Much effort has recently gone into explaining the observed broad precoalescence size distribution of droplets in cloud and fogs, because this differs from the results of condensational growth calculations which lead to much narrower distributions. A good example of droplet size-distribution broadening was observed on flight 17 (25 July) of the NRL tethered balloon during the 1987 FIRE San Nicolas Island IFO. These observations caused the interactions between cloud microphysics and turbulent mixing to be re-examined. The findings of Broadwell and Breidenthal (1982) who conducted laboratory and theoretical studies of mixing in shear flow, and those of Baker et al. (1984) who applied the earlier work to mixing in clouds, were used. Rather than looking at the 25 July case at SNI, earlier fog observations made at SUNY (6 Oct. 1982) which also indicated that shear-induced mixing was taking place, and which had a better collection of microphysical measurements including more precise supersaturation measurements and detailed vertical profiles of meteorological parameters were chosen instead

Gerber, H.

English

NASA Technical Reports Server

Nbo-e8249
SUPERSATURATION, DROPLET SPECTRA, AND TURBULENT MIXING IN CLOUDS
H. Gerber
Naval Research Laboratory, Washington, DC 20375-5000
I. Introduction
Much effort has recently gone into explaining the observed broad precoales-
cence size distribution of droplets in clouds and fogs, because this differs
from the results of condensational growth calculations which lead to much nar-
rower distributions. The correct explanation of this difference is important,
since the observed broadening has a strong influence on the optics of the
clouds, as well as on their colloidal stability. Existing explanations, which
have yet to be proved definitive, have generally dealt with the interaction of
turbulence with the cloud, such as in entrainment and mixing (Telford et al.,
1984; Jonas and Mason, 1982; Baker et al., 1980; Clark and Hall, 1979; and many
others). Most have dealt with the "favored-droplet hypothesis", which states
that due to the statistical nature of turbulence some "favored" droplets experi-
ence a lifetime in the clouds which is more conducive to condensational growth
(or evaporation) than other droplets, and hence cause the broadening (e.g.,
Cooper, 1989).
A good example of droplet size-distribution broadening was observed on
flight 17 (25 July) of the NRL tethered balloon during the 1987 FIRE San Nicolas
Island IFO. On this date a stratocumulus cloud cover formed rapidly from clear
sky conditions. The balloon was aloft at the time, and was able to penetrate
the Sc clouds 7 min. after their formation. These virgin clouds, which could
not include the complexity of upwind evolution, contained significant numbers of
32.5 _ diameter droplets (the largest size bin of the CSASP particle spectrome-
ter). Such large droplets cannot easily be explained by a standard updraft
argument. Instead, observed RH and wind shear in the vicinity of these clouds
suggested a formation mechanism which included Kelvin-Helmholtz induced mixing
of a nearly saturated layer. These observations motivated us to take another
look at the interaction between cloud microphysics and turbulent mixing. We
used the findings of Broadwell and Breidenthal (1982) who conducted laboratory
and theoretical studies of mixing in shear flow, and those of Baker et al.
(1984) who applied the earlier work to mixing in cloud. Rather than looking at
the 25 July case at SNI, we chose instead to look in detail at earlier fog ob-
servations made at SUNY (6 Oct., 1982) which also indicated that shear-induced
mixing was taking place, and which had a better collection of microphysical
measurements including more precise supersaturation measurements (see Gerber,
1980 for description of technique) and detailed vertical profiles of meteorolog-
ical parameters; see Fig. I.
In this study we address the following questions:
i. Does B-B (Broadwell-Breidenthal) mixing, or gradient diffusion control
droplet evolution in shear situations involving nearly saturated air?
2. Can B-B mixing account for the observed large transient supersaturations
which can last as long as tens of seconds?
3. Can B-B mixing account for strong droplet broadening and the appearance
of the largest droplets?
II. Broadwell-Breidenthal Mixing
B-B mixing describes a mixing process different from the usual approach in
which turbulent mixing occurs via gradient diffusion. The latter is only
reasonably successful when the scale of the turbulence is small in comparison to
145
PRECED:NG PAGE BLAr_K NOT F;LMED
https://ntrs.nasa.gov/search.jsp?R=19900018933 2020-03-19T21:11:44+00:00Z
the distance across which the diffusing quantity is changing significantly.
Broadwell and Breidenthal (1982) clearly demonstrated in laboratory measurements
that gradient diffusion between two species in shear flow does not hold, but is
instead governed by another mechanism consisting essentially of a two stage
mixing process: For most of the lifetime of a decaying eddy containing two
species, the identity of the species remains largely intact. Only as the Kol-
mogorov microscale is reached do the two species rapidly mix with each other by
molecular diffusion.
We first apply B-B mixing to an eddy with properties estimated from the 6
Oct. fog case. The eddy had a characteristic dimension L = 6.9m, it consisted
of two saturated parcels with a difference in temperature of 3°C (which is the
maximum temperature difference observed over the height of the micromet tower),
and it contained droplets with a size distribution given by curve 0715 in Fig. 1
and measured at about RH = 100Z. We derive an expression similar to the classi-
cal equation
S(t) = Qlfdh- Q2fdw (1)
giving the time evolution of supersaturation S(t) in terms of the mass balance
of total water in an ascending cloud parcel (e.g., see Squires, 1952). Instead
of the excess (supersaturated) vapor released by the change in height dh of the
parcel, we include the release of excess vapor by the B-B mixing of the satu-
rated parcels. Under isobaric conditions, and under the assumption that the
rate of release of excess moisture is proportional to the rate of formation of
interfacial area between the two saturated parcels in the eddy we find
3/2
K
-3/2
- "---!---1 E (r
L2M_ + i
R 2Cp
,+B]Ir _ dt (2)
r
where x = vapor mixing ratio of mixed eddy, x s = saturation mixing ratio of
mixed eddy, w = liquid water mixing ratio, N(r) = particle density at radius r,
C = rate constant, tk = time to reac h Kalmogorov turbulence scale, T L = time for
complete homogenization, and other symbols have the usual meaning.
The predictions of (2) are illustrated in Fig. 2. The curve labeled 0715
shows a rapid release of excess moisture at around 35 sec of the calculation,
with a maximum value of S near the maximum S = 0.43Z that can be achieved by
mixing two saturated parcels with a 3-°C temperature difference, and with a
decrease of S from the maximum corresponding to the takeup by growing droplets
and lasting tens of seconds. Curve 0719 corresponds to the use of the distribu-
tion labeled 0719 in Fig. 1 as the initial condition (initial LWC = 0.09 g/m3);
and the curve for XI0 uses the 0719 distribution with i0 times the LWC value.
We can conclude from Fig. 2 that (a) the B-B mixing mechanism causes the forma-
tion of transient supersaturations in the fog, (b) the release of excess vapor
is faster than the time scale of droplet response, and (c) the time constant of
the decay of the transient S is approximately proportional to the integral
radius of the droplets as is evident from the second term of (2).
146
These conclusions are supported by the observations in the 6 Oct. fog.
Mixing was clearly shear induced as indicated by the stable temperature profile,
gradient diffusion could not have caused the observed supersaturation transients
because of the magnitude of the temperature gradient, and the observed tran-
sients were of a magnitude consistent with the maximum temperature differences
observed with height.
III. Broadening of Droplet Size Distribution
An important consequence of B-B mixing as illustrated in Fig. 2 is that the
release of excess vapor appears to be essentially independent of the droplet
content, and thus can be treated separately from the conversion of the vapor to
liquid, which depends approximately on the integral radius. It is thus possible
to use the droplet growth equation
ffi c s - - + (3)
r r
with an average S parameterized in terms of the integral radius, if the maximum
value of S(t) is measured or estimated.
To fully exploit this possibility, the integral form of (3) was rewritten to
give a set of analytical expressions
'rd > lO-6cm
[ ] r° > rdr = I ro,S,t,rd for S -I (4)
it > 0
Lparticle type
which give one-equation solutions to the time dependence of r, given S (ro =
initial radius, rd = dry nucleus radius). This approach greatly speeds up the
calculations, because (3) is implicit in r and usually requires numerical inte-
gration, and is difficult to use for small droplets. Equation (4) is mathemati-
cally well behaved, and it gives the proper transitions between activated drop-
lets and haze particles; see Fig. 4. Equation (4) should be useful in improving
the coupling between microphysics and dynamic models.
A model was developed to evaluate the effects of B-B mixing on droplet
broadening. The approach, as in Nichols (1987), was to combine the effects of
stochastic turbulent diffusion with explicit microphysical calculations. How-
ever, in the present case only droplet growth by diffusion is considered. Fur-
thermore, this model, rather than being driven by models of turbulent diffusion,
is driven by the observed statistics of S and eddy size (see Fig. 3). High-
lights of the model include: The Monte-Carlo approach is used to randomly mix
eddies in a Lagrangian framework. The mixing process includes contributions
from mixing eddies proportional to their volume, and conserves particle concen-
tration, liquid water volume, and total nuclei volume. Calculations are done in
N(r) space using (4), and timescales are estimated from Eulerian observations of
eddy frequency and the droplet integral radius.
Some initial runs of the model are shown in Figs. 5 and 6. Using the
measured cumulative probability of S and L shown in Fig. 3 with the 0715 size
distribution to initialize the model, results in the droplet spectrum for t = 15
min shown in Fig. 5. On the average droplets have evaporated to smaller sizes
147
in this case. This is consistent with'the slightly subsaturated conditions
reflected by the means in Fig. 3. The statistics in Fig. 3 were collected 1.5-m
above a grassy surface in post sunrise fog where some warming of the surface was
taking place. The second example in Fig. 6, which illustrates the flexibility
of this model, shows a double-peaked distribution with significant broadening
and maximum droplet sizes close to the ones observed in the 6 Oct. fog. For
this case the S distribution in Fig. 3 was biased by S = + .001, the time scale
of the excess S decay was set at a constant 20 s, and 80Z of the final eddies
making up the distribution in Fig. 6 correspond to t = 3 min while the rest is
for t = 15 min. Other runs of the model (not shown), using S decay coupled to
the integral radius as suggested by B-B mixing, gave significantly broadened
size distributions, but none with peaks in the size distribution exceeding about
i0 _m diameter even for t = 60 min.
IV. Conclusions
The observations in the 6 Oct. fog and the initial results of the stochastic
condensational growth model suggest the following conclusions on the importance
of Broadwell-Breidenthal mixing in fogs and clouds:
(a) B-B mixing of nearly saturated parcels at different temperature causes
large transient supersaturations as observed in the 6 Oct. and other fogs.
(b) These transient supersaturations and turbulent mixing cause droplet
spectral broadening governed by the "favored droplet hypothesis"
(c) The effect of B-B mixing on droplet broadening is approximately propor-
tional to the temperature difference of nearly saturated mixing parcels and
inversely proportional to the integral droplet radius, thus hazes and clouds of
low LWC with strong temperature gradients will see the largest effects.
(d) B-B mixing could not explain the observation of 30-_m diameter droplets
in the 6 Oct. fog. The rapid increase of droplet size observed 1.5-m above the
surface is partially explained by the downward mixing of this fog which formed
20 min earlier aloft. It is proposed that B-B mixing contributes primarily to
the mid-size peak (about 5 _m diameter) often found in these fogs, and that the
large droplet peak is due to very large supersaturations generated by another
mechanism found higher up in the fog.
Baker, M.B., R.G. Corbin and J. Latham, 1980: The influence of entrainment on
the evolution of cloud droplet spectra: I. A model of inhomogeneous mixing.
Quart. J. Roy. Met. Soc., 106, 581-598.
Baker, M.B., R.E. Breidenthal, T.W. Choularton and J. Latham, 1984: The effects
of turbulent mixing in clouds. J. Atmos. Sci., 41, 299-304.
Broadwell, J.E., and R.E. Breidenthal, 1982: A simple model of mixing and chemi-
cal reaction in a turbulent shear layer. J. Fluid Mech., 125, 397-410.
Clark, T.L., and W.D. Hall, 1979: A numerical experiment on stochastic condensa-
tion theory. J. Atmos. Sci., 36, 470-483.
Cooper, W.A., 1989: Effects of variable droplet growth histories on droplet size
distributions. Part I: Theory. J. Atmos. Sci., in print.
Gerber, H.E., 1980: A saturation hygrometer for the measurement of relative
humidity between 95Z and 105Z. J. Appl. Meteor., 19, 1196-1208.
Jonas, P.R., and B.J. Mason, 1982: Entrainment and the droplet spectrum in cumu-
lus clouds. Quart. J. Roy. Met. Soc., 108, 857-869.
Nicholls, S., 1987: A model of drizzle growth in warm, turbulent, stratiform .
clouds. Quart. J. Roy. Met. Soc., 113, 1141-1170.
Squires, P., 1952: The growth of cloud drops by condensation. Aust. J. Sci. Res.
A, 5, 59-86.
Telford, J.W., T.S. Keck and S.K. Chai, 1984: Entrainment at cloud tops and the
droplet spectra. J. Atmos. Sci., 41, 3170-3179.
148
tll.q g.S
Ii_I
|141
T
_i le*I
u
ii "1
I0"11(!. I
\
I _II OTlI
Itli ,
,i
*i
'i
' ,
' i
',
i
IS l .... loll IS *I
DROPLET DIF_TER (pm)
Fig. I - Size distribution dN/dD el droplets measured near the
surface at indicated times in the fog on 6 Oct- Dashed curve re-
suits from classicaL condensatinnal growth calculations with
$ • 0.001 for 5 rain. with 0715 as the initial dlstrtbutinn.
ili.4
Iml.l
loll.l
IH.I
i ll.l - I
Sl.li '
ll.l l
4
II.? j
I
ii,l
I
tl.5 II II
CLI_TIYIE PROE.qBILITY (X)
Fig. 3 - Cumulative probability dtstributions of super3aturatlon
(o) and eddy width (x) measured near the surface dttTing the
6 Oct. fog.
IP
II
Is
II
13
It
II
II •
:z
Iil _l
I i0, i
'I
. le
[e°
le-)'
tO -4! ........ J .... ,
l0 "1 le ! 1O*1
rlF_)PLIr T D IP;MET[R (pm)
Fig. 5 - Predicted evolutLon of dN/dO after 15 rain. according
tO the stochastic condensational growth model usinej the st,R-
tistics in Fig. 3 and the 0315 distribution for initlal[zation,
s.4
0.3
8.2
e.i!
II I
S - S.43)k
...............................................................
0715
IS 35 4S 45 56 55
TIME iS)
Fig. 2 - Time evolution of supersaturation for mlxin_ of two
eat ttrat_l parcels with a temperature difference of 3 °C and
di_erent initial droplet size dlstributtons, according to the
Broadwell-Bretdenthal mixing mechanism.
i| I
TIME Is)
Fi T. 4 - Comparison of the growth rate of a 2-urn droplet
formed on an urban nucleus calculated with the exact growth
equation (3) and with the approximate analytical e.xpressions
(4;x) as a f_nction of the indicated supersaturations.
iS*!I
iS"1
°I
u
. IS I
iii-I
iS-I
noon
no o
oo of:%
o
iO -_ ........ I , ....... '
II -I I! • II *l II+I
DROPLET DIFIEtETER rum)
F|g. 6 * S_me as Fig. _, except that s_persalxLrat|o_ st3t|st|cs
in Fig. 3 were biased by S = + O. OOI, and parcels with d_f_rent
lime histories were averaged.
149



Supersaturation, droplet spectra, and turbulent mixing in clouds

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server