428 
[7]. Naturally the line in the 2, ¢ plane along which data 
are known need not be a vertical line corresponding to 
a fixed value of x. As in the more general case of the 
previous section, it can be any line in the 2, ¢ plane 
Fre. 9—A “squall line’ magnified in the same ratio as 
the scale of miles on a micromap showing the position of a 
pressure jump. 
which is not a characteristic line. Accordingly, Tepper 
assumes that he knows the motion of a steep cold front 
on the weather map. This cold front is adjacent to a 
region in which there is an inversion and the air under 
the inversion is in some equilibrium state with equal 
inversion height and equal west-wind components 
throughout. (The north-south wind components and 
the Coriolis parameter are neglected entirely in this 
particular study.) The cold front then begins to accele- 
rate and this pushes the inversion up in its vicinity. 
This compression wave ultimately develops into a pres- 
sure jump. In the meantime the front decelerates to its 
previous speed, or a lower speed, to the west and an 
expansion wave moves out behind the pressure Jump. 
Some successive cross sections describing this phenome- 
non appear in Fig. 10. 
Interaction of an Expansion Wave and a Pressure 
Jump. The whole area in which the characteristic lines 
in Fig. 4 are diverging is called an expansion wave. Such 
a wave has two speeds: it approaches from the left with 
a speed given by the slope of the first diverging charac- 
teristic and leaves to the right with a speed given by 
the slope of the last diverging characteristic. The ap- 
proximate speed of a pressure Jump can be computed by 
a method demonstrated to the writer by C.-G. Rossby. 
If the jump is assumed to have an unchanging shape 
and to be moving with constant speed V, the law that 
the difference in pressure force is equal to the loss in 
momentum can be expressed as 
(wu = V) pushs => (us = V) pushes 
he 
hy 
= P, dz — P, dz. 
0 0 
(21) 
Equation (21) reduces to 
p(ur — V) wha ag plus an V) ushe 
pire cn 
= $ ( — p')(ha — hi). 
DYNAMICS OF THE ATMOSPHERE 
The continuity of mass is expressed by the similar equa- 
tion: 
(uy = Vy)hy — (us = V)yhe 5 (23) 
If the value of (w2 — V) from equation (23) is substi- 
tuted in equation (22), the resulting equation can be 
solved for the following four possible values of V: 
mae he he + In |? 
wa [dre eS] 
U 4% 
nap hi he + hy |? 
ti +[o(1 “) h, 2 | ; 
where Vi, = Vo, and Vi = Vo. 
If the jump is from a low value of h, to a higher value 
of hy, it can be seen immediately that if we set 
Vig (24) 
Vos (25) 
V=u+a (26) 
to correspond to the slope of the characteristics 
ube (27) 
dt 
where 
e= [Qi - 2) 
p 
that 
a> GQ (28) 
and 
ay < Co. (29) 
This is a quantitative statement of the following rule 
expressed for gas dynamics by Courant and Friedrichs 
[5]: Each member of a sequence made up of alternate 
expansion waves and jumps in the same family of 
characteristics will move to overtake the member pre- 
ceding it. This is true because the expansion waves 
move at the speed wu + c¢ and the jumps move at the 
te 
Fic. 10.—A series of simple waves and pressure jumps 
showing how they overtake and modify each other. 
speed wu + a. This rule is illustrated in Fig. 10 where 
it can be seen that the left-hand side of expansion wave 
I moving at wu + c is being overtaken by jump II 
