DYNAMIC FORECASTING 
seem insurmountable, as the following considerations 
will indicate. 
Tt is very probable that the state of the motion at 
any given time will depend continuously upon the 
§0_170 150___130_110_90 
70__60 50. 40 50 
v SEN 
170} 
150} 
100 90 80 70. 60. 50 40 
Fic. 4—The broken lines represent the 24-hr height change 
computed by translating the baroclinically computed tend- 
ency field for 0300 GMT, January 30, 1949 in the direction of 
the mean current and with the speed of the trough. The solid 
lines represent the observed 24-hr height change. 
initial motion; if the initial conditions are varied 
slightly, the motion at a subsequent time will vary 
shghtly. Thus if the initial distribution of pressure, 
temperature, and horizontal wind is known to a rela- 
tively high degree of accuracy, the final distribution 
of these quantities will also be accurately determined 
and this will be true for whatever initial values are 
assigned to the unobservables. 
To elaborate, let us consider two motions whose 
velocity components wu, v, w and w’, v’, w’ result from 
the slightly differmg initial velocities wm, vo, wo and 
uo , 00 , Wo. If dm is the element of mass, the integral 
T= fjfiiu — wy + & — oy + w — w')| dm 
over the forecast region is a measure of error in the 
motion when ww, vp, Ww are regarded as correct. We 
shall assume that wo, vo, wo are measured with such 
accuracy that J), the initial value of J, is small in 
comparison with the initial kmetic energy 
Jf J 4¢(ws + vd + wa) dm. 
Now let us assume that the differences w — wu’, v — v’, 
w — w’ remain fairly small in comparison to wu, v, w 
but allow for the possibility of a relative increase. The 
differences will then satisfy the linearized perturba- 
tion equations for the mean flow w, v, w and reduce to 
Up — Ud, Uy) — V0, Wo — wo for t = O. If this perturba- 
tion motion is stable so that the kinetic energy does 
not imcrease with time, the error J will also not increase 
and the motion w’, v’, w’ will continue to be a good 
approximation. In practice, when forecasting for a 
limited region, the kinetic energy of the difference 
motion might conceivably increase as a result of work 
481 
done at the boundaries or by advection of kinetic 
energy at the boundary. But in view of what is known 
about influence velocities one can say that it cannot 
possibly increase by more than the kinetic energy con- 
tained in that region, lying outside the forecast area, 
from which influences can reach the boundary in the 
forecast time. Hence, if the kinetic energy within this 
region is not many times larger than the kinetic energy 
within the forecast region, the above conclusions still 
apply. If the difference motion is unstable, the question 
of the utility of the primitive equations is not easily 
decided. It is certainly not obvious that dynamic in- 
stability necessarily implies their inutility. If, for ex- 
ample, the original motion is a small perturbation on an 
unstable steady flow with wu, v, w, and w’, v’, w’ ~ 
e(y a positive number), the error J will increase ex- 
ponentially; but the kinetic energy of the actual mo- 
tion will also increase exponentially in the same pro- 
portion so that the relative error will remain constant. 
J. C. Freeman and the writer’ have carried out a 
numerical integration of the Eulerian equations for 
small perturbations in a barotropic atmosphere, taking 
care to satisfy the Courant-Friedrichs-Lewy stability 
criterion. Had the quasi-geostrophic equation been 
used, the motion would have consisted only of gener- 
alized Rossby waves; instead, the computed motion 
was found to consist of two superimposed parts, the 
first nearly identical to the Rossby motion and the 
second a gravitational wave motion of much smaller 
amplitude. Because of the necessarily erroneous values 
used for the acceleration and divergence (these were 
assumed to be identically zero initially), the changes 
in the velocity occurring after the first few time steps 
were quite incorrect. But there soon took place an 
adjustment of a kind that the motion at no time was 
found to deviate appreciably from the quasi-geo- 
strophic. The function of the acceleration and di- 
vergence fields lay apparently in the mechanism by 
which the quasi-geostrophic balance was maintained. 
In a manner of speaking, the gravity waves created by 
the slight unbalance served the telegraphic function of 
informing one part of the atmosphere what the other 
part was doing, without themselves influencing the 
motion to any appreciable extent. Thus it mattered 
little that the initial error in the unobservables changed 
the character of the gravitational waves; only their 
existence was important. 
The question arises whether the unobservables im the 
baroclinic atmosphere play an analogous role. This 
question is again not easily answered: in the case of 
barotropic motion, gravity waves are of the external 
type which move with the Newtonian velocity of sound; 
whereas in the baroclinic atmosphere they may be of 
the internal type—for which the speeds and frequencies 
are smaller—and might conceivably interfere with the 
large-scale motions. 
In the last analysis, the feasibility of using the 
primitive equations will be decided only when numeri- 
cal integrations have been carried out. For this purpose 
3. Unpublished report, Meteorology Project, The Institute 
for Advanced Study, Princeton, New Jersey. 
