ENERGY EQUATIONS 
By JAMES E. MILLER 
New York University 
The concept of energy has two applications in mete- 
orological problems: it facilitates the analysis of many 
atmospheric processes, and it makes possible a treat- 
ment of temperature changes associated with air mo- 
tion. While many analyses that are facilitated by the 
energy concept can be accomplished without it, the 
relation between characteristics of the air motion and 
temperature changes following the motion is essentially 
an energy transformation and should be so treated. 
The following discussion of atmospheric-energy equa- 
tions is based primarily upon the work of Reynolds [11], 
with extensions and interpretations suggested by the 
work of many others, especially Margules [6, 7, 8], 
Richardson [12], and Ertel [4]. The mathematical anal- 
ysis, which is presented in outline form here, can be 
found more fully developed, though with a few slight 
differences in approach, in a recent paper [9]. 
The Concept of Energy 
When discussing energy one must define the system 
with which the energy is identified. A system may be 
the gas in a cylinder; it may be a certain interconnected 
series of rods, wheels, and gears in a mechanical device; 
it may be an electronic circuit; and it may be a part of 
a fluid. In the most general sense, a system is a portion 
of space with prescribed boundaries. 
When a body moves through a distance d, opposite 
to the direction of a force F that is acting on the body, 
it is said to have done work equal in amount to the 
product F X d. An insulated system is one that can do 
work or have work done on it, but is restrained from 
any other interaction with its environment. Energy, 
then, is that property of an insulated system which 
decreases when the system does work and increases 
when work is done on the system, the amount of in- 
crease or decrease being equivalent to the work done. 
Different kinds of work are associated with changes 
of different forms of energy: When a gas system ex- 
pands, its thermal energy decreases; when the gas is 
lifted in a gravitational field, its potential energy in- 
creases. The change of any form of energy in an insu- 
lated system can be identified from the work that ac- 
companies the change. The change can be correlated 
with measurable properties of the system in a series of 
suitably controlled experiments and expressed as a 
mathematical function of those properties. It is assumed 
that the functions for various forms of energy, though 
established by a necessarily limited number of experi- 
ments, can be used to calculate energy changes in all 
circumstances—in noninsulated systems of any con- 
figuration and in complex processes where many dif- 
ferent energy forms are changing simultaneously. 
If the concept of energy went no further than the 
preceding discussion, it would have no value whatso- 
ever; “energy change” would simply be a synonym for 
“work.” There is one additional characteristic of energy 
that gives the concept its real importance. No excep- 
tion has ever been found to the rule that energy changes 
can be expressed in terms of changes of the state pa- 
rameters of a system, such as temperature, velocity, 
and position, regardless of how those changes take 
place. Thus, a change of thermal energy can be ex- 
pressed in terms of the temperature change, and has 
the same value regardless of the intermediate states of 
the system. A similar statement cannot, in general, be 
made about work. 
How, then, can the change of energy, as defined, be 
independent of intermediate states whereas work is 
not? It is true that, for an insulated system, both energy 
change and total work are completely determined by 
the initial and final states of the system. But for a 
system that can interact with its environment in other 
ways besides the performance of work, part of the 
energy change may be associated with a flow of energy 
into or out of the system. A gas that is being heated 
while it is expanding against external pressure will 
experience a change of thermal energy determined by 
its initial and final temperatures, while the work and 
the energy added through the boundary may have 
many different values, so long as their sum equals the 
energy change in the system. 
Now-if a system is restrained from the performance 
of work, but is not restrained from other external ac- 
tions, a certain form of energy may flow through the 
boundary into the system. The increase in the amount 
of that energy form present in the system can be de- 
tected from changes of the parameters that enter into 
the corresponding energy function. Such changes, asso- 
ciated with energy flux alone, can be correlated with 
measurable properties on the boundary surface in a 
series of controlled experiments and expressed as math- 
ematical functions of the boundary properties. It is 
assumed that these functions can be used to compute 
energy flux in all circumstances. 
The preceding definitions of work, energy, and energy 
flux, and the empirical evidence that they can always 
be applied consistently to natural processes, lead to a 
general law of energy, which is expressed here in terms 
of changes with time: 
Rate of in- Rate at which Rate at which 
crease of en- | _ | energy is add- | _ | work is done (1) 
ergy in the | | ed through the by the system ]* 
system boundary 
The First Law of Thermodynamics 
The special form of the energy law that is most com- 
monly used in meteorological analysis is the ‘‘first law 
of thermodynamics”: 
d bh Or ee 
gor th=a@ +Q ar GR rae (2) 
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