LARGE-SCALE VERTICAL VELOCITY AND DIVERGENCE 
By H. A. PANOFSKY 
New York University 
Influence of Scale on Divergence and Vertical Velocity 
The vertical velocity is the only component of the 
air velocity vector not generally recorded. Yet it pro- 
duces important effects of turbulence and is basic to 
the formation of precipitation. On a small horizontal 
scale it has been measured directly in thunderstorm 
and turbulence research, but the average vertical veloc- 
ity over large areas (10“ cm? or more) must be estimated 
indirectly. 
The horizontal velocity divergence, div V (called 
simply divergence in this article), is related to the verti- 
cal velocity by the equation of continuity. According 
to this equation, vertical divergence dw/dz is almost 
exactly equal in magnitude and opposite in sign to hori- 
zontal divergence, provided that the fractional change 
of density is small. The distribution of horizontal di- 
vergence therefore prescribes the distribution of dw/dz. 
If the vertical velocity w is known at any one level, 
the distribution of horizontal divergence determines 
the distribution of the vertical velocity itself. Gener- 
ally, the vertical velocity can be assumed to be known 
at the ground. Because of this close relationship between 
vertical velocity and horizontal divergence, the two 
quantities are treated together in this article. 
The magnitudes of divergence and vertical velocities 
are influenced to an extreme degree by the size of the 
areas over which they are averaged. This can be seen 
most easily for the divergence. Let « and y be two 
horizontal Cartesian coordinates and wu and v the veloc- 
ity components in the x and y directions, respectively. 
Then the divergence is given by 
j Ou ov 
SE reac (1) 
If this quantity is averaged over a square of length 
L, the result is 
(as — m1) + (Os — %e) 
LL ‘ 
where w; is the mean value of wu on side 3 of the square, 
and the other velocity terms have corresponding in- 
terpretations (see Fig. 1). 
The magnitudes of a andd are almost independent 
of scale, and are of the order of 10 m sec. Even dif- 
ferences of the horizontal velocities depend relatively 
little on scale; horizontal velocity differences of the 
order of 10 m sec appear over 100 m as well as over 
1000 km. . 
On all scales there is a tendency for a — a to be 
opposite in sign and almost equal in magnitude to 
Y% — Vo (cf. [5]). It turns out that the numerator of 
equation (2) is of the order of 1 m sec, regardless of 
div V = (2) 
scale. Thus the order of magnitude of the divergence 
is 1/Z sec if L is measured in meters. 
On the scale involved in the thunderstorm studies! 
Lis of the order of 3000 m, so that actually div V should 
be near 3 X 10-4 sec. The observed values are usually 
somewhat higher because thunderstorms occur at times 
of unusual vertical motion. On this scale, 3 X 10-4 
sec! might be regarded as a “normal” order of magni- 
tude of divergence. 
y 
x 
2 
Fic. 1.—Illustration for definition of mean divergence in 
square area. 
On the ‘‘weather-map scale” (10°m < L < 10° m) 
the divergence is between 10-> sec! and 10-* sec. 
The remainder of this article will be particularly con- 
cerned with divergence of this order of magnitude. 
On this weather-map scale we may express the equa- 
tion of continuity in the form (assuming the surface 
horizontal) : 
= jain W, (3) 
Ph 
where w is vertical velocity, h is an arbitrary level, and 
p and div V represent density and divergence, respec- 
tively, averaged over height. If h is 3000 m and if 
there is no noticeable difference between the magnitudes 
of diy V and diy V, the vertical velocity at weather-map 
scale should be between 0.3 cm sec! and 3 cm sec™’. 
Again, vertical velocities much larger than these values 
appear with smaller scales. Byers finds vertical veloc- 
ities averaging about 10 m sec in thunderstorms. 
Wi = 
Direct Measurement of Divergence 
Since the divergence is defined in terms of horizontal 
winds, and since winds can be measured with reasonable 
accuracy, it is possible in principle to determine the 
divergence directly from equation (2). Generally, the 
pseudo-Cartesian coordinates of meteorology are em- 
ployed in this procedure with « toward the east and y 
1. Consult “Thunderstorms” by H. R. Byers, pp. 681-693 
in this Compendium. 
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