ENERGY EQUATIONS 
where Kn = 0,?/2, the kinetic energy, per unit mass, 
of the averaged motion; P = gx;, which, incidentally, 
is equivalent to the unaveraged gx; or P; and E£ is the 
“eddy-transformation function”’: 
OD; 
Ox, 
E = aly; (26) 
The other terms and their superimposed symbols have 
the same meanings as in the preceding analyses. 
In the second method of deriving a kinetic-energy 
equation for averaged properties, the starting point is 
the basic equation of kinetic energy (10), transformed 
for a unit volume. Again, v; and p,; are replaced with 
0; + v; and Div + Dri 5 and the terms are averaged. 
The resulting equation contains the kinetic energy of 
eddy motions as well as averaged motion. When equa- 
tion (23) is subtracted from this equation, there is ob- 
tained, after transformation to the general system, 
= lf (apK. — pv,K.) de | 
o A 
(27) 
= Es = [pf ch [om do | ; 
o Ww 
where K, = v;2/2 and K, = v’2/2. The function E* ap- 
pears with opposite signs in (25) and (27). This explains 
why # has been called the eddy-transformation func- 
tion; when positive, it decreases the energy of the mean 
motion and increases the energy of the eddy motion. 
It is analogous to the molecular transformation func- 
tion WM. 
Dk* 
Dt 
Applications of the General Energy Equations 
Equations (22), (25), and (27) constitute an array of 
general energy equations for atmospheric processes. 
Many different special equations can be obtained from 
them. For example, if turbulent fluctuations are non- 
existent (vi = 0, = p’ = pr = 0), if the system’s 
boundary everywhere moves with the air, and if the 
system is homogeneous, equation (27) vanishes, and 
(22) and (25) can be reduced to the basic equations (9) 
and (10). 
If one wishes to establish a criterion for the increase 
of eddying motion, one starts with equation (27) and 
modifies it according to whatever conditions may be 
assumed. Richardson’s criterion [12], however, cannot 
be derived from that equation because of the omission 
of terms involving density fluctuations. One such term 
omitted from (27) is —[  gp'04 dV. If this term® is in- 
Vv 
cluded and if, following Richardson, one assumes that 
the boundary work and flux terms and the term M? 
are negligible, then (27) becomes 
3. Calder [2] derives the kinetic-energy equations for mean 
and eddy motion by a procedure similar to that followed here, 
but he retains this one term involving p’. His arguments for 
retaining it appear logical, but it would seem that the argu- 
ments apply equally to some of the terms that he does not 
retain. 
IDI: 
Di (28) 
= 5 — | go's av. 
Vv 
Richardson’s number is the ratio 
[ ge’ av 
eS So 
EH* 
According to the assumptions that are usually made in 
applications of Richardson’s criterion, the ratio reduces 
to the stability divided by the square of the vertical 
shear; when this ratio is less than a critical value Ricrit, 
turbulence is supposed to increase. Richardson assumed 
that Ricrit = il, 
It will not be argued that the terms involving p’ are 
negligibly small, for they very well may not be. They 
have been omitted from the equations developed here 
solely to permit a discussion of the general aspects of 
the equations without too much involvement with de- 
tails. But the fact that this omission eliminates the 
possibility of deriving Richardson’s criterion from the 
equations may not be a serious fault. The criterion de- 
pends upon the term gp’v3 being positive and different 
from zero; in other words, upon a positive correlation 
between density and vertical velocity. There is no ex- 
perimental evidence that such correlation is to be ex- 
pected. The occasional examples cited in support of 
the criterion are more qualitative than quantitative; 
according to Sutton [16], ‘some support can be found 
for almost any value of Ricriz between 0.04 and 1.” 
Richardson’s criterion appears to be qualitatively 
correct, because observations show that turbulence 
tends to be suppressed in a stable layer and to be in- 
creased when the vertical wind shear is great. The 
qualitative success may be explained without consider- 
ing the term gp’v;. Suppose (27) is applied to a ho- 
mogeneous system, and all terms are dropped except 
the following: 
DK* 
= —M? + E*. 
Di Me. + 
(29) 
The first term on the right represents the dissipation of 
eddy energy into thermal energy; it is probably nega- 
tive most of the time so that it consistently acts to 
decrease the eddy energy by converting it into thermal 
energy. The second term £*, for the frictional layer, is 
essentially the volume integral of 
77 Oh a Ob 
pH = — pr, 4 — pogo 2, (30) 
0X3 0X3 
where the 23-axis is vertical. According to empirical evi- 
dence, the eddy-transformation function # is predom- 
inantly positive, so it consistently acts to increase the 
eddy energy at the expense of the energy of mean mo- 
tion. The magnitude of H increases with imcreasing 
vertical wind shear, 00;/0x3 and 00/023 ; thus, the eddy 
energy tends to increase with increasing shear. Its 
magnitude increases also with increasing v3; but this 
fuctuation of the vertical velocity tends to be damped 
