NORWEGIAN WAVE-THEORY 39 
COMPRESSIBILITY 
The compressibility of water is 
comparatively small so that it can be 
neglected in most cases. Atmospheric 
air, on the other hand, has a high 
degree of compressibility. Under the 
influence of compressibility alone 
sound waves are obtained. The in- 
fluence of compressibility on atmos- 
pheric wave motions does not give 
rise to unstable waves as long as the 
lapse rate of temperature is below 
the adiabatic, or for saturated air 
the moist adiabatic. The considera- 
tions which lead to this conclusion 
are well known, and were given else- 
where in this booklet’). 
SHEARING WAVES 
If a wind discontinuity is present 
in an otherwise homogeneous fluid, 
waves are possible at this surface. 
These waves are always unstable in 
contrast to the wave types we have 
mentioned so far, which were only un- 
stable if the stratification of the fluid 
or the gas was unstable to begin with. 
The computation shows that the ampli- 
tude increases more rapidly with time, 
or in other words, the wave is more 
unstable the smaller the wave length. 
Since the existence of this type of 
wave is due to the wind discontinuity 
or to the shear of the wind, it will 
be called a shearing wave here, and 
we shall speak of shearing instability. 
In the atmosphere, hardly ever do 
we observe a shearing discontinuity 
which is not associated with a tem- 
perature inversion and therefore a 
density discontinuity.* When this is 
the case, waves at the surface of dis- 
continuity must be of a mixed type, 
since the stabilizing gravitational 
effect and the unstabilizing shearing 
effect act at the same time. For 
small waves the unstabilizing effect 
of the wind shear overcompensates 
the stabilizing effect of stratification 
and the waves are unstable. As we 
pass on to longer waves we come into 
a region where the stable stratifica- 
tion is more effectual than the shear- 
ing instability. So now we are in the 
region of wave lengths for stable 
waves. The limit between shorter 
unstable waves and longer stable 
waves lies at a wave length of at the 
most a few km under atmospheric 
conditions. It varies of course with 
the order of the wind and tempera- 
ture discontinuity. It will be seen 
from this discussion that the unstable 
waves of about 1000 km length re- 
quired by the cyclone theory are not 
obtained when only the influence of 
gravity, compressibility and wind 
shear are considered. 
INERTIA WAVES 
In order to find waves which re- 
semble the nascent cyclones of the 
weather map the effect of the earth’s 
rotation has to be taken into account. 
To understand fully the importance 
of the earth’s rotation upon atmos- 
pheric wave motions we start out 
with a simple problem which at first 
seems quite devoid of any meteoro- 
logical application. Owing to its fun- 
damental significance it will be neces- 
sary to deal with this case somewhat 
in detail.*) If a hollow cylinder is 
partly filled with water and com- 
pletely closed, then set in rotation 
around a central vertical axis, the 
surface of the fluid which at rest was 
horizontal will become parabolic. 
When the rotation is sufficiently fast, 
the water is pressed completely 
against the cylinder walls so that 
the liquid has a practically cylindri- 
*Even if no temperature inversion is present 
there is still a certain degree of stability as 
long as the vertical temperature gradient is 
less than the adiabatic. 
3Namias, this booklet, pp. 4-8. 
4Bjerknes, V. and Solberg, H.: Zellulare 
Tragheitswellen und Turbulenz, Avh. Norske 
Vid. Akad., Bd. I, Math-Nat. K1., no. 7, 1929, 
pp. 1-16. 
