424 
envelope which encloses all points in the «, t plane cov- 
ered by two or more characteristics. The meaning of 
such an envelope depends almost entirely on the physi- 
t 
CHARACTERISTICS 
ENVELOPE 
A ‘ x 
Fic. 2.—The characteristics of a compression wave. 
cal nature of the problem and of the quantity u. The 
envelope has meaning if it is possible to have two values 
of uw at a point. Usually the assumptions involved in 
deriving equations (1) and (2) are violated before the 
envelope is formed by the characteristics. If the region 
in which equations (1) and (2) cannot be used is very 
small and motion inside it is not important, it is assumed 
to be so thin that it can be drawn as a line in the z, ¢ 
plane and it is called a jump. It is so called because 
there are jumps in values of w and w on going across 
such a line. Such an assumption can be made in gas 
dynamics where the jump is called a shock, in hydraulics 
where it is called a hydraulic jump, in flow under an 
inversion where it is called a pressure gump, and to a 
certain extent in flow in a planetary jet stream where 
the jump is called a block. 
FLOW UNDER AN INVERSION 
Statement of the Problem. The changes in height 
of an inversion and the flow beneath an inversion are of 
great interest to synoptic meteorologists. The frontal 
contour charts of the Canadian Weather Service [6] 
bring home the fact that motions of the frontal surface 
are important even far behind its intersection with the 
ground. The behavior of low-level winds during and 
just after the passage of a surface cold front in the 
Western Plains is usually a topic of discussion among 
forecasters. Finally, the forecasters in tropical areas are 
interested in the variation of the winds below the trade 
wind inversion. It is the writer’s opinion that a mathe- 
matical theory of flow under an inversion will help to 
describe these phenomena and perhaps will eventually 
lead to quantitative forecasting techniques. Several 
phenomena have been described by means of the sim- 
plest such theory (see Freeman [7], Abdullah [1], and 
Tepper [19]). Flow under a widespread inversion in the 
atmosphere can be approximated, at least somewhat 
better than qualitatively, by the flow of a shallow layer 
of liquid under a deep layer of almost the same density. 
The details of the validity of this approximation can be 
found elsewhere [7, 8, 18]. The pressure in a shallow 
layer of fluid flowing under a deep layer of fluid at rest 
is given by the formula 
P = (h’ — h)p’g + (h — 2)pg, 
DYNAMICS OF THE ATMOSPHERE 
where his the height of the fluid with density p, and h’ is 
the much greater height of the surface of the fluid with 
density p’ (p’ < p);z is the height of the point at which 
the pressure is measured, and g is the acceleration of 
gravity. If this is substituted in the two-dimensional 
equations of motion (neglecting the earth’s rotation), 
we obtain the following results: 
Ce NOP ff, iN ab 
Tae eae a(1 ae (11) 
Ce We OV en Nal 
di pay ( le ~ 
If the flow under the interface at h does not depend on 2, 
the equation of continuity is 
(18) 
Equations (11), (12), and (13) are the equations of 
motion and continuity for two-dimensional unsteady 
flow of a shallow layer of liquid under a deep liquid of 
smaller density. These equations are difficult to inte- 
grate as they stand, and very likely high-speed com- 
puting machines would be required to solve even the 
simplest problems involving all three variables x, y, and 
t. If the dimensions are restricted, however, results 
can be found numerically or graphically with com- 
parative ease. 
Time-Dependent Flow under an Inversion. If there 
is no variation in the y direction during the time a flow 
is studied, it can be investigated as a one-dimensional 
unsteady flow. If v is zero initially and remains zero, 
equation (12) need not be considered and equations 
(11) and (13) become 
ou du DP \On = 
Ht ult + (1 Se 0 (14) 
and 
oh oh Ou 
respectively. This is a system of quasi-linear partial 
differential equations of the type discussed in the pre- 
vious section. If, in equations (1) and (2), the conditions 
A, = Il, B, = 4, Ch = ©, 
A, = 0, B, = h, C, = 1, 
Dy = ( - el Dy = u, (16) 
E, = 0, Uu =U, 
Es = 0, = h, 
are substituted, equations (14) and (15) result. If the 
values (16) are substituted in equation (7), the char- 
acteristics of (14) and (15) become 
dx p’ 
U + ¢C. 
(17) 
ll 
