ENERGY EQUATIONS 
Addition of (13) and (14) gives a total-energy equa- 
tion: 4 
D 
Dor + Ut + Kt + PP) 
i | (| bol dain ae eee) ee ade | (16) 
ae | _ [spade |. 
The important effect of adding the equations is the 
elimination of M*, the only term on the right-hand 
side of the equations that depends upon processes 
within the system. By means of equation (16), it is 
possible to compute the change of total energy from 
measurements on the boundary, without any knowledge 
of processes within the system. If there 1s no work or 
energy flux at the boundary, the right side of (16) is 
zero, and the equation then expresses a law of conser- 
vation of energy. 
The energy equations are sometimes derived in an 
order different from that followed here. The principle 
of conservation of energy is accepted as the basic law, 
and a total-energy equation in a form similar to (16) 
is written immediately. The law of kinetic energy is 
derived from the equations of motion, as is done here, 
and then the first law of thermodynamics follows upon 
subtraction of the kinetic-energy law from the total- 
energy law [15]. In contrast, the procedure used here 
is, first, the formulation of the law of kinetic energy 
and the first law of thermodynamics; and second, a 
statement of the total-energy law derived by combin- 
ing the two. This procedure is considered more logical 
because one cannot write the necessary energy and 
work functions in the total-energy law without recourse 
to the experimental evidence embodied in the two 
special laws. 
Effects of Averaging Processes 
The scale domain of nonturbulent variations, the 
domain for which the energy equations are presumably 
valid, may be defined as follows: If instrumental meas- 
urements of a property are not appreciably affected by 
decrease in size and increase in response of the instru- 
ment, the property is beg measured in a scale domain 
below the scale of the smallest turbulent fluctuations of 
that property. 
Meteorological measurements of the various physical 
properties are made with instruments whose size and 
period of response are certainly greater than the size 
and period of the smallest turbulent fluctuations. Sup- 
pose several anemometers of varying size and response 
were mounted near each other in the atmosphere. The 
largest, most insensitive anemometer would indicate a 
fluctuation of velocity with time; the next more refined 
instrument would indicate more rapid fluctuations su- 
perimposed on those detected by the first imstrument; 
and so on, down the scale. According to the postulate 
of a scale domain smaller than the smallest fluctua- 
tions, all anemometers smaller and more sensitive than 
a certain anemometer would give indications that were 
487 
essentially identical in period and amplitude. The veloc- 
ities indicated by these anemometers are the velocities 
that are supposed to satisfy the equations of motion 
and energy. Experimental studies suggest that the ane- 
mometers should be at least smaller than one centi- 
meter in diameter and should react to velocity changes 
within at least one second [8]. 
Because the terms in the basic energy equations can- 
not be correctly evaluated by ordinary meteorological 
observations, it is no great exaggeration to say that the 
equations are not very useful for meteorological pur- 
poses. The equations can be rendered more useful by 
an analysis that will now be demonstrated; but, as will 
be seen, the analysis introduces new terms whose eval- 
uation has long been a subject for speculation, postula- 
tion, and experimentation. 
A property s will be separated into a mean value § 
and a deviation s’ from the mean: 
G= S45 4/5 (17) 
The mean value is defined by a space-time integral: 
{| il s dt dV 
ae va t 
: me ee 
where V is the volume of the space occupied by the 
instrument, and ¢ is the period for which the mstru- 
ment, because of its inertia, automatically averages the 
(18) 
reading. If the mean value is defined arbitrarily for a 
selected volume and a selected period, the averaging 
can be carried out by numerical methods. Thus, any 
set of observations of s, at different points and different 
times, may be converted into a mean value over a 
selected volume and period by calculation of the inte- 
gral‘in (18). Whether the averaging is done automat- 
ically by the instrument, or by numerical methods, or 
by both, § refers to a central point and time in V and 
t, and it is a continuous function of space and time. 
The values of § at nearby points and times are deter- 
mined in overlapping volumes and periods. 
The various properties s are replaced in the equa- 
tions by §-+ s’ and the individual terms of the equa- 
tions are averaged by an integral of the form (18). It 
is assumed that s’, the average value of s’ for the same 
space-time as §, is negligible; and that, therefore, §, 
which represents the average of the continuously vary- 
ing § in the same space-time, is equivalent to the value 
of § at the center of the space-time.” Further, for the 
sake of simplification, it is assumed that p = p, or that 
the density fluctuations are negligible; but this does not 
necessarily mean that the fluid is incompressible. The 
consequences of the assumptions are, for example, 
Pri = 0, v; = 0;, 
a—a)s 
pU; = pd;, as (pd;v,) = 0, ete. 
Ox; 
2. Note that s’ is not the average value of deviations from 
a fixed mean, because § also varies from point to point; thus, 
57 = 0 does not follow from the definitions but must be stated 
as an assumption, 
