THE QUANTITATIVE THEORY OF CYCLONE DEVELOPMENT 
may express separately the changes in kinetic and 
“Hotential” energy in terms of arbitrary displacements 
of the form 5a = e*'5a(a, y, 2), etc. The kinetic energy 
change contains the factor 6;, while the “potential” 
energy change does not, so that the law of conserva- 
tion of energy gives an expression for 6; in terms of the 
displacements. The possible simultaneous values of the 
displacements are restricted (or constrained) by the 
equations of motion and we can to some extent delimit 
possible values of 6; by considering only some of the 
constraints, as in somewhat analogous problems in 
dynamics. In the present instance we consider only 
the equation of continuity and one momentum equa- 
tion (and suitable boundary conditions) and then ap- 
ply algebraic inequality theory to our undetermined 
displacements. We thereby obtain an upper bound to 
the possible value of 6; (corresponding to negative square 
of frequency) just as in dynamics we obtain a lower 
bound to frequency (squared) by considering a less 
constrained system. 
The value of the foregoing, described in more de- 
tail elsewhere [3], derives partly from the fact that 
we can treat a wider variety of problems than we can 
by complete solution, while the value of 6; (maximum) 
thereby obtained is usually not much greater than the 
true value of 67. But its greatest usefulness is that it 
makes the process of breakdown of unstable systems 
immediately intelligible. If we take into account only 
our limited set of constraints, it is immediately evident 
what kind of displacement field is necessary for a re- 
lease of “potential” energy. It is of course essential 
that the term ‘potential”’ energy be correctly inter- 
preted. For our purpose it comprises all forms of energy 
- other than the kinetic energy of the perturbation and 
therefore includes, besides gravitational potential en- 
ergy, the organised kinetic energy of the mean flow 
(smoothed of harmonic variation). This distinction be- 
tween two kinds of kinetic energy change is justified 
by their different roles in the turbulent motion which 
is the ultimate state in practice: turbulent energy may 
arise either from a decrease in gravitational potential 
energy or from a decrease in kinetic energy of the mean 
motion. We may classify our systems according to 
whether the “potential” energy source is (a) static 
(gravitational) or (b) dynamic (kinetic). Now the dis- 
placement field is merely the nascent form of a process 
of overturning and we shall need to consider only two 
possibilities: (1) overturning in a vertical plane; (2) 
overturning in a quasi-horizontal plane. Thus we may 
also classify our systems according to the kind of over- 
turning associated with instability. In our list of four 
simple types of instability we anticipated their classi- 
fication from both points of view. Let us consider the 
characteristics of these systems. 
la. Gravitational Instability. We consider barotropic 
conditions, so that initially there is no wind change 
with height, and for simplicity we suppose that d@/dz = 
B, where B is constant. If, to begin with, we neglect the 
rotation of the earth, then “potential” energy exists 
only in gravitational potential form. Suppose that two 
small parcels of air of equal potential volume were 
467 
slowly interchanged. Then since potential density would 
depend only on entropy, we should obtain, if B were 
negative, a net release of energy for any two parcels 
at different levels, while if B were positive no inter- 
change could release potential energy. Clearly the con- 
straints associated with the continuity equation cannot 
alter this result—the overturning process is equivalent 
to a set of such interchanges of different amplitudes. 
Since horizontal motion does not affect potential en- 
ergy we need consider only vertical overturning; calcu- 
lations by the energy method give 6; < —gB, where 
g = gravitational acceleration. Of course in this simple 
case it is easy to obtain complete solutions, represent- 
ing the nascent stage of Bénard cells, and calculations 
show that 67 (maximum) is nearly attained for narrow 
deep cells, where little energy is wasted in horizontal 
motion. We shall postpone the extension to large-scale 
convection, where the rotation of the earth is con- 
sidered, since this is really a combination of types 
la and 1b. 
1b. Centrifugal Instability. We shall suppose the mo- 
tion to be barotropic and horizontal with the initial 
velocity V, a function of y only. For simplicity we 
take dV,/dy constant and, to begin with, we put B = 0 
(isentropic conditions). Then ‘‘potential’’ energy exists 
only in the “‘kinetic” form. Let us consider the change 
due to overturning in the (vertical) y, z plane. Fila- 
ments of air in the x-direction move as a whole and we 
easily derive from the equations of motion that, during 
displacement, 6V. = foy, where f is the Coriolis parame- 
ter. If the x-axis is directed toward the east, this cor- 
responds to constancy of absolute angular momentum. 
But for our purposes, where a mean value of f is used, 
the orientation of the z-axis is arbitrary. A simple calcu- 
lation shows that potential energy is released only if 
dV./dy > f, corresponding to negative absolute vor- 
ticity, and the energy method gives 6; < f(dV./dy — f). 
Although values of dV,/dy near the critical value are 
sometimes observed in narrow bands, it is doubtful 
whether centrifugal instability ever occurs on a large 
seale except perhaps in low latitudes. The rotation of 
the earth, normally at least, has a stabilising effect so 
far as vertical overturning is concerned. 
Similar results are obtained if, instead of rectilinear 
motion, we consider a barotropic circular vortex (with 
no motion relative to the earth as a special case). 
The condition for instability is again negative absolute 
vorticity. 
We may note that in both the foregoing cases maxi- 
mum instability occurs for shallow, flat, cells since 
“potential” energy changes depend only on horizontal 
motion (no energy is wasted in vertical motion). 
lab. Gravitational-Centrifugal Instability. It is easy 
to combine the results of the previous sections for a 
system in which neither B nor dV,/dy vanishes. There 
is instability if either B or (f — dV./dy) is negative, 
for the cells may be either so deep that centrifugal 
stability is negligible or so shallow that static stability 
is negligible. The important practical case is that of no 
motion (special case of circular vortex) with B < 0. 
Instability occurs for disturbances which are sufficiently 
