EXTRATROPICAL CYCLONES 
20a cos® C0) 
ne : 
ve =e+ (2) 
In this equation c is the eastward speed of propagation 
of the wave, © is the angular velocity of the earth, a is 
the earth’s radius, ¢ is the geographical latitude, and 
n is the wave number per circumference of the earth. 
Numerical values of (2Qa cos’ ¢)/n? are given in 
Table II. This table, in conjunction with equation 
(2), shows that in short waves, such as are found to 
accompany the individual traveling cyclone, the level 
of nondivergence lies at an elevation where the un- 
disturbed current moves only a little faster than the 
wave. In long waves the air at the level of nondiver- 
gence moves eastward much faster than the wave 
itself, particularly in low latitudes. 
TasxeE II. VALUES OF (20a cos? ¢)/n? (in m sec7) 
Wave length (deg. long.) 
% 
180° ~ 120° 60° 36° 18° 
70° 9.3 4.1 1.0 0.4 0.1 
60° 29.0 12.9 3.2 1.2 0.3 
50° 61.6 27.4 6.8 2.5 0.6 
40° 104.3. 46.3 11.6 4.2 1.0 
30° 150.7 67.0 16.8 6.0 1.5 
The level of nondivergence may be determined from 
sets of aerological maps, with the aid of Table II. Its 
height differs from case to case, but according to 
Charney [4] and Cressman [5] it averages around 600 
mb for both long and short waves. Hence, with a 
given model of baroclinic westerlies, the speed vz of 
the undisturbed westerlies at the level of nondivergence 
is approximately the same parameter for long. and 
short waves. The speed of all such waves, which are 
superimposed on the same westerly current, therefore 
varies with wave length according to the formula, 
* 20a cos’ d 
c= Vz aR mene cOErEEr 
(3) 
Short waves (large n) move almost with the speed of 
the air at the level of nondivergence. Long waves move 
eastward more slowly than short ones, and may also 
retrograde. Table II, applied to the case c = 0, gives 
us a survey of the wind at the level of nondivergence 
in stationary waves. At 70° latitude the west wind 
must be quite light for waves to be stationary, and the 
180° wave length seems to be the most likely one for 
standing waves. Proceeding to lower latitudes, we find 
that the 180° stationary wave requires stronger wester- 
lies than are ever known to occur. Assuming that v* 
would never be greater than 30 m sec, we see that 
below 60° latitude the 180° stationary waves would 
never occur, below 50° the 120° stationary waves also 
become impossible, and so on. This dependence of the 
long-wave pattern on geographical latitude usually 
leads to the establishment of only two or three stand- 
ing waves per earth’s circumference near the pole and 
stationary patterns with higher wave numbers in lower 
latitudes. In the latitudes of pattern transitions, com- 
plicated cases of wave interference occur. 
579 
The waves in the westerlies associated with the 
moving extratropical cyclones are by necessity of short 
wave lengths, say thirty degrees longitude. According 
to (8) and Table II, such waves move with a speed c, 
only slightly smaller than vy , the speed of the wester- 
lies at the level of nondivergence. 
Figure 1 shows the position of the pressure minimum 
at sea level relative to that of the upper trough. The 
axis of minimum pressure of the closed low tilts towards 
the coldest side, which is usually to the west or north- 
west of the location of the surface center. The tropo- 
spheric part of the upper trough is also displaced west- 
ward with height, but not as far per unit height as the 
subjacent center of low pressure. 
The friction layer of the closed vortex (up to about 
1-km height) has horizontal convergence. Above the 
influence of surface friction the eastern half of the 
vortex has horizontal convergence of mass and the 
western half, horizontal divergence. This holds true 
also for the upper trough up to the level of nondiver- 
gence, beyond which divergence and convergence ex- 
change positions. The tendency equation (1) applied to 
the schematic cyclone cross section of Fig. 1 gives 
an answer to the two questions: How ean the cyclone 
move eastward as most middle latitude cyclones do? 
and, How can it deepen despite the frictional conver- 
gence? 
The eastward displacement of the cyclone is assured 
if the vertical integral of horizontal mass divergence 
in the tendency equation is determined as to sign by 
the atmosphere above the level of nondivergence. The 
deepening of the pressure minimum likewise depends 
on the influence from above the level of nondivergence. 
Because of the westward tilt of the axis of the cyclone 
a vertical air column located at the surface center will 
show horizontal convergence in its lower portion, where 
it passes through the forward part of the vortex, and 
horizontal divergence where it traverses the upper wave 
pattern east of the wave trough. Deepening of the 
surface center will occur only if this upper-air diver- 
gence overcompensates the low-level convergence. 
In all parts of the cyclone the surface pressure tend- 
ency represents a small change in the weight of the 
local vertical column, resulting from the difference 
between accumulation and depletion of air, each of 
which represents much greater weight changes. The 
natural adjustment of the pressure tendencies to the 
observed moderate values can be visualized as follows. 
In the low-level vortex, the convergence in the front 
and the divergence in the rear are more strongly de- 
veloped the faster the vortex is forced to move. We 
have also seen from Table II that the level of non- 
divergence in the upper wave rises to a higher eleva- 
tion, and the divergence values above that level de- 
crease, when the wave speed increases. Therefore, a 
supposed increase in speed without a change in the 
structure of the cyclone would lead to a weakening of 
the high-level contribution and a strengthening of the 
low-level contribution to the change in weight of air 
columns. This would be tantamount to a decrease in 
pressure tendencies. Quite analogously, it can be shown 
