534 
Mm, of water vapor at temperature 7’, and a mass m, of 
water at temperature 7”. Both phases are presumed to 
be at atmospheric pressure p. We will assume that the 
system undergoes only adiabatic (dQ = 0) and isobaric 
(dp = 0) transformations; in this case we have, (16), 
* * 
Ma = Ma, My + My = My + Mu, 
and 
/ y eo * * 
A(T", 1, Ma, Mr, Mu) a A(T, *) Ma y My, Mio), 
where (7’, T”, ma, Mv, Mw) and (Ty, 7%, mz, mi, ms) 
represent two states of the system at the same pressure 
p. It turns out [14] that 
(Ma Cpa + My Cpr) (T’ — Tx) 
= (ms — my) [L,(T%) + hu(T%) — hw(P”)] 
+ mis av( T's) <= i? NI 
where L,(T) = h,(T'%) — hy(T's) is the heat of vapori- 
zation of water and cy, and ¢p, are the specific heats at 
constant pressure of dry air and water vapor. 
Let us first consider a mass (ma + m,) of moist air 
at temperature 7’ = YT’. At pressure p, assumed con- 
stant, we evaporate into this system a mass of water 
mw» such that the vapor becomes saturated at tempera- 
ture 7. This temperature, which is represented by 
Tw, 1s by definition the temperature of the wet-bulb 
thermometer if the evaporating water assumes the 
temperature 7’, of the saturated gaseous phase. Thus 
we are led to put T’ = T, T; = T” = T,., and m4 =0 
in (17) from which is obtained the psychrometric for- 
mula 
(17) 
[ro(Tw) a ¥ ry (L)\Ly (Dw) 
Cpa + CpvTo(T’) ‘ 
where r,(7’) stands for the mixing ratio of water vapor 
at temperature 7’. 
Now let us consider a mass m* + m* of moist air at 
temperature 7 = T. We will assume that at constant 
pressure p one can condense the mass m; of water vapor 
contained in the mixture. The temperature 7’ which 
the system then reaches is the equivalent temperature 
T. of the moist air under consideration, provided it is 
assumed that the vapor can be condensed at tempera- 
ture 7, of the gaseous phase. Thus we are led to put 
Tt, =T’ =T 7’ = 7, and mi =m, = 0m (9, 
from which results von Bezold’s formula 
1, Oa) 
Cpa 
Lp = f= (18) 
Ny = WT (19) 
We note that this isobaric condensation is a fictitious 
transformation, since it can never be realized physi- 
cally. 
NONHOMOGENEOUS SYSTEMS 
General Remarks. The open nonhomogeneous sys- 
tem of greatest interest to meteorologists consists of a 
turbulent fluid. The physical state of the fluid is defined 
by the specific mass p and the pressure p, its state of 
motion by the components (v1, v?, v8) of velocity v in a 
DYNAMICS OF THE ATMOSPHERE 
system of rectangular Cartesian coordinates (21, x?, x), 
fixed with respect to the earth.! Scalars p and p and 
vector v are functions of 2}, x?, x? and of time ft. 
By hypothesis, a mass element? péx!6x?6a?, moving 
at velocity v*, exchanges no mass with the surroundings 
(absence of molecular diffusion). Therefore the system 
satisfies the equation of continuity in p and v*; its 
movement is governed by gravity, the Coriolis force, 
and the pressure gradient force. The equations of bal- 
ance of momentum pv defined by the components 
pv*, and of kinetic energy k = 4v’v’, can easily be de- 
duced from the Eulerian equations of motion [11]. 
We designate by X the Reynolds mean of the function 
X and by ¥ = pX/p the corresponding weighted mean. 
It should be recalled that X’ = O and pX” = pX” = 0, 
wien 20 == = amd ke? = = 5 
The mean state of the fluid is defined by p and p, and 
its mean motion by v*, ( = 1, 2, 3). Since px”* = 0 
there is no diffusion at the scale of the mean state and 
of the mean motion, and as a result the mass element 
pov'dx26x%, moving at the mean velocity v*, does not 
exchange matter with its environment. Therefore the 
mean system satisfies the equation of continuity in p 
and v'. We know that the movement is governed by 
gravity, the Coriolis force (velocity v"), the force due © 
to the gradient of mean pressure p, and the resultant 
force due to the Reynolds stresses Ri = R} = — pv’*y”3 
(7, k = 1, 2, 3). The equations of balance of the mo- 
mentum components av' and of the corresponding ki- 
netic energy k, = Lip of the mean motion can easily 
be deduced from the equations of motion [11]. We 
observe that the absence of diffusion of mass (pv”* = 
0) at the scale of the mean state does not imply the 
absence, at this scale, of diffusion of the properties of 
the fluid. In fact, if f represents the specific magnitude 
of any property whatever, we have in general pfu”? ~ 0. 
For example, by substituting f = v* we would obtain 
pv'v”7 = pv”*y”i = —R‘. Thus it appears that the 
Reynolds stresses R§ result from turbulent diffusion of 
momentum pv*. By contrast, a mass element pdx16276x° 
moving with mean velocity v* diffuses mass and for this 
reason constitutes an open system, which does not 
admit an equation of continuity. 
Finally, by subtracting the equations of kinetic en- 
ergies k and k», we find the equation of balance of tur- 
bulent kinetic energy k, = 300”? (4, 11], 
to) ~ aa ——— 
& (oh) + oF Ging! - sean A AeA = Fere20) 
t 0x! 
where 
bg eae 
R; dvi Coma 
A = wend eae S F = A ; 
Da puaahN HANI eMEE (21) 
(EP Sh), 
1. In order to simplify the discussion we have neglected 
molecular viscosity. Extension of the results of this section to 
the case of a viscous fluid does not present any difficulty [11]. 
2. By hypothesis, 6¢ = 0. 
