570 THE GENERAL CIRCULATION 
The production SS may be looked upon as the integral 
of the contributions from the various horizontal layers 
of fluid present and written as 
Gs [Uf (ot + 2) a ay | ie (7) 
In view of the fact that the surface integral 
[f (242) ao a 
must vanish if the horizontal velocity is zero across 
the fixed walls, it follows that a given horizontal stratum 
of fluid cannot give a positive contribution to S unless 
larger values of the pressure p are associated with areas 
of horizontal divergence than are associated with areas 
of convergence. Thus areas of horizontal divergence 
represent primary kinetic energy sources, while areas 
of convergence represent sinks for kinetic energy. 
Furthermore, in a mechanically closed system of the 
kind here considered it is impossible to have source 
regions for kinetic energy without at the same time 
having sinks of a hydrodynamic nature, entirely inde- 
pendent of frictional effects. 
Equations for the Atmosphere. Before embarking 
upon a discussion of the meteorological implications of 
the material presented above, it is desirable to develop 
the concepts involved in more general terms, so as to 
render it possible to perform integrations over the entire 
mass of the atmosphere. 
To a sufficiently close degree of approximation the 
shape of the geopotential surfaces may be considered 
as spherical so that we may make use of spherical polar 
coordinates in which r is the radius, ¢ is latitude, and 
d is longitude. By analogy with the Cartesian case we 
may then write the equations of motion for the hori- 
zontal directions (see Brunt [2]) in the form 
a OD oy os 2). osey(on arg cy = 9 in a) 
dt r r 
= tages +Fe,| 
p 0x 
sti (8) 
Sean ee” ae OOu she 
dt r r 
2h ten 
= Bip ae 
where wu, v, and w are the linear velocity components 
in the eastward, northward, and upward directions, 
respectively; « and y are measures of linear distance 
eastward and northward, respectively; and is the 
angular velocity of the earth. The analogous energy 
equation in this case may be written as 
Vi ne 
pa se oy ae 2pQuw cos d 
dt 2 r 
_ _ (dpu , Op _ pu 
(2 =P ay - tan ‘) (9) 
Ou ov v 
+0(% a ge 6) 4. 
If we make use of the observational fact that the last 
two terms on the left-hand side of (9) are of a very 
small order of magnitude, these terms will be dropped.! 
The manipulation of the remaining term on the left 
side may now be carried out with the aid of the con- 
tinuity equation much as before, since this operation is 
independent of the specific coordinate system used, so 
that we may write 
= + div; HV = —div2 pV, + pdiv. V;, — d. (10) 
A volume integral of (10) may now be taken and written 
in the form 
o | Bar = f Bv.as — ff pode — way) ar 
(11) 
+ | pdivevidr — [ dar, 
where dz is a volume and ds a surface element. Hqua- 
tion (11) is physically identical with (5) and has, there- 
fore, the same interpretation. In symbolic form we 
may write 
OK 
Ay Sie 
ry (12) 
which states that the rate of merease of horizontal 
kinetic energy for a fixed volume is equal to the net 
rate of advection of such kinetic energy into the region, 
plus the rate at which work is being done by the sur- 
roundings on the fluid in the region through horizontal 
motions, plus the production of kinetic energy in the 
volume, minus the frictional dissipation. For a system 
which is mechanically closed, A and W again vanish. 
This is therefore true when the entire atmosphere is 
considered. In this case the surface integral of the hori- 
zontal divergence over each closed geopotential surface 
must vanish as in the case of the chamber previously 
considered. 
Although it is possible to form other energy integrals 
for fluid motion, as pointed out in standard texts on 
hydrodynamics (e.g., [1]), the particular merit of the 
procedure followed above is that the expression for 
production of kinetic energy assumes a form which is 
of interest in meteorological problems. The implica- 
tions of equation (12) may be stated in brief as follows: 
1. The intensity of the primary source of horizontal 
kinetic energy at a given point in the atmosphere is 
given by the product of the pressure and the divergence 
of the horizontal velocity. 
2. Positive primary sources must always occur in 
combination with negative sources or sinks independ- 
1. In reality these terms represent a conversion of kinetic 
energy of horizontal motions into kinetic energy of vertical 
motions, and as such do not involve a production of kinetic 
energy. Indeed, by methods’ similar to those used in this paper 
one can investigate separately the kinetic energy of motions 
in each of the three directions, namely, zonal, meridional, and 
vertical. In that case other conversion terms of a similar nature 
arise. 
ee 
