THE QUANTITATIVE THEORY OF CYCLONE DEVELOPMENT 
By E. T. EADY 
Imperial College of Science and Technology 
Introduction 
The title of this article indicates its principal theme 
but not its full scope, for although we shall explain 
the method of formulating and solving certain theoreti- 
cal problems and shall interpret the answers in terms 
of the initial stages of development of extratropical 
cyclones and anticyclones, our analysis has also a wider 
significance. Not only does a fundamentally similar 
theoretical analysis apply to a wide variety of develop- 
ment problems (including, for example, the develop- 
ment of ‘long’? waves as well as the shorter “frontal” 
waves and even phenomena due primarily to ordinary 
convective instability) so that a comprehensive analy- 
sis is desirable, but this analysis gives results of primary 
importance in the theory of development 7m general, 
that is, from the point of view of the general forecast- 
ing problem. We shall infer from our results that there 
exist, in general, certain ultimate limitations to the 
possibilities of weather forecasting. Certain apparently 
sensible questions, such as the question of weather 
conditions at a given time in the comparatively distant 
future, say several days ahead, are in principle un- 
answerable and the most we can hope to do is to de- 
termine the relative probabilities of different outcomes. 
The full significance of our theoretical problems becomes 
apparent only when it is clear what kind of question 
we should attempt to answer. 
The science of meteorology is a branch of mathemati- 
cal physics; it can be fully understood only in a quanti- 
tative manner. Moreover, all the practical questions 
we should like to answer are of a quantitative charac- 
ter. Having discovered the relevant equations of mo- 
tion, we ought to aim at obtaining significant integrals 
(more precisely, solutions of significant boundary-value 
problems) which may be applied directly to practical 
problems. In order to obtain tractable problems, and 
at the same time to see clearly what we are doing (7.e., 
“to see the wood for the trees”), we may, for a first 
analysis, simplify the equations by omitting all factors 
not vitally affecting the nature of the answer. Later 
we may refine our solutions by taking into account 
factors previously omitted (e.g., by the method of suc- 
cessive approximations), thereby testing whether we 
have in fact included all the vital factors. This is a 
procedure with which we are familiar and, however 
laborious it may be in practice, it mtroduces no new 
difficulty in principle. The really serious difficulty is 
to discover what kind of problem ought to be solved, 
for this difficulty arises as soon as we consider the 
question of the stability of atmospheric motion. Ob- 
servation suggests that the motion may, at least some- 
times, be unstable, and we shall infer from subsequent 
analysis that instability (to a greater or less degree) 
is a normal feature of atmospheric motion. 
It is important to be quite clear as to the meaning 
of the term ‘unstable’? when applied to a system of 
fluid motion. If we suppose the initial field of motion 
to be given, the final field of motion, after a given 
interval of time, is determined precisely by the equa- 
tions of motion, continuity, radiation, etc., together 
with the appropriate boundary conditions. If we con- 
sider a slightly different (perturbed) initial state, the 
new final state, after the same interval of time, will be 
determined in a similar manner. The stability or in- 
stability of the motion depends on the behaviour of 
the resulting change (perturbation) in the final state 
as the time interval is increased. If the final perturba- 
tion remains small for all time for all possible initial 
perturbations, the motion is stable. If, on the other 
hand, the perturbation in some or all regions grows 
(initially) at an exponential rate for any possible initial 
perturbation, the motion is unstable. There is an inter- 
mediate case, conveniently described as neutral stability, 
when the perturbations grow linearly or according to a 
low-degree power law, but this need not concern us here. 
The practical significance of a demonstration that 
the motion is unstable is clear, for in practice, however 
good our network of observations may be, the initial 
state of motion is never given precisely and we never 
know what small perturbations may exist below a 
certain margin of error. Since the perturbation may 
grow at an exponential rate, the margin of error in the 
forecast (final) state will grow exponentially as the 
period of the forecast is increased, and this possible 
error is unavoidable whatever our method of forecast- 
ing. After a limited time interval, which, as we shall 
see, can be roughly estimated, the possible error will 
become so large as to make the forecast valueless. In 
other words, the set of all possible future developments 
consistent with our initial data is a divergent set and 
any direct computation will simply pick out, arbitrar- 
ily, one member of the set. Clearly, if we are to glean 
any information at all about developments beyond the 
limited time interval, we must extend our analysis and 
consider the properties of the set or “ensemble” (cor- 
responding to the Gibbs-ensemble of statistical mechan- 
ics) of all possible developments. Thus long-range fore- 
casting is necessarily a branch of statistical physics in 
its widest sense: both our questions and answers must 
be expressed in terms of probabilities. 
There are two important connections between these 
general considerations and subsequent analysis. Firstly, 
this analysis will show the existence of at least one type 
of large-scale unstable disturbance in a simplified but 
typical system, and we shall infer that instability is a 
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