HYDRODYNAMIC INSTABILITY 
simply Solberg’s equation of pulsations or orbital cir- 
cular frequencies y of inertial oscillation [31], that is, 
vo — bys — a = 0; (49) 
where a is the discriminant (26) of Klemschmidt’s 
quadratic form Q (13) and b = ay, + a2. + 4: . 
We recall that to a positive root vs of (49) there 
corresponds a periodic motion with an elliptical tra- 
jectory centered at 0, and circular frequency » , and 
to a negative root v5 there corresponds a motion with 
a hyperbolic piano ee about the same center, and of 
parameter 1/—7}. Inthe first case (vp > 0), the inertial 
oscillation of A’ around O is said to be stable, in the 
second case (v5 < 0), unstable. 
The sign of the roots vo of (49) depends on. the signs 
of a and b. But since the term az; in b is 10° times as 
large as either of the others, the sign of roots vp of (49) 
depends in the final analysis on the signs of a and a,, , 
that is, on the sign of the quadratic form Q. Conse- 
quently, the inertial oscillations of A around O are stable 
or unstable depending on whether the hydrodynamic equi- 
librium at O is stable or unstable for transverse displace- 
Ss 
ments OA’ of the perturbed particle A. 
Let v2 and v3 be the roots of (49); we have 
1g = 5 + /1 + 4a/b?), 
(50) 
(1 — V1 + 4a/b?); 
the first root is the larger when they are both real. Four 
cases can be distinguished, and they correspond to the 
four cases of Table II [31, 36, 37]: 
Case I. When vg > 0 and v3 > O, the transverse 
orbital motion (48) is the resultant of two oscillations 
of circular frequencies vg and v, around elliptical tra- 
jectories with a common center O. 
Case IT. When vg > 0 and »3 < 0, the motion is 
the resultant of a periodic motion on an elliptical tra- 
jectory of circular frequency vs and a motion on a hyper- 
bolic trajectory of parameter ./—v?,, with a common 
center O. 
Case IIT. When v3 < 0 and v3, < 0, the motion is 
the resultant of two motions on hyperbolic trajectories 
with a common center O and parameters ~/—v3 and 
Nae 
Case IV. When vg < 0 and v3 > 0, the motion is 
the resultant of a periodic motion on an elliptical tra- 
jectory of circular frequency vp and a motion on a 
hyperbolic trajectory of parameter ~/—v?, about the 
common center O. 
As a special case when w, = 0, that is, when the 
current u is zonal (a = 0), these ellipses and hyperbolas 
degenerate into straight lines passing through O in the 
meridian plane from this point. The same situation 
holds under the approximation (w, = w, = 0) agreed 
to on page 438, but in this case the straight lmes are 
transverse and no longer meridional. The straight lines 
which correspond to vg 2 0 are quasi-vertical, while 
those related to v3 2 O lie in the neighborhood of the 
441 
isentropic surface passing through O, which in general 
is very slightly inclined to the horizontal plane. In the 
latter case the straight line is a little more or a little 
less steeply inclined to the horizontal plane than the 
isentropic surface, depending on whether the inertial 
oscillation v2, is stable or unstable (vz < 0) [31). 
When the current u deviates from the west-to-east 
direction, the longitudinal component w, of w introduces 
an inertial influence which tends to substitute a trans- 
verse elliptical oscillation for the rectilinear meridional 
oscillation. The major axis of this very flattened ellipse 
is close to the vertical or to the isentropic surface de- 
pending on whether we are considering the oscillation 
of circular frequency vs or vp . In the case of instability 
these ellipses become hyperbolas elongated in the direc- 
tion of the conjugate axis, which is quasi-vertical or 
quasi-isentropic depending on whether the motion is 
characterized by parameter ./—v2 or V9; = PB « 
Finally, the longitudinal motion (47) is superimposed 
on these transverse motions in such a way that the 
resultant orbital motion is on an elliptical or hyperbolic 
trajectory whose plane, to the approximation o, = 
w, = 0, is parallel to the current u and quasi-vertical 
or nearly tangent to the isentropic surface through O 
depending on whether one considers root vs or Vp of 
equation (49) (81, 36, 37]. 
It is easy to see that b 2 —VO-VH & 10° * sec 
and a & 10” sec -. The circular frequencies of the 
inertial Ration are then approximated by the follow- 
ing values [31]: 
yeZvVbz 10° sec’ and vp & V —a/b ~ 10° sec, 
and the corresponding periods approach the values 
ts = 2n/vs & 10 min and tp = 21/yy & 20 hours 
(« = 3.14...). From this there results the proposition 
[31]: every geostrophic current allows two inertial oscilla- 
tions, one of short, the other of long period. 
We shall now show that the inertial oscillation of 
short period is associated with the stability of hydro- 
static equilibrium in the vertical direction z, and that 
the long-period oscillation is associated with the stabil- 
ity of hydrodynamic equilibrium in the isentropic sur- 
faces [34, 37]. 
To the approximation w, = wy 
Vb = M/ 6a: 
Viisili-Brunt). When hydrostatic stability prevails 
(0/dz > 0), vs is real, the corresponding displacement 
is a periodic function of time, and the short-period 
inertial oscillation is stable; on the other hand, when 
this equilibrium is unstable (00/dz < 0), vs is a pure 
imaginary, the corresponding displacement grows ex- 
ponentially with time, and the oscillation is unstable. 
Similarly we have 
ney ean 
(critical circular frequency of V. abe and 
a [1 - ah 
tp & 
ae Foy’ 
= 0, we have vs = 
= vs, (eritical circular frequency of 
