LARGE-SCALE VERTICAL VELOCITY AND DIVERGENCE 
that wz, the vertical velocity at the top of the friction 
layer, is given by 
Wz = : curl, + 
H p f v0) 
where t) is the surface stress acting on the boundary 
between air and ground in the direction of the wind, 
p the density, subscript v denotes the vertical com- 
ponent, and f is the Coriolis parameter. It is difficult 
to evaluate the surface stress quantitatively over land; 
but since the surface stress is in the direction of the 
wind and the wind blows counterclockwise about low- 
pressure centers, this formula also indicates upward 
motion above pressure minima. Since nonfrictional the- 
ories predict upward motion in the southerly flow above 
low-pressure centers, friction has the effect of mecreas- 
ing the absolute magnitudes predicted by these the- 
ories. Possibly a combination of the Bjerknes-Holmboe 
theory and the theory of friction would account for 
the observed distribution of divergence and vertical 
velocity in the troposphere. 
The relation between meridional and vertical motion 
has been observed so far only in the United States 
and England, where the isotherms run essentially from 
east to west. It may be asked whether a similar relation- 
ship would hold when the isotherms run in some other 
direction. 
Charney’s theory is based on the assumption that the 
isotherms run from east to west. According to the 
Bjerknes-Holmboe theory the distribution of diver- 
gence depends on sinusoidal air motion superimposed 
on an east-west channel; upper-level streamlines be- 
have in this way only when the isotherms run from 
east to west. Consequently both these theories show a 
relation between vertical and meridional velocities only 
if the isotherms are parallel to the latitude circles. 
In general, both theories would predict a relation be- 
tween vertical motion and the horizontal wind com- 
ponent at right angles to the isotherms, or between 
vertical motion and horizontal temperature advection. 
Only when the isotherms run from east to west is a 
relation expected between vertical and meridional mo- 
tion. 
Effects of Divergence 
Divergence and Pressure Change. One of the reasons 
why meteorologists have been interested in divergence 
is that it is related to pressure change through the 
Bjerknes pressure-tendency equation: 
a ne i 
2) = (gow), — af div Vp de — 9 [ V-Vipdz, (8) 
h h a 
or 
at 
(2 be 
at). 
where s stands for the surface which is assumed hori- 
zontal. Fleagle [8] showed that the vertical velocity 
term in equation (8) almost exactly compensates the 
- [ div Vp dz —g | V-Vupds, (9) 
643 
divergence term, and in (9) the low-level divergence 
almost exactly compensates high-level divergence. 
Therefore neither equation can be applied to determine 
pressure changes from measured divergence. Moreover, 
most theories of divergence are not sufficiently accurate 
to permit estimates of pressure changes from these 
equations. 
Divergence and Change of Vorticity. Another reason 
for interest in divergence is the effect it has on the 
individual change of absolute vorticity. The vorticity 
equation may be written (Gf we neglect solenoidal fields, 
friction, and terms depending on the horizontal varia- 
tion of vertical velocity) 
il ake v of 
c+ f dt RE + f) do 
where ¢ is the vertical component of vorticity and v 
is the meridional velocity component. 
Rossby’s trajectory method was derived from the 
vorticity equation on the basis of negligible divergence. 
Some investigators attribute the systematic errors of 
the method in certain regions to the omission of diver- 
gence [18]. Studies at New York University seem to 
indicate that the divergence term is at least of the 
same order of magnitude as the term containing the 
variation of the Coriolis parameter with latitude, even 
for very large scales (areas of the order of 101° cm’). 
The omission of the divergence term is justified only 
near the level of nondivergence or when equation (10) 
is integrated vertically through the whole atmosphere. 
Divergence and Change of Stability. Local changes of 
stability can be brought about by three factors [6]: 
(1) vertical divergence, (2) different horizontal tem- 
perature advection at different levels, and (8) vertical 
advection of lapse rate. Since vertical divergence almost 
equals the negative horizontal divergence, the latter 
may be considered as a factor producing stability 
changes. Fleagle [6] found that the effect of horizontal 
divergence on stability changes is of the same order of 
magnitude as that of differential advection; he also 
noted that accurate forecasts of local stability changes 
based on measured divergence and differential advec- 
tion are inaccurate, probably due to the large errors 
in measurements of divergence. 
Effects of Vertical Velocity 
Vertical Advection of Velocity. Many of the mete- 
orological equations contain vertical advection terms 
which have frequently been neglected. For example, 
the acceleration of the horizontal wind vector may 
be written 
— div V, (10) 
dV ov 
dt dz 
Charney [5] indicated that the term containing w should 
be one order of magnitude smaller than the other terms 
in the expression. This conclusion, based on a dimen- 
sional argument, is at variance with results obtained 
at New York University. Vertical velocities average 
about 1 em sec~; the vertical shear of horizontal wind 
