484 
Written in this way, the law can be applied to a very 
small system containing moist air and moving at the 
velocity of the air. The air is subject to viscous forces, 
but its thermal-energy function is assumed to be the 
same as that for a perfect gas. The operator d/dt is the 
rate of change following the motion of the system or 
the air; c, is the specific heat of the mixture at constant 
volume; T is the absolute temperature; and d(c,T)/dt 
is the rate of change of thermal energy per unit mass. 
The rate of change of latent-heat energy, per unit mass, 
of water within the system is dL/dt. It is not necessary 
to establish a zero point for either thermal energy or 
latent-heat energy, because only their rates of change 
appear in (2). 
The term Q on the right-hand side of equation (2) 
represents the rate at which thermal energy, per unit 
mass, is added by radiation or molecular conduction 
through the boundary of the system; and Q” represents 
the rate at which latent-heat energy, per unit mass, is 
added by the net flux of water through the boundary. 
The last term represents the rate of work, per unit 
mass, done on the system by relative motion against 
the molecular stresses. It is written with the aid of 
tensor notation which, despite its convenience in a term 
of this nature, is not widely used in meteorology. The 
indices 7 and k may have the values 1, 2, or 3, corre- 
sponding to the three directions of a Cartesian coord- 
inate system; thus, x; or x, may be 2, iy or #3. The 
velocity components v; or », may be 0, v2, OY Vs, corre- 
sponding to the directions of the 21, 22, a X3 axes, re- 
spectively. The general derivative 0v,/dx, stands for 
any one of nine different derivatives: 
Ov; th Ov; OV OV, 
OX; 0a” O25” 0x3” 
Oe 
Oxy, 3 
OVs 
0X» : 
OVs 
0x3 ; 
v3 
OX,” 
O03 ’ 
OX» J 
v3 
0x3 : 
The specific volume of the mixture in the system is a, 
and p;: 1s the molecular stress tensor. 
The components of p,; are given by the expression 
as (o+5 ie 2 » 4) a (+ =) (3) 
aU 
where p is the pressure; 6, is a unit tensor of value 1 
when 2 = k and of value 0 when 7 # k; p is the coeffi- 
cient of molecular viscosity; and 7 is an index with the 
value 1, 2, or 3. In the tensor notation, repetition of 
an index within a term means that the term is summed 
over all values of the index. Thus, 
Ov; Lv Z Ov; Le Ov, OVe 
Ox; = j=1 Ox; Chit 7 Xe r : 
Pit SS Dik = 
which is the three-dimensional velocity divergence. Any 
component of p;; 1s a force per unit area, acting in the 
plane perpendicular to the 2;-axis. If it is regarded as 
acting on the fluid lying to the negative side of that 
plane, then the force is positive in the 2;-direction. If 
the velocity gradients are vanishingly small, or if the 
fluid is nonviscous (u = 0), pe: reduces to the negative 
of the pressure, p: 
Pw = Pr = Py = Psi = Px = P32 = O, 
Din = P22) — 38) ap 
DYNAMICS OF THE ATMOSPHERE 
Both z and k are repeated in the last term of (2), 
and hence 
ey et 
7=1 Ore 0 Cn 
=1 
ape: 
When this term is summed, expanded, and rearranged, 
there is obtained: 
5 Sum of squares of terms 
Vi ° 6 Shea 
aDii — ne 8 tL w| involving derivatives |> 
Ore 2a; of velocity components 
where 0u;/dx;, the velocity divergence, is equivalent to 
6, 0v;/d2,. According to the equation of continuity, 
Ov; lee 1 da 
ae Rea (4) 
Therefore, 
Bn ves = —p— cle + [Sum of squares]. (5) 
: OUE dt 
The first term on the right, —p da/dt, is the rate of 
work by expansion, and the second term is the rate of 
work by relative motion against viscous stresses. The 
latter term, Stokes’ “dissipation function,” is always 
positive because both » and the sum of squares are 
positive. 
The first law of thermodynamics has been established 
by many experiments, controlled and interpreted with 
a philosophy of the energy concept similar to that pre- 
sented in the first part of this paper. The earliest state- 
ment of an energy law, by Leibnitz in 1693, dealt not 
with thermal energy but with kinetic and potential 
energies (the “live” and “dead forces’). This energy 
law ean be established by a mathematical transforma- 
tion of Newton’s second law of motion, and its validity 
rests upon the validity of the law of motion. 
The Law of Kinetic Energy 
Newton’s second law, for a compressible viscous fluid 
on the rotating earth, can be expressed as follows: 
=: apt (2w cos ¢)v3 — (2 sin d)m = a Opa 
Chup 
dv2 = i Ore 
dt + (2 sin $)r1 = @ Ene (6) 
dv; Oprs el 
ai = (2w cos 6) = Oe g, 
where w is the angular velocity of the earth’s rotation; 
$, the latitude; and g, gravity. The a-axis is directed 
toward east; the x-axis toward north; and the 23-axis 
toward the zenith. When the first equation is multiplied 
by v; , the second by v , the third by v; , and the result- 
ing equations are added, the Coriolis terms disappear, 
and the result is 
02, oy 2 
Dy} OX 
in which v,2 = v2 + vo? + v32. The first term on the 
— 39; 
