488 
Hesselberg [5] has defined the mean velocity by the 
formula: 
J [ev aeav i ps dt a. 
[ feaav 
as a consequence of which pu, is automatically zero 
whether p’ is negligible or not. Energy equations written 
for this density-weighted mean velocity are similar in 
form to the ones that follow, but many of the terms 
have different meanings and values. The choice of the 
averaging formula for the velocity should be deter- 
mined finally by the instrumental technique that is 
used in measuring v,;. If the instrument responds di- 
rectly to the momentum pv; , then it measures the @; 
of Hesselberg’s formula. If it responds directly to the 
velocity v;, then it measures 3; and formula (18) 
should be used. If p’ is truly negligible or is not corre- 
lated with velocity components, the two formulas are 
identical. 
Ui = 
Energy Equations for Averaged Properties 
The process of analysis for averaged properties is 
applied to the first law of thermodynamics as follows. 
By combination with the equation of continuity, equa- 
tion (9) is transformed to apply to a unit volume: 
to) Ca) 
al (pl + pL) + a (oxpl + vxpL) ag) 
= pQ’ + pQ” + pM. 
Then v% is replaced by % + ;,, and the terms are 
averaged by formula (18) over the space and time: per- 
taming to 0; : 
= = () = > me = T 
a (pL + pL) + — (ipl + depL + prxl + preL) 
ot Chia 
= pQ’ + pQ” + pM, (20) 
Now, 
— , 1 OD; - Ov; 
M = ap © cana 2 OLS dus ems OD a vs ma APki Bs, (21) 
in which the last two terms are assumed to be neg- 
ligible. The first two terms on the right will be indi- 
cated by M,, and M, , respectively. The energy equation 
is integrated over the total volume of the arbitrary 
system and becomes finally 
D & L* 
pet 
L As D) = OD 2 Ses van 
—[—M, — Mily, (22) 
where w is the normal eg of the boundary surface 
relative to 0, , or Vn — Dy 
The term ap( + L) is | the flux of the averaged 
DYNAMICS OF THE ATMOSPHERE 
thermal and latent-heat energies at the velocity of the 
boundary relative to the averaged air velocity. The 
flux pv,(I + L), averaged locally before integration 
over the surface, is associated with the “eddy” velocity 
of the air, v,, normal to the surface; it can be called 
the eddy flux of thermal and latent-heat energies. The 
terms 7, and @» are averaged values of the fluxes that 
appear in the thermal-energy equation (14); q,, for 
example, consists in part of molecular conduction pro- 
portional to the gradient of the mean temperature: 
iin = —@ = + flux of radiant energy, 
where x, is a coordinate normal to the surface and 
directed outward. The last terms, M* and M7, are the 
volume integral of the sacllecullene transformation fune- 
tion: M* for averaged stresses and velocity, M? for 
deviations of stresses and velocity from their arranged 
values. 
There are two alternate ways of deriving an energy 
equation for averaged properties from the equations of 
motion: The equations may be averaged first and then 
converted into an equation for the kinetic energy of 
averaged motion alone; or they may be converted into 
an energy equation and then averaged, yielding an 
equation for kinetic energies of both averaged and eddy 
motions. The difference between these two equations is 
the equation for the kinetic energy of eddy motions 
alone. 
In the first method, the equations (6) are converted 
to the momentum form by combining each of them 
with the equation of continuity. The velocities and 
stresses are replaced with their averaged values and 
deviations, and the terms of the equations are averaged 
by formula (18). These three equations are multiplied 
by 31, %, and 23, respectively; the averaged form of 
the equation of continuity is multiplied by — 0,?/2; and 
all four equations are added together. The final result is 
to) OF rena to) 4 02 2a 
at (08 AF pies) + Ap (we > aF isos) 
- OF x: — OPK: 
= 0; A O 
Chia oP i Chia 
(23) 
The symbol 7;,; stands for the “‘eddy stress” tensor: 
(24) 
This term, though it arises from the averaging of the 
accelerational term, is customarily regarded as a stress 
against the mean motion, acting in the x-direction in 
a plane normal to the 2;,-axis, on the fluid lying to the 
negative side of the plane. 
Equation (23) is transformed into an equation valid 
for a general system by the method already described, 
and it becomes 
Ud 
Tre = Tix = — pv'j;, - 
a 4 BY) = [ i alt, & D van ia 
-[us+ E* — / d;(Pus + Tri) ao 
