536 
(k = 1, 2, 3). Upon substituting (27) and (28) into 
(30) we find, after taking account of the equation of 
continuity of mean motion [11], 
Ap d Nba CE alte 
at (68m) ar axk Ex ar Pies = (31) 
where 
Wi + Wi dln , . (a+ Ar — «) 
o (Tj? dat + p T,, >0 (32) 
represents the rate of production of entropy. The second 
law states that the entropy production is real (¢> 0). 
We recall that A, — x = —A; from (20). 
Equation (31) shows that the flux of mean entropy 
includes a convective flux and a thermal flux due to 
radiation, conduction, and turbulence. The production 
of entropy results from the nonuniformity of the tempera- 
ture, from the production of heat, from the dissipation of 
kinetic energy of mean motion by turbulence, and from 
the destruction of turbulent kinetic energy (11). 
Finally, it should be noted that in the case of a 
viscous fluid the expression A; which appears in (25), 
(27), and (32) must be replaced by A; — A, where 
A,(>0) represents Rayleigh’s dissipation function, that 
is, the fraction of kinetic energy of mean motion which 
is dissipated by viscosity (imternal friction on the molec- 
ular scale) into heat per unit mass and per unit time. 
The Heat Flux Due to Atmospheric Turbulence in the 
Vertical Direction. If it is assumed that the mean 
motion of the air is horizontal, the adiabatic eddying 
motion in the vertical direction is governed by the 
differential equation [11] 
” 
ae = Tee (Pp — y)r, — TH, 
Y ip 
(33) 
where v% = the eddying velocity in the vertical z direc- 
tion, 
r, = the mixing length, 
T = the mean temperature of the dry air, 
y = the vertical lapse rate —d0T7'/dz, 
IT = the adiabatic lapse rate (g/c,. where g repre- 
sents gravity), 
the excess of the temperature of the parcel 
above the mean temperature of the level 
from which it originated. 
After multiplying (33) by pvz we obtain the expression 
for the energy of instability A; released during the 
eddying motion, (21), 
ll 
W 
Tx 
pa, = pol Me = Mo (ar — 4) — peel, 84) 
d ip 
where A = prv: > O is the exchange coefficient 
(Schmidt’s austausch) in the vertical direction. 
DYNAMICS OF THE ATMOSPHERE 
As for the heat flux W, resulting from the adia- 
batic eddying motion in the vertical direction, it is 
given by (26): 
I= Geshe = —Aey, (= =) Ee eae, (GB) 
The first term of the last member of (35) represents the 
flux of Schmidt and the second, the thermo-convective 
flux (Ertel, Priestley, and Swinbank) due to differences 
between the temperatures of parcels from the same 
level. These differences are initiated and maintained 
by sources of heat whose distribution is determined by 
pa. Because of this fact, parcels which are displaced 
vertically experience an additional hydrostatic buoyant 
force per unit mass equal to g./T. According to 
whether 7% < or > 0, that is to say according to 
whether the particle is more or less dense than its 
surroundings, we have vz < or > 0 and consequently 
Tue > 0. This inequality brings out the manner of 
organization of thermal convection. It turns out that 
the energy of instability due to the excess 7% of the 
parcel’s temperature above the mean temperature of 
its starting level is always positive and therefore cor- 
responds to a positive production of turbulent kinetic 
energy. This production can result only from a trans- 
formation of heat into kinetic energy, and consequently 
P22 Ee SO (36) 
T 
Moreover we observe that the thermal-convective flux 
of heat is always upward. 
Finally, by returning to equation (82), it can be 
shown that the intensity of the entropy source as- 
sociated with heat flux (85) is given by 
ACpa(T = 7) ‘ 
oO = 
: cmd pau 
Giz 
(@P 
(37) 
ke E ~ oe prea | as 
T T 
provided that 7, = T. From (36) and the inequality 
y > 0, the inequality of (37) is always verified. The 
flux W, is therefore compatible with the second law of 
thermodynamics without the necessity of assuming a 
priori that it is upward (as Ertel claims). It should be 
noted? that the heat flux of Schmidt involves an effec- 
tive production of entropy represented by the perfect 
square in the first term of the expression for o. 
Application to the General Circulation of the Atmos- 
phere. The equations which we have considered above 
can be of use in the study of the general atmospheric 
circulation. The ‘mean motion” is then a zonal current 
(westerly or easterly), and the “eddying motion” cor- 
3. Bibliographical information and further details can be 
found in [11]. 
