HYDRODYNAMIC INSTABILITY 
or < 0 depending on whether the vortex is cyclonic 
or anticyclonic relative to the earth. We shall assume 
that the vertical distribution of mass is stable. Then 
the vortex is unstable at any point for a transverse 
isentropic displacement from this point when, (80) 
and (81), 
du U 
mtpti< (82) 
since the vorticity in the atmosphere is always cyclonic 
in a fixed, geocentric reference system (u/R + f/2 > 0, 
see equation (80) where f/2 must be substituted for w). 
This being the case, to an isentropic displacement of 
radial component AR there corresponds an azimuthal 
motion of the displaced particle, whose velocity is 
given by, (7), 
7 a eee “) 
Vn =U (r++ AR. 
Here V, denotes the particle’s velocity along the par- 
allel associated with the vortex. Consequently the cy- 
elonic circulation, relative to the earth, of air particles 
subjected to a radial isentropic displacement toward 
the exterior of the vortex (AR > 0), decreases or 
increases depending on whether the vortex is stable 
or unstable. It follows that when the air particles in a 
vertically oriented, unstable, circular vortex undergo an 
outward radial, isentropic displacement, they release in- 
stability energy which reinforces their cyclonic circula- 
tion around the vertical axis of the vortex. According to 
Sawyer [380], this mechanism plays an important role 
in the formation of tropical cyclones. The probable 
truth of this opinion is enhanced by the fact that at 
low latitudes the Coriolis parameter f is relatively 
small (107° sec) so that condition (82) is more easily 
fulfilled there than in middle latitudes. 
Inertial Oscillations of an Arbitrary Flow 
Ertel [7] has given the equation of circular frequencies 
of the inertial oscillations of an air parcel in arbitrary 
motion. This more general problem has recently been 
taken up by W. L. Godson [11]. 
If the atmosphere is referred to a right-handed 
Cartesian system, «22°, chosen at random but at 
rest with respect to the earth, the equation of motion 
and the equation of continuity are, respectively, 
da | 9,0 — —9 9H _ o® 
de Yh Ome Oe (83) 
oP ode : 
=e ayn Bae (i,k = 1, 2, 3) 
and 
d ( ‘e825 x 
dp ans 
in which w,; = —w,, are the antisymmetrical com- 
ponents of the rotation » of the earth [41], and the 
repeated index k should be considered as a suramation 
index (dummy index convention). In the general case 
) = 0) with 6% = 0, 
449 
considered here, S, 0, P, II are functions of a’ and of t, 
and ® is a function of x’. As in the case of geostrophic 
motion (see p. 435), let us associate to the equilibrium 
position O(x), x, 3) which a particle occupies at time 
t, the perturbed position 
A(a* + Ag’, « + Ax’, o® + Az*) 
which this particle occupies at the later instant 
t > t) after receiving an impulse at O at time t. 
The unperturbed particle will occupy position P(2’, x”, x’) 
at time ¢. 
The coordinate components Ax’ of the displacement 
rt = PA should satisfy the variation equations de- 
duced from (83), that is, 
dAx* dAx" all all ab 
eo Vien NG eae = AS 
comes at One Ohne eae WO) 
with 
a = Sa! = div r 
Si occa 
Let us recall that d/dt represents the individual total 
derivative followmg the motion or air with respect to 
the earth. The variation A applied to any scalar 
quantity can be broken down into a local variation 
5 ee te) =e 
A, at P and a convective variation Az* Ag along PA; 
0 5 
we thus have A = A; + Aa* —. Assuming the per- 
dak P 
turbation to be adiabatic (A® = 0) and assuming an 
absence of local variation in the geopotential ®, equation 
(84) reduces to 
dA TI 
Lix(Az*) = =(§) a) 
Ox? 
(85) 
provided we set 
a d 
[bp SS Vakta pate 2 On San ik 
b= Vik Tp win aT oaks 
in which 7% (=0 or 1 depending on whether 1 ¥ k or 
i = k) are the coefficients of the metric form in rectan- 
eular coordinates 2’, x, 7 and 
aif om ab 
= - - 
i Ox'dxk ~~ dx'*dak 
oP S’ oP oP Ob 
= Taek Ty AT = : AEE Oki 
Ox dx" C2 dx Ox = 0x da* 
are the components of Ertel’s stability tensor [7]. If we 
consider the perturbation to be small, we may consider 
the coefficients of the differential system (85) as being 
constants whose numerical values are those which these 
coefficients take at the point P at the instant ¢. 
In general, we will designate by a;; the elements of 
the determinant of the third order a = || a;;|| and 
by a’ the algebraic minor of a relative to the element 
a;;; we will then have aa, = 0 or a depending on 
whether j # k or j = k. With this in mind, let us now 
multiply the two terms of (85) by L“ and perform the 
