432 
The usefulness of the method of characteristics is not 
confined to the study of flow under an inversion. Any 
problem in which certain elements are described by 
equations such as (1) and (2), with real characteristics, 
can be studied by these methods. It is hoped that this 
article will be useful to every meteorologist with such 
problems and that some may even be encouraged to 
study some of the problems mentioned below. 
The Effect of the Coriolis Force on Flow under an 
Inversion. In all of the discussions of nonlinear flow 
under an inversion that have appeared in the literature 
the Coriolis parameter is neglected. The Coriolis force 
works to turn the winds parallel to the isobars, but it 
takes time for this force to act. If a parcel of air is under 
the influence of a pressure gradient for only a short time, 
it will follow the path dictated by the equations without 
Coriolis force so there is some justification in neglecting 
this parameter in the pioneering stages of such a study. 
If the value of the theory of flow under an inversion to 
middle-latitude forecasting is to increase, however, the 
Coriolis parameter must be included in the equations. 
Some of the physical effects of this force can be seen 
immediately. Suppose that a cold front is holding an 
inversion of constant height at rest at point 2) and time 
ty. At t the front begins to recede, moving westward 
from the inversion. An expansion wave will be formed 
and the slope in the inversion height will be such that 
pressure will increase to the east. If the front continues 
to recede, the air parcels near it and beneath the inver- 
sion will be under the influence of such a pressure gradi- 
ent for a long period of time and will therefore move 
northward, giving south winds in the now lower-pres- 
sure zone to the east of the front. If, on the other hand, 
the front pushes slowly into the inversion, the new 
pressure gradients will cause north winds. The equations 
that tell how much south or north wind is to be expected 
and how long it will blow are being studied at the 
present time. 
The Characteristics of a Circular Vortex and the 
“Rings” of a Hurricane. The rings or spirals of violent 
convective activity in hurricanes have been observed 
by many authors [138, 25]. We have seen in earlier 
sections that the Mach lines and the envelopes of Mach 
lines are at least approximately the lines along which 
physical disturbances move in a flow. The Mach lines 
of a circular vortex in a fluid with a free surface are 
spirals almost tangent to the streamlines near the center 
of the vortex, making a larger angle with them as the 
distance from the center increases. If the presence of a 
stability retarding the vertical motion of air involved 
in a hurricane can be established, the theory of flow 
under an inversion can be applied and might lead to an 
explanation of these spirals. To the writer’s knowledge 
no one is working on this problem at the present time. 
Interactions and Frontogenesis. The discussion at 
the end of the previous section indicates that a vortex 
sheet (or wind shear line) is formed when two strong 
pressure Jumps are involved in an interaction. If these 
vortex sheets can bring about frontogenesis, and this 
occurrence is frequent enough to be important to syn- 
optic meteorologists, this should be a fruitful field of 
DYNAMICS OF THE ATMOSPHERE 
investigation. From synoptic experience we can say 
that if the vortex sheet persists it is very likely to bring 
about frontogenesis. A wind-shear line usually develops 
into a front because there usually are horizontal tem- 
perature gradients in an air mass. This problem is not 
being studied at the time of this writing. 
The “Expansion-Wave Storm.” Brunk [4] has made a 
complete and very interesting study of a wind storm in 
1944. The winds over the northern part of the United 
States were from the east under a quasi-stationary 
front. A small wave moved from west to east along this 
front (rather rapidly) and, as it moved, the winds to 
the north of it became very strong from the east. This 
storm is now being studied as an expansion wave moving 
from the west followed by a compression wave and a 
subsequent pressure Jump on the inversion north of the 
quasi-stationary front. The effect of the Coriolis force 
is being included in this investigation. The problems of 
flow under an inversion are very well suited for com- 
plementary work by theoreticians and synoptic mete- 
orologists. 
Blocking Waves in a Planetary Jet Stream. Rossby 
[17] has shown that an analogue to the stationary 
hydraulic jump can occur in a planetary jet stream. The 
jet stream he uses is defined as a symmetric stream of 
fast-flowing air moving through stationary surround- 
ings. He showed that if the variation of the Coriolis 
parameter is considered and the momentum transport 
for a given volume transport is expressed as a function 
of velocity, the momentum transport has a minimum 
value. This means that for any other value of the mo- 
mentum transport there are two possible flows with 
this volume transport. Thus, a necessary condition for 
the sudden change from one state of flow to another 
is satisfied. This is exactly what happens in a hydraulic 
jump. Rossby advances the theory that this analogue 
to the hydraulic jump appears on the weather map as a 
blocking wave of the type discussed by Berggren, Bolin, 
and Rossby [8]. Rossby’s paper naturally suggested 
that a system of equations of a nonstationary flow of 
this type might be developed and that possibly they 
would be similar to equations (1) and (2). If the velocity 
profile of the jet stream is such that the vorticity in the 
jetis fo (.e., w = wo + By?/2), the jet stream has width 
a, the acceleration in the y direction can be neglected, 
and the geostrophic approximation can be made out- 
side the jet stream, then the equations describing the 
flow of the stream are 
These are equations of the type (1) and (2). Simple 
waves are possible, and the compression waves lead to 
envelopes of the characteristics. Needless to say, this 
problem is being studied actively and in detail. The 
success of the method of characteristics in the study of 
this problem emphasizes the fact that the usefulness of 
this method to meteorologists is not confined to the 
