DYNAMIC FORECASTING 
quantitative estimates of the convergence criteria for 
the nonlinear difference equations—estimates that 
would be difficult if not impossible to obtain by any 
other means. Also, the study of linearized counterparts 
may be undertaken for the physical insight they give 
into the nature of the dynamical processes at work in 
the atmosphere—without which insight, numerical fore- 
casting is at best a haphazard affair. 
One problem to which at least a partial answer is 
required is the determination of the speeds at which 
influences propagate in the atmosphere. The elliptic 
character of (1) demands that the boundary conditions 
be known as functions of time along vertical surfaces 
surrounding the forecast region, as well as at the ground 
and at the ‘“‘top” of the atmosphere. Since such con- 
ditions cannot be known, there appears to be a basic 
indeterminacy in the problem of forecasting for 
a limited region. The reason for this indeterminacy is 
that the introduction of the geostrophic approxima- 
tion effectively filters out the sound and gravity waves 
whose finite speeds place an upper limit on the rate 
of propagation of a disturbance. Thus disturbances 
propagate instantaneously in quasi-geostrophic mo- 
tion, and effects from the boundaries are felt immedi- 
ately at every point in the region they enclose. One 
feels that smce sound and gravity motions interact 
negligibly with the large-scale systems, the solution 
to the difficulty must be found in the quasi-geostrophic 
equations themselves. Let us, therefore, examine these 
equations in their linearized form. 
As we shall be concerned primarily with horizontal 
propagation it will be permissible to treat the motion 
barotropically, for it has already been shown that the 
horizontal motion of a barotropic atmosphere approxi- 
mates the actual motion at a certain mean level. 
For small perturbations on a constant zonal current 
U, the barotropic equation (30) becomes 
Vi = + 
ot (45) 
o® OP 
Uf = = =, 
=) a Ox : 
The quantity 8 = df/dy will be given the constant 
value corresponding to the mean latitude 45°. Equa- 
tion (45) admits of the plane wave solution 
® = exp [i(kx + py — rt)] (46) 
in which the frequency »y is related to the wave num- 
bers k = 27/1, and p = 2z7/l, by 
A Bk 
DOE esc (47) 
The group velocity components c,, and c,, are 
Ov ie = et 
Gm = SS = U SF B 
ok k? 22 
(k? + yu) (48) 
Ov 28k 
au P+ 
Now it may be shown that the kinetic energy H of a 
point disturbance obeys approximately the law 
475 
ob ar g (Cy2 H) + a (Coy L) = 0, 
ou Oy (49) 
which states that the energy of the disturbance asso- 
ciated with a given area in the x, y plane will not 
change with time when each point of the area moves 
with the local group velocity given by (48). Since an 
arbitrary initial disturbance may be regarded as a sum 
of point disturbances, we may state that no influence 
is propagated faster than the maximum group velocity. 
An inspection of (48) shows that for increasing J, and 
l, the group velocities as well as the phase velocities 
become arbitrarily large. However, the magnitudes of 
1, and l, are limited by the finiteness of the earth’s 
surface, and even this limitation is scarcely realistic, 
for the extent of the motions with which we are con- 
cerned is far less than world-wide. To obtain a better 
estimate of the maximum group velocities, we first 
provide for the finiteness of the earth by assuming x 
to have a period equal to the circumference of the 45° 
latitude circle. If the unit of distance is taken as the 
radius of this circle and the unit of time as one day, k 
becomes an integer representing the number of waves 
encircling the earth and 6 = 2r7. 
A typical spectrum determined by Fourier analysis 
of the v component of the velocity at 45°N for 0300Z 
on January 11-13, 1946 at 500 mb is shown in Fig. 1, 
| 2 3 4 5 6 7 8 S) 10 
k — 
Fie. 1—Mean spectral distribution of the kinetic energy 
of the north-south motion at 500 mb and 45° N for 0300Z, 
January 11, 12, and 13, 1946. 
in which the square of the amplitude, A”, measuring 
the kinetic energy of the north-south motion, is plotted 
against k. One energy maximum occurs at k = 2.5, 
corresponding to a wave length of 144° long. at 45° 
lat., and the other at k = 7.5, corresponding to a wave 
length of 48° long. The first maximum is associated 
with the very long wave quasi-stationary disturbances 
which, as Charney and Eliassen [4] have shown, are 
produced mainly by topographical action. As we are 
