HYDRODYNAMIC INSTABILITY 
In the general case, as in the geostrophic case, we 
also have 
in which 
with 
— edly wy 9 G GD 
vata, In 
0 dx’ 
together define the baroclinity at point P. 
In order to enable us to interpret the formulas 
conveniently, let us orient the horizontal axis x of the 
xyz system along the tangent to the isobar at P in the 
direction of the geostrophie flow u(u, , uw, , 0); we then 
have [11]: 
6Uz : j Lae 
ie , the transversely isentropic geostrophic wind shear, 
(perpendicular to the streamlines of geostrophic 
flow within the isentropic surfaces); this wind 
shear is cyclonic or anticyclonic depending on 
whether 6u,/5y < 0 or > 0; 
6 : 
= > or < 0 depending on whether the curvature of 
streamlines of geostrophic motion in the isentropic 
surfaces is cyclonic or anticyclonic; and 
OUz : 
= > or < 0 depending on whether these stream- 
lines are converging or diverging. 
It follows from the order of magnitude and approxi- 
mate values of the coefficients 8, «, and o of equation 
(88), as well as from their classical expression as func- 
tions of the roots, that the largest root of this equation 
has a value in the neighborhood of v2. Dividing the 
first member of equation (88) by v5 — v2, we obtain 
Godson’s approximate equation [11] 
vy — vars tC=0 
_, (=) 4 ite 2] 
Ox 
~ 10 to 10°’ sec *. 
(92) 
in which 
C=a/y.= 
If we refer to the equations of the inertial oscilla- 
tions, derived from (85), it is apparent that the inertial 
oscillation corresponding to the root v2 is quasi-vertical 
(static stability) and that those corresponding to the 
two roots vo of (92) are quasi-isentropic [11]. 
The nature of the roots of the biquadratic equation 
(92) depends on the sign of the discriminant 
Abe! Wein? 2 OUy OUx . 
wy —4C=f\f+(—+— 
ox OW] 
; duz\ Win Oke 
+4(%) +o(3 -=)|. 
451 
In general, equation (92) admits of two real roots 
(4C < v3), whose approximate values are va and C/7j.. 
There are four cases to be considered: 
I. vi > 0, C > 0: hydrodynamic stability regard- 
less of the particular isentropic displacement at point P. 
Il. vi < 0, C > O: hydrodynamic instability re- 
gardless of the particular displacement at point P. 
(v3 <0,C <0 
: hydrodynamic instability at P. 
mS 0,0 <@ SEs: th 
It should be noted that C can be positive only if 
(6u;/5y)(6u,/da) < 0. In this case, a transversally 
isentropic, cyclonic shear (anticyclonic shear) of the 
geostrophic wind will necessarily correspond to a cy- 
clonic (anticyclonic) curvature of the streamlines of 
geostrophic motion within the isentropic surface at P. 
The case of cyclonic geostrophic wind shear invariably 
corresponds to the stable type-I (marked stability); 
the case of anticyclonic geostrophic wind shear corre- 
sponds to the unstable type II or to the stable type I 
(weak stability), but only if the absolute value of the 
anticyclonic vorticity du,/éx — du./dy < 0 is smaller 
than the Coriolis parameter f. 
Furthermore, C is necessarily negative if 
(6u./dy)(6uy,/dx) > 0. 
In this case, a transversally isentropic, anticyclonic 
(cyclonic) shear of the geostrophic wind will correspond 
toa cyclonic (anticyclonic) curvature of the streamlines 
of geostrophic motion in the isentropic surface at P. 
The sign of the transversally isentropic geostrophic 
wind shear and the sign of the curvature of streamlines 
of geostrophic motion in isentropic surfaces both play 
preponderant roles; the divergence or convergence of 
these lines of motion, on the other hand, are with- 
out influence over the conditions leading to hydrody- 
namic stability or instability—only the absolute 
value | 6u,/6x | = | 6u,/dy | is decisive. 
The biquadratic equation (92) admits of two con- 
jugate imaginary roots when: 
IV. 40 — vi > 0: hydrodynamic instability regard- 
less of the isentropic displacement which takes place 
at point P. This case can arise only when vi is of an 
order of magnitude less than 10 * sec °, that is, when 
the vorticity 6u,/d% — 6u,/dy is anticyclonic, and when 
its absolute value is of the same order of magnitude as 
the Coriolis parameter f. The existence in the general 
case of a second type of total instability in the isen- 
tropic surfaces is thus not excluded. Let us note that 
the first type (case II) is none other than the one we 
met previously in our discussion of geostrophic motion 
(case II of Table II) and as a consequence the results 
obtained in the geostrophic case can be extended to 
the general case of any given motion. 
Thus, the condition vi < 0 is a sufficient condition 
for instability, but is no longer (as in the geostrophic 
case) a necessary and sufficient condition. On the other 
hand, the condition v7 > 0 is a necessary condition of 
stability but is no longer (as in the geostrophic case) a 
necessary and sufficient condition [11]. Nevertheless, it 
