40 AIR MASS ANALYSIS 
cal shape. The gravitational force is 
so much smaller than the centrifu- 
gal one that its effect is unnoticeable 
when the rotation is sufficiently fast. 
Instead of the horizontal surfaces of 
liquids as ordinarily observed in 
nature under the vertical action of 
gravitation, the fluid in the rapidly 
rotating cylinder has a practically 
vertical surface due to the horizontal 
action of the centrifugal force. If 
this vertical surface is subjected to 
a small disturbance a wave motion 
will originate. This is quite analogous 
to the case of gravitational waves on 
a horizontal surface of water, but 
in that case the energy of the wave 
motion was gravitational. In the 
present case of a rotating cylinder 
the centrifugal force replaces the 
gravitational force. Since the centri- 
fugal force is due to the inertia of 
the mass these are called inertia 
waves by V. Bjerknes and H. Sol- 
berg.*) 
The existence of inertia waves in 
a rotating fluid can also be demon- 
strated by an elementary computa- 
tion. If a homogeneous incompressi- 
ble fluid like water is enclosed be- 
tween two rigid walls which are in- 
finite in horizontal direction it can 
easily be shown that no wave motion 
is possible as long as the fluid system 
does not rotate. As soon as the ro- 
tation is taken into account, as is 
necessary for the large scale atmos- 
pheric motions on the rotating earth, 
it is found that now a wave motion 
is possible with a period longer than 
half a pendulum day. (A pendulum 
day is equal to 24 sidereal hours di- 
vided by the sine of the geographic 
latitude, this being the t-:me required 
for the swinging plane of a pendu- 
lum on the rotating earth to return 
to its initial position.) Only the angu- 
lar velocity of the earth’s rotation 
and not the acceleration of gravity 
appears in the relation between velo- 
city and length of this type of waves, 
indicating that they are inertia waves. 
Some remarks on stability and in- 
stability of inertia waves are neces- 
sary in order to see their significance 
for the wave theory of cyclones. To 
choose the simplest case, which shows 
the principle clearly enough, it may 
be assumed that a fluid mass rotates 
around a vertical axis with an angu- 
lar velocity q which is constant for 
the whole fluid. If the fluid is en- 
closed between rigid horizontal boun- 
daries and situated at either pole of 
the earth we have just the case con- 
sidered in the previous paragraph. 
The constant angular velocity of the 
fluid is equal to the angular velocity 
of the earth’s rotation. An observer 
on the earth does not observe this 
“absolute” velocity since he takes 
part in the rotation of the earth. 
The “absolute” velocity v at the dis- 
tance 7 from the center is 
v=qr 
From the principles of mechanics 
it is known that the angular momen- 
tum vr = q.r of an individual parti- 
cle remains constant. Thus, if a parti- 
cle is pushed away from the axis, say 
irom the distance 7 to r + s, then the 
constancy of the angular momentum 
requires that this particle in its new 
position must have a smaller angular 
velocity q’ which is given by 
qr =q (Fa7 Ss)" 
while the angular velocity of the sur- 
rounding fluid masses at the distance 
r + s is q as before. 
The centrifugal force acting on the 
displaced particle is 
G (arS) = er 
approximately, 
while the centrifugal force in the sur- 
rounding fluid at the same distance is 
greater, namely, 
q(r+s) 
