NORWEGIAN WAVE-THEORY 43 
tion to shearing instability, gravita- 
tion and inertia act upon them. 
The mathematical analysis also 
shows that these waves have veloci- 
ties of the order of magnitude ob- 
served in nascent cyclones, that the 
velocities are generally directed east- 
wards, and that the motion of the 
air is of the type found on the 
weather chart. Therefore, we can 
say that the theory shows that the 
formation of a cyclone from waves 
not only is possible but must occur in 
the atmosphere because the waves 
are frequently unstable and there- 
fore form spontaneously. Of course, 
stable waves also occur; if they are 
much shorter than cyclonic waves 
they are observed as billow clouds, 
ceiling fluctuations, and microbaro- 
metric oscillations, and if longer than 
cyclonic waves (and thus stable again 
owing to the stabilizing influence of 
inertia) they appear as flat frontal 
waves like young cyclones, but never 
develop into mature cyclones. (See 
Fig. 9.) 
The theoretical investigation of 
cyclonic waves shows further that the 
velocity of propagation consists of 
two terms. Their physical significance 
is understood most easily by con- 
sidering a wave on the surface of a 
river. The propagation of such a 
wave is partly due to the bodily trans- 
port of the oscillating water masses 
by the flow of the river, the “con- 
vective” term, and partly due to the 
motion of the wave relative to the 
water, the “dynamic” term. Similar- 
ly, the first term in the expression 
for the wave velocity at a surface of 
discontinuity is due to the mean mo- 
tion of both air masses. It simply 
indicates the fact that the wave is 
transported passively due to the un- 
disturbed motion of both layers. This 
part of the total velocity of propa- 
gation is therefore called the ‘con- 
vective” term. The second term, 
which is due to the dynamical pro- 
cesses of the wave motion, is referred 
to as the dynamic term. 
REMAINING PROBLEMS 
While the wave theory is _ suffi- 
ciently far advanced for one to say 
with certainty that cyclonic waves 
occur in the atmosphere, much re- 
mains to be done yet. The investi- 
gations so far have been dealing with 
wave motions at the boundary be- 
tween isothermal air masses, because 
isothermy gives equations which are 
easier to handle than equations which 
take the usual linear temperature 
gradient into account. But isothermal 
lapse rate implies, obviously, a larger 
stability of atmospheric stratification 
than is ordinarily found, so the nu- 
merical results will have to be modi- 
fied. Notable advances in this direc- 
tion have recently been mode by H. 
Solberg.’) Furthermore, note that a 
sharp discontinuity of wind and tem- 
perature is assumed in the theory 
while in reality a narrow transitional 
zone exists in which the elements 
change rapidly but continuously. It 
has been shown, however, that the 
wave is practically the same whether 
there is a sharp discontinuity or a 
transitional zone, provided that the 
thickness of the transitional zone is 
small compared with the wave length’). 
This condition is always fulfilled for 
cyclonic waves. 
A certain deficiency of the wave 
theory of cyclones in its present state, 
which we tactfully have not men- 
tioned so far, is the assumption that 
the lower cold layer and the upper 
6H. Solberg: Schwingungen und Wellenbe- 
wegungen in einer Atmosphare mit nach oben 
abnehmender Temperatur, Astrophysica Norve- 
gica, vol. 2, no. 2, 1986, pp. 123-172. 
‘Haurwitz, B.: Zur Theorie der Wellenbe- 
wegungen in Luft und Wasser, Verdéffentl. d. 
Geophysikal. Inst. Univ. Leipzig, Spezialarb., 
Ser. 2, Vol. V, no. 1, 1981, pp. 52-53, 73-74. 
