ENERGY EQUATIONS 
right is recognized as the rate of work, per unit mass, 
done on a small volume of the fluid as a result of its 
motion in the direction of the net molecular stress act- 
ing on the volume. It can be transformed to 
Ov; 
0 
— ADKi ae + @ an (vipri)s 
and the last term, —v3g, can be written —d(ga3)/dt, 
neglecting the variations of gravity. The equation then 
becomes 
Ov; to} 
ODii a + a aes (vpii). ~ (7) 
Note that, because of the summation convention, each 
term on the right is summed over the three values of 
2 and k. 
This equation, though established by a mathematical 
transformation of the equations of motion, is inter- 
preted as a special form of the general energy law, equa- 
tion (1). It will be applied to the same system to which 
the first law of thermodynamics is applied: a very small 
volume containing a mixture of air and water vapor. 
The term v,?/2 is known as the kinetic energy and gz 
as the potential energy, both per unit mass. Such inter- 
pretation is consistent with the energy concept, for 
these functions are determined by the system’s state 
and not by its history, and their total increase is equiva- 
lent to work done on the system. 
For simplicity the following symbols will be used in 
writing the energy equations: 
I =c,T, the thermal energy 
K =0v;/2, the kinetic energy ;per unit mass 
P = gx;, the potential energy (8) 
Ov; 
eee OX: 
Equations (2) and (7) then become 
SC+D=W+@a-(-My, ©) 
d 0 
di (K+ P) Sait ae E ph CX, aan (0; Dri) ie (10) 
The subseripts A and W indicate that the preceding 
terms in brackets represent energy added to the system 
or work done by the system, respectively. Since P is 
understood to mean g times a distance along a line 
directed opposite to gravity, there is no required orien- 
tation of the axes in equations (9) and (10). 
The rate of work represented by M contributes to 
an increase of the sum of thermal and latent-heat ener- 
gies, and simultaneously takes the same amount from 
the sum of kinetic and potential energies. It represents 
an energy transformation through the mechanism of 
molecular stress, and hence it will be called the molecu- 
lar transformation function. 
Energy equations that may be useful in meteoro- 
logical problems will be derived from the basic equa- 
485 
tions (9) and (10) by purely mathematical methods, 
with the aid of certain simplifying assumptions about 
the physical properties of the moist air. Before this 
analysis is begun, a few remarks about the basic equa- 
tions are in order. 
Remarks on the Basic Energy Laws 
Equations (6), upon which (10) is based, express 
Newton’s second law of motion in a coordinate system 
that rotates with the earth. The rotational terms, such 
as (2w cos ¢)v3, vanish during the derivation of the 
energy equation, so that the velocity v; and the rate 
of change d/dt in equation (10) may be measured in 
any Cartesian system, whether fixed, rotating, or mov- 
ing at constant speed. It is customary to refer both (9) 
and (10) to a Cartesian coordinate system which is in- 
stantaneously fixed in the rotating earth, but which 
also follows the small air system so that two axes always 
lie in a plane tangent to the earth’s surface. The mag- 
nitudes of terms in (10) may be different in other 
coordinate systems. For example, if a 1-g air particle, 
moving from east to west at 20 m sec on the earth’s 
equator, is brought to rest, it loses 2 X 10° ergs of 
kinetic energy in the commonly used coordinate system; 
but it gains 90 X 10° ergs in a nonrotating coordinate 
system. There is a compensating difference in the work 
in the two coordinate systems. In the former, the air 
particle performs work in the amount 2 X 10° ergs; in 
the latter, work is done on it in the amount 90 X 10° 
ergs. 
The terms a 0p;;/0x; in (6) are supposed to repre- 
sent all body forces, per unit mass, on a small fluid 
element, with the exception of gravity. Whether they 
do or not depends upon the accuracy of the expressions 
for the stress tensor in (3). These expressions are pre- 
sumed to be correct, but they have been verified experi- 
mentally for only a few special cases. 
There is a further uncertainty about the equations 
that is of particular concern to the meteorologist. It is 
well known that the observed velocity of air motion 
varies with the type of instrument used. Fluctuations 
of the velocity in space and time cause a large, sluggish 
anemometer to react differently from a small, sensitive 
one. The same sort of scale effect is certainly present 
in measurements of temperature, pressure, and other 
properties. This situation poses a problem: How shall 
the properties be measured in order that the equations 
will be satisfied? 
It can be shown logically that the equations (6), 
with the stress represented by (3), if correct at all, are 
strictly correct only in a certain scale domain with time 
and space periods greater than the molecular periods 
but smaller than the periods of the smallest turbulent 
fluctuations [11]. This conclusion presupposes that there 
is a lower limit to the periods of turbulent fluctuations 
of the various properties. If the conclusion is correct 
for the equations of motion, it is also correct for equa- 
tion (10) and presumably for equation (9). 
The latter equation, the first law of thermodynamics, 
is generally applied only to a stationary, homogeneous 
fluid whose changes of state take place very slowly. The 
