572 
energy.” When the divergence is negative the sense of 
this conversion process is reversed. It should be noted 
that this particular result is independent of the physical 
nature of the “‘working substance,” which might indeed 
be partly liquid (or even solid), with the gaseous and 
liquid components undergoing changes of phase. The 
result therefore automatically embraces the conse- 
quence of all condensation phenomena insofar as they 
contribute to the horizontal kinetic energy. 
GLOBAL BALANCE OF TOTAL ENERGY 
Basic Equations. Thus far we have found it con- 
venient to deal with the kinetic energy problem alone, 
since this quantity can be changed only by mechanical 
forces, and hence may be studied separately in terms 
of the systems of such forces considered as given by 
observational data. However the problems connected 
with the total global energy balance must in the end 
be of significance in the further understanding of at- 
mospheric and oceanic circulations. For this reason 
we shall now attempt to formulate certain relationships 
involved in this more general subject. 
Proceeding along more classical lines, let us consider 
the statement of the general physical energy equation 
written in the form 
dU da dU pdp 
SS Dr ne PEL rere pide. 
Here p dq/dt is the rate of external heat addition per 
unit volume, U is the total internal energy per unit 
mass, a = 1/p is the specific volume, y is the rate of 
generation of heat by friction per unit volume, while 
the other symbols have already been defined. With the 
aid of the continuity equation 
(15) 
where c is the total vector particle velocity, we may 
write that 
dq _ al 
apt = Dare ING: (16) 
We next proceed to evaluate the last term in (16) from 
the dynamical equation of motion written in vectorial 
form as follows: 
d 
po = —Vp — pV® — 292 X c — F, (17) 
where © is geopotential energy per unit mass, @ is the 
constant angular velocity of the earth’s rotation, and F 
is the vectorial retarding force per unit volume due to 
friction. 
The scalar product of (17) with the particle velocity 
2. It is worthy of note that the present treatment gives no 
information as to whether the bulk of the kinetic energy gen- 
erated in the atmosphere represents a conversion from geo- 
potential energy or whether it represents a conversion directly 
from internal heat energy. 
THE GENERAL CIRCULATION 
yields the corresponding equation of energy which may 
be written after slight rearrangement as 
dc 
Pat 2 
In (18), c is the magnitude of c, and d= c-F is the rate 
at which work is done by the fluid against frictional 
forces per unit volume. Assuming that the geopotential - 
® is constant with time at a fixed point with respect to 
the earth (this is true except for such things as the 
small tide-producing disturbances), we may write, with 
the aid of the continuity equation in the form 
= pV-c — V-pe — V-pbc + 6V-pc — d. (18) 
—-+ V-pe = 0, (19) 
that 
®V-pc = -: (p®). (20) 
Using (18) and (20), we can now rewrite (16) in the 
form 
dq ib, eee a 
pa ty doogl T+ 
(21) 
0 
1a ey (ob) + V-(p + p®)e. 
Since with the aid of (19) it follows that 
d Oa) k 
De Ga a or STA Ce Kes 
we finally have the equation 
LU yes Ole Be at 
P it ~ Pome 
(22) 
2 
+(e +05 +b +p) 
The various considerations which have entered into 
the formulation of equation (22) are true for any fluid 
medium without significant approximation. We may 
therefore apply the equation to the entire fluid enve- 
lope of the earth or portion of it, making no distinction 
between the atmosphere and the hydrosphere. We can 
thus integrate it over an equatorial belt between lati- 
tudes —¢ and +¢ and include all bodies of water such 
as the oceans, rivers, lakes, ete. Considering again the 
average conditions so that local time variations disap- 
pear we have, with the aid of the divergence theorem, 
that 
rs dq 
n= | (of +y -a)ar 
3) 
= if (ou + Ps + p& + p)on ds 
where dr is a volume element. Equation (23) has of 
course a very simple interpretation and could have 
indeed been written directly from general considera- 
tions. If we include under friction only the effects of 
