THERMODYNAMICS OF OPEN SYSTEMS 537 
responds to perturbations embedded in this current. 
In this case it is clearly a question of turbulence on a 
very large scale (synoptic scale), essentially different 
from the small-scale turbulence which is usually studied. 
At the scale of the general circulation it becomes 
difficult indeed, if not impossible, to admit the con- 
servation of potential temperature and of momentum 
or angular momentum of the large eddies along their 
trajectories. 
On the other hand nothing allows us to state a 
priori that a fraction of the kinetic energy of the zonal 
current is really transformed into kinetic energy of 
large-scale turbulence (macroturbulence); in this case, 
the opposite might very well occur and probably does 
occur. In short, a thermomechanical theory of macro- 
turbulence (Grossturbulenz of A. Defant) remains to be 
worked out. 
The equations of balance of momentum and kinetic 
energy can be used to obtain qualitative indications of 
the production and transport of these quantities in the 
atmosphere [8, 12]. Unfortunately, in the case of large- 
scale turbulence we do not have the exact values of 
the Reynolds stresses R%, the fraction A, (> or < 0) 
of the kinetic energy of mean motion dissipated by 
turbulence, and the energy of instability A; released 
in the course of the eddying motion. It is therefore 
impossible to make a quantitative theoretical study of 
the distribution of sources of zonal momentum, of 
kinetic energy, of internal energy, and of entropy, as 
well as of their manner of redistribution in the atmos- 
phere. 
Knowledge of the tensor of large-scale turbulence 
would permit us to achieve a better comprehension of 
the processes and the evolution of the general circulation 
of the atmosphere. New investigations, both theoretical 
and synoptic, into the sources and the flux of mo- 
mentum, of energy in its various forms [13], and of 
entropy appear desirable. 
Final Remarks. When the system under considera- 
tion is homogeneous, the fundamental equations of 
thermodynamics can be written for the case of arbi- 
trary addition of mass (dm 2 0) to the system. These 
equations have manifold applications not only in 
meteorology (clouds with precipitation, evaporation of 
precipitation, mixing of air masses of different tempera- 
tures, psychrometry), but also in chemistry, in biology, 
and in industry. 
The situation is not quite the same when the system 
is nonhomogeneous (gradients of 7’ and p), for in this 
case the material points of the system are necessarily 
in motion. Now the notion of movement can be defined 
and studied only if the moving element retains its 
identity in the process of its displacement; in other 
words, a system of dynamical equations can give a 
true representation of motion only if it includes the 
equation of continuity of mass, expressing the in- 
variance of the mass of an element along its trajectory. 
Furthermore, in the case of a nonhomogeneous system, 
it is necessary to adopt in succession two scales of 
observation: first the microscopic scale, that is, the 
scale of the individual elementary eddy (the scale of 
the molecule, if one envisions diffusion and molecular 
viscosity), then the macroscopic scale, that is, the scale 
of the mean motion. It must be observed that the 
latter scale cannot be adopted arbitrarily, for there 
must be no diffusion of mass at this scale (pv”* = 0). 
If this were not true, there would be no equation of 
continuity on the macroscopic scale, and it would be 
impossible to define and study the mean motion. There- 
fore the extension of the laws of thermodynamics for 
nonhomogeneous systems, to which mass can be added 
or removed arbitrarily, encounters difficulty in the 
very beginning unless it is possible to satisfy the con- 
dition pv”* = 0. We shall assume this condition to be 
met; then for an observer who is carried along with 
the mean motion and follows the evolution of the 
system on a microscopic scale, the flux of the diffusion 
of mass is expressed by means of pv”* ~ 0 and the 
pk 
ae a 0k = 
1, 2, 3). However, as we have said before, the absence 
of diffusion at the scale of mean conditions (pv”* = 
Opv’™ 
dak 
diffusion of any arbitrary property f (intensive quan- 
tity) of the fluid, within the flow of which the mean 
flux of f is given by pfv”* ~ 0. Thus it is possible to 
carry out a study of the turbulent diffusion (eddy 
diffusion) in the atmosphere of water vapor (f = the 
specific humidity e«) or of any other substance in sus- 
pension in the air, of sensible heat (f = ¢paZ’), of 
latent heat (f = eZ), of internal energy (f = e), of 
entropy (f = s), of kinetic energy (f = k), of mo- 
mentum (f = v), ete. 
One final remark: The necessary condition pu’* = 
0 shows that v”* is the fluctuation relative to a weighted 
mean, and consequently the components of the mean 
flow are necessarily the weighted means ve of the com- 
ponents v® of the velocity of the elementary eddies. 
The weighted mean has the unique property of de- 
composing, in an additive fashion, the average of the 
total kinetic energy k into kinetic energy of mean 
motion and mean kinetic energy of turbulence (k = 
Kim + hi). No other mean possesses this same property. 
Therefore, it is impossible to avoid the necessity of 
introducing the weighted mean into the study of tur- 
bulence of fluids. 
intensity of diffusion by means of — 
= ()) does not imply the absence at this scale of 
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2. pgp Donner, T., “L’affinité.” Paris, Gauthier-Villars, 1927. 
3. —— and Van RyssevBercue, P., Thermodynamic Theory 
of Affinity. Palo Alto, Calif., Stanford University Press, 
1936. 
4. Deray, R., “Introduction 4 la thermodynamique des 
systémes ouverts.’? Bull. Acad. Belg. Cl. Scz., 5° sér., 
15: 678-688 (1929). 
5. Gipss, J. W., ‘‘Equilibrium of Heterogeneous Substances” 
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