468 
deep relative to their breadth, the condition being that 
there should be a net release of “potential” energy. 
Tn low latitudes not only is —B sometimes (temporar- 
ily) relatively large but the stabilising effect of the 
earth’s rotation is small, so that convection cells of 
relatively enormous diameter (nascent hurricanes) can 
develop. 
In general, maximum growth rate corresponds either 
to very deep or to very shallow cells but this result is 
not of great significance because practical systems are 
very inhomogeneous. 
2a. Baroclinic Instability. We suppose the initial mo- 
tion to be rectilinear and take the initial velocity V, 
to be a function of z only. For equilibrium this implies 
a horizontal gradient of entropy A = d®/dy, where, 
approximately (A not too small), dV,/dz = —gA/f. 
For simplicity we suppose A to be constant. Pure 
baroclinic instability should correspond to B = 0, but 
it will be convenient to consider directly the more 
general case B ~ 0. In practice we usually have (at 
least in the mean) B > 0, and this is the only case we 
need examine, for when B < 0 the system is obviously 
unstable. The isentropic surfaces have an angle of 
slope a(<7/2) given by tan a = | A/B| and in prac- 
tice we normally have tana < 1. 
Consider first the change in gravitational potential 
energy resulting from interchange of parcels of air in 
the manner of subsection la. The result is no longer 
independent of the y-displacement. If the direction of 
displacement lies outside the acute angle a, there is 
an increase of energy, but if it lies inside, energy is re- 
leased. There is zero change for displacement either 
along the isentropic surfaces or horizontally, and calcu- 
lation shows that maximum release of energy occurs 
(approximately, assuming a < 7/4) for displacement 
in the direction of the bisector of a (in the y, 2 plane), 
which we shall call the s-axis. Now consider the change 
in kinetic “potential” energy. If the overturning were 
in the vertical plane, this would occur in the manner 
of subsection 1b, with increase of energy. But if over- 
turning occurs in thez, s plane (7.e., quasi-horizontally), 
with the perturbations varying harmonically in the 
x-direction, there is no change in the mean motion and 
no change of energy (correct to the appropriate order 
of small quantities). Hence our energy method gives 
instability in all cases and on calculation: 
2 alse 
01 < gsB; —_ 4 he 
where g. and B, are the components of gravity and 
entropy gradient, respectively, along the s-axis (note 
analogy to 1a) and h? is the Richardson number defined 
by h? = gB/(dV/dz)?. 
This result is of course provisional but, as in other 
cases, is verified by complete solution. With artificial 
(but physically possible) boundary conditions we can 
obtain nearly the maximum value of 6,, but a more 
realistic model gives 6, = 0.31 f/h. The reduction in 
the coefficient from 0.5 to 0.31 is due to the additional 
constraints imposed by the boundary conditions, as a 
result of which displacements cannot everywhere be 
(ES Si), 
DYNAMICS OF THE ATMOSPHERE 
in the optimum s-direction: for one particular wave 
length (more precisely, for one ratio of horizontal to 
vertical “‘wave length’’), the displacements are as near 
optimum as possible and it is to this dominant wave 
that the coefficient 0.31 applies. Longer waves grow 
more slowly while very short waves are stable. Thus 
there is “natural selection” for one particular wave 
structure. It has been shown elsewhere how, by con- 
sidering compound systems containing a region where 
h? is a minimum, realistic models of both nascent wave 
eyclones and long waves may be constructed. Briefly, 
the smaller disturbances develop in frontal regions, 
where cloud masses reduce the effective static stability 
and therefore also h?. The long waves occupy the whole 
troposphere, and secondary modifications, caused by 
constraints associated with the variability of the Cori- 
olis parameter, are then significant. 
lab. Generalised Vertical-Overturning Instability. We 
may consider from the point of view of vertical over- 
turning an initial system similar to that of subsection 
2a, but it will be convenient to generalise by supposing 
Vz to vary with y as well as with z. Then using the 
same general method as before, we obtain 
wi < [os + s(0-)] 
/ [w+ s- 9] 
+4 | (oay — gBf (i - = 
where the surd has always to be taken as positive. 
This is the general formula for vertical overturning, - 
including the examples previously given as special cases. 
If either B < 0 or dV;/dy > f, the system is certainly 
unstable. If neither condition is satisfied, we require 
for instability (gA)? > gBf(f — dV./dy), which is 
equivalent to 1/h? > [1 — (1/f)(dV./dy)|. In the im- 
portant special case when dV,/dy vanishes, this con- 
dition becomes simply h? < 1, equivalent to the well- 
known condition of negative absolute vorticity in the 
isentropic surfaces. 
2b. Helmholtz Instability. Consider the system of two 
barotropic air masses with uniform horizontal motion 
V. = U, in one, and V, = Uz in the other, separated 
by a vertical “front” at the x, z plane. If the earth 
were not rotating, we could apply the well-known 
results of Helmholtz (complete solutions) which show 
this system to be unstable for all perturbation wave 
lengths, growth-rate being inversely proportional to 
wave length, but it is instructive to apply the energy 
method. Helmholtz’s solutions involve only horizontal 
motion, associated with corrugation of the “front,” 
so we need consider only horizontal overturning. The 
“potential” energy is entirely “kinetic” and the manner 
of its release is clear from the flow pattern, obtained 
by considerations of continuity and boundary condi- 
tions alone. Outside the y-limits to which the corruga- 
tions of the “front” extend, the mean motion is un- 
altered, but inside these limits there is a change in the 
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