HYDRODYNAMIC INSTABILITY 
to the earth and having the axis Ox directed parallel 
to the geostrophic current u and the axis Oz directed 
toward the zenith above point O. In this system we 
thus have uz = u(y, x) > 0, wu = u. = 0, P = Py, 2), 
© = Oly, z), © = H(z), and the components of the 
earth’s rotation @ are given by w, = cos ¢ sin a, 
®y = w» Cos ¢ COS a, and w. = w sin ¢, where ¢ is the 
latitude of poiit O and a is the angle between the vector 
u and the west-to-east direction. Let us now introduce 
Exner’s function Il = c,(P/100)"/” (in which P is 
expressed in centibars), where c, stands for the specific 
heat at constant pressure and # for the specific ideal 
gas constant for air. Finally, to facilitate the presenta- 
tion, any orientation parallel to the current u will be 
called longitudinal and any orientation perpendicular 
to this current will be called transverse. 
The law of geostrophic motion expresses the equi- 
librium of the transverse forces —V6,—OVII, and 
— 20 X u, that is, 
20 Xu + OVI + Ve = 0. (1) 
By taking the curl of equation (1) of hydrodynamic 
equilibrium, we find the equilibrium condition of Mar- 
gules, 
20-Vu = [VO X VI]. = N, (2) 
which expresses the integrability of equation (1) with 
respect to Il. In the equilibrium condition, therefore, 
the gradient of the air speed is proportional to the 
number N of isobaric-isosteric solenoids per unit of 
transverse area. 
We shall now consider a certain particle, in the in- 
terior of the air mass which is in hydrodynamic equi- 
librium, carried along by the current u and occupying 
at any imstant t the position of an arbitrarily chosen 
point O; at the instant f we apply to it a transverse 
impulse; then at the same instant we take away the 
impulse. Let A(x, y, z) be the position occupied by the 
perturbed particle at stant ¢ > fo ; let v(vz, vy, vz) 
be the velocity of this particle relative to current u at 
A at the instant ¢, and let V(V., Vy, Vz) be the velocity 
of this same particle relative to the earth at the same 
instant; we then have 
. oe i 
Vi pes Oia Uap Vy == Wy, 
5 (3) 
7 a tes 
OS ie 
where the operator d/dt stands for the substantial de- 
rivative with respect to time ¢ in the reference system 
Oxyz, at rest with respect to the earth. 
We now propose to study the motion of the per- 
turbed particle A around its equilibrium position O. 
This relative motion will be assumed to be a small 
motion; furthermore, we shall neglect the effects of 
friction, conduction, and radiation on the displaced 
particle, and we shall assume that its displacement in- 
volves no perturbation of the pressure field. Conse- 
quently the perturbed particle retains the potential 
temperature @,) which it had at pomt O, and at any 
435 
instant ¢ > %& its pressure is equal to that at the point 
A which it is occupying at that mstant. The motion 
of the perturbed particle A(x, y, z) relative to the 
earth is then determined by the vector equation 
Ot + 20 XV + OVI+ VS = 0 (4) 
and by the initial conditions 
P—=7=2=0) =n Wo = th = WO,0) 
4 A (5) 
Vn = Dy» Ve = Uby 
where vy and vz are the transverse components of the 
initial velocity communicated at O. By eliminating 
V® between (4) and (1), we obtain 
- + 20 X v = (0 — ©)VI. (6) 
We observe that if, at the initial instant ft, we 
apply the same impulse to all the particles which occupy 
the points of the Ox axis at this stant, all these par- 
ticles will perform the same adiabatic, mertial motion 
(4) or (6). Consequently, the quantities which describe 
this motion will be mdependent of the longitudinal 
coordinate «, like all those which define geostrophic 
motion. It is thus natural to separate the motion (6) 
of point A into transverse and longitudinal parts. 
In order to obtain the equation of longitudinal mo- 
tion at A, it suffices to project equation (6) on the 
axis Ox; if we use (8), we obtain the equation of motion 
of the projection A” (x, 0, 0) of A on the longitudinal 
axis through O: 
dV, + 2w, dz — 2w.dy = 0. (7) 
This equation gives the change dV,, relative to the 
earth, which takes place in the longitudinal velocity 
of the perturbed particle when it undergoes transverse 
displacement (dy, dz). It is mtegrable at once; by virtue 
of (5) and of the relation uw = w + (Ou/dy)oy + 
(du/dz)oz, we obtain 
Ou Ou 
pe (= LN) gp l(a, oe EN 
v ; ( We an) y ( Wy a) (8) 
where the subscript zero serves as a reminder that the 
quantities im parentheses must be replaced by their 
values at point O. We see, therefore, that the longitu- 
dinal motion of point A is fixed, as soon as we know its 
transverse motion. The latter is governed by the equa- 
tions obtained by projecting relation (6) on the trans- 
verse axes Oy and Oz; following (8), the equations of 
motion of the projection A’(0, y, z) of A on the trans- 
verse plane through O are 
dv: 
— — Qwvz = Wy, — + 20.y = Wz, (9) 
where W, and wz are the components of the transverse 
force 
wt = (0 — O)VIL — 20 X Vv; (10) 
which the surrounding medium exerts on the unit mass 
of the displaced particle. This transverse force con- 
