UDC 551.511.33:551.658.1
laboratory Simulation of Wake Effects on Second
and Third Thermals in a Series
EUGENE M. WlLKlNS', YOSHIKAZU SASAKI, and ERNEST W. MARION
University of Oklahoma, Norman, Okla.
ABSTRACT-Individual thermals rising through the at-
mosphere may encounter the wake of one or more thermals
that started their rise at an earlier time. Laboratory simu-
lations show that the growth rate of second and third
thermals in a series is enhanced relative t o that of a solitary
thermal. The enhancement is due to a gain in momentum
from the wake of a preceding thermal, and equations are
developed for predicting the amount of enhancement as a
function of the delay time between thermals. Agreement
with expcrimental data is good for first and second ther-
mals. The third thermal was found to have a growth rate
w r y similar to that of a second thermal. Theory agrees to
the extent that only a slight increase is predicted, and this
amount could have been lost within the experimental error.
Thus, it appears that by the time the third thermal in a
series occurs, an equilibrium condition has been reached
as far as the thermal growth rate and velocity field are
concerned.
1. INTRODUCTION
Free convection is convective motion that is due
entirely to buoyancy forces that arise from variations in
temperature or density. In the atmosphere, for example,
free convection occurs when an air parcel has a temperature
slightly warmer than its environment and rises because
of its buoyancy. Cumulus-type clouds that are formed by
free convection are often referred to as thermals. Elements
of free convection have been simulated in the laboratory
by the release of a buoyant element to form a solitary
thermal in a neutial environment. The solitary thermal
has been studied in great detail in an effort to understand
convection in the atmosphere. The probable occurrence
of successive convective elements originating from the
same point due to terrain irregularities, for example, has
stimulated an interest in the investigation of the effect of
the wake of a preceding thermal on a following thermal.
The purpose of this work is to investigate the wake effects
on second and third thermals in a series. We will utilize
results obtained by previous investigators of solitary
thermals.
In the study of free convection, thermals from instan-
taneous point sources and plumes from maintained point
sources have been investigated extensively. Batchelor
(1954) investigated free convection in fluids and showed
how experiments of this type were related to atmospheric
convection by describing the common ground between
them. He used dimensional analysis and similarity argu-
ments in his treatment of plumes and thermals for both
laminar and turbulent flow. Morton et al. (1956) devel-
oped two separate sets of equations to describe the con-
servation of volume, momentum, and buoyancy for the
'Also at Advanced Technology Center, Inc., Dallas, Tex.
2 Now affiliated with the National Severe Storms Laboratory, National Oceanic and
Atmospheric Administration, Norman, Okla.
plume and thermal. Variables of the plume were given as
a function of height and for the thermal as a function of
time. The conservation equations of Morton et al. (1956),
modified to include wake effects, form the basic theory
for the solitary thermal used in this work. The modifica-
tion for the second and third thermals presented in this
paper is based on the assumption that the wake velocity
decays according to the same law as the cap velocity.
Scorer (1957) conducted experiments to determine the
proportionality constants that arise from dimensional
analysis. The experimental technique used was to inject
a small amount of cloud material, a negatively buoyant
dyed fluid, into a tank of water and to iecord the growth
photographically. Woodward (1959) used the same experi-
mental technique, along with tracer particles, to deter-
mine the circulation in a thermal. Streak photographs
showed that the vertical velocity a t the center of the
thermal was twice as great as a t the top; and the down-
ward velocity at the edge was about half as great as the
vertical velocity of the top (thermal cap). A visible cir-
culation pattern was observed that is a result of cloud
elements moving upward in the center and downward a t
the periphery of the thermal. This resembles a ring or
toroidal-type vortex.
The effect of a rotation field on a thermal, although not
considered in this work, has been investigated. Wilkins
et al. (1969) modified the conservation equations of Morton
et d . (1956) to include the effect of a rotation field on a
thermal. The rotation was shown to suppress the growth
of the thermal, and this effect was verified experimentally.
Wake effects on second thermals in a series were in-
vestigated both experimentally and theoretically by
Wilkins et al. (1971a, 19716), who modified the momentum
conservation equation of Morton et al. (1956) to state the
conservation of relative momen tum of a wake-imbedded
May 1972 / 399
second thermal. Schauss (1970) developed analytic solu-
tions for the second thermal through the assumption of an
exponential decay of the wake velocity. Wilkins et al.
(1971~) also modified the momen tum conservation equa-
tion to include the interaction of forces due to buoyancj
and rot ation -suppression. They developed analy ti c a1 solu-
tions for the second thermal by assuming that the wake
velocity is proportional to the vertical velocity of the pre-
ceding thermal and the magnitude of the effect is de-
pendent on the amount of the wake intercepted by the
second thermal (increasing with growth). The vertical
velocity for both the first thermal and its wake was as-
sumed to decay according to the same law (inverse square
root of time).
The theoretical development of wake effects on second
and third thermals in a series, presented in this paper, is
based on the same modified conservation equations except
that the assumption concerning the effect on the thermal
of intercepting only a portion of the wake is dropped. The
dropped assumption seemed rather artificial. It made the
problem tractable, but the solutions obtained do not fit
the new experimental data nearly as well as the numerical
solutions given in this paper. The wake velocity encoun-
tered by a thermal will simply be assumed to equal the
vertical velocity of the preceding thermal, corrected for
the time delay between thermals. The momentum equa-
tion will also be generalized for the nth thermal in a series.
The time interval between the successive thermals is of
basic importance in the investigation of wake effects on
succeeding thermals. A “short” time delay will result in
a second thermal rapidly overtaking the preceding thermal.
A “long” time delay can be described as one that allows
the wake velocity to decay to the point where i t has little
or no effect on succeeding thermals. An “intermediate”
time delay results in an increased vertical velocity of the
second thermal but does not necessarily result in over-
taking. Overtaking, if it occurs, is a problem in this
investigation of wake effects because the second thermal
is no longer entirely in the wake of a preceding thermal.
This mould invalidate one of our assumptions. This in-
vestigation is limited to the consideration of wake effects
and, for this reason, an intermediate time delay was
selected to assure that overtaking did not occur in the
experiments.
Wilkins et al. (1971~) encountered an experimental dif-
ficulty that hampered the investigation of second t,hermals
in a series. The difficulty was that, after a short period of
time, the second thermal began to be obscured by cloud
elements trailing behind the preceding one. This made it
difficult to distinguish the boundaries of the second
thermal. This, coupled with the fact that the lifetime of
their second thermals was only about 5s, limited the
opportunity to study the growth of the second thermal.
The experimental technique for the present investigation
is to use a thermal with a longer lifetime and to provide an
unobstructed view of the thermal under investigation.
The lifetime of the thermal id increased by using a small
quantity of low-buoyancy dyed fluid as cloud material in
the simulation. An unobstructed view of the wake-affected
TABLE 1.-Symbols used in this paper
E
F
V
b
9
h
t
W
ff
P
PO
7
entrainment rate
buoyancy force per unit inass
thermal volume
vertical velocity of thermal
sphere-equivalent radius of thermal
gravitational acceleration
height of the thermal
time
entrainment constant
density of buoyant elcment
density of environment
time delay between thermal injections
thermal is provided by making the preceding thermal
transparent (by simply omitting the dye).
2. GOVERNING EQUATIONS
The development of the governing equations for a
second and third thermal in series involves modifying the
basic conservation equations of Morton et al. (1956) for
a solitary thermal to account for the momentum gained
from a wake. The key assumptions of Morton et al. are:
(I) entrainment is proportional to vertical velocity; (2)
lateral profiles of the mean vertical velocity and mean
buoyancy force are similar a t all levels; and (3) local
variations of density are small when compared with the
reference density, which is defined here as the density of
the environmental fluid. The same assumptions are made
in the development to follow.
The symbols used in the development are given in
table 1. Throughout the paper, subscripts of 1, 2, and 3
will be used to denote terms applying to first, second, and
third thermals, respectively. The units of all terms will
be in the cgs system.
The three basic conservation equations developed by
Morton et al. (1956) are:
and
A set of solutions for the conservation eq (1)-(3) was
given by Morton et al. (1956). These are given below with
some change of notation:
and
400 / Vol. 100, No. 5 / Monthly Weather Review
The boundary conditions are b=O when t=O and
momen tum = 0 when t = 0.
Second Thermal in Series
The equation for the vertical velocity, w,, of a preceding
thermal, corrected for lapsed time, is used to estimate a
wake velocity in the development of the theory for the
second thermal in a series. This procedure entails the
assumption that the first thermal in a series will behave as
a solitary thermal and will not be affected by the thermal
following in its wake. Probably such an assumption is
reasonable up to the time that the first thermal is over-
taken by the second one. We also assume that the ambient
density, po, is unaffected by the mixing of preceding
thermals into the environment. This assumption is
justified in the single-source convection discussed here
but probably not in the case of multiple sources, as shown
by Sasaki (1967).
Equation (2) is the only one of the set [eq (1)-(3)]
developed by Morton et al. (1956) that is changed. The
change, it will be seen, is to account for an increase in
momentum of the second thermal that is gained from the
wake velocity of the preceding thermal. I n the case of
the second thermal, the wake velocity a t time t i s assumed
to be equal to the vertical velocity of the preceding thermal
at time t+7 where 7 is the time delay of the second thermal
injection.
The conservation equations for the second thermal are:
(7)
momentum (" nb:wz)=F+% d 4 (5 nbiwl)j (8 )
dt 3
and
where
The additional term in the momentum eq (8) has been
underlined.
From the integration of the momentum eq (8) and the
initial condition bp=O at t=O, we have
From eq (7), the relation
1 dbz wz=- -9
dt
when substituted into eq (11) along
for w l from eq (lo), gives
(12)
with the expression
Equation (13) is nonlinear but is easily solved for bz by
numerical methods. It is then possible to solve numerically
for the other variables; that is, vertical velocity from eq
(12), height from
and entrainment rate from
1 dVz 3 db
"V , d t bz d t E -_ __=- -'.
Equation (13) may be written in the form
y3( y' - 1) =x(x2- 1) (16)
where the variables are nondimensionalized as
and
The prime denotes a derivative with respect to x. The
assumption a=al=az has been made to obtain eq (16).
This equation may be solved easily by numerical methods.
The solution is valid for all values of T , a, and F provided
only that overtaking does not occur. Thus, b2 versus t for
various combinations of values of the parameters T , a,
and F may be determined from a single curve of y versus 2.
Third Thermal in Series
The theoretical development for the third thermal in a
series parallels that of the second. The momentum con-
servation equation for the third thermal is
The new term in eq (19) (underlined) contains the vertical
velocity, w,, of the second thermal a t time t + r where T is
the delay between the second and third thermals. The
value of wz is assumed to be equal to the wake velocity
encountered by the third thermal. I t should be noted that
any possible momentum gain from the wake of the first
thermal in the series is also included because wz was
computed from eq (12) and (13).
The momentum eq (19) can be simplified by integration
and by application of the boundary condition b3=0 a t
t=O to
b:w3=g+ b:wz. (20)
Further simplification can be made by substituting for
w2 from eq (12) and for w3 from
which is obtained from the volume equation in the form
of eq (7) but for the third thermal (Le., all subscripts=3).
May 1972 / Wilkins, Sasaki, and Marion / 401
This changes eq (20) to the form
Equation (22) is easily solved by numerical methods.
Other characteristics of the third thermal may be deter-
mined from equations that have the same form as those
employed for the second thermal.
Comparison of eq (22) with eq (13) shows that the
generalized momentum equation for the nth thermal in a
series must be
which, for any n, can be iterated stepwise from a previously
solved form such as eq (13) for n=2. Thus, we can in
principle determine numerically the growth properties for
any given number of thermals in a series, but, practically
speaking, we need not continue beyond the point a t which
an equilibrium condition is reached. Some insight into
this can be obtained from the generalized momentum
equation mitten in the form
The last term gives the amount of momentum gained from
the wake of the preceding thermal, which is not the same
as its total momentum because it includes only that
amount of its mass that falls within the volume of the
nth thermal. This is much smaller at any time t than the
volume of the (n- 1) th thermal. By iterating eq (24) back
to n=2, we see from eq (13) that there is a contribution
to the wake velocity from wl, which, by the time the nth
thermal is released, is proportional to
[t+ (n- 1)7]-1’2.
This shows that the contribution of earlier thermals to
the wake velocity may become comparatively small after
a few have evolved, and thus the system does approach
an equilibrium condition.
The validity of our assumption that the wake velocity
is the same as the cap velocity of the preceding thermal
must become more suspect as the spacing between ther-
mals increases. The other constraint is that r must be
sufficiently long that overtaking does not occur. Thus, we
see that the range of r over which so simple a theory may
be applied is quite limited.
3. EXPERIMENTS AND DATA ANALYSIS
The experimental investigations of second and third
thermals in series mere conducted in a Plexiglas tank
183 cm deep and 75 cm in diameter? The tank was filled
with water to a depth of 158 cm.
To avoid the problem encountered by Wilkins et al.
(1971~) of not being able to distinguish second thermals
3 Mention of a commercial product does not constitute an endorsement
402 1 Vol. 100, No. 5 Monthly Weather Review
from first thermals, me devised a new technique of cloud
injection in which a clear salt solution was used for all
thermals preceding the one under observation. The blue-
dyed cloud material consisted of 50 percent mater and
50 percent Sheaffer’s Scrip Writing Fluid (Washable Blue
#432) made into a 6.6-percent salt solution. Both clouds
had a density of 1.045 g .~m -~. Sufficient quantities of both
cloud materials were made to insure uniformity through-
out all the experiments conducted.
The cloud elements \\*ere injected into the tank of water
from the top by quickly inverting a small beaker of mate-
rial into the water. The injections consisted of 15 cm3
each. The time interval between injections in series was
20 s in all experiments. Data mere recorded a t 2-s intervals
for 20 s with a Nikon F Photomic-T 35-mm camera
equipped with motor drive and remote triggering. A
28-mm F3.5 wide-angle lens was used to increase the
viewing area. The use of backlighting with fluorescent
lamps diff used through white tracing paper eliminated
unwanted reflections.
The data were extracted using photogrammetric
methods that have been described by Wilkins et al.
(1969). The two cloud features that can be measured
quantitatively for comparison with theory are cloud
volume and height of rise. The volume estimate was made
by assuming axial symmetry and dividing the thermal
image into clisks 2 cm in height. The diameter of each
disk was measured for the purpose of calculating its
volume. The widest distance from edge to edge of the
disk \vas considered to be the diameter, introducing a
negligible error. A volume estimate was obtained by
summing up the volumes of all disks within the outline
of the cloud. The height, h, of the thermal was measured
directly from each frame, as simply the distance from the
point of injection to the top of the thermal.
The values of b and h a t 2-s intervals were obtained
for each run for the three types of thermals: solitary
(10 runs), second (11 runs), and third (9 runs). Mean
values of b and h a t 2-s intervals for each type of thermal
were calculated a t each data point.
The theory was developed on the assumption that the
thermal originates from an instantaneous point source
of buoyancy. The point source cannot be produced experi-
mentally because it has as one of its boundary conditions
b=O when t=O. The experimental data were, therefore,
subjected to a small virtual origin correction to make a
satisfactory comparison with theory. A straight line was
fitted to plotted values of h2 versus t , using the method of
least squares. The point at which the straight line inter-
cepted the t axis (h2=0) defined the location of the virtual
origin, which was then translated to the intercept point
to satisfy the boundary condition h2=0 when t=O. The
amount of translation was the time correction (about
1.5 s) to be applied to the data.
The entrainment rate, E, was calculated using the
finite-difference form
1 AV
v At E=- -
where AV is the incremental change of volume in a n
L
.*A_." .
FIGURI.: 2.-Photographs of t.he throe types of t.hcrnials a t data
time 1= 10 s.
FIGURIC 1.-Sequential photographs of a second thermal in a series,
tho prcceding one being tranqmrent . Time delay txtwecn thermals
is 20 s.
increment of time At. Values of V t d A P were taken
from smoothed rncan value curves of volunic versus
time.
Ninet,--five-pcrcent confidence limits for thc populntion
mean were computctl iising the saniplc nienns ~i i i d :is-
sinning A t-distribution. The confidence limits provitlc
boiiiids to the estiriiiite of the populiition mean. In this
case, wc can be 95-percent, certain tlirit thc trnc inc~in
of the populntion will lic bct \\-em thc limits shoir-n. Thc
limits werc plottccl to give some idea of the reliability of
the mean values for comparison with theory.
4. COMPARISON OF LABORATORY
SIMULATION WITH THEORY
The algorithm used in solving t,he theoretical equat.ions
for the second and third t,l-icrrnnls in series \viis :I. foiirdi-
order Runge-Kut,ta niet_hocl using Gill cocfficient.s (Rnl-
ston and Wilf 1960). The s t i d i i g point, usctl in t.his
rnetliod was selected by t,lie inetliod shown in t,he :iiq)eiidis.
The met,hod docs not permit, st,art,ing with t= 0 bccmse
thc method clcpcnds on an evalrintion of the first cleriva-
tive, and, at this point, it is undefined. The starting
point i.: iniportnnt because, if it is tinomaloiis, the whole
(wrw will be crroneous.
A value of the entrainincnt constant, a, wis deter-
niinecl for each t-pc of therninl using the csperiment8al
rnlucs of sI)licrc-equi\-iileIit rutliiis tind height. The values
of (I! found were 0.244, 0.220, antl 0.217 for the solitary,
second, nntl third tlicrin:ils, respectively. These values
rire consistcnt with valrics found by other investigators :
0.25 (Scorer 1957), 0.20-0.25 (Turner 1963), and 0.20
(JTilkinq ct ail. 1969). Values of a tletermined esperi-
rncnttilly u c usri:illy in the range of 0.20-0.25. The
theorcticiil ciilciilations for the various t>?pes of thermals
\\-ere cnrrietl out using the valne of CY that was tlctcrmiried
cspcrimentall!- for tlitit i ,~ pc, tis this sceiiis to provide
thc best test of thc theory.
The seqiientid phot ogriiphs in figurc I show the second
thcrnitd in :I, serics tis seen in the cspcriments. This figure
illiistratcs the iinohstrrictetl view of thc second thermal
when the transptircnt cloud matcrial is used for thc
prccctlirig therin:il. It shoultl bc notctl also that the
swontl tlieriniil is vcrj- sirriilrir in nppeiirririce to :t solitary
thermnl, :is scen 1)~- comptirison with figure 2, antl thut it
May 1972 / Wilkins, Sasaki, and Marion / 403
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FIGURE 3.-Theoretical curve of sphere-equivalent radius VS. time
for a solitary thermal compared with mean values of experimental
data. experimental data.
FIGURE 5.-Theoretical curve of sphere-equivalent radius vs. time
for a second thermal in series compared with mean values of
thermals also resemble photographs of the second ther-
mal in series. For comparison, photographs of the three
types of thermals at t= 10 s are shown in figure 2. The
solitary thermal can be easily distinguished from the
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FIGURE 4.-Theoretical curve of height vs. time for a solitary ther-
mal compared with mean values of experimental data.
FIGURE 6.-Theoretical curve of height vs. time for a second thermal
in series compared with mean values of experimental data.
SOLITARY
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thermal.
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FIGURE 9.-Theoretical curve of sphere-equivalent radius vs. time
for a third thermal in series compared with mean values of ex-
perimental data.
SOLITARY
SECOND
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The approximate point of overtaking is easily deter-
mined by plotting the height versus time curves of the
solitary and second thermals. The nondimensional curve
is used to determine values for the second thermal for any
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FIGURE 10.-Theoretical curve of height VS. time for a third ther-
mal in series compared with mean values of experimental data.
0' ' ' I ' " ' I ' ' ' 1 0 2 4 6 8 IO 12 14 16 18 20 22 24
TIME (5 )
FIGURE 11.-Theoretical curves of sphere-equivalent radius vs.
time and experimental mean values for the solitary, second, and
third thermals.
particular time delay, T , and the solitary thermal is put
into proper time frame by the value of T selected. The
intersection of the two curves will show the time at which
overtaking occurs. The point of overtaking for time delays
of 5 and 10 s are shown in figure 8 for thermals of the
same buoyancy as those used in this investigation. It is
May 1972 / Wilkins, Sasaki, and Marion / 405
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0 0 2 4 6 8 IO 12 14 16 18 20 22 24
TIME (S I
FIGURE 12.-Theoretical curves of height vs. time and experimental
mean values for the solitary, second, and third thermals.
interesting to note that the time required for overtaking
to occur in the case of a 10-s delay is about twice as long
as for the 5-s delay, suggesting a possible linear
relationship.
Figures 9 and 10 compare experimental data with
theoretical values of sphere-equivalent radius and height
of rise of a third thermal. The theoretical values of sphere-
equivalent radius and height do not agree especially well
with the experimental values after about t=6 s. It is
obvious that the growth of the third thermal is slightly less
than the theory would predict.
Figures 11 and 12 show the sphere-equivalent radius
curves and height curves for the three types of thermals
simultaneously. These show enhancement in the growth
of both the second and third thermals relative to that of
the solitary thermal. The enhancement of the third thermal
over the second thermal is not predicted to be as great as
the enhancement of the second thermal 0~7er the solitary
one. Prom the experimental data, we find that there is no
significant difference between the second and third ther-
mals. This is a t least in the right direction to agree with
theory, although the fit is less than perfect.
Figure 13 shows the theore tical curves and experimental
values of entrainment versus time. The theoreticnl curves
406 1 Vol. 100, No. 5 Monthly Weather Review
0.9 I I J I I I I I I l l
1
THEORETICAL
- SOLITARY
SECOND
THIRD
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0.4
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.
SOLITARY
SECOND
A THIRD
b
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. 0' ' I I ' ' ' ' ' ' ' ' 1 0 2 4 6 8 IO 12 14 16 18 20 22 24
TIME (S I
FIGURE 13.-Theoretical curves and experimental values of en-
trainment vs. time.
are all grouped very closely and show very little difference
for t8he three types of thermals. The same is true for the
experimcntal data points, and the agreement is also good
for the magnitudes of the en trainmeiit rates.
5. CONCLUSIONS
The main contributions from the research reported here
lie in the improvements in cloud simulation technique and
in the theory for successive thermals. The new technique
has provided more acciirate measurements of cloud growth
and has avoided the obscuration of successive clouds due
to merging. As more accurate measurements became avail-
able, it became obvious that the earlier theory (Wilkins
et al. 1971~) for wake eflects needed to be changed, and
this was accomplished.
The theoretical development for second and third
thermals in a series predicts that successive enhancement
of growth rate will occur due to momentum gains from the
wakes of preceding thermals. The theory developed for the
second and third thermals was based on conservation of
total momentum to account for the enhancements.
The close agreement of the theoretical and experimental
results for the solitary thermal reaffirms the theory of
AZorton et al. (1956). The importance of this agreement to
the present analyeis is that our predictions for successive
thermals are based on an exten ;ion of that theory.
Experimental results of the second thermal do show,
correctly, the amount of enhancement of the growth rate
of the second thermal over that of the solitary thermal.
This agreement tends to verify the theory regarding wake
effects 011 a second thermal. The nondimensional solution
of the momentum equation provides n descriptio11 of the
second thermal for various values of cloud buoyancy and
time delay, and the curve can be used with the height ver-
sus time curve of the solitary thermal to predict when
overtaking will occur for various time delays.
The theoretical and experimental values for the third
thermal do not agree especially well after about t=6 s.
The experimental values for the second and third thermals
show that there is no significnnt difference between them.
This indicates that the third thermal is encounteriug a
wake similar to that encountered by the second thermal.
It may be that an equilibrium condition is reached with
only a very few thermals in a series. Another possible
explanation is that, while the theory for the third thermal
ipcludes momentum gain from the wake of the first ther-
mal, experimentally, the wake was too dissipated (possibly
due to viscosity) to be detected. During the observation
of the third thermal, the age of the first thermal increases
from 40 to 60 s. It is possible that tank wall effects may be
present by thk time, although the solitary thermal is not
observed to contact the bottom of the tank within 60 s.
The results of this investigation suggest other experi-
men tal variations that might be made to improve further
the understanding of successive thermals. It would be of
interest to learn whether or not the theory for the third
thermal would be validated by a statistically more
significant set of experimental data. Variation of the time
delay between injections is of interest to test the validity
of the theory for shorter time delays. An attempt should
be made also to derive a theory describing the overtaking
of thermals and to test the theory by simulation experi-
men ts. Finally, the new theory and laboratory simulations
should be extended to the more complex case of successive
thermal interactions in a rotating environment.
APPENDIX: METHOD OF CHOOSING.AN INITIAL
VALUE FOR THE COMPUTATIONAL SCHEME
The initial value for the Runge-Kutta method is esti-
mated from a solution that is an approximation to the
true curve. We begin by letting t=s-l in eq (16) to
displace the curve to the origin. The new equation becomes
(25) y3(f - 1 ) =t( E2+35 + 2)
where the prime now denotes a derivative with respect to E .
A trial solution of the form y=A[” gives
A3E3“(mAp-’-l)=F(t2+3[+2), (26)
which has the proper form as [-to only if m= 112. We then
have
(27) 1 - A4-A3t”2=[2+3[+2 2
and note that, very near the origin (very small [), we can
evaluate the coefficient A as A4=4. Thus, the required
approximate solution is y2=2t, and the only question that
remains is how near the origin we must choose tl to compute
an acceptably accurate curve. We satisfied this require-
ment by computing the curve from [,, y, sufficiently near
the origin that a trial computation using a starting value
of t one-tenth as large as E l would make a difference of no
more than 2 percent a t the fnrthest point on our curve.
ACKNOWLEDGMENTS
The authors cxpress thcir gratitude to RCX L. Inman and Claudc
E. Duchon for suggestions and for critical rcview of the manuscript.
We thank Wallace H. Chaplin, U.S. Air Force, for assistance with
the siniulation cxperimcnts. This research was supported by thc
Atmosphcric Sciences Section, National Science Foundation, under
grants Nos. GA-16350 and GA-27665.
REFERENCES
Batchelor, G. K., “Heat Convection and Buoyancy Effects in
Fluids,” Quarterly Journal of the Royal Meteorological Society.
Vol. 80, No. 344, London, England, July 1954, pp. 339-358.
Morton, B. R., Taylor, Geoffrey I., and Turner, J. S., “Turbulent
Gravitational Convection From Maintained and Instantaneous
Sourccs,” Proceedings of the Royal Society of London, Ser. A,
Vol. 234, No. 1196, England, Jan. 24, 1956, pp. 1-23.
Ralston, A., and Wilf, H. S., Mathematical Methods for Digital
Computers, John Wiley & Sons, Inc., New York, N.Y., 1960,
387 pp. (see pp. 110-120).
Sasaki, Yoshikazu, “Some Dynamical Aspects of Atmospheric
Convection,” Tellus, Vol. 19, No. 1, Stockholm, Sweden, 1967,
Schauss, Roger H., “Numerical and Laboratory Simulations for
the Investigation of Wake Effects on Successive Thermals,”
hI. S. thesis, University of Oklahoma, Norman, 1970, 65 pp.
Scorer, Robert S., “Experiments on Convection of Isolated Masses
of Buoyant Fluid,” Journal of Fluid Mechanics, Vol. 2, Pt. 6,
Taylor & Francis, Ltd., London, England, Aug. 1957, pp. 583-594.
Turner, J. S., “Model Experiments Relating t o Thermals with
Increasing Buoyancy,” Quarterly Journal of the Royal Metcorologi-
cal Society, Vol. 89, No. 379, London, England, Jan. 1963, pp.
Wilkins, Eugene M., Sasaki, Yoshikazu, Friday, Elbert W., Jr.,
i\lcCarthy, John, and McIntyre, James R., “Properties of
Simulated Thermals in a Rotating Fluid,” Journal of Geophysi-
cal Research, Oceans and Atmospheres, Vol. 74, No. 18, Aug. 20,
1969, pp. 4472-4486.
Wilkins, Eugene M., Sasaki, Yoshikazu, and Schauss, Roger H.,
‘[Interactions Between the Velocity Fields of Successive
Thermals,” Monthly Weather Review, Vol. 99, No. 3, Mar. 1971a,
Wilkins, Eugene M., Sasaki, Yoshikazu, and Schauss, Rogcr H.,
“Vortex Formation by Successive Thermals: A Numerical Simu-
lation,” Monthly Weather Review, Vol. 99, No. 7, July 1971b,
Woodward, Betsy, “The Motion in and Around Isolated Ther-
mals,” Quarterly Journal of the Royal Meteorological Society,
Vol. 85, No. 364, London, England, Apr. 1959, pp. 141-151.
pp. 45-53.
62-74.
LIP. 215-226.
pp. 577-592.
[Received August 6, 1.971; revised February 66, 19721
May 1972 / Wilkins, Sasaki, and Marion / 407
UDC 551.551.5:551.657.5:551.576.1:561.507.362.2(084.1)"1970,12.28/2~'
PICTURE OF THE MONTH
A Turbulent Region
ARTHUR H. SMITH, JR.-Environmental Technical Applications Center,
U.S. Air Force, Washington, D.C.
Satellite photographs of certain atmospheric conditions
can frequently be used in locating specific regions of high
risk of turbulence occurence (high risk areas). In particular,
the cloud patterns associated with polar and subtropical
jet streams, which are known as areas of high turbulence
probability, are distinguishable on satellite photographs.
These high risk areas are easily verified when they occur
over data-rich areas and, if similar cloud patterns are
identified over little-traveled, data-sparse regions, the
identification and forecasting of turbulence can most
certainly be improved.
One such example of cloud patterns associated with a
high risk area occurred on Dec. 28-29, 1970. The jet stream
cloud patterns associated with the turbulent areas can be
seen in figure 1. The clouds associated with the sub-
tropical jet stream originate in the intertropical conver-
gence zone (A) and sweep northeastward in an anticy-
clonically curved arc (B,C). According to Anderson (1969),
the polar jet stream can be located where there is a change
from unstable clouds to stable clouds (D,E). While it be-
comes difficult to pinpoint an exact location of the polar
jet stream as it continues eastward (F,G,H), careful ex-
amination does reveal a slight shadowline (F to G) from
the jet stream cirrus appearing on the lower clouds.
Mountain-wave clouds associated with both the polar
jet stream (I) and the subtropical jet stream (J), plus the
FIGURE 1.-ITOS (improved TIROS operational satellite) 1 view, pass 4246 at 2230 GMT, Dec. 28,1970, and pass 4247 at 0000 GMT, Dec. 29,
1970.
408 f Vol. 100, No. 5 Monthly Weather Review
45,000 F T 45.000 F T
43.000
4 1,000 41,000
39.000 39.000
STJ 43,000 ~
--.
FIGURE 2.-Turbulence reports for the period 0000-0600 GMT,
Dec. 29, 1970. Height is in hundreds of feet. Jet streams are
shown as solid arrows. Areas of cirrus (taken. from fig. 1) are
enclosed by scalloping.
area of transverse bands within the subtropical jet cloud
pattern, are other areas indicative of a high risk of turbu-
lence observable on meteorological satellite pictures.
These two jet streams tend to converge in a manner
similar to that described by Kadlec (1966) with the
northern (polar) jet stream in a trough-ridge pattern and
the subtropical (southern) jet in a broad anticyclonically
curved pattern. I n general, it has been noted that, if the
jets converge to within 400 mi or less, a cirrus sheet
associated with both jets is continuous from the upper
trough to the next downstream ridge associated with the
polar jet. It is in and near portions of this cirrus sheet
that regions of moderate or greater clear air turbulence
(CAT) are encountered. I n this particular case, the two
jets converge to within about 750 mi and the cirrus
sheet is not continuous between the two jets. Figure 2
shows that most of the actual turbulence is in or near the
cirrus, not in the clear air between the two jets.
The distribution of turbulence both in the horizontal
and vertical compares favorably with the “type B”
37.000
35,000
33.000
31.000
29.000
27.WO
37,000
35,000
33,000
31.000
29,000
27,000
2 5,000
23.000
2 1,000
25,000
23,000
21,000
WlLMlNGTON NORFOLK NEW VORK MONTREAL MIAMI
FIGURE 3.-Vertical cross-section of polar and subtropical jet
streams, cirrus pattern, and associated turbulence areas for model
type B (Kadlec 1966). PJ is the polar jet core, STJ is the
subtropical jet core, the stippled area represents polar and sub-
tropical jet stream cirrus, and the crosshatched area indicates
moderate or greater turbulence.
turbulence model by Kadlec (1966) depicted in figure 3.
In the horizontal, there is a concentration of moderate and
severe turbulence reports, associated with the subtropical
jet stream in the dense-traffic area near Los Angeles,
Calif., and a lesser number of reports further north and
east as traffic density decreases. North of the subtropical
jet over central California, Nevada, and Utah, the air
is relatively smooth with negative and light turbulence
reports prevailing. A secondary maximum of moderate
turbulence appears over northern California, southern
Oregon, and extreme northern Nevada in association
with the polar jet and the mountain-wave clouds (I in
In the model, turbulence associated with the polar jet
is a t a higher altitude (32,000-37,000 ft mean sea level)
than that with the subtropical jet (26,000-33,000 ft).
The present example has a similar vertical structure. The
turbulence with the polar jet stream is generally above
30,000 f t while, with the subtropical jet stream, the
moderate to severe turbulence is concentrated below
30,000 ft.
The north-south vertical cross-section (fig. 4) shows a
pattern similar to figure 3 with the subtropical jet at a
higher altitude than the polar jet. An area of coincident,
strong, horizontal and vertical wind shear is a favored
region for moderate to severe turbulence (George 1960).
The area from San Diego, Calif., to just south of Vanden-
berg Air Force Base, Calif., between 21,000 and 26,000 f t
is such an area and was characterized by a large number of
severe turbulence reports. The horizontal and vertical wind
shear with the polar jet stream at this time is not as strong
as with the subtropical jet. Also, in the polar jet area, the
shear has a much greater vertical extent, from near 25,000
to over 35,000 ft. The turbulence associated with this
weaker shear is reported as light to moderate and has a
large vertical extent with a slight concentration between
about 33,000 and 37,000 ft above mean sea level over the
jet core. Wind speeds are low and shear is small from north
of Vandenberg Air Force Base to northern California.
No turbulence is reported in that area.
fig. 1).
May 1972 J Smith J 409
SdN VBG I
290 393
w)
>
w
1’ :-IO
SLE UIL YS
694 797 896
FIGURE 4.-A north-south vertical cross-section for 0000 GMT, Dec. 29, 1970.
REFERENCES
Anderson, Ralph K., e t al., “Application of Meteorological Satellite
Data in Analysis and Forecasting,” A WS Technical Report 212, Kadlec, Pa.ul, “Flight Observations of Atmospheric Turbulence,”
U.S. Air Weather Service, Washington, D.C., June 1969, sec- Final Report, Contract No. FA66WA-1449, Federal Aviation
tions separately paged.
George, J. J., A Method for the Prediction of Clear A i r Turbulence,
Eastern Air Lines, Atlanta, Ga., Aug. 1960, 17 pp.
Agency, Washington, D.C., 1966, 52 pp.
410 / Vol. 100, No. 5 / Monthly Weather Review