THE SOLUTION OF NONLINEAR METEOROLOGICAL PROBLEMS 
BY THE METHOD OF CHARACTERISTICS 
By JOHN C. FREEMAN 
The Institute for Advanced Study 
INTRODUCTION 
The Method of Characteristics. The method of char- 
acteristics is only one means of solving hyperbolic 
differential equations or systems of differential equa- 
tions with real characteristics. Problems of this kind 
are also solved by related methods, such as simple 
wave theory, the Riemann method of integration, and 
the marching method. In addition, in a nonlinear sys- 
tem, shocks or jumps occur which cannot be studied 
by means of systems of equations with real characteris- 
tics, but must be studied by other methods. Loosely 
speaking, all methods mentioned above can be grouped 
under the single heading—the method of characteristics. 
If this definition is used, the method of characteristics 
is not new to meteorology. Richardson used the march- 
ing method correctly in the initial example in his 
monumental work on numerical weather prediction 
[15, pp. 6-10]. Rossby [16] used the Riemann method 
of integration to discuss the effect of a line source of 
planetary waves in a barotropic atmosphere. In all of 
its phases, however, the method of characteristics has 
been applied most often and most thoroughly to prob- 
lems in gas dynamics and the flow of a shallow layer of 
fluid with a free surface. Riemann devised his method 
of integration to solve the problem of one-dimensional 
unsteady flow of a gas in a pipe, and most other methods 
have been developed with such problems in mind. Cour- 
ant and Friedrichs [5] give a bibliography of this work. 
Early Work on Hydraulic Jumps in the Atmosphere. 
Mathematically speaking, the theory of flow of a shal- 
low fluid with a free surface is, to a certain degree of 
approximation, the same as the theory of flow of air 
under a shallow (1000-10,000 ft) inversion in the atmos- 
phere [7]. A very striking phenomenon in the flow of 
shallow water with a free surface is the bore or jump. 
A model of the jump is the breaker on a gently sloping 
beach. A breaker moving over a surface, whose hori- 
zontal dimensions are about ten times the height of the 
breaker, may be viewed as a line along which the height 
and speed of water change abruptly. In fact, in some 
cases there is a very shallow outgoing flow near the 
shore, and the jump or breaker is a transition to a very 
deep incoming flow. When such a jump is stationary in 
a channel, it is called a hydraulic jump. If the force of 
gravity is modified by a buoyancy factor (which is unity 
in the case of flow in water, and depends on the ratio 
of the densities above and below an inversion in the 
atmosphere), the laws governing flow under an inver- 
sion are the same as those governing flow of shallow 
water. Thus, jumps can occur in the atmosphere. 
A jump in the atmosphere was first recognized as such 
(to the writer’s knowledge) and discussed by M. Mc- 
421 
Gurrin [14] of the San Bruno Weather Bureau Forecast 
Center. He recognized that the jump might be the cause 
of several weather phenomena and gave a rather com- 
plete discussion of the equations for a steady-state jump 
in an inversion. His work has not been published. Mr. 
McGurrin has addressed local meetings of the American 
Meteorological Society in California, and probably be- 
cause of his work the concept of a Jump in an inversion 
is not new to many meteorologists. It should be empha- 
sized here that McGurrin’s paper [14] shows that he 
was completely aware that the steady-state jump could 
occur in the atmosphere on many scales and in many 
synoptic situations. He describes a stationary Jump in 
the height of a fog bank. The winds blow through this 
jump as they blow into the San Bruno valley in Cali- 
fornia. 
The method of characteristics is the application of 
methods related to those employed by Richardson [15] 
and Rossby [16] to problems related to those considered 
by McGurrin. Rossby [16] predicted that a study of 
the internal waves in the atmosphere would show that 
they develop sharp forward boundaries because of their 
dispersive character and the existence of a maximum 
value of energy propagation. The waves on an inversion 
are a limiting case of internal waves and have the 
properties which he predicted. 
MATHEMATICAL FOUNDATIONS 
Quasi-linear Differential Equations. We can use the 
method of characteristics, under certain conditions, on 
equations of the following type: 
Ou Ou ow ow 
D 
br ieaiees gamle ran Nes or 
ou Ou ow Ow 
Ag Baa tO at + Doo Ep. 
In these equations (for the moment) the independent 
variables x and ¢ are Cartesian coordinates in the 2, ¢ 
plane. The dependent variables w and w are unknown 
functions of x and ¢. The coefficients A; , 5, ete., are 
known functions of 2, t, u, and w. Any of them can be 
constant or zero. These equations are nonlinear if any of 
the coefficients, A,, By, etc., are functions of wu or w. 
They are called quasi-linear because they are linear in 
the derivatives of w and wand the method of solution 
does not depend strongly on their nonlinearity. This 
type of equation has been studied by many authors, 
and bibliographies concerning such studies appear in 
Courant and Friedrichs [5] and Isenberg [12]. The dis- 
cussion here is very much like that of Isenberg. 
Definition and Determination of Characteristics. The 
