490 
out when the stability is great. Hence, with smaller 
stability v3 is greater on the average, EH is greater, and 
the eddy energy tends to increase with decreasing sta- 
bility. 
The array of general energy equations may be com- 
bined in four ways: any two or all three may be added 
together. When all three are combined, a total-energy 
equation is obtained: 
D+ D+ Rit P+ RD 
=| [+E +k +P +R) (31) 
= pin +L + KDI ae | 
A 
= |-/ Os(Dni + Tri) + Vi Pnil do| 
Tf no energy flux or work occurs at the boundary sur- 
face, the total energy within the system remains un- 
changed. This is the law of conservation of energy. 
Margules’ models of energy transformations in the 
atmosphere were based upon an equation which can be 
obtained from (81) by neglecting turbulent fluctuations 
and boundary activity: 
at + It + K* + PY) =0. (32) 
Margules computed the changes of latent-heat, poten- 
tial, and thermal energies for closed, stationary systems 
which progressed, in certain special ways, from a state 
of instability to a state of stability. He was then able 
to compute the associated change of kinetic energy by 
means of (32). His masterful analysis of the models has 
deeply influenced meteorological thought on atmos- 
pheric-energy transformations. Margules, however, was 
apparently well aware that his models were far from 
realistic and that, at best, they provided nothing more 
than general suggestions of the source of atmospheric 
motions. Furthermore, Spar [13] has investigated the 
energy changes in two deepening cyclones and has 
stated that the results “lend no support to Margules’ 
energy theory of cyclones.” 
To reduce the total-energy equation to Bjerknes’ 
generalization of Bernoulli’s equation [1], one must 
omit a number of terms and undo most of the analysis 
that produced the total-energy equation. Turbulent 
fluctuations, viscous stresses, and latent-heat energy 
are omitted; and the equation is applied to a system 
consisting of a small volume of air moving at the air 
velocity, instead of a large volume moving arbitrarily. 
With these conditions, equation (30) reduces to 
() _ op 
a, Wl + K + P+ pa)] at 
; (33) 
Si ae [ou.(I + K + P + pay]. 
After application of the equation of continuity this be- 
DYNAMICS OF THE ATMOSPHERE 
comes, for a steady pressure distribution, 
2 
OT + 5 + gx3 + pa = const, (34) 
along a trajectory. 
The eddy flux of thermal and latent-heat energies is 
the first of the following three eddy terms from the 
right. side of (81): 
—pin(I + L) — pu,Ke + vipai. 
It differs from the expression for the vertical eddy flux 
of heat derived by Montgomery [10], partly because 
he defined the mean velocity by Hesselberg’s formula, 
and partly because he combined the pressure part of 
the work term v;p,; with the flux term. The first dif- 
ference is unimportant if p = p. As for the second dif- 
ference, the term vipn; becomes either — vnp’ or 
— vip (since vnp = 0) if the viscous stresses are ignored; 
and this, combined with the flux, gives Montgomery’s 
expression for the x3-direction: 
— pv3(I + L + pa) = — prshe, 
where kh: =J+L-+ pa. For dry air, hk =h= 
cy I’ + pa, the enthalpy. 
The preceding examples are only a small sample of 
the numerous studies of atmospheric energy. Many 
others can be found without difficulty in the literature. 
Since no general approach to the philosophy of atmos- 
pheric energy has been adopted by meteorologists, the 
problems discussed in the literature may sometimes 
seem to have very little to do with the energy laws dis- 
cussed here. It is unfortunate that the experimentally 
determined facts are not always kept distinct from the 
definitions, assumptions, and speculations. If this dis- 
tinction were preserved, the work of different investi- 
gators could be more readily fitted into a single phi- 
losophy. 
The general energy equations permit the use of aver- 
aged values of velocity and stresses, but they require 
also a knowledge of eddy terms that are not measured 
directly. These terms have been the subject of many 
investigations during the past forty years. The most 
common approach to their evaluation seems to be 
guided by their analogy to terms associated with mo- 
lecular motions. The molecular stress p;, is a function 
of the pressure p, viscosity «, and velocity gradients. 
By analogy, one might wish to express the eddy stress 
T,.: by a similar formula in terms of an “eddy pres- 
sure” [12], ‘“eddy viscosity,”’ and gradients of the aver- 
aged velocity. The molecular flux of heat is a function 
of a coefficient of heat conduction and the temperature 
gradient; and by analogy, the eddy flux of thermal 
energy may be expressed in terms of a coefficient of 
eddy conduction and the gradient of mean temperature: 
or 
OXK i 
nl=—C 
The trouble with the analogy is that the coefficients of 
eddy viscosity and thermal conductivity are too varia- 
ble and unpredictable, and are not positive at all times; 
