THE METHOD OF CHARACTERISTICS 
moving at w% + a; because a > c and that jump II, 
in its turn moving at w + a2, is being overtaken by 
the right-hand side of expansion wave III moving at 
U2 + C2 because a2 < c.. This phenomenon was first 
demonstrated synoptically by Tepper [19], who showed 
that on May 16, 1948 the pressure maximum which 
coincides with point A in Fig. 10 had an average speed 
of 54 mph over the network stations at Wilmington, 
Ohio, while the jump had a speed of 45 mph. This phe- 
nomenon of overtaking leads to a modification of the 
expansion waves and jumps involved. The exact nature 
of this modification in all cases and the results of all 
‘“nteractions” as they are called are not known. In 
some simple cases an approximation to the actual 
conditions can be made that has some validity. For- 
tunately, one of these simple cases is important in the 
study of squall lines. One manner of dissipation of a 
pressure jump is for it to be modified by a following 
expansion wave (Fig. 10). 
Map Analysis with a Pressure Jump. The small 
amount of space and time occupied by a pressure jump 
has been discussed previously. The small dimensions in 
space of the pressure jump are emphasized in Fig. 9, 
where the line marking a squall line from the Daily 
Weather Map of the U.S. Weather Bureau is magnified 
in the same proportion as the scale of miles. These lines 
068 
J 105 7088 
085 a f 
Fig. 11.—Proposed method of drawing isobars in the vi- 
cinity of a pressure jump. The map is for 2230 EST May 
16, 1948 and the squall line is through northwestern Pennsyl- 
vania and southeastern Ohio. 
are four miles wide and it can be seen that this one 
covers most of the zone of rising pressure. With this 
chart in mind it has been proposed [9] that the same 
approximation used in the mathematical analysis be 
429 
made, and that the pressure gump be analyzed as a dis- 
continuity in the field of pressure. This method of analy- 
sis would have the following advantages: (1) more of 
the winds would blow nearly parallel to the isobars, 
(2) the analysis would fit more of the pressures than is 
usually possible with conventional methods (as can be 
seen from Fig. 11), and (3) the analysis would corres- 
pond more closely to the proposed mathematical theory 
of pressure jumps. This type of analysis has the dis- 
advantage that the isobaric field is not correct in the 
immediate vicinity of the pressure Jump, but Fig. 9 
shows that the area that is not accurately represented is 
almost covered by the line marking the pressure jump. 
Pressure jumps can be located rather accurately on a 
well-plotted synoptic map by this method. An example 
of such an analysis is given in Fig. 11. Pressure jumps 
could be located very accurately indeed if the sugges- 
tion were followed of sending a special report on the 
hourly network when a pressure jump passed the sta- 
tion. 
Transmission of Energy by Means of Finite Disturb- 
ances in Inversion Height. Abdullah [1], working inde- 
pendently, established the existence of a pressure Jump 
on the inversion behind a cold front. He advanced the 
hypothesis that the pressure jump in cool air was the 
medium that carried the energy of distant cold air to 
the vicinity of a cyclone and that this energy was then 
available to aid the formation or deepening of a cyclone 
on the front separating the cool air from the warm air. 
Abdullah also suggested that pressure jumps of finite 
lateral extent could be one of the “‘disturbances”’ needed 
to start the formation of a wave on a front in the frontal 
theory of cyclones. By means of an energy computation 
that can be found in his paper, he showed that for the 
usual dimensions of waves on a front a pressure jump 
of ordinary dimensions will supply more than ten times 
the amount of energy that is found in the frontal wave. 
Abdullah used the Riemann method of integration. The 
method is particularly well adapted for the study of a 
situation in which the boundary conditions appear in 
closed form. Assuming that the coldest air begins moving 
at t = 0, x = 0 with a constant acceleration a, he showed 
that a compression wave moving into air with a constant 
height ho of the inversion will break into a pressure 
jump (i.e., the envelope of characteristics will form) 
at a point 
_2 p 
To =F 4/ sto(1 -*), 
_2 ( =) 
co = — gh 1—=). 
3a p 
Using these formulas, he constructed a table showing 
computed and observed distances («) from the front 
at which breaking into a pressure jump occurred. The 
essential information from his table is presented in 
Table I in which T is the mean temperature of the cool 
air, and AT is the difference in temperature between 
the cool and the warm air. The best synoptic example 
of a pressure jump on a large scale to appear in the 
(30) 
(31) 
