18 
MR, LEWIS F. RICHARDSON ON 
Comparison with Taylor’s expression for the Dijfusivity and with W. Schmidt’s 
“ AustauschC 
In Taylor’s remarkable investigation (‘ Phil. Trans.,’ A, vol. 215) from which the 
present research took its stimulus, the diffusivity K is given, in the present notation, 
as the mean value of Vn(h — h 0 ) over a large horizontal plane; and it is stated that 
h — h 0 is the height through which an eddy moves from the layer at which it was at 
the same temperature as its surroundings, to the layer with which it mixes. This 
definition of h — h 0 is puzzling, for it seems impossible to reconcile the supposed 
starting and stopping of the air, with the ceaseless motion which we observe in nature, 
except in the case of cumulus eddies. Happily we are now in a position to clear away 
the mystery. For it has been shown independently in the present paper that the 
diffusivity is given by ______ 
K = (h 2 -hf 2 /2 (4-h) 
where (t 2 — t x ) is a time long compared to the fluctuations in the wind, and where the 
bar implies an average taken over a still much longer time. As ( t 2 —t x ) is the same 
for all the quantities which are averaged, we may remove it from under the * bar, 
writing 
K (t a -tj) = i(/i 2 -/q) 2 . 
Differentiate this equation with respect to t 2 , 
K = (h.-hf ~ {ha-h^) = (ha-hi) • v n at C 
thus K is expressed as the mean of the product of the rise in height during a long¬ 
time into the vertical velocity at the end of that time. It may also be taken at the 
beginning. Comparing with Taylor’s form quoted above we see a strong resemblance, 
and we are led to suppose that Taylor’s theory makes two unnecessary and 
unnatural restrictions : (1) that the portion of air should start at the same temperature 
as its surroundings ; (2) that the portion of air should finally mix with its surroundings. 
But that if these restrictions be removed, then another becomes necessary, namely 
that {t 2 —t^) should be sufficiently long (several minutes). Whether the average be 
taken over a large horizontal plane, or over a very long time (6 hours), appears to be 
a matter of indifference. 
The extent to which Taylor assumes viscosity to be independent of height in his 
general theory (‘Phil. Trans.,’ A, vol. 215, pp. 11 to 13) is this: he neglects the terms 
due to the initial eddying in his equation (6). That is a doubtful proceeding, unless 
the initial eddying is zero : but zero is independent of height. 
The “Austausch” of W. Schmidt is defined by him (in ‘ Sitz. Akad. Wiss.,’ Wien 
(1917), pp. 4 to 5) as 
2 (element of mass crossing horizontal plane) x (vertical displacement of element) 
(whole area) x (time of motion) 
