SOME MEASUREMENTS OF ATMOSPHERIC TURBULENCE. 
17 
minute mark, and working out the standard deviation, in height about the mean 
height, after a time a. 
Table III. 
a, secs. 
1 x 60 
3 x 60 
5 x 60 
7 x 60 
9 x 60 
^ = 1(P x 
2 a 
cm. 2 sec. -1 
| 0-93 
0-82 
0-55 
0-55 
0-33 
(T 2 
p-\ 
a. z 
dyne cm. -2 
e 
j> 0-345 
o-ioo 
0-041 
0-029 
0-014 
Number of points 
14 
12 
10 
8 
6 
Now o- H 2 /2a would be the diffusivity if a were a “ long” time ; and the criterion of 
sufficiency in length is that <r H 2 /2a should not vary with a. The small number of 
points makes the probable errors large, so that the decrease of o- H 2 /2a between 
a = 7 mins, and a = 9 mins, is not significant. It looks as though the diffusivity K 
were here of the order of 0'5 x 10 4 cm. 2 sec. -1 . 
Again — pa-f/ a 2 would be the eddy-stress, hh, if a were so small that further decrease 
made no further change in the quantity. This stage is not reached at a. — 1 minute. 
All that we can say is that the eddy-stress is probably greater than 0'345 dynes cm. -2 . 
The mean height of this observation is 1 km. above ground, and the mean velocity 
1270 cm. sec. -1 . 
The photograph of a smoke trail in fig. 2 suggests that the eddying is partly 
random but also partly sinusoidal. Let us therefore see what would happen if the 
path of the particle were an exact sine curve without any random variations. Let 
the height h of the particle be given by h — B = A sin qt where A, B and q are 
constants. The increase in height in a time a would be 
A 
sin 
A . 2 cos qt . sin 
by trigonometry. So that the standard deviation <x H after a time a would be given by 
L, 
(cos qt) 2 dt 
4A 2 ( sin— 
2 V 2 
0- H = “ 
L 
* = L 
( 
t = 0 
where L is a very long time. The integral is equal to fL plus a negligible oscillatory 
part. Consequently erf = 2A 2 (sin . It follows that the stress hh, which is the 
limit of —perfl a 2 when a is small, comes to —pq 2 A 2 /2 ; whereas the diffusivity K, 
which is the limit of <x H 2 /2a when a is long, comes to zero. 
VOL. CCXXI.—A. 
D 
