SOME MEASUREMENTS OF ATMOSPHERIC TURBULENCE. 
27 
that p and K are independent of height, it may be as well to remove these restrictions. 
It is then found that £ as defined by equation (l), of Part I., is given by 
^ = A JJa mH dxdy, .(2) 
where p' is the pressure at the initial level, m n the vertical momentum per volume. 
Now let us insert numerical values. The surface air was seen to begin to move at 
or a little before 7|-h. The cumuli appeared to cease rising before 16h. and the height 
of their tops is known to average about 2 km. (vide HAnn’s ‘ Meteorology,' Illrd edn., 
p. 280). Now if we suppose, as seems reasonable, that the top of the cumulus is formed 
from the damp air which was initially close to the ground, then the displacement, 
measured by pressure, is about 2 decibars, so that (p'—p) = 2 x 10 5 dyne cm. -2 . The 
vertical velocity is 2 km. in 8*5 hours, that is 6'5 cm. sec. -1 . So the momentum per 
volume = m = 7 x 10 -3 grm. cm. -2 sec. -1 in the rising current. The rising current 
covered 0‘4 of the sky, so that averaging over the area A, as is done in (2), is 
equivalent to taking 0'4 of m H (p' —p) for the rising current. But the invisible 
descending currents contribute an equal amount to the integral. So 
^ = (/x0’8x2x10 5 x7x 10 -3 
= 11 x 10 5 grm. 2 cm. -2 sec. -5 . 
This figure is about ten times greater than measures of f at a height of a few 
hundred meters, deduced by various authors. If the air which forms the top of the 
cumulus had really started from a height of 1 km. instead of from the ground, as we 
have supposed, then the numerical value of £ would have to be divided by four. 
Reasons have already been given (Part VII.) for supposing that £ derived in this 
way from cumulus clouds is a measure of frictional effects, but not of the diffusion of 
entropy, because the linear term in (Part V., 3) does not vanish on taking the mean, 
owing to the fact that the eddies are produced by variations of entropy. To put it in 
another way : In G. I. Taylor’s deduction of formula (l) the vertical gradient of the 
diffusing quantity is treated as not correlated with the vertical velocity. When we 
are dealing with cumulus clouds that assumption is probably justified if the diffusing 
quantity is horizontal velocity, but not if it is potential temperature. 
To find ^ in the sense of diffusivity for potential temperature we should have to 
employ formula 32 of Part VII., namely 
K 
aev 
dh 
V 
H 
a 2 e/a h 2 
For insertion in this we require lapse rates in cumulus clouds and in the clear air 
between them. Such have recently been obtained by airmen. 
