76 JOSEPH SMAGORINSKY 
and the precipitation rate is 
dW p 
= | vow) SF? ap 6) 
dt 
The potential temperature 6 change due to 
such condensation or evaporation is given by 
d\iné T 
=d-wH (7) 
dt Pp 
The notation is as follows: p is the pressure, g is 
the acceleration of gravity, pw is the density of 
water, w = dp/dt, r, is the saturation mixing 
ratio, 6 is the fraction of mass undergoing moist 
adiabatic processes, and 6* is the fraction of con- 
densing water being precipitated. Also 
oh Shove 
yn(T', p) = p 
dp @g=const 
(8) 
_ +a-—1 <5 
ay-atl 
d\iné@ 
I(T, p) = p 
dp 8g=const 
(9) 
ka —xk— 1 
= -a |———— } <0 
(= —at+ ') 
where 
Lr s 
eT, p) = — (10) 
Cpl 
L 
VT) = 11 
RT (11) 
and T is the absolute temperature, #z the equiva- 
lent potential temperature, L is the latent heat 
of condensation or sublimation, (1 — k) = ¢,/¢p 
is the ratio of the specific heat of air at constant 
volume to that at constant pressure, R* is the 
gas constant for water vapor ~4.62 x 10° cm? 
sec deg.-! 
Unlike 6* which is zero for w > 0, 6 does not 
depend on w since downward motion of a cloud 
parcel must result in dynamic evaporation so 
that dW /dt < 0 and dr/dt > 0. We are not free 
to specify arbitrarily 6 since (1) and (2) must be 
satisfied simultaneously by (3) and (4). Ignoring 
variations in a, B, and W,, then (1) and (2) 
require that 
(12) 
dw a3 _ dh 
a BW. = 
“dé = 9P : Vier 
Since by definition 
(18) 
then 
dh 1 fdr. drs 
dt 7, (é y dt ) ao 
The first term, the change of h caused by a 
change in water vapor, is given by (3); the second 
term also depends on the fraction of mass under- 
going moist adiabatic changes 6, but of course 
does not vanish for purely dry adiabatic proc- 
esses, since it changes with temperature and may 
be written as 
drs 
= rwld-¥m 
a (15) 
ar dl = 5) yal ) 
where 
1lnr 
vat) = »(§ =) =xy—1>0, (6) 
dp Joxconst 
and it may easily be verified that 
Vm = Xd Yn = Va 
= = 
Y 1+ ya 
(17) 
Hence 
dh 
dt 
2 E h)é . = (i —=5)R =| » (18) 
Inserting (4) and (18) into (12) yields 
= a* + xh Vc va/ym 
1+x Weld — h) + hya/yml 
(19) 
where 
= 1.58W,/r; (20) 
For ¢ < «, we have that 6* = 0 so (19) and 
(1) uniquely define 6 as a function of ¢ or h. Also, 
for c. < ¢ < c,andw > 0, we have that 6* = 0, 
again uniquely defining 6. On the other hand, 
when w < 0, we have 6,;* = 0 as before, and since 
all condensing water vapor must be precipitating 
when ¢ = ¢2 then by (4) 62 = 6.* = 1. Assuming 
6* to vary linearly in this range then 
c—t c—1 
6* = ———__ = 
C2 — C1 0.3 
(21) 
> Ofora <c<a,w <0 
We furthermore see that a maximum liquid wa- 
ter is attained for ¢ = c., which by (2) is 
We = Wie28/? (22) 
Hence the maximum liquid water content is 50% 
