486 
equation, which has been limited here to a system small 
enough to be considered homogeneous, will later be in- 
tegrated for large, nonhomogeneous systems. Whether 
the fluid is stationary does not seem to make any dif- 
ference. If it moves, so that the system contains kinetic 
energy, the changes of this energy form are completely 
accounted for in equation (10); and transformations of 
kinetic into thermal energy are accounted for by the 
transformation function 17, common to both equations. 
Equation (9) contains some terms that are not always 
found in expressions of the first law of thermodynamics. 
In the first place, the heat released in phase changes is 
frequently included in a general Q term, implying that 
the latent heat is supplied by some source outside the 
system. In equation (9) the latent-heat energy L and 
its net flux Q¥ are written specifically, in order to make 
a clear distinction between energy transformations 
within the system and flux of energy across the 
boundary. 
In the second place, equation (9) includes, within 
the function M/, not only the expansional work but also 
Stokes’ dissipation function, which represents work 
done in relative motion against viscous forces. Since 
this function usually has a magnitude much smaller 
than that of expansional work, its neglect is of no con- 
sequence in most meteorological problems. But if it is 
omitted from a discussion of energy transformations, 
there can be no explanation of the frictional conversion 
of kinetic energy into thermal energy [14]. 
The General System 
The basic energy equations can be integrated for a 
large, nonhomogeneous system, whose boundary sur- 
face has any shape and moves and changes shape in 
any prescribed manner, by the following procedure. 
Each equation is transformed to apply to a unit volume 
instead of a unit mass and is integrated over the vol- 
ume of the large system, yielding an equation for the 
local rate of change of the energy. A term representing 
energy flux through the boundary surface, associated 
with the arbitrary and variable motion of the surface, 
is added to both sides of each equation. The equations 
then give the rate of change of energy following the 
large, irregularly shaped, irregularly moving system, 
the shape of whose boundary may also be changing. 
This procedure will be illustrated by the transforma- 
tion of equation (10). 
The equation of continuity (4) in the form 
Op to) va 
apa a (ov,) = 0, 
where p is the density of the small volume of moist air, 
is multiplied through by K + P. Equation (10) is 
multiplied through by p and the resulting two equa- 
1. The term Z can be expanded into hysw, + hiswi, where 
hys and his are the latent heats of sublimation and freezing, 
and w, and w; are the mass proportions of water vapor and 
liquid water to the total mass of moist air. This method of 
representation is not completely satisfactory because the 
system, stipulated to consist of moist air only, may also con- 
tain some liquid water and ice. 
DYNAMICS OF THE ATMOSPHERE 
tions are added; each term in the sum now refers to a 
unit volume. When this equation is integrated over the 
volume of the large system, there results 
5 (K* + P*) = — If AUS Se Pra |, 
— [10° a [ vvpasde . 
o Ww 
Here an asterisk indicates that a term has been multi- 
plied by p and integrated over the volume; thus, K* = 
(11) 
if pK dV. Two volume integrals on the right-hand side 
Vv 
of the equation have been transformed to surface in- 
tegrals by the divergence theorem. An element of the 
boundary surface is do; the component of air motion 
normal to the boundary and directed outward from the 
system is v, ; and the three components of the stress 
tensor acting in a plane tangent to the boundary sur- 
face are Dri. 
Finally, the net flux of the energy K + P into the 
system through the boundary surface, associated with 
the normal, outwardly directed velocity V, of the sur- 
face, is written as a surface integral. The rate of change 
following the system is, then, the local change plus the 
net flux due to the motion of the system’s boundary 
surface: 
De ps) = 2 (eee) 
Di ot 
; (12) 
+ | Vae(K + Pda, 
and (11) becomes 
= (K* 4+ P*) = [i an Pie |, 
= Es cet [ evnsde | > 
o Ww 
where w is equal to V, — v,, or the velocity of the 
boundary relative to the velocity of the air. 
Similarly, equation (9) becomes 
=r + L*) = | [v0 gee Cl el, (14) 
[ow], 
where g7 and qg% are the components of the fluxes qj 
and q# of thermal energy and latent-heat energy at the 
boundary surface, and are directed normal to the sur- 
face out of the system. The fluxes q; and gi , whose 
units are energy per unit area and time, are defined in 
terms of Q? and Q” : 
(13) 
I 0 I L 0 L 
= — — d = — — 5 (tS 
PQ) ae (q%) and pQ aa (a). (15) 
For example, the part of gt associated with molecular 
conduction is — c 07'/da, , where c is a constant, and 
the net inward flux per unit volume is c 0?7'/dm;? , a 
summation over the three values of k. 
