440 
on their vertical separation. These two quantities in- 
crease or decrease depending on whether those distances 
decrease or increase. When the isentropic surfaces ap- 
proach one another while maintaining constant slope, 
the baroclinity N and the static stability v= both in- 
crease in such a way that N/v; remains constant, so 
that by virtue of (40) or (41), packing of the isentropic 
surfaces can involve instability when a sufficient baro- 
clinity is attained. On the other hand, when the slope 
of the isentropic surfaces increases while their hori- 
zontal separation remains constant, the static stability 
decreases while the baroclinity remains constant; this 
baroclinity can become critical for a sufficiently small 
value of the static stability. It follows that in a frontal 
zone where the static stability and the baroclinity are 
relatively large, instability can occur only if the baro- 
clinity is particularly pronounced, that is, when the 
isentropic surfaces are not only very close to one another 
but also markedly inclined to the horizontal (case of 
active frontogenesis). On the other hand, in the tropical 
air immediately above the frontal surface and the polar 
air immediately below, where the static stability is 
much less pronounced, a much smaller baroclinity is 
enough for the threshold of hydrodynamic instability 
to be attained. : 
In the stratosphere, where the static stability is large 
(5 X 10 “sec °), but where the baroclinity is relatively 
weak (quasi-horizontal isentropic surfaces), the state 
of geostrophic motion is stable [8, 9]. 
In summary, strong baroclinity and weak static 
stability favor hydrodynamic instability (see (40)). 
The Lower Latitudes. At the equator, ¢ = 0, and 
therefore w, = Sin a, w, = w cos a, and w, = 0. By 
virtue of (1) dIl/dy = dP/dy = 0; furthermore we have 
(Table I) 0/dz2 = —0/0Y and d/dy = 0/dZ. It follows 
that in (12), if (1) is taken into account, a, = Qyz = 
Ay = Azz = Ayz = Azy = O and 
2 2 
Azz = dyy = 4w COS a 
g 00 (43) 
0 fu 
+ 02 (2) 2 cose mois 
All these formulas hold in the neighborhood of the 
equator, providing we assume that, as we approach the 
equator, 0P/dy approaches zero as does sin ¢. 
In equatorial latitudes the quadratic form Q reduces 
to the degenerate form a..rz , which is thus independent 
of 7, ; consequently, at the equator, out of all the pos- 
sible transverse displacements only the vertical ones 
permit an energy exchange between the displaced par- 
ticle and its environment. At lower latitudes the force 
acting on a displaced particle is vertical. 
Finally, the hydrodynamic equilibrium in equatorial 
regions is stable or unstable depending on whether 
Qzz2 > or < O, that is, whether 
00 Qu cos a(Qw cos a + du/dz) 
G22 
Bape 2wu COS a — g 
2 
Vs 
(44) 
Ou 
=> —20 — cosa. 
OZ 
DYNAMICS OF THE ATMOSPHERE 
Since | 2w(dw/dz) cos a| S 10° sec, we observe that 
the threshold of hydrostatic stability in equatorial regions 
practically coincides with the threshold of hydrostatic in- 
stability in the vertical direction [18]. 
Inertial Oscillations of the Geostrophic Current 
The orbital motion of perturbed particle A(z, y, z) is 
the motion of this particle relative to the geostrophic 
current u. Referring to the coordinate system Oxyz 
considered earlier, in which the origin O is the equi- 
librium position of A, the orbital velocity V — u of A 
is defined by its coordinate components (3) 
Wein a io 
pew atch ane 
; d 
g=4=Vy=—s, (45) 
; dz 
DES Mia 0 
The dot indicates the substantial derivative with re- 
spect to time ¢ in a reference system embedded in the 
geostrophic motion. Deriving (45) for time ¢ im the 
reference system Oayz, at rest with respect to the earth, 
we obtain 
aV, i ou. ou 
dt Eh ag Fe ; 
aVy _ ay _ 
Flea taki cata eo) 
Te Oe 
CGT E REL 
account having been taken of the fact that v., v,,and vz 
are independent of the longitudinal coordinate x. By 
referring to equations (8) and (9) and to (11), (12), 
and (15), we obtain the equation of longitudinal orbital 
motion for the perturbed particle A, 
& = y curl? U — z curl} U, (47) 
and the differential system of transverse orbital motion 
is 
os : 0 0 
Y — 2022 = —Ay,Y — Ayzz, 
Z2+ 20,y = 
This system defines the motion of A’, the projection 
of A on the transverse plane through O. We see at once 
that the transverse orbital motion is independent of 
the longitudinal motion and that the latter follows from 
the former. It is therefore sufficient to determine the 
motion defined by (48). 
The problem of integrating a system with constant 
coefficients (48) is classical; we know that its general 
integral is a linear combination of circular and hyper- 
bolic functions of time ¢. The form of this general in- 
tegral depends on the nature of the roots of the char- 
acteristic equation of the system (48); this latter is 
(48) 
0 0 
—AzyY — azze. 
