HYDRODYNAMIC INSTABILITY 
potential energy per unit mass ® is an extremum; depend- 
ing upon whether this extremum is a minimum or a 
maximum, the equilibrium at O between gravity, the pres- 
sure gradient, and the Coriolis force is stable or unstable 
[38]. 
The Quadratic Form of E. Kleinschmidt 
Let us now consider the case of vertical displacement 
(r, = 0, r- # 0), the equilibrium being hydrostatic 
(u = 0). In this case the earth’s rotation can be 
neglected; by referring back to (12), we obtain a, = 
Qyz = Az = 0 and a.. = (g/9)(00/0z), consideration 
having been taken of the hydrostatic equation in the 
vertical (2) direction. We thus rederive the classical 
criterion: the vertical distribution of air in the field 
of gravity g is stable or unstable depending upon 
whether 00/dz > or < 0. We observe that the energy 
of instability released per unit mass in the course of 
vertical displacement r. of a particle in hydrostatic 
equilibrium is given by —14(g/@)(00/0z)r? . 
In the case of a geostrophic current (wu ~ 0), the 
vertical equilibrium is stable or unstable depending 
upon whether 
dll 00 
ou) NS or <0. 
22 = 2 2 
os Bui az) eae 
In the atmosphere, however, the second term is always 
at least one hundred times larger than the first and 
therefore determines the sign of a... 
We consider next the case of transverse isentropic 
displacement, which is defined at any point by r; = 
rd@/dz2 ~ + rdO/dy 
ee an fe! 
form (14) at this point for this displacement can be writ- 
ten independently of the chosen system of coordinates as 
follows [37] 
Q(r;, r2) = 2(@-VO)(VO-curl U)r’/(Ve)*. (21) 
In order to give (21) a convenient analytical form, it 
is useful to adopt Kleinschmidt’s rectangular, right- 
handed reference system XYZ [18], where axis X co- 
incides with the longitudinal axis x and the transverse 
axis Y is perpendicular to the earth’s axis and points 
toward the earth’s interior. The direction cosines be- 
tween the new axes X, Y, Z and the former axes 2, y, z 
are as given in Table I. We can set x sin 6 = sin d 
k cos B = cos ¢ Cos a, and k = sin” @+ cos’ ) cos; a. 
Tasie I. Drrection Cosines ror KLEINSCHMIDT’S 
CoorDINATE System 
. The value of the quadratic 
x iv Z 
a 1 0 0 
y 0 sin 6 cos B 
z 0 —cos B sin 6 
In this coordinate system we have wx = w cos ¢ sin a, 
wy = 0, wz = xw and 
(Je) ou 
V6-curl U = =, (2 — =) 
aZ oY 2) 
437 
where the operator 
Db © | (EQ a 
iy a Gaz OZ 
represents the isentropic derivative in the direction of 
the transverse axis Y. Following (21), 
= 140s. pe) = ee ) if 
WQ(r,, 72) = kW Ss ay aKQ) (ve)2” 
It turns out that geostrophic motion at any point, for 
a transverse isentropic displacement from this point, 
is stable or unstable depending on whether 
(23) 
(24) 
= < or > 2kw. 
Furthermore we note that the right side of (24) gives 
the energy of instability released per unit mass during 
an isentropic displacement. 
Finally, we come to the general case. Study of the 
sign of the quadratic form Q is accomplished with the 
help of its diseriminant [18, 37] 
@ = Aye — Ay Azz = 2(@-VI1)(VO-curl U) 
(25) 
(26) 
and of the coefficient of one of the squared terms. First 
we observe that, (1), 
20(@-VI) = —20-Vb = —fg (27) 
where g represents gravity and f the Coriolis parameter 
20, = 2» sin ¢. After substituting (22) and (27) into 
(26), we obtain 
— fg 08 (du _ 
O= © BUNGE oo) 
We next find by reference to equations (12) that the co- 
efficient of the term in (rz) can be written 
dl08 _ 1 000m _ gsing 00 
OZdZ =OdZ AZ KO dZ 
Since the sign of quadratic form @ depends upon the 
signs of a and azz, we see in the final analysis that this 
sign depends upon the sign of 00/dZ and the sign of 
6u/65Y —2«w; whence we can construct Table II [18, 37]. 
TasBiLe IJ. GENERAL CONDITIONS FOR THE STABILITY OF 
GrostropHic Motion 
(28) 
azz = 
(29) 
Case | 00/0Z Lhed a Nature of the hydrodynamic equilibrium 
Stable whatever the transverse dis- 
placement. 
6 oe ea tee |b 
mW) +} + |+ 
Unstable for transverse displace- 
ments close to the isentropic 
surface. Stable for transverse 
displacements close to the Z 
direction. 
III = + — | Unstable whatever the transverse 
displacement. 
IV =- = + | Unstable for transverse displace- 
ments close to the Z direction. 
Stable for transverse displace- 
ments close to the isentropic sur- 
face. 
