442 
where 7 = 2x/f represents the Foucault pendulum 
half-day and the direction y is horizontal and perpen- 
dicular to current u, pointing toward the left of this 
current (low-pressure region). 
When the hydrodynamic equilibrium in the isentropic 
surfaces is stable (6u/éy < f), vp is real, the correspond- 
ing transverse displacement is a periodic function of 
time, and the inertial oscillation of long period is stable; 
on the other hand, when that equilibrium is unstable 
(6u/6y > f), vp is a pure imaginary, the corresponding 
displacement increases exponentially with time, and the 
oscillation is unstable. We also observe that the period 
Tp is greater or less than the Foucault pendulum half- 
day depending on whether 6u/éy > O or < 0. In the 
first case, which is the one usually encountered in the 
atmosphere, the period rp is of the same order of magni- 
tude as the period of cyclonic waves. When, through 
an increase in the baroclinity, 6w/dy finally exceeds the 
threshold value f (0 < f < du/dy) the quasi-isentropic 
inertial oscillations of period greater than the Foucault 
pendulum half-day and the hydrodynamic equilibrium 
for corresponding transverse displacements become un- 
stable simultaneously. It 7s worth noting that the mertial 
oscillation which may become unstable in the atmosphere 
is a quasi-isentropic oscillation whose period is precisely 
of the order of the period of atmospheric disturbances 
[31], and that this instability can appear only where these 
disturbances would normally occur [84, 37]. 
In summary, there exists a unique and reciprocal 
correspondence between the stability and instability 
of inertial oscillations in the atmosphere on one hand, 
and the stability and instability of hydrostatic and 
hydrodynamic equilibrium on the other [34, 37]. 
Hydrodynamic Instability and Cyclogenesis in Extra- 
tropical Latitudes 
Far from any perturbation, the vertical component 
of the velocity and the horizontal and vertical com- 
ponents of the acceleration of the air are practically 
zero. We can then reasonably assume that the state 
of geostrophic motion in these regions is that which 
precedes the initial stage of cyclogenesis. If we suppose 
that the geostrophic current u is zonal (a = 0, w, = 0), 
any particle displaced in the interior of this current 
from point (2, y, 2) to pomt (« + rz, y + 1,2 + 72) 
acquires an orbital motion according to equations (47) 
and (48) 
‘ =| Ou 
he = = Py = == Pras 
( oy Oz 
Here y, and w, are the meridional and vertical com- 
ponents of the force which acts on the displaced par- 
ticle because of the nonuniformity of the field of flow 
u and of the field of potential temperature © associated 
with it [20]. These components are given by 
Wy & vir, + Nrz, py. & Nr, — (52) 
where v: , v3, and N have been defined in (36), (37), and 
(34). 
Let us first envisage a vertical displacement (7, = 
Po = We. (Gl) 
Ty = Wy; 
2 
VsTz, 
DYNAMICS OF THE ATMOSPHERE 
0, r, # 0). In this case the earth’s rotation can be 
neglected (f & 0); therefore y, = 0 and y, = —vir.. 
Since the vertical distribution of mass is assumed to be 
stable, the vertical force y. is a restoring force which 
tends to bring the displaced particle back to its pomt 
of departure, so that after several oscillations of short 
period (approximately 10 min) and of an amplitude 
which is smaller the greater the stability, the particle 
returns to its equilibrium position in the geostrophic 
current [39, 40]. Therefore, in an atmosphere possessing 
vertical hydrostatic equilibrium in which the air under- 
goes adiabatic transformations only, we can assume as 
a first approximation that on a synoptic scale the air 
particles are displaced along isentropic surfaces. The 
existence of these isentropic displacements, favored by 
vertical stability, led Rossby [29] to the concept of 
lateral mixing and Raethjen [27] to the similar idea of 
Gleitaustausch of the air. 
To the approximation adopted above, the inertial 
oscillations of long period take place in the isentropic 
surfaces, and we are led to consider isentropic displace- 
ments. In this case we have (00/dy)r, + (00/0z)r. = 0, 
and consequently the transverse force w which the sur- 
roundings apply to the displaced particle is horizontal 
to a first approximation; this gives ¥, & —vgr, and 
vy. & 0. The horizontal orbital motion of the displaced 
particle is then defined by the differential system 
i, — (va/f)ry = 0, fy + vary = 0, (58) 
and the initial conditions r, = ry = O and 7, = 0, 
Ty = 0), fort = t) = 0. Two cases can be distinguished 
depending on whether the meridional force is a restoring 
force (Wr, < 0, stability) or a force which carries the 
displaced particle always farther from its point of de- 
parture (Wr, > 0, imstability) [39, 40]. 
First case: vg = f (f — du/dy) > 0. In this case the 
integral of (53) is an ellipse with parametric equations 
fp = 2G = cos mi) Py = "0 sin vat. (54) 
i va 
Having received at the initial instant a transverse im- 
pulse in the isentropic surface that contains it, the 
particle oscillates in this surface around an ellipse for 
which the longitudinal semiaxis is vo/f and the hori- 
zontal projection of the transverse semiaxis 1S Uo/Va . 
The center of the ellipse lies on the longitudinal axis 
through the initial position, located with respect to that 
initial position in the same sense as current u or in the 
opposite sense depending on whether the particle rises 
(v > 0) or descends (vw < 0) along the isentropic sur- 
face. The major axis of the ellipse is longitudinal or 
transverse depending on whether the vorticity, pro- 
duced by the geostrophic current in the isentropic sur- 
face, is cyclonic (6u/éy < 0, great stability) or weakly 
anticyclonic (0 < 6u/dy < f, slight stability). The 
ellipse reduces to an inertial circle when 6u/dy = 0. 
The elliptical motion of the perturbed particle has a 
period 27/vg = 20 hours. Since 7, > or < 0 as 7, > or 
< 0, the motion takes place in the anticyclonic sense. 
The surrounding medium, however, opposes this anti- 
cyclonic circulation, since in the stable case the dis- 
