438 
When a > 0, that is, when the quadratic form Q at 
the poimt considered is positive for certain displace- 
ments and negative for others, it becomes zero for two 
transverse directions d’ and d’’. For these directions 
the hydrodynamic equilibrium at the point is neutral 
(—Q = Yr, + rz = 0). It is simple to show that the 
angle between these two directions is given by 
2y/ 2(w- VII) (VO-curl U) 
2(@-curl U) — (VII-Ve) 
Since the sign of Q at any point is that of 00/dZ for a 
displacement parallel to Z and is that of 2x — 6w/6Y 
for a transverse isentropic displacement, and since these 
signs are different because of the inequality a > 0, the 
directions d’ and d’ separate the transverse tangent 
to the isentropic surface at this point from the line 
parallel to the Z axis. The directions d’ and d’’ separate 
the transverse plane at this point into four regions, 
opposite to one another in pairs across the vertex. The 
two regions containing the displacements for which 
Q > 0 (<0) constitute the stable (unstable) sector 
of the transverse plane at the point considered. More- 
over, the isentropic surface passing through this point 
lies in the unstable sector in Case II and in the stable 
sector in Case IV [8, 9, 14, 15, 17, 18]. 
When a = 0, the directions d’ and d’’ both coincide 
with direction 
tan (d'd’) = = 10m 20) 
ey 00/0Y 
d0/0Z’ 
that is, with the transverse tangent d to the isentropic 
surface at the point considered. When a becomes posi- 
tive, in passing from Cases I and III to Cases II and 
IV, directions d’ and d’” separate from this tangent 
and include between them an angle defined by (80). 
This angle is very small, averaging at the most on the 
order of a few tenths of a degree, and the isentropic 
surface at the point considered lies in the acute angle 
between directions d’ and d’’ from this point [87]. 
In the atmosphere we generally have 00/dZ > 0, 
so that only Cases I and II are ordinarily possible, 
Case I being the most common. Therefore, in general, 
at any pomt in the atmosphere the state of geostrophic 
motion is stable whatever may be the transverse dis- 
placements from this point, with the exception of isen- 
tropic and quasi-isentropic displacements, when [37] 
pe eazy, 
azz 
Ve-curl U < 0. (31) 
Hydrodynamic Instability as a Function of Latitude 
Middle and High Latitudes. Let us return to the co- 
ordinate system Oxyz introduced earlier. In the 
Northern Hemisphere, we have in this system dII/dy < 
0 and 06/dy < 0. At the pole ¢ = 90°, so that w, = 
w, = 0. In the polar regions ¢ is close to 90° and the 
two horizontal components of w are practically zero. 
Furthermore, we know that in middle latitudes we can 
neglect all the terms of the dynamic equations which 
contain w cos ¢; when thus simplified the equations are 
exact, at least to the nearest hundredth. As a first 
DYNAMICS OF THE ATMOSPHERE 
approximation, we can thus eliminate the terms con- 
taining components w, and w, of the earth’s rotation 
from all equations in the preceding sections. In par- 
ticular we have, (12), [20, 37] 
— 62 u a _ du ZS —8 —2 
tn = if a2 ()|=7 au) 72 10 sec, 
() =! at ny 
= fo 2 (2) 7 10 " to 107° sec 2, 
0z\9 Oz 
le) (2) 
g ij —6 —2 
el 
Qzy on 0 to 10° sec, 
ime) =A =) 
z= -— Vl s 
a Bn 0 sec, 
account having been taken of the simplified equation 
(1), and we have from (28), 
fg 08 [du 
0 Of — yee = 20 (2 ), (33) 
as well as from (2), 
jot _ 20 aI _ 20 all 
Oz Oy Oz Oz O74 
y @ oy (34) 
g 00 = = =? 
i ea 
N= © ay 0 to 10 sec 
The operator 
Sh Oy a a 
by ay c dz (35) 
gives the transverse isentropic derivative in the direc- 
tion of the horizontal y-axis. The sign of form Q for 
a transverse isentropic displacement (rj , r>) is that 
of the difference f — = and for a vertical displacement 
r, 1s that of az,, that is, that of 00/dz. Thus we see 
that at any point the hydrodynamic equilibrium for a 
vertical displacement from this point is stable or un- 
stable depending on whether the hydrostatic equi- 
librium in the field of gravity is stable or unstable. 
Therefore, to the approximation agreed upon here, the 
vertical gradient 00/dz of the potential temperature 
® reassumes its classical significance. We can also say 
that 
2 _ Wf le) as Ves 
FAS GE rE 
defines the static stability in middle and high latitudes. 
In (86), 7 represents the absolute air temperature; 
y, the vertical lapse rate of temperature; and y., the 
adiabatic lapse rate. However, in the general case con- 
sidered in the preceding section, 06/dz must be re- 
placed by 00/0Z to be rigorous. 
This having been established, we can adopt without 
change the reasoning of the previous section and sub- 
stitute 00/dz and 6u/dy — f for 00/dZ and 6u/6Y — 
2xw in Table II. Since we generally have 00/dz > 0 
above the convective level, geostrophic motion i middle 
and high latitudes is stable for any transverse displace- 
ment, except in the neighborhood of the isentropic 
surfaces when 6u/dy > f [17. 18, 37]. In the extra- 
Y ~ 10* sec (36) 
