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Water supersaturation in convective clouds
Atmospheric Research, 30 (1993) 105-126
105
Elsevier Science Publishers B.V., Amsterdam
Water supersaturation in convective clouds Norihiko Fukuta
Department of Meteorology, Universityof Utah, Salt Lake City, UT84112, USA (Received June l, 1992; revised and accepted January 21, 1993)
ABSTRACT The microphysics-dynamics interaction of clouds was theoretically studied in the zone after maximum supersaturation ( S - 1 )me where the droplet number concentration remains nearly constant. The analytic solution obtained, employing the Maxwellian droplet growth theory, describes (S-1 ),~c=Iwau/an-i/2, where I is the proportionality constant, wu the updraft velocity and n the number concentration of the droplets. This solution agrees well with previous studies. Factor I increases with altitude in the adiabatic atmosphere, decreases with temperature under constant pressure and increases with pressure under constant temperature. For the zone sufficiently after ( S - 1 )me, an approximate relationship ( S - 1 ) oc r - i oct - i/3 is shown to hold, where ( S - 1 ) is the supersaturation, r the average droplet radius and t the time. Using the diffusion-kinetic theory of droplet growth, which includes the effects of thermal accommodation and condensation coefficients, numerically soluble relationships are derived for ( S - 1 ), r and t. Application of this theory is shown to increase ( S - 1 )me considerably. The Maxwellian analytic solution that is obtained, the variation of Factor I under different atmospheric conditions and the effect of condensation and thermal accommodation coefficients through the use of the diffusion-kinetic droplet growth theory suggest that maximum supersaturation may reach as high as 10% and beyond in convective clouds. RI~SUMI2 On 6tudie throriquement les interactions entre la microphysique et la dynamique des nuages dans la zone au-delh de la sursaturation maximale ( S - 1 )me o/l la concentration en nombre des gouttelettes est pratiquement constante. La solution analytique obtenue, qui utilise la th6orie de Maxwell de croissance des gouttes, donne ( S - 1 )mc=Iw3J4n-i/2, oil Iest la constante de proportionnalitr, wu la vitesse du courant ascendant et n la concentration en nombre des gouttelettes. Cette solution s'accorde bien avec de prrcrdentes 6tudes. Le facteur I augmente avec l'altitude dans l'atmosph+re adiabatique, diminue avec la temprrature sous pression constante, et augmente avec la pression sous temprrature constante. Dans la zone suffisamment au-delh de ( S - 1 )me, on trouve une relation approchre ( S - 1 )oc r - l o t t-1/3, o/l ( S - 1 ) est la sursaturation, r le rayon moyen des gouttelettes, e t t le temps. En utilisant la throrie cinrtique de croissance des gouttelettes, qui prend en compte les effets de l'accommodation thermique et des coefficients de condensation, on 6tablie des relations pouvant se rrsoudre n u m r r i q u e m e n t pour ( S - 1 ), r et t. L'application de cette throrie montre une augmentation considrrable de ( S - 1 )me. La solution analytique de Maxwell obtenue, la variation du facteur I sous diffrrentes conditions atmosphrriques, et l'effet des coefficients de condensation et d'accomodation thermique par le biais de l'utilisation de la throrie cin6tique de
0169-8095/93/$06.00
© 1993 Elsevier Science Publishers B.V. All rights reserved.
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N. FUKUTA
croissance des gouttelettes, sugg6rent que la sursaturation maximale peut atteindrc el marne d6passer 10% dans les nuages convectifs.
INTRODUCTION
A rising thermal is known to cool through adiabatic expansion and raise the relative humidity. As the thermal passes through the lifting condensation level (LCL) or the convective condensation level (CCL), the relative humidity exceeds 100%, or the air space becomes supersaturated, and cloud condensation nuclei (CCN) begin to activate. The activation continues with the growth of formed droplets. The number and the size of the droplets thus increase, while the cloudy parcel sustains its upward motion and raises the level of effective supersaturation further. Under this condition, the population of formed cloud droplets abstract moisture from the immediate and limited environment and return the latent heat of condensation generated during their growth to it. Both of these processes serve to reduce supersaturation in the vicinity of growing droplets. The larger the droplet size and the number concentration, the faster the supersaturation reduction. Eventually, the supersaturation reduction rate, by the population of growing droplets, comes to balance the production rate by the cloud parcel lifting, and there the supersaturation reaches the maximum. At this point, CCN activation ceases and the number of droplets in a given cloud parcel remains unchanged thereafter, at least for some period of time. Cloud droplets, however, will continue to grow after this stage, for a substantial period of time, under slowly reducing supersaturation. In the real clouds, entrainment of outside air by the cloudy parcel distorts this simplified picture. Nevertheless, treatment of such an idealized system will provide a clear footing for understanding the complex system of real clouds, including their properties of radiation energy transmission. This nucleation-growth-dynamics interaction at and above the cloud base was studied numerically by Howell ( 1949 ), with the help of a hand calculator. Later Neiburger and Chien (1960) and Mason and Chien ( 1962 ) researched the process using an electronic digital computer. More sophisticated numerical studies of cloud processes were later done by Clark (1973, 1974), Young (1974), Clark and Hall ( 1979 ) and Grabowski ( 1989 ). However, as typical for numerical studies, the relationship among factors directly pertinent to the cloud supersaturation remained unexposed. Twomey (1959), on the other hand, took an analytical approach to the problem. By expressing the CCN activity spectrum with an exponential function of supersaturation ( S - 1 ), where S is the saturation ratio, and introducing an approximation scheme for the nucleation-growth interaction in the rising cloudy air, he obtained a range in which the solution is confined. Works of Neiburger and Chien, Mason and
WATER SUPERSATURATION IN CONVECTIVE CLOUDS
107
Chien and Twomey involved the so-called "Maxwellian" condition at the droplet surface, which assumes that the temperature and the vapor density just above the droplet surface are the same as those on the real surface. Therefore, the vapor density at the droplet surface is saturated at the temperature. It is well accepted that exchange of mass, heat and momentum between the gaseous and condensed phases takes place not necessarily with full efficiency. The factors describing the efficiency are known as the condensation (for G L interface) and the deposition (for G-S interface ) coefficients for mass exchange, the thermal accommodation coefficient for heat, and the momentum accommodation coefficient for momentum, where G stands for gas, L for liquid, and S for solid, respectively. This makes the Maxwellian condition no longer applicable at the droplet surface, and the condition of the droplet system changes to satisfy the steady state and continuity requirements of fluxes involved (Fukuta and Walter, 1970). A recent experimental study (Hagen et al., 1989) showed that, although the condensation coefficient appeared to be close to unity or at the theoretical upper limit at the beginning of the laboratory experiment on cloud processes, the value went down towards 0.01 as the growth process continued. This seems to support the importance of accommodation coefficients involved in growth kinetics of cloud droplets (diffusion-kinetics, Smirnov, 1971 ). Under the condition of forming clouds where supersaturation is constantly being generated, the vapor and temperature fields of each droplet have a limited space restricted by boundaries of cells or spaces between neighboring ones. This deviates from the concept of fields commonly applied in Cloud Physics. Such fields are assumed to have infinite expanse. However, as shown in Fukuta ( 1992 ), this effect in the kinetics of droplet growth has been identified as small in the range of normal cloud conditions for both Maxwellian and diffusion-kinetics. Supersaturation plays an important role in determining the characteristics of liquid phase clouds through the microphysics-dynamics interaction. If the process occurs in a sub-freezing temperature zone, it is also expected to influence corresponding ice phase processes, including nucleation and growth. After the maximum, supersaturation generation by the updraft and the abstraction by the droplet system become interlocked in this time sequence. In a time sequence of processes, it is well known that the slower process dominates. The shortness of the relaxation time of the droplet system thus leads to a condition where the slow process of supersaturation generation or reduction in the cloud, i.e., updraft or downdraft, governs the balancing level. Under this condition after the maximum supersaturation, the droplet number concentration remains nearly constant and the supersaturation, including the maximum, can be estimated retrospectively, as long as the entrainment process is slow, solely from the properties of the droplet system and the thermodynamics as well as the dynamic condition of the corresponding portion of the cloud. It is the
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N FUKUTA
purpose of this paper to analytically treat the cloud supersaturation after the maximum, with proper approximations and to clarify the relationships among the contributing factors by applying the Maxwellian and the diffusion-kinetic theories of droplet growth in a back-tracking manner. ENVIRONMENTAL
PARAMETERS
IN C L O U D Y U P D R A F f
The supersaturation is defined as S-l=
e_l,
(1)
es
where e and es are water vapor pressure of the environment surrounding the droplet in question and that saturated at the environmental temperature, T, respectively, e depends on the air pressure p and the temperature T, while es is a function of T alone. Under the condition e - es, the change in supersaturation may be expressed as d ( S - 1) _ 1 (de-des). es
(2)
The supersaturation is generated due to cloud air lifting and is depleted by droplet growth. The balancing level of supersaturation is hence achieved by these two processes. Figure 1 describes the continuing CCN activation and the system of growing droplets shortly after the moist air rises through the LCL (or CCL) level. In the figure, an arbitrary position of the cloud system path (T, e) is described by combination of a dry adiabatic path from the cloud base ( To, ec) to a point Q and an isobaric process of droplet growth from Q to ( T, e). For the dry adiabatic process, the pressure change with altitude is given by the hydrostatic equation
dp -~z=p,g,
(3)
where z is the vertical distance, Pa the density of air and g the gravitational acceleration. The temperature change involved during the dry adiabatic process and the droplet growth may be described by a modified pseudo-adiabatic equation dT=-
_1 (gdz+Ldw),
(4)
Co
where cp is the specific heat at constant pressure and w the mixing ratio. Note that w is not the saturation mixing ratio, ws, because, as can be seen from the figure, the point ( T, e) has not yet reached the saturation or is not on the
WATERSUPERSATURATIONIN CONVECTIVECLOUDS
109
e
DRY / AD IA B A T ~ / / J e
-*(Tc,ec)
des
Ida2
j/ jf //
-I-2
dT
"
- T
d T ~ - -
Fig. I. The cloud microphysics-dynamics interaction above the cloud base. An arbitrary point ( T, e) belonging to actual cloud which contains growing droplets under generating supersaturation is expressed by combination of dry adiabatic process from (To, e¢) to Q, and droplet growth process from Q to ( T, e). For notations, see the text.
saturation vapor pressure curve es. The system still holds supersaturation. In Fig. l, the first term on the right-hand side corresponds to dT~ and the second dT2. We now proceed to describe e and es terms in Eq. (2), which are a function ofp and T, thus far described. As is evident in Fig. l, de = del + de2.
(5)
From the figure, and using Eq. (3) and w ~- ee/p under e << p, where E= M~ Ma=0.622, M and Ma being the molecular weight of water and that of air, respectively, we obtain de1 = _Pvg dz,
(6)
where Pv is the density of water vapor. This is because, under the condition of adiabatic expansion without condensation, w remains constant and therefore eocp and for a small temperature variation of the process in question, eocpv. Since the de2 term is due to vapor removal by droplet growth de2 =P- dw. Combining Eqs. (6) and (7) with Eq. ( 5 ), we have
(7)
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N. FUKUTA
d e = _Pvg d z + P dw.
(8)
The saturation vapor pressure, es, may be expressed as the sole function of temperature by the Clausius-Clapeyron equation in the differential form as e~L
des = R~. ~- d T,
(9)
where Rv is the specific gas constant for water vapor. Combination of Eqs. (2), (8) and (9) yields d ( S - 1)=Adz+Bdw,
(10)
where
~Lg ( 1 T ~ , A-RaT~ cp eLJ
(l l)
and B-
eL 2
q
p
(12)
cpRa T 2 eel"
Therefore the changing rate of supersaturation may be expressed as (Mason, 1971 )
d(S-l)_AdZ dt
dw
~+B
(13)
dt"
The first term on the right-hand side of Eq. ( 13 ) describes supersaturation generation rate by cloudy air without the influence of droplet growth (dry adiabatic process). The second term describes its reduction rate due to the growth. EQUATIONS FOR DROPLET GROWTH
When the radius is small compared with the distance between its two nearest neighbors (Fukuta, 1992) and disregarding the Kelvin effect of surface tension on the curved surface and the solute effect, the growth rate of pure water droplet in the cloud may be expressed by the diffusion-kinetic equation as (Fukuta and Walter, 1970 )
dm_
4rtr(S- 1 ) L2
-~
(14) 1
'
WATER S U P E R S A T U R A T I O N IN CONVECTIVE C L O U D S
111
where m, t, K and D are the droplet mass, the time, the thermal conductivity of air and the diffusivity of water vapor in air, respectively. Ps~ is the water vapor density in the surrounding air environment at the temperature, T~ ( = To), r
f~=r+t,~'
(15)
[,
(16)
K(27tRaT~) 1/2 ( 2 - o r )
~= ~
2a
'
Cvand a being the specific heat at constant volume and the thermal accommodation coefficient of air, respectively, and r
fp=r+6,
(17)
_
xl/2
fl being the condensation coefficient of water vapor. When r -, oo, f , ~ 1, fa --' 1 and Eq. (14) converges into the Maxwellian,
dm
4rtr(S- 1 )
( 19 )
SUPERSATURATION
The microphysics-dynamics interaction at and just above the cloud base determines the initial characteristics of cloud droplets. They are complex, and only numerical modeling of processes involved can provide a proper estimate. On the other hand, an analytic approach, if possible, normally gives quick illumination to the processes within the assumption and approximation applied. In this regard, the work of Twomey ( 1959 ) is of value although it includes a number of approximations and stays within the range of Maxwellian theory. In addition, these deductive approaches suffer from the uncertainty caused by entrainment processes in general. In natural clouds, the zone below the maximum supersaturation is shallow compared with the rest, and as long as the cloud parcel is moving in the vertical direction, supersaturation (or undersaturation) keeps changing. However, after the maximum supersaturation, activation of CCN ceases, and the
l 12
N. FUKUTA
droplet number concentration remains nearly constant, an advantage for analytic treatment. Therefore, we shall first examine the behavior of this zone after the supersaturation m a x i m u m in terms of droplet-growth-cloud-dynamics interaction and retrospectively come back to the m a x i m u m for its assessment.
Features of droplet population before and after the supersaturation maximum While the cloudy air mass rises towards the zone of m a x i m u m supersaturation, the heterogeneous condensation nucleation continues. As a result, the droplet population in this zone becomes polydisperse, and the number concentration n increases with time. Whereas, in the zone after the maximum, the nucleation ceases while droplets continue to grow. There, n practically remains constant provided that entrainment of outside air is negligible, and the droplet growth leads closer to the monodisperse condition. In the following sections, we shall treat the population of growing droplets after the supersaturation m a x i m u m and derive the m a x i m u m by reversing the course through the use of the above-mentioned two equations of droplet growth.
The Maxwellian droplet growth theory The change of liquid water content in Eq. (13 ) may be expressed as the product between the number of droplets in unit weight of air n/p~* and the droplet mass increase dm, or dw = - n dm, Pa
(20)
where
dm = 4nr2pLdr.
(21 )
By integrating Eq. ( 13 ) with Eqs. (20) and (21 ) under constant updraft velocity, we have
( S - 1) = E t - F r 3
(22)
where
E=Awu,
(23)
and *Droplets are assumed to be completely monodisperse. While the actual population is polydisperse, the condition after the supersaturation m a x i m u m becomes closer to the assumed. In a numerical treatment, s u m m a t i o n of appropriate term within unit volume will replace n.
WATERSUPERSATURATION INCONVECTIVE CLOUDS F - 4npL Bn 3pa
1 13
(24) '
wu = d z / d t and PL being the updraft velocity and the density of liquid water, respectively. Equation (22) is, in essence, conservation o f supersaturation and means that the supersaturation in the system o f growing droplets is that generated, minus that depleted by droplet growth. On the other hand, combination o f Eqs. ( 13 ), ( 19 ) and (20), considering n as a constant, yields d(S-1) dt =E-GM(S-1)r,
(25)
where
-
4rtBn
.
(26)
aM pa [KRL T 2ooI-~s~D] Equation (25) describes conservation o f the rate o f change o f supersaturation; i.e., the rate o f actual supersaturation change is given by the difference
between the generation rate of the environment and the depletion rate by the droplet system. Equations (22) and (25) describe the actual system. We hence apply the latter to the supersaturation m a x i m u m ( S - l ) m with respect to t=tm or Eq. (25 ) = 0. Then, E 1 ( S - 1 ) m - - -GM rm"
(27)
Inserting Eq. (27) into Eq. (22), with the condition ( S - 1 ) = ( S - 1 )m, we obtain F 3 _ 1 1 t m = ~ r m - t GM rm"
(28)
Thus, Eq. (28) conveys the restriction of Eq. (27) and is valid only at one point which corresponds to the real m a x i m u m supersaturation, ( S - 1 )rnc, and the matching radius, rmc. The task now is to find out this solution point.** "Equation (27) is valid by itself for the supersaturation maximum and describes the condition for the given time sequence of the event. However, if we consider (S- 1)m and rm as variables, the equation covers the area which does not correspond to the real system in question except the point given by Eq. ( 27 ). Equation (28 ) has the same meaning as that of Eq. ( 27 ) except that it contains tm term. Nevertheless, the period tm covers carries the same condition as that after the supersaturation maximum, i.e., monodisperse droplets with constant n (no nucleation in the midcourse) which is a large deviation from reality. In this regard, at the point of (S - 1)me and rmcwhich corresponds to a real (solution) value, tmcis still under restrictions and associated errors, and therefore is not realistic.
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N. FUKUTA
The behavior of Eq.(28) is shown in Fig. 2. In this figure, the rm curve covers the entire range of r-coordinate, while the curve does not exist between 0 and/me on t-coordinate. This is because, in the gap, the condition of maxim u m supersaturation, i.e., balancing between the supersaturation generation rate o f the rising cloudy air parcel and the abstraction rate of the growing droplets, has not yet been met under the constant droplet n u m b e r concentration retrospectively applied. As can be seen in Fig. 3, the real supersaturation increases towards the m a x i m u m monotonically. This is to say that as t in-
It,.'=r~'
/.
' , ~
-t
tme
Fig. 2. Relationship among the terms in Eq.(28). t,,~ l/rm relationship represents the condilion of m a x i m u m supersaturation and lm VCrm3 relationship describes the condition for total depletion of supersaturation by droplets. The real supersaturation maximum corresponds to (t .... r.,~,) point. For details of discussion, see the text.
(S-~)
/s
~\\
REAL
PROCESS
(s -l)~,
~
-
tmc Fig. 3. The relationship between ( S - 1 )m and the real supersaturation (dashed line). Two terms of ( S - 1 )m (thin solid lines) are also shown. For notations, see the text.
WATERSUPERSATURATION IN CONVECTIVECLOUDS
115
creases and the system reaches the supersaturation maximum in Fig. 3, the same increase moves t to tm~ in Fig. 2 where, for the first time, t comes in contact with the rm curve that satisfies the condition of the supersaturation maximum. Thus, it is clear that the point corresponding to the true supersaturation maximum is that of the smallest t on rm curve, or (tmc, rmc). Thus, (tmc, rmc) is the minimum of the rm curve with respect to r variation, and we can obtain the point by differentiating Eq. (28) with rm and setting the derivative to zero, i.e., (E~ 1/4 rmc = \3--ff~MJ (29) Replacement of Eq. (29) into Eq. (28) results in
4
F 1/4
tm~ - - 3 3 / 4 EI/4G3M/4.
(30)
From Eqs.(27) and (28), we have ( S - 1)m~--
31/4E3/4F 1/4 G~ 4
(31)
Equations (29) through (31) satisfy Eqs.(22), (25) and (27), the conditions of supersaturation and supersaturation generation rate conservation, as well as the maximum supersaturation. In regard to the formulation of (S-1)mc, two main system variables Wuand n are involved. Using Eqs.(23), (24), (26), (29), (30) and (31), one finds
W1/4 rmcOC nl/2, U
(32)
1
tmc °c
!/4
Wu
n
1/2'
W3/4 U
( S - 1 )m¢OC nl/~.
(33)
(34)
The relationship Eq. (33), i.e., tmcOCW u 1/4 is inverse to the tendency of the actual process (Howell, 1949 ). It has come in the course of combining Eq. (22) and Eq. (25 ) = 0 for estimation of ( S - 1 ) me. Equation (25 ) shows that r and ( S - 1 ) are sensitive to the time rate of change of supersaturation, viz. d ( S - 1 )/dt, and not to t itself. Under the condition of Eq. (25) =0, i.e. at the supersaturation maximum, even this influence disappears. As a result, rm and ( S - 1 )m determine mac and ( S - 1 )mc explicitly regardless of t. As discussed in the preceding section, tmc is under full influence of the unreal term, 1/rm,
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N. FLIKUTA
which has been introduced into Eq. (28) as a part of the supersaturation maxi m u m condition, the condition never met in the range of t < t,,,. It is evident in Figs. 2 and 3 that there exist neither rm nor the corresponding ( S - 1 )m in the zone between the coordinate origin and tmc. In other words, the solutions for rmc and ( S - 1 )me have been obtained by describing the real time rate of change with that for tin.
The diffusion-kinetic droplet growth theory To include the effects of condensation and thermal accommodation coefficients, we replace GM in Eq. (27), with GDK representing the effect of diffusion-kinetic treatment of droplet growth, i.e.
4zeBn GDK--
L2
1
'
(35)
to obtain E 1 ( S - 1 )m --GDK rm"
(36)
GDK is a function of rm and smaller than GM, implying that ( S - 1 )m value becomes larger for the diffusion kinetic treatment than for the Maxwellian case. For the treatment in detail, rm terms in GDK have to be exposed as Hi rm GDK -- H2 rm + H3'
( 37 )
where Hi -
4 zcBn Pa
,
(38)
p ~ DJ
(39)
[ Z2[a H3 = LK/~ T 2 +ps~D]"
(40)
Combining Eqs. (22) and ( 36 ), together with Eqs. ( 38 ) through (40), we obtain
WATERSUPERSATURATIONIN CONVECTIVECLOUDS
(S-1)m
1 EH2 1 ÷ EH3 --HI rm Hi r 2'
117
(41)
and tm /-/2 1 //3 1 F 3 --Hi rm 4"~-1 ~ m + E rm"
(42)
Comparing the above two equations with Eqs. (27) and (28), it becomes clear that the second terms on the right-hand side of both Eqs. (41 ) and (42) have appeared due to the effects of condensation and accommodation coefficients, or the diffusion kinetic treatment. Also, Eq. (41 ) does not have any extrema, while Eq. (42) has one extremum. By differentiating Eqs. (41) and (42) with rm and dividing the latter result with the former, and setting the derivative dtm/d(S- 1 )m=0, we arrive at E 3 fH2
H3
r2mc- 3FH-~---~c + r3~) = 0,
(43)
which is analytically insoluble but is easily soluble with the numerical method. Application of rm~, thus obtained, to Eqs. (41 ) and (42) will yield ( S - 1 )m~ and tmc for the diffusion kinetic droplet growth theory, but the latter term will not correspond to the real cloud condition as explained above. DISCUSSIONS
In the above treatments, we have dealt with the growth processes of the population of cloud droplets in the zone after the maximum supersaturation where the droplet number concentration remains virtually unchanged and droplets are more uniform in size. Analytic solutions of tmc (unreal), rmc and ( S - 1 )me, for the Maxwellian droplet growth theory, were obtained. A set of equations for the solutions with the diffusion-kinetic theory, easily solvable by the numerical method, were described. The underlying assumption for the treatment is that the entrainment effect is negligible relative to the droplet growth process. In real clouds, such a condition does exist, like the weak echo region (WER) of the thunderstorms where virtually no dilution occurs. We now examine the results obtained above in comparison with natural conditions.
The system with Maxwellian droplet growth When the cloud base condition is given, by exposing the main variables, n and Wu, we can write Eq. (31 ) as
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N. FUKUTA W3/4
(S-
1 )mc = I - - F/I/2'
(44)
where
I { pLp2aA3~''4( L2 =\i6~-~]
\KRJL
1
'~/4
+ps~-D)
(45)
Figure 4 shows the relationship among ( S - 1 )me, Wuand n, computed from Eq. (45) which uses the Maxwellian droplet growth theory, for the case of Tc=0°C and Pc (the cloud base pressure) =500 mb and I = 1.749 x 10 .3 c m -9/8 s 3/4. In the figure, one can see that there exists a substantial zone on n-Wu plane where ( S - 1 )me exceeds a few percent, the previously believed upper limit. It should be pointed out that, for a given condition of n and Wu, ( S - 1 )me is achieved due to balancing of the supersaturation depletion rate by growing droplets with the generation rate by adiabatically cooling cloudy air. The Maxwellian droplet growth and the resultant supersaturation depletion rate are faster than those of the diffusion-kinetic rate. From this viewpoint, the ( S - 1 )me values in Fig. 4 represent the practical lower limit, and the actual values may well be higher as we shall discuss later. Thus, by examining the previously unexplored zone in the figure, it has become clear that,
I
-7
O. 0.0
Fig. 4. Relationship among the maximum supersaturation ( S - 1 )mc, the updraft velocity w,, and the number concentration of cloud droplets n, computed from Eq.(45) for Maxwellian droplet growth theory. The temperature and pressure of the cloud base are taken as 0°C and 500 mb, respectively.
W A T E R S U P E R S A T U R A T I O N IN C O N V E C T I V E C L O U D S
119
contrary to the c o m m o n belief o f ( S - 1 )mc < 1% (Mason, 1971; Pruppacher and Klett, 1978 ), ( S - 1 )mc can reach m u c h higher values under realistic cloud conditions. Figure 5 describes rmc values u n d e r the same condition as in Fig. 4. Since the Maxwellian droplet growth theory is the faster o f the two dealt with here, rmc in the figure also represents the lower limit. If n is unusually low and wu very high, rm~ can reach 10/tm and beyond. Whereas, if n is high, like in polluted air, and wu low, rmc will reduce to submicron level. As can be seen in Eq. (44), F a c t o r / c o n t r o l s ( S - 1 )mc, and it m a y be worthwhile to examine the effect o f variables describing the cloud base on the term. Figure 6 shows a variation o f Factor I at the cloud base in a dry adiabatic atmosphere with potential temperature 0 = 20°C. In the figure, it can be seen that a higher cloud base tends to produce a higher ( S - 1 )mc under a given n and Wu. This is due to the fact that variation o f t h e r m o d y n a m i c factors o f the adiabatic cooling, such as A and B, exceeds that o f kinetic factors o f droplet growth in the atmosphere. Figure 7 describes a variation o f Factor I as a function o f temperature T at three different atmospheric pressures, p = 500, 700 and 900 mb. It m a y be
10
E
:k
v
I
0.1
Fig. 5. Relationship among the radius of droplet at the maximum supersaturation, rmc,the updraft velocity wu, and the number concentration of cloud droplets n, computed from Eq. (29). The cloud base condition is same as that in Fig. 4.
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N. FUKUTA
T 61
-40 -30 -20
(°C) -I0 0
I0
20
]
5 ,o,~,, 4
% --
3
400
560
600 p
7oo 800 (mb)
960
,600
Fig. 6. Variation o f Factor 1 in Eq. (45) in a dry adiabatic a t m o s p h e r e with a potential temperature 0 = 20 ° C. The curve represents the cloud base.
8~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6r ~
5
3
[
i
i
i
I
-°4o -~;o -~o 40 T
6
I'0
20
(°C)
Fig. 7. Variation of Factor I in Eq. ( 4 5 ) as a function o f t e m p e r a t u r e T at atmospheric pressure p = 500, 700 a n d 900 mb, respectively.
WATER SUPERSATURATIONIN CONVECTIVECLOUDS
121
seen in the figure that factor/, therefore ( S - 1 )me, tends to become larger for a given set of n and wu when T lowers. At 700 m b level, a cloud base temperature shift from 10 ° to - 2 0 ° C increases the factor by nearly 2. Again, the thermodynamic terms like B, instead of kinetic terms of droplet growth, dominate the change and produce this tendency. Figure 8 shows a variation of Factor I as a function of atmospheric pressure p at temperatures T = - 20 °, 0 ° and 20 ° C. Low pressure gives smaller Factor /, for the same reason as stated above. At 0°C, a pressure increase from 500 mb to 700 m b raises the factor by 22%. It is not possible at this m o m e n t to compare the present analytic result with observed data since an accurate measurement m e t h o d of cloud supersaturation does not exist. Results of numerical cloud modeling are now available, but they too are not suitable for comparison as they contain other factors which are not included in the present treatment. Results of two computations, one numerical (Howell, 1949) and the other analytic (Twomey, 1959), appear to provide a comparison basis for the present treatment. Table 1 compares the result of the numerical computation by Howell (1949) and the present Maxwellian treatment under p = 8 0 0 mb and T = 0 ° C . For computation numbers 1 and 2, ( S - 1 )me agreement is complete within the limit of errors for the values used in the computation. For computation number 3, the present treatment gives a value larger by a factor of 2.2 than that of Howell's. However, Howell's result for this computation is not consistent with 4
5 o~1¢o
% -
2
i
/
I
0
400
5(30 660 700 p (mb)
8()0
900
Fig. 8. Variation o f F a c t o r I in Eq. ( 4 5 ) as a f u n c t i o n o f a t m o s p h e r i c pressure p at t e m p e r a t u r e s T = - 2 0 °, 0 ° a n d 20°C, respectively.
122
N. FUKUTA
TABLE 1 Comparison between the result of numerical computation by Howell (1949) and t he present Maxwellian treatment under p = 800 mb and T = 0 °C Computation Number Howell (1949)
n (cm -3)
wu (cms -~ )
Howell
Eq. (31)
~r"~ (llm) Howell
Eq. (29)
1
249 39.97 416
60.4 1.51 30.2
0.36 0.056 0.073
0.354 0.0555 0.163
1.78 2.88 2.65
1.55 1.53 1.01
3
( S -- 1 ) ~C
TABLE2 Comparison for the Maxwellian maximum supersaturation between the "overestimation" value ( S - 1 ) " of Twomey (1959) and the present treatment ( S - l ) m c (Eq. 31 ) under p = 8 0 0 mb and T= IO'~C [~'u (ems -~)
tl (cms -3)
( S - - I )" ( S - l )me • (%) (%) (cm 3)
( S - l )" (%)
( S - I )me tl (%) (cm 3)
{,~)'-- 1 ) ' (~T_ ] )m, t%) (%)
10 100 1000
37 61 100
0.17 0.75 3.3
0.13 0.61 2.8
0.139 0.644 2.98
0.06 i).26 ~.1
0.178 0.778 3.42
60 89 131
310 554 985
0.0614 0.258 1.09
the rest, such as ( S - 1 ) after ( S - 1 ) m c is not proportional to t j~ as we shalle see later. Therefore, this result of his may be discarded. The droplet radius rmc at ( S - 1 )me is larger with Howell's computation. This is apparently because he took a droplet growth equation formulated by Langmuir, which resembles the diffusion kinetic equation, but was shown to be erroneous (Fukuta and Walter, 1970 ). The diffusion-kinetic effect is known to persist in the growth process of droplets whose radii are far beyond the normal range of Kelvin and solute effects (Fukuta and Walter, 1970). The latter effects are also included in Howell's work. At any rate, the diffusionkinetic-like equation of Howell has seemingly raised the rmc value, as droplet growth computed with such an equation is slower than that of the MaxweUian theory. The necessary kinetic balance between supersaturation generation and the depletion is achieved only when rmc has reached a larger value. Table 2 gives a comparison for Maxwellian m a x i m u m supersaturation between the "overestimation" value ( S - 1 )" o f T w o m e y (1959) and the present treatment ( S - 1 )me (Eq.31) under p = 8 0 0 mb and T = 10°C. In his analytic treatment, Twomey estimated the range of supersaturation in which the real value is expected to fall. He claims that the upper and lower constraints of supersaturation never differed by more than 30%. In Table 2, it is seen that ( S - 1 )me is slightly larger than ( S - 1 )" in most cases, but never more than 7%, which is within the limit of error arising from the uncertainty of values
WATER SUPERSATURATION IN CONVECTIVE CLOUDS
123
used in the computations. Therefore, there is good agreement between the two computations. The reason why the present result agrees with the upper limit of Twomey's work remains unclear. In the preceding section, we have seen that the supersaturation depletion rate of the population of growing droplets balances for the first time with the supersaturation generation rate of the environment, and the supersaturation reducing capability of the population continues to increase with the droplet size [see Eq. (25) ]. As the droplets grow and pass the point of ( S - 1 )me,the supersaturation depleting capability of the population becomes larger than that at ( S - 1 )me. Nevertheless, the droplet population cannot totally deplete the existing supersaturation immediately after passing the ( S - 1 )me point, because the depletion rate also depends on ( S - 1 ) [see the second term of Eq. (25)]. This is to say that there exists a feedback effect proportional to supersaturation in the supersaturation depleting capability of the population. As a result, the real supersaturation can change only slowly after ( S - 1)me. If we ignore this slow rate of supersaturation change, we have a relationship identical to Eq. (27) which applies after ( S - 1 )me,i.e., E 1 ( S - 1 ) ~ G M r"
(27')
The same condition applies to the differential form of Eq. (22), or
E = 3Fr 2 dr dt"
(46)
After integration, Eq. (46) yields
r3w.t.
(47)
Then, from Eqs. (27') and (47), we obtain l l
( S - 1)~ octl/3.
(48)
As can be seen in Fig. 3, to establish this relationship, in reality, a transitional period is required after passing the ( S - 1 )me point. Thus, except for this transitional period, Eq.(27') holds with reasonable accuracy. In fact, Howell's numerical computations 1 and 2 explicitly demonstrate the above relationship.
The system with diffusion-kinetic droplet growth Under the influence of small thermal accommodation coefficient ot or small condensation coefficient of water vapor fl, or both, droplet growth is known to become slower than what the Maxwellian theory predicts (Fukuta and
124
N. FUKUTA
Walter, 1970). Since ( S - 1 )mc is achieved as a result of competition between the supersaturation generation rate of the cloud environment, which is normally constant, and the depletion rate due to droplet growth, the slowed droplet growth rate pushes the supersaturation to a high level. For an environment of T = 0 ° C and p = 1 atm, if o~= 1 and fl=0.03, the ratio for droplet growth rate between the diffusion-kinetic and the Maxwellian theories reaches 4.7 for r = 1 #m and 1.8 for r = 5/~m. Although rmc values for the diffusion-kinetic treatment are larger than those for the Maxwellian one, the latter shown in Fig. 5 gives an idea of their possible range under the former treatment. The rmc values are most likely within the range of this diffusion-kinetic effect. Clark (1974) reported a factor of 2.1 increase in ( S - 1 )mc from the corresponding Maxwellian value in his numerical model computation using an erroneous diffusion-kinetic droplet growth equation given by Rooth ( 1957 ) (see Fukuta and Walter, 1970). Although the equation used had this problem and the values of accommodation coefficients he employed seemed to be too small, which tend to result in a slower droplet growth rate, his computational result clarified the importance of the effect. A recent cloud model (Grabowski, 1989 ) uses a more realistic constant which is based on a combination of ~ = 0.9 and fl=0.03. Saxena and Fukuta (1982) reported 1.8 for the ratio of m a x i m u m supersaturation between the diffusion-kinetic and the Maxwellian treatments under a = 1, fl=0.04, wu= 10 cm s-1 and n = 100 c m - 3 While detailed analysis of the diffusion-kinetic effect is beyond the scope of the present paper, it is clear that this will increase the level of m a x i m u m supersaturation considerably and it deserves special attention. The derived relationships in the section describing the diffusion-kinetic droplet growth theory will provide a basis for the numerical estimation.
Field application In some clouds, the entrainment process occurs significantly, and the above treatment becomes difficult to apply. If, however, the process is moderate, by observing two main variables, n and wu, as well as the cloud base condition, solving the "would have been" ( S - 1 )me as a function of height and finding the largest, one may estimate retrospectively the real ( S - 1 )me and its location. The relationship (27'), which requires additional measurement of droplet radius, may be useful for assessing the supersaturation level after ( S - l)mc. CONCLUSION
The microphysics-dynamics interaction above the cloud base of convective clouds has been analytically studied in the zone after the supersaturation maximum. Following are the main findings.
WATER SUPERSATURATION IN CONVECTIVE CLOUDS
125
( 1 ) By realizing that the number concentration of cloud droplets remains nearly constant and the supersaturation changes rather slowly after the supersaturation maximum, and by reversing the time, analytic solutions for the maximum supersaturation ( S - 1 )me and the corresponding droplet radius were obtained for the Maxwellian growth theory with the help of fictitious growth time [see Footnote for Eq.(28)]. For the diffusion-kinetic growth theory, the relationship of those two real values as well as the fictitious growth time, analytically insoluble but numerically easy to solve, were also derived. (2) For the Maxwellian growth theory, the analytically solved maximum supersaturation agreed well with the computed values of other studies. (3) ( S - 1 ) mcbased on the Maxwellian droplet growth theory was found to be proportional to w ~ 3/4 n-1/2, where wu is the updraft velocity and n the number concentration of cloud droplets after the maximum. (4) The proportional factor for this relationship includes the cloud base condition which consists of thermodynamic terms as well as kinetic terms of droplet growth, and the former dominate the latter. The proportional factor was found to increase with height in the adiabatic atmosphere, decrease with temperature under constant pressure and increase with pressure under constant temperature. ( 5 ) A relationship ( S - 1 ) ocr - t oct - 1/3, where r is the droplet radius and t the time, applicable to the condition sufficiently after ( S - 1 )mc was derived. (6) Application of the diffusion-kinetic theory of droplet growth, which deals with the effects of thermal accommodation and condensation coefficients, was shown to yield ( S - 1 )me values substantially larger than those of the Maxwellian theory. (7) Considering ( S - 1 )m~ values estimated from the analytic solution based on the Maxwellian droplet growth theory, the effect of condensation and thermal accommodation coefficients through the use of the diffusion-kinetic droplet growth theory, and variation of the proportional constant under different environmental conditions, it is concluded that ( S - 1 )mc as high as 10% and beyond may be possible in convective clouds. ACKNOWLEDGMENT This work was supported by the Division of Atmospheric Sciences, National Science Foundation under Grant ATM-8616568. REFERENCES Clark, T.L., 1973. Numerical modelingof the dynamics and microphysicsof warm cumulus convection.J. Atmos. Sci., 30: 857-878. Clark, T.L., 1974. On modellingnucleation and condensation theory in Eulerian spatial domain. J. Atmos. Sci., 31:2099-2117.
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Clark, "I-.L. and Hall, W.D., 1979. A numerical experiment on stochastic condensation theory. J. Atmos. Sci., 36: 470-483. Fukuta, N., 1992. Theories of competitive cloud droplet growth. J. Atmos. Sci., 49:1107-1114. Fukuta, N. and Walter, L.A., 1970. Kinetics of hydrometeor growth from a vapor-spherical model. J. Atmos. Sci., 27:1160-1172. Grabowski, W.W., 1989. Numerical experiments on the dynamics of the cloud- environment interface: Small cumulus in a shear-free environment. J. Atmos. Sci., 46:3513-3541. Hagen, D.E., Schmitt, J., Trueblood, M., Carstens, J., White, D.R. and Alofs, D.J., 1989. Condensation coefficient measurement for water in the UMR cloud simulation chamber. J. Atmos. Sci., 46: 803-816. Howell, W.E., 1949. The growth of cloud drops in unitormly cooled air. J. Meteorol.. 6: 134149. Mason, B.J., 1971. The Physics of Clouds. 2nd Ed. Oxtbrd University Press, London, 071 pp. Mason, B.J. and Chien, C.W., 1961. Cloud-droplet growth by condensation in cumulus. Q. J. R. Meteorol. Soc., 88:136-142. Neiburger, M. and Chien, C.W., t 960. Computation of the growth of cloud drops by condensation using an electronic digital computer. Physics of Precipitation. Geophys. Monogr., 5: 191-210. Pruppacher, H.R. and Klett, J.D., 1978. Microphysics of Clouds and Precipitation. Reidel, Boston, 714 pp. Rooth, C., 1957. On a special aspect of the condensation process and its importance in the treatment of cloud particle growth. Tellus, 9: 372-377. Saxena, V.K. and Fukuta, N., 1982. The supersaturation in fogs. J. Rech. Atmos., 16: 327-335. Smirnov, V.I., 1971. The rate of quasi-steady growth and evaporation of small drops in a gaseous medium. Pure Appl. Geophys., 86:184-194. Twomey, S., 1959. The nuclei of natural cloud formation, ll. The supersaturation in natural clouds and the variation of cloud droplet concentrations. Geofis. Pura Appl., 43: 243-249. Young, K.C., 1974. The evolution of the drop spectra through condensation, coalescence, and breakup. Am. Meteorol. Soc. Conf. Cloud Physics, Tucson, Arizona, pp. 95-98.
105
Elsevier Science Publishers B.V., Amsterdam
Water supersaturation in convective clouds Norihiko Fukuta
Department of Meteorology, Universityof Utah, Salt Lake City, UT84112, USA (Received June l, 1992; revised and accepted January 21, 1993)
ABSTRACT The microphysics-dynamics interaction of clouds was theoretically studied in the zone after maximum supersaturation ( S - 1 )me where the droplet number concentration remains nearly constant. The analytic solution obtained, employing the Maxwellian droplet growth theory, describes (S-1 ),~c=Iwau/an-i/2, where I is the proportionality constant, wu the updraft velocity and n the number concentration of the droplets. This solution agrees well with previous studies. Factor I increases with altitude in the adiabatic atmosphere, decreases with temperature under constant pressure and increases with pressure under constant temperature. For the zone sufficiently after ( S - 1 )me, an approximate relationship ( S - 1 ) oc r - i oct - i/3 is shown to hold, where ( S - 1 ) is the supersaturation, r the average droplet radius and t the time. Using the diffusion-kinetic theory of droplet growth, which includes the effects of thermal accommodation and condensation coefficients, numerically soluble relationships are derived for ( S - 1 ), r and t. Application of this theory is shown to increase ( S - 1 )me considerably. The Maxwellian analytic solution that is obtained, the variation of Factor I under different atmospheric conditions and the effect of condensation and thermal accommodation coefficients through the use of the diffusion-kinetic droplet growth theory suggest that maximum supersaturation may reach as high as 10% and beyond in convective clouds. RI~SUMI2 On 6tudie throriquement les interactions entre la microphysique et la dynamique des nuages dans la zone au-delh de la sursaturation maximale ( S - 1 )me o/l la concentration en nombre des gouttelettes est pratiquement constante. La solution analytique obtenue, qui utilise la th6orie de Maxwell de croissance des gouttes, donne ( S - 1 )mc=Iw3J4n-i/2, oil Iest la constante de proportionnalitr, wu la vitesse du courant ascendant et n la concentration en nombre des gouttelettes. Cette solution s'accorde bien avec de prrcrdentes 6tudes. Le facteur I augmente avec l'altitude dans l'atmosph+re adiabatique, diminue avec la temprrature sous pression constante, et augmente avec la pression sous temprrature constante. Dans la zone suffisamment au-delh de ( S - 1 )me, on trouve une relation approchre ( S - 1 )oc r - l o t t-1/3, o/l ( S - 1 ) est la sursaturation, r le rayon moyen des gouttelettes, e t t le temps. En utilisant la throrie cinrtique de croissance des gouttelettes, qui prend en compte les effets de l'accommodation thermique et des coefficients de condensation, on 6tablie des relations pouvant se rrsoudre n u m r r i q u e m e n t pour ( S - 1 ), r et t. L'application de cette throrie montre une augmentation considrrable de ( S - 1 )me. La solution analytique de Maxwell obtenue, la variation du facteur I sous diffrrentes conditions atmosphrriques, et l'effet des coefficients de condensation et d'accomodation thermique par le biais de l'utilisation de la throrie cin6tique de
0169-8095/93/$06.00
© 1993 Elsevier Science Publishers B.V. All rights reserved.
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N. FUKUTA
croissance des gouttelettes, sugg6rent que la sursaturation maximale peut atteindrc el marne d6passer 10% dans les nuages convectifs.
INTRODUCTION
A rising thermal is known to cool through adiabatic expansion and raise the relative humidity. As the thermal passes through the lifting condensation level (LCL) or the convective condensation level (CCL), the relative humidity exceeds 100%, or the air space becomes supersaturated, and cloud condensation nuclei (CCN) begin to activate. The activation continues with the growth of formed droplets. The number and the size of the droplets thus increase, while the cloudy parcel sustains its upward motion and raises the level of effective supersaturation further. Under this condition, the population of formed cloud droplets abstract moisture from the immediate and limited environment and return the latent heat of condensation generated during their growth to it. Both of these processes serve to reduce supersaturation in the vicinity of growing droplets. The larger the droplet size and the number concentration, the faster the supersaturation reduction. Eventually, the supersaturation reduction rate, by the population of growing droplets, comes to balance the production rate by the cloud parcel lifting, and there the supersaturation reaches the maximum. At this point, CCN activation ceases and the number of droplets in a given cloud parcel remains unchanged thereafter, at least for some period of time. Cloud droplets, however, will continue to grow after this stage, for a substantial period of time, under slowly reducing supersaturation. In the real clouds, entrainment of outside air by the cloudy parcel distorts this simplified picture. Nevertheless, treatment of such an idealized system will provide a clear footing for understanding the complex system of real clouds, including their properties of radiation energy transmission. This nucleation-growth-dynamics interaction at and above the cloud base was studied numerically by Howell ( 1949 ), with the help of a hand calculator. Later Neiburger and Chien (1960) and Mason and Chien ( 1962 ) researched the process using an electronic digital computer. More sophisticated numerical studies of cloud processes were later done by Clark (1973, 1974), Young (1974), Clark and Hall ( 1979 ) and Grabowski ( 1989 ). However, as typical for numerical studies, the relationship among factors directly pertinent to the cloud supersaturation remained unexposed. Twomey (1959), on the other hand, took an analytical approach to the problem. By expressing the CCN activity spectrum with an exponential function of supersaturation ( S - 1 ), where S is the saturation ratio, and introducing an approximation scheme for the nucleation-growth interaction in the rising cloudy air, he obtained a range in which the solution is confined. Works of Neiburger and Chien, Mason and
WATER SUPERSATURATION IN CONVECTIVE CLOUDS
107
Chien and Twomey involved the so-called "Maxwellian" condition at the droplet surface, which assumes that the temperature and the vapor density just above the droplet surface are the same as those on the real surface. Therefore, the vapor density at the droplet surface is saturated at the temperature. It is well accepted that exchange of mass, heat and momentum between the gaseous and condensed phases takes place not necessarily with full efficiency. The factors describing the efficiency are known as the condensation (for G L interface) and the deposition (for G-S interface ) coefficients for mass exchange, the thermal accommodation coefficient for heat, and the momentum accommodation coefficient for momentum, where G stands for gas, L for liquid, and S for solid, respectively. This makes the Maxwellian condition no longer applicable at the droplet surface, and the condition of the droplet system changes to satisfy the steady state and continuity requirements of fluxes involved (Fukuta and Walter, 1970). A recent experimental study (Hagen et al., 1989) showed that, although the condensation coefficient appeared to be close to unity or at the theoretical upper limit at the beginning of the laboratory experiment on cloud processes, the value went down towards 0.01 as the growth process continued. This seems to support the importance of accommodation coefficients involved in growth kinetics of cloud droplets (diffusion-kinetics, Smirnov, 1971 ). Under the condition of forming clouds where supersaturation is constantly being generated, the vapor and temperature fields of each droplet have a limited space restricted by boundaries of cells or spaces between neighboring ones. This deviates from the concept of fields commonly applied in Cloud Physics. Such fields are assumed to have infinite expanse. However, as shown in Fukuta ( 1992 ), this effect in the kinetics of droplet growth has been identified as small in the range of normal cloud conditions for both Maxwellian and diffusion-kinetics. Supersaturation plays an important role in determining the characteristics of liquid phase clouds through the microphysics-dynamics interaction. If the process occurs in a sub-freezing temperature zone, it is also expected to influence corresponding ice phase processes, including nucleation and growth. After the maximum, supersaturation generation by the updraft and the abstraction by the droplet system become interlocked in this time sequence. In a time sequence of processes, it is well known that the slower process dominates. The shortness of the relaxation time of the droplet system thus leads to a condition where the slow process of supersaturation generation or reduction in the cloud, i.e., updraft or downdraft, governs the balancing level. Under this condition after the maximum supersaturation, the droplet number concentration remains nearly constant and the supersaturation, including the maximum, can be estimated retrospectively, as long as the entrainment process is slow, solely from the properties of the droplet system and the thermodynamics as well as the dynamic condition of the corresponding portion of the cloud. It is the
108
N FUKUTA
purpose of this paper to analytically treat the cloud supersaturation after the maximum, with proper approximations and to clarify the relationships among the contributing factors by applying the Maxwellian and the diffusion-kinetic theories of droplet growth in a back-tracking manner. ENVIRONMENTAL
PARAMETERS
IN C L O U D Y U P D R A F f
The supersaturation is defined as S-l=
e_l,
(1)
es
where e and es are water vapor pressure of the environment surrounding the droplet in question and that saturated at the environmental temperature, T, respectively, e depends on the air pressure p and the temperature T, while es is a function of T alone. Under the condition e - es, the change in supersaturation may be expressed as d ( S - 1) _ 1 (de-des). es
(2)
The supersaturation is generated due to cloud air lifting and is depleted by droplet growth. The balancing level of supersaturation is hence achieved by these two processes. Figure 1 describes the continuing CCN activation and the system of growing droplets shortly after the moist air rises through the LCL (or CCL) level. In the figure, an arbitrary position of the cloud system path (T, e) is described by combination of a dry adiabatic path from the cloud base ( To, ec) to a point Q and an isobaric process of droplet growth from Q to ( T, e). For the dry adiabatic process, the pressure change with altitude is given by the hydrostatic equation
dp -~z=p,g,
(3)
where z is the vertical distance, Pa the density of air and g the gravitational acceleration. The temperature change involved during the dry adiabatic process and the droplet growth may be described by a modified pseudo-adiabatic equation dT=-
_1 (gdz+Ldw),
(4)
Co
where cp is the specific heat at constant pressure and w the mixing ratio. Note that w is not the saturation mixing ratio, ws, because, as can be seen from the figure, the point ( T, e) has not yet reached the saturation or is not on the
WATERSUPERSATURATIONIN CONVECTIVECLOUDS
109
e
DRY / AD IA B A T ~ / / J e
-*(Tc,ec)
des
Ida2
j/ jf //
-I-2
dT
"
- T
d T ~ - -
Fig. I. The cloud microphysics-dynamics interaction above the cloud base. An arbitrary point ( T, e) belonging to actual cloud which contains growing droplets under generating supersaturation is expressed by combination of dry adiabatic process from (To, e¢) to Q, and droplet growth process from Q to ( T, e). For notations, see the text.
saturation vapor pressure curve es. The system still holds supersaturation. In Fig. l, the first term on the right-hand side corresponds to dT~ and the second dT2. We now proceed to describe e and es terms in Eq. (2), which are a function ofp and T, thus far described. As is evident in Fig. l, de = del + de2.
(5)
From the figure, and using Eq. (3) and w ~- ee/p under e << p, where E= M~ Ma=0.622, M and Ma being the molecular weight of water and that of air, respectively, we obtain de1 = _Pvg dz,
(6)
where Pv is the density of water vapor. This is because, under the condition of adiabatic expansion without condensation, w remains constant and therefore eocp and for a small temperature variation of the process in question, eocpv. Since the de2 term is due to vapor removal by droplet growth de2 =P- dw. Combining Eqs. (6) and (7) with Eq. ( 5 ), we have
(7)
I 10
N. FUKUTA
d e = _Pvg d z + P dw.
(8)
The saturation vapor pressure, es, may be expressed as the sole function of temperature by the Clausius-Clapeyron equation in the differential form as e~L
des = R~. ~- d T,
(9)
where Rv is the specific gas constant for water vapor. Combination of Eqs. (2), (8) and (9) yields d ( S - 1)=Adz+Bdw,
(10)
where
~Lg ( 1 T ~ , A-RaT~ cp eLJ
(l l)
and B-
eL 2
q
p
(12)
cpRa T 2 eel"
Therefore the changing rate of supersaturation may be expressed as (Mason, 1971 )
d(S-l)_AdZ dt
dw
~+B
(13)
dt"
The first term on the right-hand side of Eq. ( 13 ) describes supersaturation generation rate by cloudy air without the influence of droplet growth (dry adiabatic process). The second term describes its reduction rate due to the growth. EQUATIONS FOR DROPLET GROWTH
When the radius is small compared with the distance between its two nearest neighbors (Fukuta, 1992) and disregarding the Kelvin effect of surface tension on the curved surface and the solute effect, the growth rate of pure water droplet in the cloud may be expressed by the diffusion-kinetic equation as (Fukuta and Walter, 1970 )
dm_
4rtr(S- 1 ) L2
-~
(14) 1
'
WATER S U P E R S A T U R A T I O N IN CONVECTIVE C L O U D S
111
where m, t, K and D are the droplet mass, the time, the thermal conductivity of air and the diffusivity of water vapor in air, respectively. Ps~ is the water vapor density in the surrounding air environment at the temperature, T~ ( = To), r
f~=r+t,~'
(15)
[,
(16)
K(27tRaT~) 1/2 ( 2 - o r )
~= ~
2a
'
Cvand a being the specific heat at constant volume and the thermal accommodation coefficient of air, respectively, and r
fp=r+6,
(17)
_
xl/2
fl being the condensation coefficient of water vapor. When r -, oo, f , ~ 1, fa --' 1 and Eq. (14) converges into the Maxwellian,
dm
4rtr(S- 1 )
( 19 )
SUPERSATURATION
The microphysics-dynamics interaction at and just above the cloud base determines the initial characteristics of cloud droplets. They are complex, and only numerical modeling of processes involved can provide a proper estimate. On the other hand, an analytic approach, if possible, normally gives quick illumination to the processes within the assumption and approximation applied. In this regard, the work of Twomey ( 1959 ) is of value although it includes a number of approximations and stays within the range of Maxwellian theory. In addition, these deductive approaches suffer from the uncertainty caused by entrainment processes in general. In natural clouds, the zone below the maximum supersaturation is shallow compared with the rest, and as long as the cloud parcel is moving in the vertical direction, supersaturation (or undersaturation) keeps changing. However, after the maximum supersaturation, activation of CCN ceases, and the
l 12
N. FUKUTA
droplet number concentration remains nearly constant, an advantage for analytic treatment. Therefore, we shall first examine the behavior of this zone after the supersaturation m a x i m u m in terms of droplet-growth-cloud-dynamics interaction and retrospectively come back to the m a x i m u m for its assessment.
Features of droplet population before and after the supersaturation maximum While the cloudy air mass rises towards the zone of m a x i m u m supersaturation, the heterogeneous condensation nucleation continues. As a result, the droplet population in this zone becomes polydisperse, and the number concentration n increases with time. Whereas, in the zone after the maximum, the nucleation ceases while droplets continue to grow. There, n practically remains constant provided that entrainment of outside air is negligible, and the droplet growth leads closer to the monodisperse condition. In the following sections, we shall treat the population of growing droplets after the supersaturation m a x i m u m and derive the m a x i m u m by reversing the course through the use of the above-mentioned two equations of droplet growth.
The Maxwellian droplet growth theory The change of liquid water content in Eq. (13 ) may be expressed as the product between the number of droplets in unit weight of air n/p~* and the droplet mass increase dm, or dw = - n dm, Pa
(20)
where
dm = 4nr2pLdr.
(21 )
By integrating Eq. ( 13 ) with Eqs. (20) and (21 ) under constant updraft velocity, we have
( S - 1) = E t - F r 3
(22)
where
E=Awu,
(23)
and *Droplets are assumed to be completely monodisperse. While the actual population is polydisperse, the condition after the supersaturation m a x i m u m becomes closer to the assumed. In a numerical treatment, s u m m a t i o n of appropriate term within unit volume will replace n.
WATERSUPERSATURATION INCONVECTIVE CLOUDS F - 4npL Bn 3pa
1 13
(24) '
wu = d z / d t and PL being the updraft velocity and the density of liquid water, respectively. Equation (22) is, in essence, conservation o f supersaturation and means that the supersaturation in the system o f growing droplets is that generated, minus that depleted by droplet growth. On the other hand, combination o f Eqs. ( 13 ), ( 19 ) and (20), considering n as a constant, yields d(S-1) dt =E-GM(S-1)r,
(25)
where
-
4rtBn
.
(26)
aM pa [KRL T 2ooI-~s~D] Equation (25) describes conservation o f the rate o f change o f supersaturation; i.e., the rate o f actual supersaturation change is given by the difference
between the generation rate of the environment and the depletion rate by the droplet system. Equations (22) and (25) describe the actual system. We hence apply the latter to the supersaturation m a x i m u m ( S - l ) m with respect to t=tm or Eq. (25 ) = 0. Then, E 1 ( S - 1 ) m - - -GM rm"
(27)
Inserting Eq. (27) into Eq. (22), with the condition ( S - 1 ) = ( S - 1 )m, we obtain F 3 _ 1 1 t m = ~ r m - t GM rm"
(28)
Thus, Eq. (28) conveys the restriction of Eq. (27) and is valid only at one point which corresponds to the real m a x i m u m supersaturation, ( S - 1 )rnc, and the matching radius, rmc. The task now is to find out this solution point.** "Equation (27) is valid by itself for the supersaturation maximum and describes the condition for the given time sequence of the event. However, if we consider (S- 1)m and rm as variables, the equation covers the area which does not correspond to the real system in question except the point given by Eq. ( 27 ). Equation (28 ) has the same meaning as that of Eq. ( 27 ) except that it contains tm term. Nevertheless, the period tm covers carries the same condition as that after the supersaturation maximum, i.e., monodisperse droplets with constant n (no nucleation in the midcourse) which is a large deviation from reality. In this regard, at the point of (S - 1)me and rmcwhich corresponds to a real (solution) value, tmcis still under restrictions and associated errors, and therefore is not realistic.
114
N. FUKUTA
The behavior of Eq.(28) is shown in Fig. 2. In this figure, the rm curve covers the entire range of r-coordinate, while the curve does not exist between 0 and/me on t-coordinate. This is because, in the gap, the condition of maxim u m supersaturation, i.e., balancing between the supersaturation generation rate o f the rising cloudy air parcel and the abstraction rate of the growing droplets, has not yet been met under the constant droplet n u m b e r concentration retrospectively applied. As can be seen in Fig. 3, the real supersaturation increases towards the m a x i m u m monotonically. This is to say that as t in-
It,.'=r~'
/.
' , ~
-t
tme
Fig. 2. Relationship among the terms in Eq.(28). t,,~ l/rm relationship represents the condilion of m a x i m u m supersaturation and lm VCrm3 relationship describes the condition for total depletion of supersaturation by droplets. The real supersaturation maximum corresponds to (t .... r.,~,) point. For details of discussion, see the text.
(S-~)
/s
~\\
REAL
PROCESS
(s -l)~,
~
-
tmc Fig. 3. The relationship between ( S - 1 )m and the real supersaturation (dashed line). Two terms of ( S - 1 )m (thin solid lines) are also shown. For notations, see the text.
WATERSUPERSATURATION IN CONVECTIVECLOUDS
115
creases and the system reaches the supersaturation maximum in Fig. 3, the same increase moves t to tm~ in Fig. 2 where, for the first time, t comes in contact with the rm curve that satisfies the condition of the supersaturation maximum. Thus, it is clear that the point corresponding to the true supersaturation maximum is that of the smallest t on rm curve, or (tmc, rmc). Thus, (tmc, rmc) is the minimum of the rm curve with respect to r variation, and we can obtain the point by differentiating Eq. (28) with rm and setting the derivative to zero, i.e., (E~ 1/4 rmc = \3--ff~MJ (29) Replacement of Eq. (29) into Eq. (28) results in
4
F 1/4
tm~ - - 3 3 / 4 EI/4G3M/4.
(30)
From Eqs.(27) and (28), we have ( S - 1)m~--
31/4E3/4F 1/4 G~ 4
(31)
Equations (29) through (31) satisfy Eqs.(22), (25) and (27), the conditions of supersaturation and supersaturation generation rate conservation, as well as the maximum supersaturation. In regard to the formulation of (S-1)mc, two main system variables Wuand n are involved. Using Eqs.(23), (24), (26), (29), (30) and (31), one finds
W1/4 rmcOC nl/2, U
(32)
1
tmc °c
!/4
Wu
n
1/2'
W3/4 U
( S - 1 )m¢OC nl/~.
(33)
(34)
The relationship Eq. (33), i.e., tmcOCW u 1/4 is inverse to the tendency of the actual process (Howell, 1949 ). It has come in the course of combining Eq. (22) and Eq. (25 ) = 0 for estimation of ( S - 1 ) me. Equation (25 ) shows that r and ( S - 1 ) are sensitive to the time rate of change of supersaturation, viz. d ( S - 1 )/dt, and not to t itself. Under the condition of Eq. (25) =0, i.e. at the supersaturation maximum, even this influence disappears. As a result, rm and ( S - 1 )m determine mac and ( S - 1 )mc explicitly regardless of t. As discussed in the preceding section, tmc is under full influence of the unreal term, 1/rm,
116
N. FLIKUTA
which has been introduced into Eq. (28) as a part of the supersaturation maxi m u m condition, the condition never met in the range of t < t,,,. It is evident in Figs. 2 and 3 that there exist neither rm nor the corresponding ( S - 1 )m in the zone between the coordinate origin and tmc. In other words, the solutions for rmc and ( S - 1 )me have been obtained by describing the real time rate of change with that for tin.
The diffusion-kinetic droplet growth theory To include the effects of condensation and thermal accommodation coefficients, we replace GM in Eq. (27), with GDK representing the effect of diffusion-kinetic treatment of droplet growth, i.e.
4zeBn GDK--
L2
1
'
(35)
to obtain E 1 ( S - 1 )m --GDK rm"
(36)
GDK is a function of rm and smaller than GM, implying that ( S - 1 )m value becomes larger for the diffusion kinetic treatment than for the Maxwellian case. For the treatment in detail, rm terms in GDK have to be exposed as Hi rm GDK -- H2 rm + H3'
( 37 )
where Hi -
4 zcBn Pa
,
(38)
p ~ DJ
(39)
[ Z2[a H3 = LK/~ T 2 +ps~D]"
(40)
Combining Eqs. (22) and ( 36 ), together with Eqs. ( 38 ) through (40), we obtain
WATERSUPERSATURATIONIN CONVECTIVECLOUDS
(S-1)m
1 EH2 1 ÷ EH3 --HI rm Hi r 2'
117
(41)
and tm /-/2 1 //3 1 F 3 --Hi rm 4"~-1 ~ m + E rm"
(42)
Comparing the above two equations with Eqs. (27) and (28), it becomes clear that the second terms on the right-hand side of both Eqs. (41 ) and (42) have appeared due to the effects of condensation and accommodation coefficients, or the diffusion kinetic treatment. Also, Eq. (41 ) does not have any extrema, while Eq. (42) has one extremum. By differentiating Eqs. (41) and (42) with rm and dividing the latter result with the former, and setting the derivative dtm/d(S- 1 )m=0, we arrive at E 3 fH2
H3
r2mc- 3FH-~---~c + r3~) = 0,
(43)
which is analytically insoluble but is easily soluble with the numerical method. Application of rm~, thus obtained, to Eqs. (41 ) and (42) will yield ( S - 1 )m~ and tmc for the diffusion kinetic droplet growth theory, but the latter term will not correspond to the real cloud condition as explained above. DISCUSSIONS
In the above treatments, we have dealt with the growth processes of the population of cloud droplets in the zone after the maximum supersaturation where the droplet number concentration remains virtually unchanged and droplets are more uniform in size. Analytic solutions of tmc (unreal), rmc and ( S - 1 )me, for the Maxwellian droplet growth theory, were obtained. A set of equations for the solutions with the diffusion-kinetic theory, easily solvable by the numerical method, were described. The underlying assumption for the treatment is that the entrainment effect is negligible relative to the droplet growth process. In real clouds, such a condition does exist, like the weak echo region (WER) of the thunderstorms where virtually no dilution occurs. We now examine the results obtained above in comparison with natural conditions.
The system with Maxwellian droplet growth When the cloud base condition is given, by exposing the main variables, n and Wu, we can write Eq. (31 ) as
118
N. FUKUTA W3/4
(S-
1 )mc = I - - F/I/2'
(44)
where
I { pLp2aA3~''4( L2 =\i6~-~]
\KRJL
1
'~/4
+ps~-D)
(45)
Figure 4 shows the relationship among ( S - 1 )me, Wuand n, computed from Eq. (45) which uses the Maxwellian droplet growth theory, for the case of Tc=0°C and Pc (the cloud base pressure) =500 mb and I = 1.749 x 10 .3 c m -9/8 s 3/4. In the figure, one can see that there exists a substantial zone on n-Wu plane where ( S - 1 )me exceeds a few percent, the previously believed upper limit. It should be pointed out that, for a given condition of n and Wu, ( S - 1 )me is achieved due to balancing of the supersaturation depletion rate by growing droplets with the generation rate by adiabatically cooling cloudy air. The Maxwellian droplet growth and the resultant supersaturation depletion rate are faster than those of the diffusion-kinetic rate. From this viewpoint, the ( S - 1 )me values in Fig. 4 represent the practical lower limit, and the actual values may well be higher as we shall discuss later. Thus, by examining the previously unexplored zone in the figure, it has become clear that,
I
-7
O. 0.0
Fig. 4. Relationship among the maximum supersaturation ( S - 1 )mc, the updraft velocity w,, and the number concentration of cloud droplets n, computed from Eq.(45) for Maxwellian droplet growth theory. The temperature and pressure of the cloud base are taken as 0°C and 500 mb, respectively.
W A T E R S U P E R S A T U R A T I O N IN C O N V E C T I V E C L O U D S
119
contrary to the c o m m o n belief o f ( S - 1 )mc < 1% (Mason, 1971; Pruppacher and Klett, 1978 ), ( S - 1 )mc can reach m u c h higher values under realistic cloud conditions. Figure 5 describes rmc values u n d e r the same condition as in Fig. 4. Since the Maxwellian droplet growth theory is the faster o f the two dealt with here, rmc in the figure also represents the lower limit. If n is unusually low and wu very high, rm~ can reach 10/tm and beyond. Whereas, if n is high, like in polluted air, and wu low, rmc will reduce to submicron level. As can be seen in Eq. (44), F a c t o r / c o n t r o l s ( S - 1 )mc, and it m a y be worthwhile to examine the effect o f variables describing the cloud base on the term. Figure 6 shows a variation o f Factor I at the cloud base in a dry adiabatic atmosphere with potential temperature 0 = 20°C. In the figure, it can be seen that a higher cloud base tends to produce a higher ( S - 1 )mc under a given n and Wu. This is due to the fact that variation o f t h e r m o d y n a m i c factors o f the adiabatic cooling, such as A and B, exceeds that o f kinetic factors o f droplet growth in the atmosphere. Figure 7 describes a variation o f Factor I as a function o f temperature T at three different atmospheric pressures, p = 500, 700 and 900 mb. It m a y be
10
E
:k
v
I
0.1
Fig. 5. Relationship among the radius of droplet at the maximum supersaturation, rmc,the updraft velocity wu, and the number concentration of cloud droplets n, computed from Eq. (29). The cloud base condition is same as that in Fig. 4.
120
N. FUKUTA
T 61
-40 -30 -20
(°C) -I0 0
I0
20
]
5 ,o,~,, 4
% --
3
400
560
600 p
7oo 800 (mb)
960
,600
Fig. 6. Variation o f Factor 1 in Eq. (45) in a dry adiabatic a t m o s p h e r e with a potential temperature 0 = 20 ° C. The curve represents the cloud base.
8~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6r ~
5
3
[
i
i
i
I
-°4o -~;o -~o 40 T
6
I'0
20
(°C)
Fig. 7. Variation of Factor I in Eq. ( 4 5 ) as a function o f t e m p e r a t u r e T at atmospheric pressure p = 500, 700 a n d 900 mb, respectively.
WATER SUPERSATURATIONIN CONVECTIVECLOUDS
121
seen in the figure that factor/, therefore ( S - 1 )me, tends to become larger for a given set of n and wu when T lowers. At 700 m b level, a cloud base temperature shift from 10 ° to - 2 0 ° C increases the factor by nearly 2. Again, the thermodynamic terms like B, instead of kinetic terms of droplet growth, dominate the change and produce this tendency. Figure 8 shows a variation of Factor I as a function of atmospheric pressure p at temperatures T = - 20 °, 0 ° and 20 ° C. Low pressure gives smaller Factor /, for the same reason as stated above. At 0°C, a pressure increase from 500 mb to 700 m b raises the factor by 22%. It is not possible at this m o m e n t to compare the present analytic result with observed data since an accurate measurement m e t h o d of cloud supersaturation does not exist. Results of numerical cloud modeling are now available, but they too are not suitable for comparison as they contain other factors which are not included in the present treatment. Results of two computations, one numerical (Howell, 1949) and the other analytic (Twomey, 1959), appear to provide a comparison basis for the present treatment. Table 1 compares the result of the numerical computation by Howell (1949) and the present Maxwellian treatment under p = 8 0 0 mb and T = 0 ° C . For computation numbers 1 and 2, ( S - 1 )me agreement is complete within the limit of errors for the values used in the computation. For computation number 3, the present treatment gives a value larger by a factor of 2.2 than that of Howell's. However, Howell's result for this computation is not consistent with 4
5 o~1¢o
% -
2
i
/
I
0
400
5(30 660 700 p (mb)
8()0
900
Fig. 8. Variation o f F a c t o r I in Eq. ( 4 5 ) as a f u n c t i o n o f a t m o s p h e r i c pressure p at t e m p e r a t u r e s T = - 2 0 °, 0 ° a n d 20°C, respectively.
122
N. FUKUTA
TABLE 1 Comparison between the result of numerical computation by Howell (1949) and t he present Maxwellian treatment under p = 800 mb and T = 0 °C Computation Number Howell (1949)
n (cm -3)
wu (cms -~ )
Howell
Eq. (31)
~r"~ (llm) Howell
Eq. (29)
1
249 39.97 416
60.4 1.51 30.2
0.36 0.056 0.073
0.354 0.0555 0.163
1.78 2.88 2.65
1.55 1.53 1.01
3
( S -- 1 ) ~C
TABLE2 Comparison for the Maxwellian maximum supersaturation between the "overestimation" value ( S - 1 ) " of Twomey (1959) and the present treatment ( S - l ) m c (Eq. 31 ) under p = 8 0 0 mb and T= IO'~C [~'u (ems -~)
tl (cms -3)
( S - - I )" ( S - l )me • (%) (%) (cm 3)
( S - l )" (%)
( S - I )me tl (%) (cm 3)
{,~)'-- 1 ) ' (~T_ ] )m, t%) (%)
10 100 1000
37 61 100
0.17 0.75 3.3
0.13 0.61 2.8
0.139 0.644 2.98
0.06 i).26 ~.1
0.178 0.778 3.42
60 89 131
310 554 985
0.0614 0.258 1.09
the rest, such as ( S - 1 ) after ( S - 1 ) m c is not proportional to t j~ as we shalle see later. Therefore, this result of his may be discarded. The droplet radius rmc at ( S - 1 )me is larger with Howell's computation. This is apparently because he took a droplet growth equation formulated by Langmuir, which resembles the diffusion kinetic equation, but was shown to be erroneous (Fukuta and Walter, 1970 ). The diffusion-kinetic effect is known to persist in the growth process of droplets whose radii are far beyond the normal range of Kelvin and solute effects (Fukuta and Walter, 1970). The latter effects are also included in Howell's work. At any rate, the diffusionkinetic-like equation of Howell has seemingly raised the rmc value, as droplet growth computed with such an equation is slower than that of the MaxweUian theory. The necessary kinetic balance between supersaturation generation and the depletion is achieved only when rmc has reached a larger value. Table 2 gives a comparison for Maxwellian m a x i m u m supersaturation between the "overestimation" value ( S - 1 )" o f T w o m e y (1959) and the present treatment ( S - 1 )me (Eq.31) under p = 8 0 0 mb and T = 10°C. In his analytic treatment, Twomey estimated the range of supersaturation in which the real value is expected to fall. He claims that the upper and lower constraints of supersaturation never differed by more than 30%. In Table 2, it is seen that ( S - 1 )me is slightly larger than ( S - 1 )" in most cases, but never more than 7%, which is within the limit of error arising from the uncertainty of values
WATER SUPERSATURATION IN CONVECTIVE CLOUDS
123
used in the computations. Therefore, there is good agreement between the two computations. The reason why the present result agrees with the upper limit of Twomey's work remains unclear. In the preceding section, we have seen that the supersaturation depletion rate of the population of growing droplets balances for the first time with the supersaturation generation rate of the environment, and the supersaturation reducing capability of the population continues to increase with the droplet size [see Eq. (25) ]. As the droplets grow and pass the point of ( S - 1 )me,the supersaturation depleting capability of the population becomes larger than that at ( S - 1 )me. Nevertheless, the droplet population cannot totally deplete the existing supersaturation immediately after passing the ( S - 1 )me point, because the depletion rate also depends on ( S - 1 ) [see the second term of Eq. (25)]. This is to say that there exists a feedback effect proportional to supersaturation in the supersaturation depleting capability of the population. As a result, the real supersaturation can change only slowly after ( S - 1)me. If we ignore this slow rate of supersaturation change, we have a relationship identical to Eq. (27) which applies after ( S - 1 )me,i.e., E 1 ( S - 1 ) ~ G M r"
(27')
The same condition applies to the differential form of Eq. (22), or
E = 3Fr 2 dr dt"
(46)
After integration, Eq. (46) yields
r3w.t.
(47)
Then, from Eqs. (27') and (47), we obtain l l
( S - 1)~ octl/3.
(48)
As can be seen in Fig. 3, to establish this relationship, in reality, a transitional period is required after passing the ( S - 1 )me point. Thus, except for this transitional period, Eq.(27') holds with reasonable accuracy. In fact, Howell's numerical computations 1 and 2 explicitly demonstrate the above relationship.
The system with diffusion-kinetic droplet growth Under the influence of small thermal accommodation coefficient ot or small condensation coefficient of water vapor fl, or both, droplet growth is known to become slower than what the Maxwellian theory predicts (Fukuta and
124
N. FUKUTA
Walter, 1970). Since ( S - 1 )mc is achieved as a result of competition between the supersaturation generation rate of the cloud environment, which is normally constant, and the depletion rate due to droplet growth, the slowed droplet growth rate pushes the supersaturation to a high level. For an environment of T = 0 ° C and p = 1 atm, if o~= 1 and fl=0.03, the ratio for droplet growth rate between the diffusion-kinetic and the Maxwellian theories reaches 4.7 for r = 1 #m and 1.8 for r = 5/~m. Although rmc values for the diffusion-kinetic treatment are larger than those for the Maxwellian one, the latter shown in Fig. 5 gives an idea of their possible range under the former treatment. The rmc values are most likely within the range of this diffusion-kinetic effect. Clark (1974) reported a factor of 2.1 increase in ( S - 1 )mc from the corresponding Maxwellian value in his numerical model computation using an erroneous diffusion-kinetic droplet growth equation given by Rooth ( 1957 ) (see Fukuta and Walter, 1970). Although the equation used had this problem and the values of accommodation coefficients he employed seemed to be too small, which tend to result in a slower droplet growth rate, his computational result clarified the importance of the effect. A recent cloud model (Grabowski, 1989 ) uses a more realistic constant which is based on a combination of ~ = 0.9 and fl=0.03. Saxena and Fukuta (1982) reported 1.8 for the ratio of m a x i m u m supersaturation between the diffusion-kinetic and the Maxwellian treatments under a = 1, fl=0.04, wu= 10 cm s-1 and n = 100 c m - 3 While detailed analysis of the diffusion-kinetic effect is beyond the scope of the present paper, it is clear that this will increase the level of m a x i m u m supersaturation considerably and it deserves special attention. The derived relationships in the section describing the diffusion-kinetic droplet growth theory will provide a basis for the numerical estimation.
Field application In some clouds, the entrainment process occurs significantly, and the above treatment becomes difficult to apply. If, however, the process is moderate, by observing two main variables, n and wu, as well as the cloud base condition, solving the "would have been" ( S - 1 )me as a function of height and finding the largest, one may estimate retrospectively the real ( S - 1 )me and its location. The relationship (27'), which requires additional measurement of droplet radius, may be useful for assessing the supersaturation level after ( S - l)mc. CONCLUSION
The microphysics-dynamics interaction above the cloud base of convective clouds has been analytically studied in the zone after the supersaturation maximum. Following are the main findings.
WATER SUPERSATURATION IN CONVECTIVE CLOUDS
125
( 1 ) By realizing that the number concentration of cloud droplets remains nearly constant and the supersaturation changes rather slowly after the supersaturation maximum, and by reversing the time, analytic solutions for the maximum supersaturation ( S - 1 )me and the corresponding droplet radius were obtained for the Maxwellian growth theory with the help of fictitious growth time [see Footnote for Eq.(28)]. For the diffusion-kinetic growth theory, the relationship of those two real values as well as the fictitious growth time, analytically insoluble but numerically easy to solve, were also derived. (2) For the Maxwellian growth theory, the analytically solved maximum supersaturation agreed well with the computed values of other studies. (3) ( S - 1 ) mcbased on the Maxwellian droplet growth theory was found to be proportional to w ~ 3/4 n-1/2, where wu is the updraft velocity and n the number concentration of cloud droplets after the maximum. (4) The proportional factor for this relationship includes the cloud base condition which consists of thermodynamic terms as well as kinetic terms of droplet growth, and the former dominate the latter. The proportional factor was found to increase with height in the adiabatic atmosphere, decrease with temperature under constant pressure and increase with pressure under constant temperature. ( 5 ) A relationship ( S - 1 ) ocr - t oct - 1/3, where r is the droplet radius and t the time, applicable to the condition sufficiently after ( S - 1 )mc was derived. (6) Application of the diffusion-kinetic theory of droplet growth, which deals with the effects of thermal accommodation and condensation coefficients, was shown to yield ( S - 1 )me values substantially larger than those of the Maxwellian theory. (7) Considering ( S - 1 )m~ values estimated from the analytic solution based on the Maxwellian droplet growth theory, the effect of condensation and thermal accommodation coefficients through the use of the diffusion-kinetic droplet growth theory, and variation of the proportional constant under different environmental conditions, it is concluded that ( S - 1 )mc as high as 10% and beyond may be possible in convective clouds. ACKNOWLEDGMENT This work was supported by the Division of Atmospheric Sciences, National Science Foundation under Grant ATM-8616568. REFERENCES Clark, T.L., 1973. Numerical modelingof the dynamics and microphysicsof warm cumulus convection.J. Atmos. Sci., 30: 857-878. Clark, T.L., 1974. On modellingnucleation and condensation theory in Eulerian spatial domain. J. Atmos. Sci., 31:2099-2117.
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Clark, "I-.L. and Hall, W.D., 1979. A numerical experiment on stochastic condensation theory. J. Atmos. Sci., 36: 470-483. Fukuta, N., 1992. Theories of competitive cloud droplet growth. J. Atmos. Sci., 49:1107-1114. Fukuta, N. and Walter, L.A., 1970. Kinetics of hydrometeor growth from a vapor-spherical model. J. Atmos. Sci., 27:1160-1172. Grabowski, W.W., 1989. Numerical experiments on the dynamics of the cloud- environment interface: Small cumulus in a shear-free environment. J. Atmos. Sci., 46:3513-3541. Hagen, D.E., Schmitt, J., Trueblood, M., Carstens, J., White, D.R. and Alofs, D.J., 1989. Condensation coefficient measurement for water in the UMR cloud simulation chamber. J. Atmos. Sci., 46: 803-816. Howell, W.E., 1949. The growth of cloud drops in unitormly cooled air. J. Meteorol.. 6: 134149. Mason, B.J., 1971. The Physics of Clouds. 2nd Ed. Oxtbrd University Press, London, 071 pp. Mason, B.J. and Chien, C.W., 1961. Cloud-droplet growth by condensation in cumulus. Q. J. R. Meteorol. Soc., 88:136-142. Neiburger, M. and Chien, C.W., t 960. Computation of the growth of cloud drops by condensation using an electronic digital computer. Physics of Precipitation. Geophys. Monogr., 5: 191-210. Pruppacher, H.R. and Klett, J.D., 1978. Microphysics of Clouds and Precipitation. Reidel, Boston, 714 pp. Rooth, C., 1957. On a special aspect of the condensation process and its importance in the treatment of cloud particle growth. Tellus, 9: 372-377. Saxena, V.K. and Fukuta, N., 1982. The supersaturation in fogs. J. Rech. Atmos., 16: 327-335. Smirnov, V.I., 1971. The rate of quasi-steady growth and evaporation of small drops in a gaseous medium. Pure Appl. Geophys., 86:184-194. Twomey, S., 1959. The nuclei of natural cloud formation, ll. The supersaturation in natural clouds and the variation of cloud droplet concentrations. Geofis. Pura Appl., 43: 243-249. Young, K.C., 1974. The evolution of the drop spectra through condensation, coalescence, and breakup. Am. Meteorol. Soc. Conf. Cloud Physics, Tucson, Arizona, pp. 95-98.