606 
fined as the intersection between the sloping polar-front 
layer (surface) and the 500-mb surface. The polar front 
at the 500-mb level thus shows latitudinal displace- 
ments of the same extent as the belt of strongest west 
wind. 
These considerations lead to the assumption that 
there must be a very close connection between the 
formation and further development of the long waves 
and the latitudinal displacement of the tropical and 
polar air masses in the free atmosphere. The upper 
troughs are regions for the maximum southward dis- 
placement of polar air, and the upper ridges are regions 
for the maximum northward displacement of tropical 
air. The existence of a really well-marked polar-front 
surface in cross sections drawn through formations of 
this type has been shown by a large number of detailed 
studies of characteristic situations. 
The horizontal dimensions of upper long waves are 
usually considerably larger than the dimensions of po- 
lar-front cyclones on surface maps. Upper long waves, 
as defined by Rossby [55], correspond more to the 
individual cyclone families first described by J. Bjerknes 
and H. Solberg [7] in their original paper on the polar- 
front theory. The mdividual members of a cyclone 
family appear at the 500-mb level as “minor” wavelike 
perturbations superposed upon the pattern of the long 
waves. The scheme is presented in Fig. 3, which con- 
taims four similar upper long waves at the 500-mb 
surface, the 500-mb polar front (here considered as a 
continuous line around the hemisphere), and the four 
polar fronts at the ground, with their minor cyclonic 
disturbances. The minor irregularities in the upper 
polar front and in the contour lines at the 500-mb sur- 
face correspond to cyclonic disturbances at the surface. 
The surface fronts are placed approximately in the same 
regions as in Fig. 124 in Petterssen’s textbook [47, 
p. 269]. 
Figure 3 represents a combination of the original 
polar-front scheme of Bjerknes and Solberg with the 
scheme for the long waves in the upper westerlies. 
Both the long waves and the polar fronts on the sur- 
face move, on the average, slowly in a west-east direc- 
tion. The motion of the air at the 500-mb level is faster 
than the movement of the mdividual cyclone waves, 
and the movement of the latter is faster than the east- 
ward displacement of the long waves. 
It might be pomted out that the number of four long 
waves in Fig. 3 is an arbitrary one. Actually the wave 
number varies from time to time. Also the eastward 
speed of the waves varies considerably; statistical stud- 
ies by Cressman [16] indicate that on the average the 
speed is in fair agreement with Rossby’s formula, 
= 20a cos* Co) 
G=U 
n2 
, (11) 
for long waves in a barotropic atmosphere (n is the 
number of hemispheric waves), if U is determined as 
the mean velocity of the west wind at the level of 600 
mb (average level of nondivergence). From equation 
(11) it can also be concluded that there must be a 
tendency for a larger wave number n at lower latitudes 
MECHANICS OF PRESSURE SYSTEMS 
than at higher latitudes. Thus the scheme in Fig. 3, 
with four long waves at all latitudes, cannot be main- 
tamed; at lower latitudes the waves must show a 
tendency to be split into a greater number. 
The air movement is not strictly horizontal. A com- 
plicated vertical movement is superposed upon the 
horizontal movement expressed approximately by the 
contour lines in Fig. 3. This vertical movement deter- 
mines a vertical circulation pattern which is essential 
for the understanding of the process of development. 
As a consequence the polar front at the 500-mb surface 
undergoes continuous changes. These processes will be 
discussed more in detail in a later section. 
Fie. 3.—Schematic circumpolar chart for the 500-mb level 
showing four long waves. Heavy line is the polar front at 
500 mb and thin lines are contours of 500-mb surface. Fronts at 
the earth’s surface are indicated by the usual symbols. 
The changes of the upper polar front and the upper 
flow pattern can be attributed to some kind of in- 
stability of the perturbations in the westerlies. The 
instability appears, for example, in a rapid increase of 
the amplitude of the polar-front contour and of the 
streamlines. At the 500-mb surface, the band of strong- 
est isotherm concentration, which during the coldest 
season is normally situated in the vicinity of latitude 
50°N, may be pushed southward to latitudes 40°-30°N. 
In the lower troposphere (between about 1000 and 
800 mb) this process is connected with a pronounced 
anticyclonic outflow of polar air masses. The surface 
polar air thus forms a cold anticyclone, whereas the 
upper polar air forms an intensified upper trough or, in 
extreme cases, a closed upper cyclone. The process can 
be regarded as a combined consequence of increase of 
cyclonic vorticity due to latitudinal displacement and 
to convergence in the upper parts of the subsiding 
cold air. 
An opposite process takes place during the formation 
and strengthening of upper ridges. These ridges are 
