448 
[18], large-scale condensation is such a cause. The 
gradient 6u/dy in a cloud layer should be determined 
in surfaces of equal wet-bulb potential temperature 
rather than in surfaces of equal dry-bulb potential 
temperature. Inasmuch as the slope of the former is 
much more marked than that of the latter, it follows 
that 6w/édy undergoes, at the moment of condensation, 
a sudden increase which is capable of bringing about 
hydrodynamic instability (6w/éy > f). We then have 
R?, > 0 and, as a result, the current w will spontane- 
ously become a cyclonic current of the stable domain 
(0 < Rx <€ Re) (cyclogenesis according to Wipper- 
mann [46]). 
The Permanent Circular Vortex. Application to Trop- 
ical Cyclones 
The problem of the stability of this state of motion 
has been treated in several different ways [2, 5, 6, 8, 9, 
13-18, 21, 31]; we will consider it here as a special case 
of permanent, horizontal, isobaric motion. To this end 
we compare the atmosphere to a circular vortex around 
the polar axis and at some point O we adopt Klein- 
schmidt’s coordinate system OXYZ, having axis OX 
tangent to the parallel through O and directed east- 
ward, and axis OY perpendicular to the vortex axis 
and directed toward the earth’s interior. Under these 
conditions, dY = —dR, R, = R, R. = «; where F is 
the radius of rotation from O; there follows from (76), 
wraor B+) 85 
Bip aelty/Ainoln olan 
u \ du dll 00 
=-2 =~|— - > a 
pe (. = =) ae aY a2’ | co) 
peti OOS) eae KS) 
= ORO? ti ka OA 
where @yz = Gzy, and uw = wu(R, Z) represents the 
zonal velocity of the air. The discriminant a reduces 
here to 
Sy t0) 
a = ayz — Ayy azz 
u \ gz 90 [3 u 
2 a=) |) Se SS Dy S = 
(0+ 4)ue Simea muple 
where gz is the absolute value of the component of 
gravity parallel to the earth’s axis. 
Since the inequality 00/0Z > O generally holds in 
the atmosphere, the permanent circular vortex is stable 
or unstable depending on whether 
(80) 
ou 
5Y 
The circulation along the parallels being cyclonic or 
anticyclonic as w > or < OQ, we observe that anti- 
cyclonic circulation favors the instability of the vortex. 
Consequently an easterly zonal circulation will become 
unstable more easily than a westerly one. Since the 
zonal circulation is easterly in tropical latitudes and 
westerly in temperate latitudes, and the Coriolis param- 
<or> 2a + (81) 
DYNAMICS OF THE ATMOSPHERE 
eter increases with latitude, it is evident from (78) 
that the zonal circulation in tropical regions can reach 
the threshold of instability more easily than that of 
temperate regions. However, inasmuch as the isentropic 
surfaces in tropical latitudes are nearly horizontal, 
stabilization of the zonal circulation is favored there. 
Finally it is easy to show, using criteria (81), that 
the permanent circular vortex is stable or unstable 
depending on whether its absolute angular momentum 
Om = R(u + Rw) increases (69/6R > 0)-or decreases 
(691/SR < 0) toward the exterior of the vortex in the 
isentropic surfaces [2]. 
The state of neutral equilibrium (691/6R = 0), 
characterized by the invariance of absolute angular 
momentum 91 in the isentropic surfaces, would be 
the state of exchange equilibrium (Austauschgleichge- 
wicht) according to Raethjen [27] and Kleinschmidt 
[18]. In other words it is that equilibrium condition 
toward which atmospheric air would tend if subjected 
to isentropic lateral mixing, Raethjen’s Gleztaustausch, 
or Rossby’s lateral mixing. If this condition prevailed, 
the lateral Reynolds stress T% due to isentropic turbu- 
lence would be proportional to 6u/éy — f (neglecting 
the curvature term in (78)), where x represents the 
eastward tangent to the parallel and y the northward 
tangent to the meridian. However, this is true only if 
the isentropic turbulence is purely transverse. In reality 
the turbulent particles move in the isentropic surfaces 
parallel to the mean flow as well as perpendicular to it. 
Priebsch [25] has been able to show that the stress 
T is actually proportional to 6u/dy and not to 
bu/sy — f. As a result, the absolute angular momentum 
relative to the earth’s axis of rotation is not an in- 
variant property of isentropic exchange of air [44]. 
We observe that the permanent circular vortex satis- 
fies an extremum principle [88] identical to that stated 
on page 436. 
Moreover, the method we followed in studying the 
inertial oscillations of a geostrophic current (pp. 032— 
040) will naturally extend itself to cover the case of a 
stationary circular vortex [31]. We thus obtain the 
results which we had established earlier for the case 
of zonal geostrophic motion (, = 0 or a = 0, 7). 
Similarly, the equations of adiabatic perturbations of 
a permanent circular vortex can be derived in the 
same way as those for a zonal geostrophic current 
(pp. 444-446) provided cylindrical or spherical coordi- 
nates are used. This problem has recently been treated 
in terms of cylindrical coordinates, in a slightly differ- 
ent fashion, by Queney [26], who has in addition pro- 
posed a method for the simplification of the equations 
which differs from Charney’s [4]. 
Let us further note that Fjgrtoft [10] has recently 
taken up the study of the stability of stationary circu- 
lar vortices with the aid of a new method based on the 
primary integral of the equations of motion (integral 
of the equation of mechanical energy) and utilizing 
the calculus of variations. 
Finally, we consider a permanent circular vortex 
whose vertical axis z is located at latitude ¢. If we 
designate by wu the linear velocity of the vortex, u > 0 
