808 WEATHER FORECASTING 
Compendium.?:* We shall refer to such studies only 
inasmuch as they have current and direct application. 
The first milestone in the new aerological era was the 
increase of knowledge of the interrelationship between 
upper-level and sea-level circulations. Perhaps no one 
school of thought can be given principal credit for the 
discoveries, for many groups working along similar 
lines were converging to the truth. In this development, 
the centers of action observed at sea level became in- 
terrelated with the long waves in the middle and upper 
westerlies. 
A clue that the movement and development of these 
upper waves might be predicted by physical methods 
was given by Rossby [17], who showed that such large- 
scale waves, when treated on the basis of conservation 
of the vertical component of total absolute vorticity, 
seemed to fit into a reasonable theory. This theory re- 
sulted in the now classical frequency equation from 
which is given the eastward speed c of a long wave in 
the upper tropospheric westerlies as a function of the 
zonal wind speed U, the wave length L, and the north- 
ward variation of the Coriolis parameter 6: 
c = U — BL?/4r’. (1) 
From this expression it develops that the stationary 
wave length L, (7.e., when c = 0) is a function only of 
L, = 20/U/B. (2) 
In the practical application of this formula one is 
confronted first with the question of how the proper 
values of zonal speed U and the wave length L are to 
be computed. A great deal of work on these and other 
questions of application has been done at the U. 8. 
Weather Bureau, and some of the more important re- 
sults are described elsewhere in this Compendium.* 
These results, and indeed this entire article, should be 
considered a part of this treatment of extended-range 
forecasting. The principal results of these studies which 
relate primarily to the 700-mb level indicate that: 
1. Zonal wind speeds for use in equation (1) are best 
obtained by computing the geostrophic wind in mid- 
troposphere averaged over large areas (perhaps of the 
order of 100° to 180° of longitude) for the latitude bands 
in which the waves are found. There may be, and 
usually are, two or more out-of-phase wave trains 
present in different latitude bands. 
2. It is difficult to identify the primary long waves on 
a daily map because of the irregularities (small cyclone 
waves) embedded in the flow. The identification, as 
pointed out earlier, may be accomplished by some form 
of smoothing such as averaging over intervals of time, 
considering high tropospheric flow, or considering the 
patterns of isotherms associated with the contours. In 
the latter case those troughs and ridges in which iso- 
therms and contours are definitely in phase are gener- 
ally the primary long waves. 
3. Consult ‘“The Physical Basis for the General Circula- 
tion” by V. P. Starr, pp. 541-550. 
4. Consult ‘‘Observational Studies of General Circulation 
Patterns” by J. Namias and P. F. Clapp, pp. 551-567. 
3. Equation (1) may then be applied by choosing a 
trough or ridge whose subsequent motion is desired and 
scaling off its distance from the next primary ridge or 
trough upstream. 
When this is done for many cases it becomes clear 
that while there is reasonably good positive correlation 
between computed and observed values of ¢ for periods 
up to three or four days, the computed motions gener- 
ally give eastward speeds which are too small. For this 
reason empirical corrections must be developed for dif- 
ferent areas and seasons as described elsewhere.* From 
such a coordination of theoretical and empirical data 
one may construct graphs (for different areas and 
seasons) which are useful in predicting the motion of 
long waves. 
Another less restrictive method of using the vorticity 
principle in extended-range forecasting consists of con- 
structing (on daily or mean charts) inertia trajectories 
based on the conservation of the vertical component of 
the total absolute vorticity. Here again, empirical, areal, 
and seasonal modifications are necessary for optimum 
success. 
Both of the methods described above suffer from 
limitations imposed by the many assumptions underly- 
ing the development of the simplified vorticity equa- 
tions. Among the most serious of these are perhaps the 
assumptions of no divergence, no solenoids, and no 
change in speed of air particles. It is indeed surprising 
that in spite of these and other restrictive assumptions, 
the waves behave in such good accordance with the 
theoretical-empirical formula. Part of this agreement 
must be ascribed to the quasi-barotropic nature of the 
atmosphere, and to fields of divergence which are nor- 
mally operative in a fairly consistent manner within a 
given season over a given area. 
However, the use of empirically derived wave-length 
formulas and constant vorticity trajectories leave nu- 
merous forecasting problems unsolved. One of the prin- 
cipal of these involves the development of new troughs 
and the disappearance of old ones. Wave-length ex- 
pressions generally fail to distinguish between retro- 
gression (westward motion of the long waves) and the 
possible development of a new trough which may in- 
crease the hemispheric wave number, thereby taking 
up the slack indicated by a wave length greater than 
the stationary wave length. On the other hand, com- 
puted vorticity trajectories may suggest a new trough 
development in an area currently occupied by a ridge 
or a flat westerly flow. 
The problem of trough formation is closely allied with 
the process of “development” which has recently re- 
ceived considerable attention from British meteorolo- 
gists, particularly Sutcliffe [20]. This approach makes 
considerable use of hemispheric thickness patterns for 
the layer 1000-500 mb. Naturally, these charts are simi- 
lar to contour charts and thus bring to light the major 
long waves in the westerlies. Sutcliffe and his group 
attempt to apply wave-length concepts to them, at 
least qualitatively. But in addition, Sutcliffe has de- 
veloped certain criteria for the deepening and filling of 
systems, formation of secondary disturbances, etc., 
