640 
toward the north. In that case equation (1) has to 
be modified to 
ou 
wv 
Ox 
div V = R 
so 2 ng (4) 
dy 
where R is the radius of the earth and ¢ is the latitude. 
Usually, separate maps of wu and v are constructed 
and analyzed, and the partial derivatives are approxi- 
mated by ratios of finite differences. The differences 
of velocity components are taken over distances of the 
order of 100 miles. This method frequently leads to 
small, irregular patterns [13], the reality of which is 
doubtful. 
A superior technique is based on equation (2). Here 
it is necessary to find averaged wind components along 
each side of a square. If the square is so small that 
only about one wind observation falls along each side 
of the square, the average wind along each side will be 
but poorly determined, particularly since the winds 
are strongly affected by local eddies. In practice, values 
of divergence can be trusted only if they have been de- 
termined from squares with sides 500 km long or longer. 
Even then, the measurements lead to doubtful results 
unless the wind coverage is complete in an area some- 
what larger than the square. 
Another technique of measuring horizontal di- 
vergence is based on the expression of mean divergence 
in “natural” coordinates instead of in Cartesian co- 
ordinates: 
div V = a ; 
where S; and S_ are the distances between two stream- 
lines, measured along orthogonals to the streamlines; 
V2 and V, are the average velocities of outflow and 
inflow across S, and S,; and A is the area enclosed by 
the two bounding streamlines, S; and S, (Fig. 2). 
Fig. 2.—Illustration for definition of mean divergence in 
natural coordinates. 
Mantis (see [16]) showed that both the “‘component 
technique” (Cartesian coordinates) and the streamline 
technique (natural coordimates) are about equally in- 
accurate; even for large areas and good wind coverage 
the sign of the divergence can be trusted only for rela- 
tively large observed magnitudes, for example, greater 
than. 5 X 10-* sec. Recently, an objective method 
proposed by Bellamy [2] has been used extensively. 
This procedure is based on a determination of diver- 
gence from observations at the corners of a triangle. 
(For another, much more tedious, objective method 
see [21].) 
MECHANICS OF PRESSURE SYSTEMS 
Determination of Vertical Velocities 
Direct Measurements of Vertical Velocities. Vertical 
velocities can be measured directly when they are 
larger than 10 em sect. Some of these methods of meas- 
urement are described in Byers’ article on thunder- 
storms.! In addition, instantaneous vertical velocities 
near the ground have been measured by many investi- 
gators of turbulence. 
Large-scale vertical velocities (horizontal area > 
10“ cm?) must be determined indirectly. The two prin- 
cipal methods which have been suggested for estimating 
these small vertical velocities involve computations 
(1) from horizontal velocities and the equation of con- 
tinuity (kinematic method), and (2) from the effect 
of vertical velocities on temperature, potential tempera- 
ture, or wet-bulb potential temperature (adiabatic 
method). 
The Kinematic Method of Determining Vertical Veloc- 
ities. On the scale under discussion here, the vertical 
velocity at level h can be found from the equation of 
continuity in the form 
h 
pdivVaze + wu, 2. 
Ph 
Ph vs 
Wi, = (5)? 
Here s stands for surface. The integral can he approxi- 
mated by a sum, and w, can be estimated from the 
horizontal wind field and the slope of the terrain. 
A more elegant form of this method is obtained if 
equation (5) is transformed to 
= h 
w, = —£ div i V dz. (6) 
Ph 8 
The vector J V dz is the “resultant vector” which is 
equal to the horizontal distance from the observer 
to the position of the pilot balloon at level h, multiplied 
by the speed of ascent of the balloon. This distance is 
available directly from the original pilot-balloon runs. 
The divergence of the resultant vector is computed by 
one of the techniques described above. The kinematic 
method yields instantaneous values of vertical veloc- 
ities, usually averaged over considerable areas. A dense 
network of actual wind observations is needed, so 
vertical velocities have been computed by this method 
only up to 10,000 ft and in regions of good weather. 
Radar and radio direction-finder pilot bailoons should 
increase the range and applicability of the method. 
The Adiabatic Method of Computing Vertical Veloc- 
ities. The adiabatic method, which is completely in- 
dependent of the kinematic method, is based on the 
assumption that rismg and sinking air will cool or warm 
at a known rate. Then a measured temperature change 
will permit computation of the vertical velocity. 
We start from the mathematical identity 
aT oT oT 
where 7’ is temperature and Vy is the vector differential 
2. Equation (3) is essentially the same as equation (5), 
but applies only for horizontal ground. 
