466 
between the actual entropy lapse and that of the ap- 
propriate wet-adiabatic. Thus there is a sharp, and 
usually a large, reduction in static stability when air 
becomes saturated. The effective horizontal entropy 
gradients are also modified as a result of saturation, 
but to a much smaller extent. Normally a cloud mass 
behaves, to a sufficiently close approximation, as if 
the air were unsaturated except for the appropriate 
modification in static stability. 
Our equations are complicated by the fact that air 
is a compressible fluid, but it is clear that this feature 
is not, for our purposes, a significant one. We are 
concerned essentially with a particular type of ‘‘vibra- 
tion” problem though our disturbances have mathe- 
matically complex wave velocities. The moduli of these 
wave velocities are in all cases, as our calculations 
verify, small compared with the velocity of sound. It 
is not surprising therefore that the forces associated 
with compressibility are negligible, that is, that the 
air behaves dynamically as if 1t were incompressible. 
The static effect of compressibility, involving a large 
change of density with height, is a complication which 
prevents atmospheric motion from being quite the same 
as that of an incompressible fluid. Nevertheless, even 
for deep disturbances, the behaviour differs little from 
that of an incompressible fluid of similar mean density: 
the modifications are essentially of the nature of distor- 
tions of wave structure without much change in more 
significant features like growth rate. We shall therefore 
confine our attention to “‘equivalent”’ incompressible 
fluid systems. Our results are, of course, more directly 
applicable to analogous oceanographic problems. 
Lack of space prevents a discussion of these points 
in mathematical terms. It has been shown elsewhere 
[3] that the basic equations may be further simplified 
by the elimination of the pressure field, so that we 
have finally four equations connecting the four de- 
pendent variables V,, V,, V- (velocity components), 
and ® (entropy). But our present concern is the physical 
interpretation and practical significance of certain calcu- 
lations rather than the calculations themselves. 
The General Theory of the Instability of Fluid Motion 
The various types of instability occurring in dynami- 
cal meteorology merge into one another so that most 
systems encountered in practice are, to a greater or 
less degree, hybrid. Nevertheless it is not only simpler 
but theoretically more instructive to consider certain 
ideal limiting cases where one or another unstabilising 
factor acts alone. Four simple types of instability will 
interest us: 
la. Gravitational instability (ordinary convection or 
static instability). 
1b. Centrifugal instability (dynamic instability). 
2a. Baroclinic instability (with thermal wind). 
2b. Helmholtz instability (at a velocity discontinu- 
ity). 
Instability of type 1a is, in middle and high latitudes, 
nearly always a small-scale phenomenon, but in low 
latitudes a modified form, taking into account the 
DYNAMICS OF THE ATMOSPHERE 
rotation of the earth, is intimately concerned with the 
development of tropical cyclones. 
Instability of type 1b has been the subject of much 
recent investigation, usually under the heading ‘‘dy- 
namic instability,” but despite its theoretical impor- 
tance it is probably rare for large-scale motion. The 
name “‘centrifugal” has been preferred to “‘dynamic”’ 
because it is more descriptive and less confusing—other 
types of instability may reasonably be called ‘dy- 
namic.” 
Instability of type 2a is probably the most important, 
on a large scale, in middle and high latitudes. It is to 
this type that our earlier remarks regarding the normal- 
ity of instability and the existence of ‘‘natural selec- 
tion” directly apply. 
Instability of type 2b was investigated by Helmholtz 
and Rayleigh for nonrotating barotropic fluids. The 
Norwegian wave theory of cyclones was a partially 
successful attempt to extend the theory to rotating 
barotropic fluids. 
Although we shall choose our initial systems so that 
only one type of instability is in question, the same 
general method of analysis applies in every case. Using 
the method of small perturbations, we obtain a set of 
simultaneous, linear, partial differential equations in- 
volving the perturbations as dependent variables. By 
elimination we obtain a partial differential equation 
with only one dependent variable and look for simple 
solutions satisfying appropriate boundary conditions. 
Usually these solutions involve only circular or expo- 
nential functions in the horizontal (a and y directions) 
and all contain the factor e*!’ where ¢ represents time 
and 6; is a constant called the growth rate. For 6; to 
be real we usually have to use a moving coordinate 
system. Fortunately these solutions for unstable waves 
are, practically, the most important ones and the dis- 
turbance of maximum growth rate, when it exists, is 
probably dominant relative to one of arbitrary initial 
structure. In any case a study of these particular solu- 
tions enables us to understand the process of breakdown 
of the initial system and to estimate the relative im- 
portance of various factors. 
The method of analysis outlined above is necessary 
if we require precise results and is the only one which 
is completely unequivocal. But it is mathematical in 
form and usually rather involved so that significant 
physical principles, which give us insight into our prob- 
lems and immediately suggest generalisations, tend to 
be obscured. Now, except that our interest is centred 
in the unstable region, we are concerned with what are 
essentially vibration problems and we may expect to 
find that energy considerations are of paramount im- 
portance. For, by the law of conservation of energy, 
the kinetic energy associated with any perturbation 
must be equal to the decrease in ‘“‘potential” energy of 
the system, and a necessary condition for instability 
is that it should be possible to find displacements which 
will decrease “potential” energy; the condition will be 
sufficient only if these displacements are consistent 
with all the equations of motion and boundary condi- 
tions. More precisely, using Rayleigh’s method, we 
