Turbulent entrainment at zero mean shear stratified interfaces has been studied extensively in the laboratory and theoretically for the classical situation in which density is a passive tracer of the mixing and the turbulent motions producing the entrainment are directed toward the interface. It is the purpose of the numerical simulations and data analysis to investigate these processes and, specifically, to focus on the following questions: (1) Can local cooling below cloudtop play an important role in setting up convective circulations within the cloud, and bringing about entrainment; (2) Can Cloudtop Entrainment Instability (CEI) alone lead to runaway entrainment under geophysically realistic conditions; and (3) What are the important mechanisms of entrainment at cloudtop under zero or low mean shear conditions

Baker, M. B.

Bretherton, C. S.

Siems, S. T.

English

NASA Technical Reports Server

Simulations and Observations of Cloudtop Recesses 
S. T. Siems*, C. S. Bretherton*, and M. 8. BakeP 
*Applied Marhematics Department **Geophysics Program 
University of Washington 
Seattle WA 98195 
Turbulent entrainment at zro  mean shear stratified interfaces has been studied extensively in the labora- 
twy and theoretically for the classical situation in which density is a passive tracer of the mixing and the hlr- 
bulent motions producing the entrainment are directed toward the interfxe (see Turner, 1986, for a comprehen- 
sive review). For gases at high Reynolds number, the-entrainment velocity w, at the interface separating a lower 
- tutbulent fluid of mean virtual potential temperature 8, from a nonturbulent fluid of virtual potential temperature 
8,-A8, depends only on a Richardson’s number, which can be defined as 
whem w and h are turbulent velocity and length scales. Carmthers and Hunt (1986). Linden (1973). StuU 
(1976). and others have described the entrainment mechanism at intermediate Ri in terms of eddies impinging 
on the interface and dragging down wisps of upper layer fluid Lidar pictures (Atlas et al., 1986) show evidence 
of dome-wisp like entrainment in the inversion topping the dry atmospheric boundary layer. 
In the layer cloud situation, radiative and evaporative cooling tend to drive convective motions away from 
the interface, instead of toward it as in the classical flows. Moreover, the function 8,(F) is piecewise linear in 
the cloud case. where F is the fraction of cloudy air in a parcel formed by mixing cloud and upper air. This 
behavior introduces another parameter into the problem, which we define as D.: 
A. 
D. 1E - 
AeJ;; 
where A. I 9,(F.) - 6 and F. is the mixing fraction of exactly saturatul air. 
The effects of these properties of the cloud-clear interface on the entrainment process are not well under- 
stood. Observations (Nicholls and Leighton, 1986; Caughey et al.. 1985) in stratocumulus show that evaporative 
coolihg enhances entrainment over that expected in the dry, linearly mixing case, even if D. > 0. On the other 
hand, observations (for example, Hanson. 1984) show that even in cases for which the “cloudtop entrainment 
instability.” or CEI. criterion holds (D. < 0.0). clouds do not necessarily thin or breakup, as had been suggested 
by Deardorfl(1980). Randall (1980) and others. This suggests that kinetic energy due lo evaporative cooling is 
a secondary effect in determining cloudtop motions in these cases. 
It has been suggested (Caughey et al., 1985; Nicholls and Leighton, 1986) that detachment of small, 
cooled parcels from the base of the optically active layer below cloudtop gives rise to convective circulations 
which resull in turbulent motions at cloudtop and thus in turbulenl entrainment. Mahn and Paumier (1982) 
showed that, in the case of some mean shear at cloudtop, mixing of cloud and clear air was in the main confined 
lo the downwind side of penetrative cloudy elements. They found significant heat fluxes carried by non-cloudy. 
yet mixed, air parcels. 
It is the purpose of our numerical simulations and data analysis to investigate these processes and. 
specifically. to focus on the following questions. 
(1) Can local cooling below cloudtop play an important role in (a) setting up convective circulations within the 
cloud, and (b) bringing about entrainment? 
(2) Can CEI alone lead to runaway entrainment under geophysically realistic conditions? 
(3) What are the important mechanisms of entrainment at cloudtop under zero or low mean shear conditions? 
11. Numerical Simulations 
representation of a Boussinesq layer. We use the following set of governing equations: 
We have numerically simulated the stratocumulus inversion layer by a two-dimensional fluid dynamic 
234 
https://ntrs.nasa.gov/search.jsp?R=19910001183 2020-03-19T20:29:36+00:00Z
3 = - (d*V)C + vV2c - b, 
at 
&=-($V)r+vV2y at 
where c is the vorticity, Y is the stream function. d is the velocity, v is an eddy diffusivity, and b is the buoy- 
ancy, which is a linear function of the two conserved quantities yl, y2 (the equivalent potential temperature and 
total water content or saturation point temperature and pressure (Beas, 1982; Brefherton, 1987)). These equa- 
tions are solved with a finite differencing scheme in time and space. Periodic boundary conditions are assumed 
in x and Dirichlet boundary conditions in z. The two-layer stratified fluid is initially at rest, and at t = 0 we 
apply a small perturbation in the conserved variables. This stratified flow would remain stable without the 
incaporation of additional physics D the model. We discuss here simulations in which evaporative cooling 
drives the circulations. Note that all mixing, turbulent and diffusive, is represented by the eddy diffusivity term 
vV2. Values used here are orders of magnitude larger than values appropriate for thermal and molecular 
diffusion. Stability considerations prevented use of lower values; thus the sirnulaled flow is laminar rather than 
tubdent. While the upper boundary did not appear to have an important effect, the lower boundary forced an 
artificial circulation in the fluid and gave rise to spurious fluxes of the conserved quantities. 
Figure 1 displays O,(F) for b e e  cases: namely. the measurements discussed below and two situations we 
have simulated. Preliminary results of our simulations are shown in Figures 2 and 3. In the right hand figure of 
each pair, we show contours of a variable which is proportional to the difference between the saturation point 
pressure (Betts. 1982) and the actual pressure; thus the zero contour indicates cloudtop and the most dense mix- 
tures of the upper and bwer  fluids. The area below this contour represents &e cloud; small contour values 
represent mixtures which are close to saturation and, therefore, relatively dense. The left hand figure of each 
pair shows the stream function. Initially, the inversion layer is at height 1, and the fluid is at rest. The simula- 
tions represent the situation at later times, when the circulations are established. 
Density of Yixed Fluids 
Figure 1. e@), where F is the fraction of cloud- 
base air in a mixture of cloudbase and upper level 
air. Solid and dashed lines: numerical simulations 
(see text); dotted line: computed from 19:06 sound- 
ing. July 5. 1987. 
232 
Figure 2 shows a geophysically realistic case (solid curve. Figure 1; D. = -0.04.) The initial cooling 81 
cloudtop has led m the formation of a steady circulation within the moist. cool cloud, indicating that evaporative 
cooling is a significant sowre of turbulent kinetic energy. The dry, lighter, invasion fluid is affected only by 
diKusion. The circulation produces "cool pools" of unsatmled air of approximafely the same buoyancy as the 
cloud. Penetrative plumes with high evaporation raes form under these pools. A sharp interface between layers 
was maintained by the circulation. instead of the gradient layu predicted by Turner and Yang (1%3). Under 
these physically reasonable initial conditions satisfying the CEI condition. no indication of runaway entrainment 
evolves. 
Figure 3 corresponds to a laboratory simulation (Shy and Breidenthal. personal communication) in which 
CEI is a greater effect (dashed curve, Figure 1; D. = -2. ). Under these extreme conditions runaway entrain- 
ment occurs. The contamination of the lower fluid with the newly created mixed fluid becomes important before 
a semi-steady circulation is established. 
m 
.................................. .................. .................. .................. 
3. Jb 
Figure 2. Vcdcal cross section of simulated strato- 
cumulus deck De = -.a, at a time when steady cir- 
culation has developed. a) Stream function. b) 
Negative values of the contoured variable 
cornspond to positive buoyancy: zero contour is 
cloudtop. 
Figure 3. As in Figure 2. but for "laboratory" 
sounding, D. = - 2.0. 
111. Data Analysis 
We report here on a preliminary analysis of high resolution measurements made aboard the NCAR Elcctra 
on July 5. 1987. as part of the FIRE project For averaged I-& thermodynamic measurements we used the 
static (fuselage) pressure, temperature measured by the the Rosemount thermometer, liquid water from the 
Johnson-Williams device. and vapor content compuced by averaging, filtering and applying a 2-second lag (Aus- 
tin. personal communication) to the signal from the dewpoint sensors, and we used the 20 Hz FSSP and Lyman 
a records as high resolution tracers of the presence of cloud and pockets of moistened air. 
Figure 4 displays the sounding made at 19:06-19:lO GMT in saturation point coordinates (Bells, 1982), 
showing the thermodynamic structure of the cloud and above cloud layers. The mixing line lies close to a 0, 
isopleth. There was some mean shear within the layer. permitting the identification of air parcel origin by hor- 
izontal velocity, as well as thermodynamic properties. The sounding values at 950 and 910 mb were used to 
calculate thc dotted curve in Figure 1. The CEI criterion is marginally meC D. = -.006 and the cloud was quile 
solid. 
233 
i, 
H Figure 4. 19%-1910 GMT. July 5 sounding cam- 
puted in saauauim p o d  Coordinalcs. hued lints 
are 8. isopluhs computed at cloudrop, 920 mb. 
Figure 5 shows portions of che lime series of liquid waler. w. 8, and the buoyancy flux (calculared as Ihe 
simple covariance, unfiltered, of w and e,) o v u  an in-cloud segmurt ~1 about 936 mb. The cloud was highly 
iurbulent and the most norable features of the trdces are h e  d i s h 1  events. tens of m e l a  in scale. 81 inlervals 
of about 4-5 kilometers in which the liquid water decreased sharply and thuc were large excursions in vertical 
velocity. This is the scale of the major undulations in cloudtop shown by lidar measurements later on this flight. 
Examination of the radiative fluxes and vapor content (not shown) suggests that the events were passage through 
columns of clear, and in some cases unsaturared. air. Note the highly intermittent character of the buoyant 
fluxes; convection is driven from below 11s well as from the LOP. and the breaks between cloud segments appear 
LO play an important dynamic role. 
Cloudtop rose before h e  cloudlop wavers, a segmeni of which is shown in Figure 6. The aircdt flew in 
and out of cloudy segments at about 912 mb. The cloudy segments. which were more turbulent than h e  inler- 
vening clear air but much less ulrbulent ban the cloud at lower levels, consist of regions of high liquid water 
content. separated by shomr. more dilute regions referred LO by Mahn and Paumier as "wisps." Noe the high 
turbulence levels marking small urrbules of cloudy air and the high vapor content in some of the "wisps." Mix- 
ing appears fairly localized to the edges of Ihe cloudy segments, as evidenced by the spikes in horizontal *and 
vertical velocity there. Evaporative cooling is probably responsible for the velocity spikes at Lhese interfaces. 
The spatial scales of these mixing regions are comparable w i h  the scales of the Liquid-fnx regions seen I lower 
levels. but the cooling is noi sufficient to explain he descent of cloudtop air LO the lower level wiihout s~me 
dynamic forcing. The nonturbulent air between &e cloudy segments is the result of prior mixing of cloud and 
upper level air. and may rcpreseni evidence of "cool pmls" as shown in the calculations. (See Fig. 2). 
=I 
P I  
2ue 7 
sue O 
uti 
Figure 5. lncloud measurements made at 936 mb. 
18:40-18:45 GMT, Average aircraft vclocity about 
105 m/s. see leX1. 
E O  
-2 4 
5 2  zw 
2YO 
35 
00 
0 GO 
Tirrtc in Stv orids 
Figure 6. Cloudtop measurements made a( 912 mb. 
19:20-1925 GMT. See text. 
234 
IV. Discussion 
From these preliminary results. it appears that mixing of cloudy and clear air takes place primarily in 
highly localized regions, both at the edges of large cloudy segmena, and in the vicinity of small cloudy turbules 
rising into the cloudtop region. The descent of the mixed air is appmntly acceleravd by evaporative cooling. 
However, our simulations, as well as the observations. show that relatively quiescent. mixed, clear air Bccumu- 
lates between the upward penetrating cloud elements, and they suggest Lhat it is not sufficiently negatively buoy- 
ant to be efficiently drawn down into the cloud. Moreover, it appears fmm the simulations that. under physi- 
cally realistic conditions, evaporative cooling alone does not lead to runaway enuainment. even when the CEI 
criterion is met, unless D. is a very large negative number. 
The relatively low efficiency of evaporative cooling in generating turbulent kinetic energy at the interface 
may be due partially to the fact that O,(F) <$,(I) only f a  relatively high F values. Parcels with these mixture 
ratios are probably characteristic only of air at the base of the inversion laya.  The displacement of the acclem- 
tion due to evaporative cooling fmm the upper layer fluid, and thus from the region at which it could most 
efficiently promote entrainment, is further suggested by recent laboratory experiments (Johari. 1988). showing 
that no molecular mixing occurs in a buoyant (or in this case, negatively buoyant) thennal until it has moved a 
distance comparable to its diameter. The evaporatively cooled parcels seen at cloudtop have dimensions on the 
order of tens of meters, so they may be roughly that far from their 6rst encounter with upper level fluid before 
they receive the additional downward acceleration due to the evaporation. For small De, the mixed fluid simply 
accumulates at cloudlop, reshaping the density profile and thus slightly facilitating further entrainment there. 
Acknowledgement: This research was supported by NSF grant ATM-8620165. 
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Simulations and observations of cloudtop processes

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