68 C. E. ANDERSON 
tion applies to all portions of the fluid along the 
streamline. From (1) we may write for this point 
Ow 
— = —p%z (3) 
ot 
or 
Pw 
= —rw (4) 
ot 
When —v? <0, w = A sinh vt, and when —vr? > 
0,w = A sinh vtifw = Oatt = 0. 
When —v > 0, we have an exponential in- 
crease in circulation with time and when —v? < 
0, harmonic oscillations about an equilibrium 
value of period 22/v are expected. In this latter 
case, vy may be regarded as a frequency factor. 
An expanded form of (2) may be employed to 
demonstrate the influence of horizontal scale on 
the stability of the circulation. Petterssen [1956] 
and Beers [1945] have shown that the frequency 
factor can be expressed 
—v = (g/To)[ (ys — y) + (ya — y)A+/A-] (5) 
Here y, = saturated adiabatic lapse rate 
T,) = temperature of the environment 
A,/A_ = ratio of updraft area to downdraft 
area 
ll 
ll 
The horizontal scale of the convection cell may 
make its influence felt through the parameter 
A,/A_. Ordinarily, ya > y > Ys, so that 
oer aA ll = ae || 
and stable solutions are to be expected in (4) if 
A,/A_ ~ 1. Unstable solutions are possible if 
| (va — ys)A3/A_| < | se — v) | 
and this may be realized whenever the ratio 
A,./A_ is small enough to make this possible. 
It is apparent that this parameter has a criti- 
cal bearing on the stability of convection and 
varies with different meteorological conditions. 
One object of the cloud studies made in Arizona 
was to get an estimate of this ratio. For the 
clouds studied, a low frequency oscillation was 
noted in the vertical growth rate. This oscillation 
was manifested by periods of rapid upward 
growth followed by periods of slow upward 
growth. By using the observed frequency to- 
gether with Raob data, (5) was solved for A,/A_. 
The results are given in Table 1. The ratio 
A,/A_, thus computed, suggests for these some- 
what isolated Cumulus clouds a convective cell 
whose updraft area is about equal to its down- 
draft area. The clouds are individual ones whose 
fields of motion do not extend much beyond their 
visual borders. This concept should be reasonable 
for ordinary fair-weather Cumulus development 
and air-mass showers. 
It is conceivable that in a meteorological situa- 
tion of synoptic-map scale low-level convergence 
and high-level divergence, A,/A_ may be very 
small if the updrafts are confined to a few very 
intense storms. This provides for rather narrow 
regions of strongly ascending air surrounded by 
broad regions of gently descending air. Under 
these conditions one might expect to find un- 
stable solutions to (4). If the circulation increased 
exponentially with time, one would expect to 
find rather spectacular results in the local 
weather. Although there are no data which will 
allow a direct verification of (5), as in the case 
of stable oscillations, one might turn to instances 
of record-breaking rainfalls to determine if the 
meteorological conditions were right to expect 
A,/A_ to be small and thus lead to runaway con- 
vection. In Table 2 is a list of outstanding cloud- 
burst rainfalls. 
The two greatest recorded rainfalls in the 
United States occurred at Thrall, Texas, in 1921 
and at Hallett, Oklahoma, in 1940. Lott [1953ab] 
describes these storms and the attending meteo- 
rological situations. The Thrall storm was most 
remarkable in that the bulk of the rain fell in two 
bursts only lasting about four hours each. Lott 
attributes the bursts to the passage of an isallo- 
baric low-pressure center which was the remnant 
of a small hurricane which entered the Mexican 
coast. The conditionally unstable lapse rate, high 
moisture content, and convergence-divergence 
pattern agree very well with the conditions one 
should expect for runaway convection. 
TaBLE 1—Computed values of A,/A~ 
es 
Peo 
Ply 
Be | A,/A 
Cloud location Date os | (com- 
ane puted 
zs a 
minute 
Tueson, Arizona July 23, 1956 | 11 1.0 
Tucson, Arizona July 24, 1956 | 11 0.7 
Flagstaff, Arizona Aug. 22, 1958} 10 1.0 
Mt. Withington, | Aug. 16, 1957) 10.3 | 1.34 
N. M.2 
Mt. Withington, | Aug. 20, 1957} 8.5 | 1.33 
N. M.s 
® Vonnegut and others [1959]. 
