THE QUANTITATIVE THEORY OF CYCLONE DEVELOPMENT 
normal feature of atmospheric motion. Although the 
unstable disturbances are continually tending to estab- 
lish a new stable state, radiative processes are continu- 
ously tending to restore the initial system, which there- 
fore remains permanently unstable. Secondly, for such 
a system the study of the ensemble of all possible 
perturbations is relatively simple. In each system there 
exists a disturbance of maximum growth-rate (so that 
we can determine the growth of the margin of error 
and estimate the limited time interval referred to above) 
which eventually becomes dominant in subsequent de- 
velopments by a process analogous to Darwinian natu- 
ral selection. Almost any initial disturbance tends event- 
ually to resemble the dominant, which is therefore 
the most probable development. The “‘ensemble”’ pos- 
sesses at least some strongly marked statistical proper- 
ties which may be utilised to extend the range of 
forecasts. In spite of inaccuracies due to oversimplifi- 
cation this result is practically significant. The dis- 
turbances referred to are approximations to nascent 
cyclones, long waves, etc.; were it not that “natural 
selection” is a very real process, weather systems would 
be much more variable in size, structure, and beha- 
viour. 
The Basic Equations 
We shall regard as basic equations the three dynami- 
cal equations, the thermal equation, and the equation 
of continuity; others, such as the gas laws and the 
laws of radiation, will be regarded as subsidiary. The 
number of dependent variables we need, or that we find it 
convenient to use, depends on the nature of the prob- 
lem and the degree of accuracy aimed at. In the prob- 
lems with which we shall be concerned it is possible to 
express the basic equations in terms of the three com- 
ponents of velocity, pressure, and entropy (or density) 
alone so that the five basic equations, together with 
appropriate boundary conditions, form a complete set. 
Clearly these equations can be appropriate only for a 
limited range of problems when certain approximations 
are justified; we shall in fact make further approxima- 
tions, our aim being to retain only those terms which 
are of prime importance in the range in which we are 
interested. 
A completely realistic theory of the stability of at- 
mospheric motion should deal with nonsteady initial 
conditions, but for simplicity we shall confine our at- 
tention to the case in which the initial motion is 
steady, and in fact we shall be concerned mainly with 
rectilinear horizontal motion. Our analysis will be ap- 
proximately true even when the very-large-scale distri- 
bution is slowly changing. 
The relative importance of the terms in our equa- 
tions depends partly on the scale of the phenomena 
with which we are concerned. Here we are interested 
in disturbances of the order of magnitude of nascent 
cyclones, say 1000 km in horizontal extent and occupy- 
ing a large part (or the whole depth) of the troposphere. 
From our point of view ordinary or gravitational con- 
vection, originating from static instability (7.e., super- 
adiabatic lapse rate), and ordinary turbulence of fric- 
465 
tional or convective origin are small-scale phenomena. 
The epithet “ordinary” is appropriate in each case 
because the disturbances whose nascent form we are 
studying may be regarded as elements of a large-scale 
convective process and this process, regarded statisti- 
cally, is a kind of large-scale turbulence. From our 
point of view the significance of small-scale turbulence, 
including ordinary convection, lies in its statistical 
properties, such as ability to transport heat, momen- 
tum, ete. Now frictionally induced turbulence is most 
effective near the earth’s surface, and rough calcula- 
tions (which we have not space to describe) using 
empirical estimates of skin friction indicate that fric- 
tional dissipation of energy usually has a relatively 
small effect on the development of large-scale disturb- 
ances (especially over a sea surface) in their nascent 
stage, provided the unstabilismg factors are not too 
weak. Since we are most interested in those regions 
where the unstabilising factors are relatively strong, 
we may obtain a useful first approximation during the 
nascent stage if we neglect the frictional terms in the 
equations of motion. It is not possible to neglect fric- 
tional terms throughout the whole life-history of a 
disturbance because in the long run the kinetic energy 
destroyed by friction must equal that generated as a 
result of instability. 
Surface friction transports heat vertically through 
a shallow layer, but since we are interested in the be- 
haviour of deep layers this effect will be neglected. 
Moreover, surface turbulence is partly convective in 
origin, and we may regard shallow convection as in- 
cluded in this argument. But sometimes (e.g., in strong 
polar outbreaks) deep and widespread convection trans- 
ports heat to great heights at a great rate. We shall 
ignore this possible complication and concentrate our 
attention on systems in middle and high latitudes 
which are statically stable in their initial stages. 
Just as, in the long run, we cannot ignore skin fric- 
tion so, in the long run, we cannot ignore radiative 
processes. Large-scale turbulence (the statistical aspect 
of our disturbances) appears to be a major factor in 
transporting heat poleward to compensate the unbal- 
anced radiation flux. But during the nascent stage, 
development (measured by the time for growth of 
the disturbance by a given factor) is relatively rapid 
and it is precisely for this reason that we are able to 
neglect frictional terms. Hence it is reasonable to sup- 
pose that in the nascent stage we may, for a first ap- 
proximation, neglect the change in the radiation bal- 
ance caused by the disturbance and use for our thermal 
equation the adiabatic equation. 
Consider first the case of unsaturated air. To a close 
enough approximation the entropy of dry air is meas- 
ured, in suitable units, by & = (1/7) In p— In p, where 
p = pressure, p = density, y = specific heat ratio, and 
& is conserved during the motion. In this case we shall 
define the static stability, which measures the restor- 
ing force due to gravity on a particle displaced verti- 
cally, as d&/dz (2 = vertical coordinate). Now con- 
sider the case of saturated air in contact with a cloud. 
The static stability is now measured by the difference 
