THE QUANTITATIVE THEORY OF CYCLONE DEVELOPMENT 
mean flow of each air mass (in opposite directions) 
resulting in a decrease in organised kinetic energy (see 
Fig. 1). 
Fic. 1.—Helmholtz waves. Arrows with a single head and a 
single shaft denote perturbation velocities; those witha single 
head and double shaft, perturbation mean velocities; those 
with a double head and a double shaft, initial velocities. 
Now, as we have seen, the earth’s rotation has a 
stabilising effect only when there is vertzcal overturning. 
Hence we should expect results similar to those men- 
tioned above when the earth’s rotation is taken into 
account. Complete solution of the problem confirms 
this, and 6} = (\?/4)(U, — U2)?, where 2/) is the 
wave length for a rotating, as well as for a nonrotating, 
system. 
In practice, B is usually strongly positive so that 
any vertical motion would decrease the net release 
of ‘“‘potential’’ energy. Moreover, frontal surfaces are 
not usually vertical and in general the boundary 
conditions cannot be satisfied by purely horizontal 
motion. Hence a sloping front is less unstable than a 
vertical one. Since the unstabilising effect of a velocity 
discontinuity is inversely proportional to the wave length, 
whereas the stabilising effect of static stability is m- 
dependent of the scale of motion, it follows that only 
waves shorter than a critical wave length are unstable. 
Very short waves are always unstable because static 
stability may be neglected. 
469 
Future Developments 
The discussion above is merely an outline of ele- 
mentary principles. Although our analysis shows that 
the equations of dynamical meteorology are by no 
means intractable from the point of view of computing 
future developments, the results so far obtained are 
only of limited applicability. Our calculations give only 
the initial form of the most probable (dominant) new 
development and we need to compute the further de- 
velopment when the perturbations are no longer small. 
As the period of the forecast is extended, analytical 
methods become increasingly involved and clumsy and 
sooner or later we have to resort, at least partly, to 
numerical methods. An adequate degree of accuracy 
is practically attainable only with the use of computing 
machines, and electronic large-memory computers will 
play an important part in extending and generalising 
the elementary theory. 
The development of numerical methods, even to the 
extent of a direct attack using observed data, does not 
absolve us from the necessity of understanding the 
precise significance of our solutions. Not only do we 
have to know how and where to approximate, but the 
reliability of our solutions varies with time, place, and 
forecast period. In fact for long forecast periods what 
is significant is not the detail, which is usually partially, 
perhaps entirely, accidental (7.e., dependent on minu- 
tiae below the margin of error), but the general nature 
(e.g., persistently settled or unsettled) of the majority 
of possible solutions. We need to develop the statistical 
theory referred to earlier. Not all the questions about 
future weather we should like to answer are in fact 
answerable, but it may well be that the growth of un- 
certainty in some directions is compensated by statisti- 
cal regularity in others. 
REFERENCES 
1. BuerKnes, V., and others, Hydrodynamique physique. Paris, 
Presses Universitaires de France, 1934. 
2. Cuarney, J. G., ““The Dynamics of Long Waves in a Baro- 
clinic Westerly Current.” J. Meteor., 4:135-162 (1947). 
3. Eapy, E. T., ‘““Long Waves and Cyclone Waves.” Tellus, 
Vol. 1, No. 3, pp. 338-52 (1949). 
4, Lams, H., Hydrodynamics, 6th ed. Cambridge, University 
Press, 1932. 
