DYNAMIC FORECASTING BY NUMERICAL PROCESS 
By J. G. CHARNEY 
The Institute for Advanced Study* 
Introduction 
As meteorologists have long known, the atmosphere 
exhibits no periodicities of the kind that enable one to 
predict the weather in the same way one predicts the 
tides. No simple set of causal relationships can be found 
which relate the state of the atmosphere at one instant 
of time to its state at another. It was this realization 
that led V. Bjerknes in 1904 [1] to define the problem 
of prognosis as nothing less than the integration of the 
equations of motion of the atmosphere. But it remained 
for Richardson [12] to suggest in 1922 the practical 
means for the solution of this problem. He proposed to 
integrate the equations of motion numerically and 
showed exactly how this might be done. That the 
actual forecast used to test his method was unsuccessful 
was in no sense a measure of the value of his work. 
In retrospect, it becomes obvious that the inadequacies 
of observation alone would have doomed any attempt 
however well conceived, a circumstance of which Rich- 
ardson was aware. The real value of his work lay in the 
fact that it crystallized once and for all the essential 
problems that would have to be faced by future workers 
in the field and that it laid down a thorough ground- 
work for their solution. 
For a long time no one ventured to follow in Richard- 
son’s footsteps. The paucity of the observational net- 
work and the enormity of the computational task 
stood as apparently unsurmountable barriers to the 
realization of his dream that one day it might be 
possible to advance the computation faster than the 
weather. But with the increase in the density and 
extent of the surface and upper-air observational net- 
work on the one hand, and the development of large- 
capacity high-speed computing machines on the other, 
interest has revived in Richardson’s problem and at- 
tempts have been made to attack it anew. 
These efforts have been characterized by a devotion 
to objectives more limited than Richardson’s. Instead 
of attempting to deal with the atmosphere in all its 
complexity, one tries to be satisfied with simplified 
models approximating the actual motions to a greater 
or lesser degree. By starting with models incorporating 
only what are thought to be the most important of the 
atmospheric influences, and by gradually bringing in 
others, one is able to proceed inductively and thereby 
to avoid the pitfalls inevitably encountered when a 
great many poorly understood factors are introduced 
all at once. 
A necessary condition for the success of this stepwise 
method is, of course, that the first approximations bear 
* Most of the work described in this article was performed 
under contract N6-ori-139, Task Order I, between the Office of 
Naval Research and The Institute for Advanced Study. 
a recognizable resemblance to the actual motions. For- 
tunately, the science of meteorology has progressed to 
the point where one feels that at least the main factors 
governing the large-scale atmospheric motion are 
known. Thus integrations of even the linearized baro- 
tropic and thermally inactive baroclinic equations have 
yielded solutions bearmg a marked resemblance to 
reality. At any rate, it seems clear that the models 
embodying in mathematical form the collective experi- 
ence and the positive skill of the forecaster cannot fail 
utterly. This conviction has served as the guiding 
principle in the work of the meteorology project at 
The Institute for Advanced Study with which the 
writer has been connected. 
The Geostrophic Approximation 
In the selection of a suitable first approximation, 
Richardson’s discovery that the horizontal divergence 
was an unmeasurable quantity had to be taken mto 
account. Here a consideration of forecasting practice 
gave rise to the belief that this difficulty could be 
surmounted: forecasts were made by means of geo- 
strophic reasoning from the pressure field alone—fore- 
casts in which the concept of horizontal divergence 
played no part. And indeed, this belief was substan- 
tiated when it was shown by Charney [2] and Eliassen 
[7] that the geostrophic approximation could be used 
to reduce the equations of motion to a single dy- 
namically consistent equation in which pressure appears 
as the sole dependent variable. 
If, in addition to the geostrophic approximation, one 
makes the assumptions that frictional and nonadiabatic 
effects may be ignored, the equations of motion are 
found to be contained in the statement that the po- 
tential temperature and the potential vorticity, as de- 
fined by Rossby [13], are conserved, and that both the 
quasi-hydrostatic and the geostrophic approximations 
may be used for evaluating the terms involving density 
and horizontal velocity in the conservation equations. 
A partial justification for this procedure was given by 
the author [2], but the ultimate test must depend upon 
comparison of theory with observation. 
The potential vorticity is defined as specific volume 
times the scalar product of the absolute vorticity and 
the entropy gradient. For the large-scale atmospheric 
motions the isentropic surfaces are quasi-horizontal, 
and we may to a first approximation replace the ab- 
solute vorticity and entropy gradient by their vertical 
components. The pressure equation then becomes [3] 
© ate a \|2- ; 
[Stet Se me ee) lig =v © 
in a rectangular coordinate system with x pointing 
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