is for a continental
u
Ž .
0.5
Ž .
0.7
cloud with N s 600 S-1 and Fig. 1b for a maritime cloud with N s 100 S-1
Ž .
Twomey and Wojciechowski, 1969 . Cloud base conditions for both figures are the Ž
. same: p s 900 hPa, T s 158C, and S-1 s 0. In these figures, the solid line shows
b b
b
the result predicted by using the DK equation and the dashed line the result using the Ž
Maxwellian equation. In the DK equations, a s 1 and b s 0.03 are used Hagen et al., .
1989 . The maximum supersaturation of the latter is about 1.5–2.0 times higher than that of the former.
Fig. 2a and b illustrate the relationship between the maximum cloud supersaturation and updraft velocity in clouds, predicted by the present model for DK theory with
varying level of b and by the Maxwellian theory. Fig. 2a is for continental clouds, and Ž
Fig. 2b for maritime clouds. As we have seen before Fukuta, 1993; Fukuta and Xu,
Ž . Fig. 1. a Supersaturation change in continental cloud predicted by the present cloud model. The dashed line
is for Maxwellian droplet theory, and the solid line for DK theory with a s1 and b s 0.03. In both cases, Ž
.
0.5
Ž . N s600 S-1
, and p s900 hPa and T s158C, w s 2 mrs. b Supersaturation change in maritime cloud
b b
u
Ž . Ž
.
0.7
predicted by the present cloud model. Conditions are the same as in a , except for N s100 S-1 .
. 1996 , these figures show that a smaller b results in a markedly larger supersaturation
maximum, and use of the Maxwellian equation results in the lowest supersaturation maximum. It reveals that the ratio of supersaturation maximum between the DK model
with a s 1 and b s 0.03, and the Maxwellian is between 1.5–3.5.
3.4. Comparison of the maximum cloud supersaturation predicted by the present cloud model and those by the analytical formulas
In order to evaluate the accuracy of the analytical formulas of Twomey and of Fukuta–Xu, it is necessary to compare their predictions of the maximum supersaturation
with those of the present model.
Ž . Fig. 2. a Comparison of the maximum supersaturation in continental clouds predicted by the present cloud
model with Maxwellian and diffusion-kinetic droplet growth theories for different condensation coefficient as Ž
.
0.7
Ž . a function of updraft velocity. T s158C, p s900 hPa, a s1, and N s 500 S-1
. b Comparison of the
b b
maximum supersaturation in maritime clouds predicted by the present model with Maxwellian and diffusion- Ž .
kinetic droplet growth theories for different condensation coefficient. Other conditions are the same as in a , Ž
.
0.5
except N s100 S-1 .
Ž .
Ž As Twomey 1959 is far more widely cited and used than his later work Twomey,
. Ž
. 1977 , we will use the former formula, Eq. 37 , despite its predictions being smaller
than those of the latter. As pointed out above, supersaturation should be a dimensionless Ž
. number. So, we will convert the output of Eq. 37 in the dew point temperature, T , in
d
Ž .
8C into supersaturation a dimensionless number by e T
Ž .
s d
S y 1 s
y 1, 41
Ž .
Ž .
max
e T
Ž .
s b
where T is the temperature at cloud base. Here we let T s 108C, and
b b
17.67 T
b
e T s 6.112 exp
,
Ž .
s b
ž
T q 243.5
b
17.67 T
d
e T s 6.112 exp
,
Ž .
s d
ž
T q 243.5
d
Ž . Ž
. where e T
and e T are in hPa.
s d
s b
Ž .
Ž .
Ž .
The choice of C and k in Eqs. 37 and 40 follow Twomey 1959 , or
1r3
N s 125 S y 1
Ž .
1
maritime ,
Ž .
1r4
5
N s 160 S y 1
Ž .
2 2r5
N s 2000 S y 1 continental .
Ž .
Ž .
3
Ž .
Twomey and Wojciechowski 1969 also determined through observations that C s 100 cm
y3
, k s 0.7 for maritime clouds and C s 600 cm
y3
, k s 0.5 for continental clouds. Ž
However, in cloud modeling, some others Clark, 1974; Grabowski, 1989; Chen,
. 1994a,b use
0 .5
N s 100 S y 1 maritime
Ž .
Ž .
0 .7
N s 500 S y 1 continental
Ž .
Ž .
Ž .
Ž .
In the present model and in Eqs. 37 and 39 , T s 283.15 K and p s 800 hPa are
b b
taken. Figs. 3–5 compare the maximum supersaturation predicted for maritime clouds using
Ž Ž
. Ž
.. analytical formulas Eqs. 37 and 39
and the present cloud model. Fig. 3 shows the
Ž .
1r 3
Fig. 3. Comparison of the maximum supersaturation in clouds for maritime CCN spectrum, N s125 S-1 .
Supersaturation predicted by the present cloud model and Twomey’s and Fukuta–Xu’s analytical formulas are given, as a function of updraft velocity, all using the Maxwellian droplet growth theory. p s800 hPa and
b
T s108C.
b
Ž .
1r 4
Fig. 4. Comparison of the maximum supersaturation in clouds for maritime spectrum, N s160 S-1 ,
predicted by the present cloud model and analytical formulas of Twomey and Fukuta–Xu with the Maxwellian droplet growth theory. Other conditions are the same as in Fig. 3.
result using maritime CCN activity spectrum with C s 125 cm
y3
and k s 1r3. Fig. 4 gives the result with another maritime CCN activity spectrum with C s 160 cm
y3
and k s 1r4. Fig. 5 is for the typical maritime CCN spectrum in which C s 100 cm
y3
and k s 0.5. In these three figures, both analytical formulas show the supersaturation
maximum discrepancy with the cloud model. In Fig. 3, the prediction of Fukuta–Xu is about 2.3 times larger than that by the cloud model, and that of Twomey is about 1.4
times. In Fig. 4, the maximum supersaturation of the Fukuta–Xu formula is about 1.9 times larger than that of the cloud model, and that of the Twomey formula about 1.4
Ž .
0.5
Fig. 5. Comparison of the maximum supersaturation in clouds for typical maritime spectrum, N s100 S-1 ,
predicted by the present cloud model and Twomey’s and Fukuta–Xu’s analytical formulas as a function of updraft velocity in the Maxwellian droplet growth system. Other conditions are the same as in Fig. 3.
Fig. 6. Comparison of the maximum supersaturation in continental clouds for continental spectrum, N s Ž
.
0.4
2000 S-1 , predicted by the present cloud model and Twomey’s and Fukuta–Xu’s formulas in the
Maxwellian droplet growth system. Other conditions are the same as in Fig. 3.
times larger than that of the cloud model. In Fig. 5, the prediction of Fukuta–Xu is about 2.9 times larger than that of the cloud model, and that of Twomey about 1.3 times.
The gap between the two analytical equations of the Maxwellian growth system appear Ž .
to be due to the difference for r term handling in Eq. 3 . Fukuta–Xu does not follow the actual history of CCN-droplet growth and tends to underestimate the term making
the supersaturation maximum larger relative to Twomey’s treatment. Figs. 6 and 7 show the predicted maximum supersaturation as a function of updraft
velocity for continental clouds. In Fig. 6, C s 2000 cm
y3
, k s 0.4, and in Fig. 7, C s 500 cm
y3
, k s 0.7 are used. Each compares the results of two analytical solutions
Ž .
0.7
Fig. 7. Same as in Fig. 3, except for the typical continental spectrum, N s 500 S-1 , in the Maxwellian
droplet growth system.
with that of the cloud model. In Fig. 6, the predicted maximum supersaturation of Fukuta–Xu is about 2.5 times larger than those predicted by the cloud model, and the
values of Twomey about 1.5 times. In Fig. 7, Fukuta–Xu gives the maximum supersatu- ration about 3.5 times larger than the cloud model does, and Twomey’s formula about
1.5 times larger. The result of these computations shown in Figs. 5–7 suggests that the ratio is a function of C and k but only slightly dependent on updraft velocity.
3.5. Correction factor and modified Fukuta–Xu formula 3.5.1. Difference of maximum supersaturation between analytical formulas and cloud
model We have shown in Section 3.4 that the maximum supersaturation predicted by Eq.
Ž .
Ž .
37 shows some discrepancy with that predicted by the present cloud model. Eq. 39 gives an even larger difference. There are three main reasons for such differences as
explained in Section 3.2. What the analytical solutions fail to include are: Ž .
i the DK effect of droplet growth, Ž .
ii the finite size of haze particles at the moment of nucleation, which is due to the Kelvin and solute effects of the Kohler theory, and
¨
Ž . iii the delay of growth to cause nucleation sometime after passing through the
critical supersaturation observed in the environment. 3.5.2. Correction factor for Fukuta–Xu formula
Although the analytical formulas show considerable difference for the maximum cloud supersaturation prediction compared with the results of the present cloud model, in
practical cloud physics research, their clarity and convenience outweigh a complicated cloud model, which is time-consuming to build. Analytical formulas are still useful as
Ž .
long as the above gap is properly corrected. Though Eq. 37 predicts smaller values Ž
. Ž
. than that of Eq. 39 , the beta function B kr2, 3r2 in the former also causes some
Ž .
inconvenience in practical use. Eq. 39 , on the other hand, includes the variation of cloud base temperature and pressure change. Therefore, we will focus on improving the
accuracy of the latter. In the previous discussion, we have seen that the ratio of the maximum supersatura-
Ž .
tion predicted by the cloud model over that predicted by Eq. 39 is a function of b, a , C, and k, but nearly independent of w . Therefore, we define a ratio
u
S y 1
Ž .
max , DK , mod
J C, k , a , b s ,
42
Ž .
Ž .
S y 1
Ž .
max , FX
Ž .
where S-1 is the maximum supersaturation of cloud model using DK theory
max, DK, mod
Ž .
Ž .
and S-1 is that of Eq. 39
ma x, FX
Ž .
Ž .
Eq. 42
shows that once the ratio J C, k, a , b is determined, the maximum
Ž .
Ž .
supersaturation can be accurately calculated by multiplying J C, k, a , b to Eq. 39 .
Ž .
Ž .
S-1 or Eq.
39 contains no inconvenient parameters; it just requires the
ma x, FX
temperature and pressure data at the cloud base. For practical use, a f 1 and k s 0.5 for
Ž .
maritime clouds and k s 0.7 for continental clouds are selected. Then J C, k, a , b is
reduced into a two-variable function, J C,0.5,1, b s J C, b
for maritime clouds
Ž .
Ž .
1 1
J C,0.7,1, b s J
C, b for continental clouds
Ž .
Ž .
2 2
Ž .
Ž .
Ž .
Using the present cloud model and Eq. 39 , J C, b and J C, b
have been
1 2
computed, and their relationship is shown in Figs. 8–11, in which b changes from 0 to Ž
. 1. As J C, b varies very rapidly with decrease of b, we plot the relationship in two
separate figures, one with b from 0 to 0.1 and the other with b from 0.1 to 1. Figs. 8 and 9 show the correction factor for maritime clouds. Fig. 8 ranges from
y3
Ž .
C s 100–1000 cm and b s 0.01–0.1, and shows that J C, b
changes smoothly
1
with the C, and very slowly with b when b 0.08. However, it varies rapidly when b - 0.04. Fig. 9 describes the relationship for the range with C s 100–1000 cm
y3
and b s 0.1–1.
Ž .
y3
For continental cloud, Fig. 10 gives J C, b for the range C s 500–2500 cm and
2
b s 0.01–0.1, and in Fig. 11, C s 500–2500 cm
y3
, but b s 0.1–1.0. Ž
. Ž
. From Figs. 8–11, we see that both J C, b and J C, b increase with the increase
1 2
Ž .
of C but decrease with the increase of b. J C, b changes very rapidly as b ™ 0 but very slowly as b ™ 1. This means that smaller b values are more sensitive to the
changes of the supersaturation maximum in cloud. In all of the above, we are
Ž .
Ž .
k
Fig. 8. Correction factor, J C, b , for Fukuta–Xu formula as a function of C of N sC S-1 and
1
condensation coefficient for a s1, k s 0.5.
Fig. 9. Same as Fig. 8, except with a different range of b : b s 0.1–1.0.
Ž .
Fig. 10. Correction factor J C, b for Fukuta–Xu analytic formula as a function of C and condensation
2
coefficient for a s1, k s 0.7.
Fig. 11. Same as in Fig. 10, except for b s 0.1–1.0.
Ž .
Ž .
Ž .
particularly interested in two special cases of J C, b ; J 100, 0.03 and J 500, 0.03 .
1 2
Ž .
Computations for J C, k, a , b show that J 100, 0.5, 1, 0.03 s 0.881,
Ž .
1
J 500, 0.7, 1, 0.03 s 0.916.
Ž .
2
Ž .
Then Eq. 39 is modified under the above conditions as S y 1
s 0.881 S y 1
for maritime clouds,
Ž .
Ž Ž .
max , DK , mod max , FX
and S y 1
s 0.916 S y 1
for continental clouds.
Ž .
Ž Ž .
max , DK , mod max , FX
These provide simple but more accurate analytical formulas for practical use in calculating cloud maximum supersaturation.
3.6. Atmospheric Õariation of condensation coefficient A major consequence of anthropogenic air pollution is the contamination of cloud
liquid water and the air in which cloud forms. The emitted sulfur dioxide is particularly water soluble. Experiments show that water vapor and water droplets can strongly
absorb the SO in air, and that causes serious difficulty in atmospheric SO measure-
2 2
ments since water vapor condenses on the wall of the intake tubing of the SO
2
Ž .
measurement devices Zou et al., 1992 . Actually, the polluted air contains not only SO
2
but various chemical substances that further cause the contamination of water droplets,
the surface in particular. Once these chemical substances are contained in droplets through condensation growth or gas phase absorption, the surface property of droplets
will change. One of the surface properties of droplets to affect the growth is the condensation coefficient and to an unknown extent the thermal accommodation coeffi-
cient. Laboratory experiments demonstrated that some chemical substances in the atmosphere are able to drastically reduce the condensation coefficient of droplets to
y4 y3
Ž .
values as low as 10 –10
Mazin, 1974 . Experiments in the room air showed that Ž
. the condensation coefficient reaches as low a value as 0.01 Hagen et al., 1989 . Most
frequently, the value of condensation coefficient b s 0.03 is used in cloud physics Ž
. Warner, 1969; Fitzgerald, 1972, 1978; Young, 1993; etc .
Ž .
According to Twomey 1977, 1991 , if clouds are able to cause a 2 change in global albedo, the heating effect of doubled CO
increase can be countered by the
2
cooling effect of aerosol and clouds. For assessing the cloud albedo effect in the radiative balance, what is needed is detailed knowledge of factors that contribute to the
increase of negative forcing or the effect of DK droplet growth, together with that of CCN increase, against the positive forcing caused by various emissions of anthropogenic
origins.
4. Conclusion
