EXTRATROPICAL CYCLONES 585 
surface sometimes satisfies this instability criterion, 
as will be shown later. 
If we drop the initial conditions of dv,/d% = 0 and 
dv,/dt = 0, no exact treatment can be offered, because 
then the fundamental current is not a steady-state one. 
The following reasoning should however be valid, pro- 
vided that the long-wave deformations of the funda- 
mental current are much slower than the short-wave 
developments on the isentropic surface. This condition 
is usually fulfilled. 
In Fig. 6 an element of the isentropic surface is 
shown in #y-coordinates. Isobars on that surface are 
parallel to the z-direction, while the distribution of the 
speed of the geostrophic wind v, is shown by slanting 
scalar curves. Thus v, has a gradient m the «-direction 
in addition to the much stronger gradient in the - 
direction. The disturbed path of a sample particle along 
the isentropic surface is supposed to go from A to B 
during the time differential. The «-component of the 
acceleration (parallel to the isobars) of the sample par- 
ticle is again, to a first approximation, dv,/dt = 2Q.v, , 
while the change of geostrophic wind encountered along 
the path is 
dv 
dt 
The acceleration of the particle in the 7-direction is 
supposed to be zero at the initial point of the trajectory 
A. The acceleration in the y-direction will also be zero 
at the end of the trajectory B if dv,/dt = dv,/dt, or 
ov Ov Ov 
= Vy an + vz om + ae (10) 
Ov Ov Ov 
20,0, = V, a + 0, Oe + a7 9 
that is, 
305 8 
Ox ot 
vy, = ov (11) 
= 
20, ay 
Specializing now for the condition 20, — dv,/dn > 0 
and for v,dv,/0% + dv,/dt > 0, we find that the 
particle given an initial speed component v, greater 
than the value found in (11) will have an acceleration 
component dy,/dt opposite to v, . Given a smaller posi- 
tive initial v, , or a negative initial v, , the particle would 
accelerate towards the value for v, given in (11). This 
value of v, therefore represents the y-component of a 
stable upgliding motion in which all the particles of the 
isentropic surface may jom. The y-component of the 
stable upglidmg motion approaches infinity when 
20, — dv,/dn goes to zero. In other words, in the case 
of imertial indifference any finite initial v, would in- 
crease exponentially. 
Quite analogous reasoning in the case 20, — 
dv,/dn > O and vzdv,/dx + dv,/dt < O reveals the 
existence of stable downgliding motion in which the 
n-component is also given by (11). 
The difference between the cases v:dv,/de + 
dv,/dt = 0 and 2 0 is thus the following: In the former 
case the departures from the geostrophic wind remain 
of a stable oscillatory nature until the anticyclonic isen- 
tropic shear reaches the critical value of —2Q,. In the 
latter cases there is a sustained stable departure from 
the geostrophic wind, represented by v,, which has 
finite values also when 22, — dv,/dn > 0. In addition 
there may be inertial perturbations superimposed on 
the current representing the vector sum of geostrophic 
motion v, and isentropic motion v, ; such perturbations 
will be stable as long as 20, — dv,/dn > 0. 
The demonstration of the occurrence of anticyclonic 
shear in the upper atmosphere, which approaches or 
even surpasses the critical limit of dynamic instability, 
is due to recent research work at the University of 
Chicago. The first profile of the westerlies showing such 
conditions was analyzed by Palmén [23] in 1948. In 
the same paper it is also shown how we must treat zonal 
flow as curved flow (radius of curvature r = a cotan ¢) 
in order to arrive at a satisfactory accuracy of an 
isovel profile which is to correspond to an observed 
meridional pressure profile. In the following discussion 
we shall use the model of straight baroclinic westerlies 
with a stationary polar front published by Palmén and 
Newton [24] and reprinted here as the left part of Fig. 
7. It represents an isotherm-isovel profile based on an 
averaging of twelve eastern North American meridional 
profiles made during December 1946. In the right-hand 
part of Fig. 7 we have added a diagram of the com- 
puted quantity 20. — dv,/dn, with n interpreted as the 
curvilmear isentropic coordinate (positive direction 
northwards). In most of the field 20, is greater than 
dv,/0n, indicating inertial stability; but in a narrow zone 
south of the maximum upper westerlies, 20. — dv,/dn 
is negative, indicating inertial instability. Furthermore, 
in the frontal zone below 600 mb, where dv,/dn has been 
measured along saturation isentropes, the values of 
20, — dv,/dn indicate only a slight amount of inertial 
stability. We shall focus our attention first on that 
part of the profile. 
The small positive (or in some individual cases nega- 
tive) values of 20, — 0v,/dn are located in a narrow 
frontal zone, while in the adjacent parts of the warm 
and cold air masses, 20, — dv,/dn is positive and far 
from zero. Since in (11) the component of stable up- 
gliding or downgliding is inversely proportional to 
20, — dv,/0dn , it follows that the air in the narrow frontal 
zone has a much greater possibility for isentropic up- 
or down-displacements than the air masses on either 
side. 
The quantity v.0v,/d% + dv,/dt, representing the 
numerator in the expression for v, in (11), cannot be 
judged from the data of one profile alone. It will be 
large and positive (1) where the isobars of the hori- 
zontal pressure distribution converge, and (2) where 
the gradient wind increases locally with time. The first 
condition is fulfilled, for example, along the axis of 
kinematic dilatation extending eastward from a col of 
the pressure field. This synoptic situation is known to 
be frequently associated with frontogenesis and sub- 
sequent maintenance of a sharp front. The second condi- 
tion, local increase of gradient wind parallel to the 
frontal zone, frequently occurs during frontogenesis, 
