HYDRODYNAMIC INSTABILITY 
placed particles do negative work at the expense of the 
energy of the neighboring air; displacement along the 
are of the ellipse ceases as soon as the kinetic energy of 
the initial impulse is dissipated. Because of our original 
hypotheses, the displaced particles cannot describe their 
entire inertial trajectory, for they lose their individual- 
ity in a time less than 27/y_. The particles descending 
the slope of an isentropic surface, following an elliptical 
are, acquire a zonal velocity w + 7., less than that of 
the current w; those climbing the slope acquire a zonal 
velocity wu + 7., greater than w. This transverse isen- 
tropic exchange, if it can be maintained, thus involves 
a weakening of the cyclonic vorticity (—éu/éy > 0) 
or an intensification of the anticyclonic vorticity 
0 > —éu/sy > —f), that is, a decrease of the 
hydrodynamic stability in the isentropic surfaces or 
even the appearance of hydrodynamic instability. 
Second case: va = f (f — du/dy) < 0. In this case 
the integral of (53) is a hyperbola with the parametric 
equations 
ry = —* sinh| va| t. (55) 
Vo 
Tx } (1 — cosh| v2 | ¢), isi 
Once the particle receives its velocity vo, its distances 
7, and r, from initial position (r. = ry = 0) increase 
exponentially with time. Moving in the isentropic sur- 
face which passes through its point of departure, the 
particle traces a branch of a hyperbola, for which the 
longitudinal semiaxis is %/f and the horizontal projec- 
tion of the transverse semiaxis is vo/ | va|. The center 
of the hyperbola lies on the longitudinal axis through 
the initial position, located with respect to that initial 
position in the same or in an opposite sense as the 
current u, depending on whether the particle rises 
(v > 0) or descends (vo < 0) along the isentropic sur- 
face. The longitudinal axis is the transverse! axis of the 
hyperbola and is the major or minor axis depending on 
whether 6u/éy > or < 2f. The hyperbola is equilateral 
when 6u/dy = 2f. We observe that the curvature of the 
branch of the hyperbola increases with the hydrody- 
namic instability. However, here 7, is > or < O as 
Ty < or > 0 and consequently the unstable isentropic 
inertial oscillation generates cyclonic circulation. Once 
set in motion, this circulation is maintained by the 
energy of instability released in the course of the isen- 
tropic displacement of the air particles. As before, it 
could be shown that the transverse isentropic exchange 
of air involves in this case a weakening of anticyclonic 
vorticity (0 > —f > —éw/dy), that is, a diminution 
of the hydrodynamic instability. The cyclonic circula- 
tion which results from transverse isentropic displacement 
of the air in an unstable geostrophic flow tends to re- 
establish a state of stable dynamic equilibrium in the 
isentropic surfaces. The result of this stabilizing action 
is that isentropic hydrodynamic instability is only a 
transitory state of short duration, which, according to 
what we have seen earlier, can develop only in regions 
of pronounced baroclinity. 
1.As used here, the term “transverse” refers to the axis 
which passes through the vertices of the hyperbola. 
443 
This being established, it follows directly from the 
definition of the isentropic meridional gradient d6w/dy 
of a westerly geostrophic current w and from the cri- 
terion for hydrodynamic instability that this instability 
can appear only when the surfaces © = const and u = 
const are closely packed and intersect one another at 
a large angle. By referring to the vertical meridional 
cross sections of Palmén [22-24], we can verify that 
these conditions are established in only two regions 
[40]: (1) in the upper troposphere between 200 and 
500 mb south of the belt of maximum westerlies, where 
du/dy — f is positive and of the order 10 ° sec ’, and 
(2) in the lower troposphere between 600 and 900 mb 
in the zone of the polar front and immediately north 
of it. We observe that in the latter region the isopleths 
of © should be replaced by isopleths of the wet-bulb 
potential temperature, whose slopes are necessarily 
larger; actually therefore, the hydrodynamic instability 
in this portion of the lower troposphere is greater than 
Palmén’s cross sections make it appear. In these two 
regions, hydrodynamic instability may occur. 
On the other hand, (1) in the higher troposphere 
north of the belt of maximum westerlies the isopleths 
uw = const and © = const are practically parallel, and 
(2) in the lower troposphere south of the belt of maxi- 
mum westerlies the isopleths © = const are nearly 
horizontal. Hence these two regions are normally char- 
acterized by pronounced hydrodynamic stability [40]. 
In the higher troposphere, consequently, meridional 
isentropic displacements of air particles bring about the 
formation of cyclonic circulation at low latitudes in the 
temperate zone and anticyclonic circulation at higher 
latitudes [40]. The existence of these two circulations 
has been demonstrated by Rossby, Palmén, and their 
collaborators [32]. On the other hand, in the lower 
troposphere, meridional isentropic displacements of air 
particles bring about the formation of cyclonic circula- 
tion at high latitudes in the temperate zone and anti- 
cyclonic circulation at low latitudes in this zone. The 
existence of this latter circulation was shown by Wexler 
and Namias [45]. The cyclonic circulations at middle 
latitudes in the lower troposphere are simply those 
characterizing the normal activity of the polar front. 
In summary, when the geostrophic motion becomes 
unstable for isentropic and quasi-isentropic transverse 
displacements in a baroclinic region of the atmosphere, 
the surrounding medium favors these displacements; 
the displaced particles set free energy of instability 
which augments the kinetic energy of the initial im- 
pulse [35-37]. Moreover, they acquire cyclonic circula- 
tion relative to the geostrophic current along branches 
of hyperbolas whose conjugate axes are perpendicular 
to the geostrophie flow [89]. 
Thus we perceive that an isentropic inertial oscillation 
of period greater than the Foucault pendulum half-day 
(2r/f), which becomes unstable in a region of large baro- 
clinity, may give rise to a cyclonic circulation. A nascent 
cyclonic circulation hence seems to be no more than 
an unstable inertial oscillation of the atmosphere [31, 
37]. But such a circulation inevitably involves major 
displacements of air and consequently a perturbation 
