CUMULUS CONVECTION AND ENTRAINMENT 
Under the assumption of the parcel method the lapse 
rate of temperature within the cloud would be given 
by AF. 
This graphical procedure has been followed for two 
different moisture distributions and mixing regimes 
(Fig. 3). The computed values of the lapse rate of 
temperature within the cloud provide an insight into 
the effect of mixing upon the growth of the cloud. 
Thermodynamic Analysis. As a saturated cloud mass 
m rises a vertical distance dz, through a pressure change 
dp, let it mix with a mass dm of its environment. The 
ascent and mixing may be considered to consist of the 
following processes: an adiabatic temperature decrease 
of the cloud mass m; an isobaric cooling of the mass m 
and isobaric heating of dm; and evaporation of a portion 
of the liquid water of m in order to saturate dm so that 
the final mixture is saturated and has a uniform hori- 
zontal distribution of temperature, water vapor, and 
liquid water. The lapse rate of temperature in the 
cloud, y., may be derived by considering the heat 
changes occurring in the rising mass. Austin and Fleisher 
[2] have shown that 
m(Cp dT — RT din p) = — c,(T — T.)dm 3) 
— L(w — we) dm — mL dw, ~* 
wh 
Cp ac a rn | 
Ve 1+ 0.621 wl?/c, RT” 
aa — T.) +o (w - w) | 
m dz 
1 + 0.621 wh? ik RT? 2 
and that 
(4) 
+ 
where c, is the specific heat at constant pressure, L the 
latent heat of evaporation of water, w the saturation 
mixing ratio at the temperature 7’, w, the actual mixing 
ratio of the environment, and fF the constant of the gas 
equation for dry air. If there is no mixing, dm/dz = 0 
and the lapse rate of temperature in the cumulus cloud 
is the same as the moist-adiabatic lapse rate of tem- 
perature ym. Since 7, << T and w. < w, Ye > Ym, that 
is, the lapse rate of temperature within a cloud which is 
mixing with its environment is greater than the moist- 
adiabatic lapse rate. 
The cumulus cloud ceases to grow either when the 
liquid-water content is zero or when the cloud becomes 
sufficiently colder than the environment so that the 
vertical velocity is zero. The variation of the liquid- 
water content w; can be ascertained by equating the 
total water content before and after mixing: 
—dw +- a (w, — Ww — Wi). (5) 
dw, = 
With the substitution of (5) in (3) an expression is ob- 
tained for the variation of the total mass of liquid 
ee 
£ (avi) = 
a rT (ae — Ye) + = = (= 1.) | (6) 
697 
where ya = g/Cp, the dry-adiabatic lapse rate of tem- 
perature. Consider the usual situation where ya 2 Yc- 
Since dm/dz > 0, then d(mw,)/dz > O as long as 
T. < T, and the cloud does not dry through mixing. 
Therefore, whenever the lapse rate of temperature of 
the environment is less than the dry-adiabatic lapse 
rate, the cloud temperature becomes less than the 
environmental temperature before the liquid-water con- 
tent reaches zero. It can be concluded that the factor 
which controls the termination of the cloud growth is 
the temperature of the cloud. 
Cloud Properties. Because there is little information 
on the mixing process, equations (3)—(6) are of limited 
practical use. However it is possible to gain an insight 
into some of the cloud characteristics by considering 
the orders of magnitude of the various terms of these 
equations. The observations analyzed by Barrett and 
Riehl [3] and Stommel [13] show that the temperature 
difference between the inside and outside of a cumulus 
cloud is very small. Figure 4 is reproduced from the 
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HORIZONTAL SCALE 
Vig. 4—Cireulation within a typical cell in cumulus stage. 
Inflow and vertical motion beneath the cloud are light and 
are not shown. (After Byers and Braham [6].) 
work of Byers and Braham [6] who arrived at this 
typical picture from the analysis of extensive obser- 
vational data. If the variation with height of the 
average vertical velocity is considered and if allowance 
is made for the force required to set the entrained air 
in motion, it follows that a virtual temperature dif- 
ference of less than 1C is ample to explain the develop- 
ment of the cloud. Even when friction is considered it 
is probable that an average virtual temperature dif- 
ference of 1C or less is sufficient to account for the 
upward force which causes the vertical development of 
most cumulus clouds. From this empirical information 
it may be concluded that the lapse rate of temperature 
within the cloud differs only slightly from the lapse 
rate of temperature within the environment. 
