476 
not here concerned with these disturbances, we may 
assume a cutoff in the wave number at k = 4. The 
maximum value of c,, then occurs at » = O and is 
equal to U + B/k? or (U + 22.5) deg long. day “, and 
the minimum value occurs at »2 = 3k and is equal to 
U — g/8k = (U — 2.8) deg long. day *. Since it is 
also unrealistic to take 1 = 0, we shall suppose that 
p > 4 also, in which case max ¢,, and min ¢,; are equal 
to U + B/8k = (U + 2.8) deg long. day *. The corre- 
sponding values of max and min ¢,, occur at w = +k 
and are equal to +11.2 deg long. day for k, p > 4. 
These group velocity estimates can be regarded only 
aS approximations in the asymptotic sense; they are 
accurate only for large values of time. Moreover, the 
derivation of the energy propagation equation (49) 
presupposes a continuous variation of frequency, where- 
as the frequencies are here restricted to discrete values 
by the finiteness of the earth. The following alterna- 
tive approach to the problem avoids these difficulties. 
If we assume a sine dependency on the y-coordinate 
(0°b/dy’ = —m’®), equation (45) becomes 
do (a® a® 208 o, OB 
Sel (sees gies ey = 0. 
H(S+0%) a pear (e) — Ura) = (Us. (80) 
Its solution may be written [8]: 
@(x, t) = d(@ — Ut, 0) 
+3 (51) 
+ [ ie = Ui = a, \ale,O) de 
where 
1S , in ike ‘ 
ip == 2G" Se (52) 
Qa —o 
and 
ea : 
e k? + m2” (53) 
The “influence function” J,2(x, t) is shown in Fig. 2 
Im2(X,1) 
2 
X 
-180° -150°-120°- 90° -60° -30° 0° 30° 60° 90° [20° 150° 180° 
-4 
Fie. 2—Influence function for m? = 18, ¢ = 1. 
DYNAMICS OF THE ATMOSPHERE 
form’ = 18 and ¢ = 1. This function is seen to become 
very small as the absolute value of x is increased. Let 
us assume that it is negligible for > a, anda < —m. 
Then the upper and lower limits of integration may be 
replaced by « — Ut — a and « — Ut + a», respec- 
tively. This means that with respect to the mean 
zonal current, influences propagate eastward with a 
speed less than x, degrees per day and westward with 
a speed less than x, degrees per day. From the figure, 
one may estimate 2, & 25° and a 40°. These esti- 
mates for the influence velocities are greater than those 
obtained for the group velocities because no restric- 
tions have been placed on the energy spectrum of the 
initial disturbance ®(z, 0). Influences from only very 
long wave disturbances propagate rapidly, and it is 
likely that a disturbance with most of its energy lying 
in the middle and short wave length region of the 
spectrum would again be found to have influence veloci- 
ties more in conformity with those given by the ex- 
treme group velocities. 
In summary, one may say that influences have a 
maximum rate of propagation not much im excess of 
U (the particle speed) and a minimum rate not much 
less than U, so that the propagation relative to the 
ground is eastward. These conclusions have been borne 
out by several recent mtegrations of the nonlmear 
barotropic equations. 
The solution for the linearized barotropic equations 
given by (51) has a certain value in itself. It has been 
used with some success for forecasting the actual wave- 
like perturbations of the westerlies [4]. The initial 
distribution of ® along a fixed latitude was determined 
from the 500-mb map, and the integral in (51) was 
evaluated by means of Simpson’s rule. It was shown 
that the topographical motions become quasi-station- 
ary when m = 18, so that the relative variation in 
the height of an isobaric surface at a fixed latitude 
could be predicted for a 24-hr period without taking 
these motions into account. Frictional effects were like- 
wise found to be negligible in forecasts for so short a 
period. These results have been taken to justify the 
neglect of topography and friction in first attempts 
at the integration of the nonlinear barotropic equations. 
Tt has also been shown [3] that influences are propa- 
gated at an effectively finite speed in the vertical direc- 
tion. The maximum vertical group velocity for small 
plane wave disturbances in a baroclinic atmosphere 
with fronts perpendicular to the x, z plane has been 
calculated at approximately 4.5 km day * at 45° lat. 
Thus in principle it is unnecessary to know the entire 
vertical structure of the atmosphere initially in order 
to predict the motion of the lower layers for short 
periods. This result is perhaps of no great practical 
importance for numerical computation, since the low 
energy of the very high level motions renders their 
influence negligible in any case. It does, however, have 
a theoretical significance in stating a property of the 
atmospheric motion that has not generally been recog- 
nized. It is this property which may constitute the 
most serious criticism of the advective model. Accord- 
ing to equation (19) the calculation of the geopotential 
