CUMULUS CONVECTION AND ENTRAINMENT 
case of saturated rising air in a nonsaturated environ- 
ment the two methods differ. The details of the slice 
method will not be treated here (see the discussion in 
Petterssen [11]). The principal difficulty in applying the 
slice method to the cumulus problem resides in the fact 
that the slice method considers slices of infinitesimal 
thickness whereas cumulus clouds are of finite extent. 
When it is assumed that the rising air cools moist- 
adiabatically there is generally some temperature differ- 
ence between the cloud top and its surroundings and, 
therefore, the slice-method analysis cannot be applied 
beyond the initial stage at the cloud base. 
One of the major defects of these two methods is the 
assumption that the vertically moving currents are 
isolated from their surroundings and that only adiabatic 
temperature changes occur within the cloud and its 
environment. The questionable nature of this assump- 
tion is well illustrated by the temperature observations 
within cumulus clouds. For example, the observations 
analyzed by Stommel [13] and Byers and Braham [6] 
failed to reveal moist-adiabatic lapse rates of tempera- 
ture within the cloud. Instead it was observed that the 
cloud temperature was only slightly different from that 
of the environment. Also measurements indicate that 
the liquid-water contents are only a fraction of those 
which would be expected from the assumption of the 
moist-adiabatic ascent of air. The empirical data then 
demonstrate that there must be continual mixing be- 
tween the rising current and the environment. This 
mixing of environmental air into the rising current of 
cloud air is referred to as entrainment. 
Concept of Entrainment and Mixing 
The concept of entrainment as the turbulent mixitig 
of fluid from an environment into a jet stream has been 
analyzed by hydrodynamicists. Its application to the 
cumulus problem is discussed by Schmidt [12] and 
Stommel [13]. The difficulty of applying hydrodynami- 
cal jet-stream theories to the cumulus problem resides 
in the concept of the jet. In the atmosphere there is no 
mechanism to eject air into its surroundings and there- 
fore the moving air column, or the jet, must originate 
from the vertical accelerations which are produced 
within the air itself. Such accelerations can be deduced 
from an assumption that the rising cloud air cools 
adiabatically. However, such an assumption is not very 
realistic. If air commences to rise so that its tempera- 
ture decreases at the moist-adiabatic rate in an environ- 
ment where the lapse rate of temperature exceeds the 
moist-adiabatice lapse rate, the virtual temperature 7’ 
of the rising air is higher than the virtual temperature 7’ 
of the environment. As a consequence of the assumption 
that the pressure on the rising particle is equal to the 
pressure of the environment, the rising particle is ac- 
celerated upward and the acceleration is given by 
Dies Ly Ay Sl 
dt ap 
where g is the acceleration of gravity. Since the accelera- 
tion is positive the upward velocity of the rising air 
increases with height. At the same time the density of 
q; (1) 
695 
the rising air decreases with height and is determined 
by the variation of the virtual temperature 7’ and by 
the pressure of the environment. For any appreciable 
value of 7’ — T the density decrease is less than the 
velocity increase. If the outer boundaries of the column 
of rising air form a cylinder, it follows that more mass 
is leaving the top portion of any section of the cylinder 
than is entering the base of the section. This state is 
impossible because it violates the condition of con- 
tinuity. Continuity of mass requires that air be brought 
in through the sides of the cylinder, but this is not 
possible in view of the assumption that the pressure 
inside the cylinder is equal to the pressure outside. 
Alternatively, continuity may be satisfied by allowing 
the horizontal dimensions of the rising stream to de- 
crease with height. This case does not appear physically 
real as a model for a cumulus cloud, since it leads to a 
very narrow current moving with a large velocity. 
Hence, when it is considered that the jet develops 
through the accelerations produced by the moist-adi- 
abatic ascent of cloud air, it must be recognized that 
there has to be a horizontal inflow of air into the rising 
saturated current in order to maintain continuity of 
mass within the ascending stream. 
Up to the present time a common approach to the 
entrainment problem has been to consider the extent 
to which the mixing changes the lapse rate of tempera- 
ture within the cloud from the moist-adiabatic rate. 
With this approach three different aspects of mixing 
between the cloud and its environment should be con- 
sidered, namely, 
1. The horizontal inflow of environmental air into a 
cloud column which is assumed to grow upward in a 
number of adiabatic steps. 
2. The hydrodynamical 
mental air. 
3. The eddy diffusion of cloud particles into the 
environment. This aspect of mixing is present even 
when there is no general rising motion within the cloud. 
The analysis has been restricted so far to a vertically 
moving stream in a stationary environment. In an 
actual weather situation the horizontal wind speed 
varies with elevation and it is now necessary to con- 
sider the effect of the vertical wind shear on the entrain- 
ment. This aspect of cumulus convection has been 
discussed by Malkus [9]. It is shown that the horizontal 
translation of an ascending cloud column relative to its 
surroundings affects the growth and motion of the 
cloud. For the usual case of an increase in the hori- 
zontal wind with elevation, the entrainment is greatest 
on the upwind side of the cloud while on the downwind 
side the liquid cloud is detrained from the field of rising 
motion. Thus the entrainment of outside air 1s asym- 
metrical. A further consideration is the horizontal varia- 
tion through a cloud section of the mixing of environ- 
mental air with cloud air even in the absence of a vertical 
wind shear. This problem has been analyzed for the hy- 
drodynamical jet stream, but it has not been solved 
for the other aspects of mixing which have been dis- 
cussed here. 
Finally, it is recognized that, as a consequence of the 
entrainment of environ- 
