578 
by Holmboe [3]. He defined a “critical eccentricity” 
which would balance the exchange of air between the 
two halves of the vortex. Values of the quantity | | — 
|v’| — 2¢ = 4Qac?, cos ¢, evaluated in a narrow iso- 
baric channel of critical eccentricity, are given in Table 
I @ and v’ are the wind velocities at southernmost and 
northernmost points, respectively, of the isobaric chan- 
nel, c is the eastward speed of displacement of the 
vortex, 2 the angular speed of the earth, a the earth’s 
radius, o, the angular radius of the isobaric channel, 
and @¢ the geographical latitude). In the case of the sta- 
tionary vortex, the exchange of air between the eastern 
and western halves of the vortex is balanced if the 
wind velocity in the southernmost point of the isobaric 
ring exceeds that in the northernmost point by the 
tabulated amount. Near the center the flow can be 
almost constant all around the isobaric ring, but the 
greater the radius the more will the west wind in the 
south have to exceed the east wind in the north, par- 
ticularly in low latitudes. 
Tase I. Vauuzs oF |v| — |v’| — 2c = 4Qa03 cos 
FOR CRITICAL HccENTRICITY (in m sec) 
Angular radius of isobaric channel 
¢ 
1° 5° 10° 20° 
90° 0 0 0 0 
80° 0.1 2.5 9.9 — 
70° 0.2 4.9 19.4 78 
60° 0.3 Goll 28.4 114 
50° 0.4 9.1 36.6 146 
40° 0.4 10.9 43.6 175 
30° 0.5 12.3 49.4 198 
In applying the table to a moving vortex, twice the 
speed of displacement of the vortex must be added to 
the tabulated speed to give|v| —|v’| . For the east- 
ward moving vortex the critical eccentricity is thus 
stronger than for the stationary one. Even slightly 
greater eccentricity would be needed to bring about 
accumulation of air in the western half of the pressure 
minimum and depletion of air in the eastern half, a 
condition which would seem necessary to make the 
system move eastward. A check with measured ec- 
centricities shows, however, only ‘‘suberitical’”’ cases; 
in other words all observed cyclonic vortices accumulate 
air in their front parts at the expense of the rear parts. 
The pressure change in such a moving vortex can 
therefore be explained only if other flow patterns prevail 
above the vortex. This conclusion is corroborated by 
the experience of synoptic aerology, and the typical 
upper flow pattern above moving vortices is that of 
the atmospheric wave. In the composite extratropical 
cyclone, the vortex part resists the eastward motion 
by pilmg up air in the front half; this resistance in- 
creases the faster the vortex is forced to move. 
The atmospheric wave? superimposes a quasi-hori- 
2. The fundamental properties of the atmospheric waves 
of synoptic meteorology were first sketched by J. Bjerknes 
[2] and later developed mathematically by Rossby [26, 27] 
and Haurwitz [10]. In this article we follow the more recent 
treatment by J. Bjerknes and Holmboe [8]. 
MECHANICS OF PRESSURE SYSTEMS 
zontal oscillation upon the fundamental current of 
straight westerlies. This is accompanied by a periodic 
distribution of horizontal mass divergence, which in 
first approximation depends on the relative strength 
of the “curvature and latitude effects” upon the wind 
speed. The curvature effect makes the air move super- 
geostrophically while overtaking the wave crest and 
subgeostrophically while overtaking the wave troughs. 
The latitude effect upon the wind speed comes from 
the fact that each flow channel is in a higher latitude 
at the anticyclonic bend than it is at the cyclonic 
bend, so that, the horizontal pressure gradients being 
equal, the geostrophic wind would be stronger at the 
cyclonic than at the anticyclonic bend. The result of 
UPPER LEVEL 
ISOBARS 
SURFACE ___ 
ISOBARS 
SINUSOIDAL 
ISOBARS 
CLOSED ISOBARS 
FRICTION LAYER 
4 
SUPERIMPOSED UPPER AND LOWER MAPS 7 
B CENTER 
——— 
PROPAGATION 
Fic. 1.—Schematic model of cyclone with lower-vortex and 
upper-wave part. West-to-east vertical profile shows location 
of horizontal divergence and convergence of mass. 
these two opposite effects on horizontal divergence is 
in favor of the curvature effect when wave lengths 
are short and the fundamental current is strong. In 
this case the wave crests are preceded by horizontal 
convergence and the wave troughs by horizontal diver- 
gence. In the case of long waves and/or a weak funda- 
mental current, the opposite distribution of horizontal 
divergence is established. The same conditions are 
found in air layers which move eastward more slowly 
than the wave. 
In the usual baroclinic westerly current of middle 
latitudes a wave extending from the slow-moving lower 
layers to the fast-moving upper layers would have 
opposite patterns of horizontal divergence in the upper 
and lower part (see Fig. 1). At the level of transition 
between the upper and lower pattern a wave motion 
with zero horizontal divergence will exist. This “level 
of nondivergence” will be found at the height where 
the speed of the undisturbed current, vz , is given by 
