346 SECTIONAL TRANSACTIONS.—A™*. 
shearing motion through all layers. In these cases, if 5p is the difference in 
density between each fluid and that immediately above it, and if h is the thickness 
of all layers except the bottom and top layers which extend to + ©, then B! = 8p/ph 
is a quantity similar to 8, and it is found that when there is only one intermediate 
layer the flow becomes unstable when «? > 2 gf, while if there are two such layers 
the criterion is ? > 2°11g8'. It seems possible that with a large number the criterion 
might be a? > 4 gf. 
Turbulence.—The stability for infinitely small disturbances has not necessarily 
any definite relationship with the question of whether or no turbulence can be formed 
in fluids with a stable distribution of density. On the other hand, disturbances of a 
stably stratified fluid of variable density necessarily involve an increase in gravitational 
potential energy. If py, is the coefficient of virtual viscosity due to turbulence, and 
if us is the virtual coefficient of diffusion, then the rate of supply of energy by the 
‘Reynolds stress’ is great enough to supply all the energy necessary to maintain 
turbulence against gravity if a?/g8>s/u,. Li we accept the Reynolds-Prandtl 
hypothesis concerning the relationship between the transfer of momentum and of 
other properties in a turbulent fluid, then yp; =u, so that the criterion for the 
maintenance of turbulence is «?/g8 > 1, a relation which was obtained by L. F. 
Richardson and by the present writer many years ago. 
The only observations of a fluid of variable density which I have been able to find 
from which py, Us, « and 8 can be calculated are those of Dr. J. P. Jacobsen on the 
mixing of salt and fresh water in Randersfjord and in the Kattegat. The figures 
given in Tables I and II are taken from Dr. Jacobsen’s measurements. It will be 
observed that, as Dr. Jacobsen has pointed out, the Reynolds-Prandtl relationship 
[4s = Uy is not found to hold when salt and fresh water mix, indeed p, is generally 
many times as great as u,. On the other hand, it will be seen that «?/g8 is of the 
same order but greater! than w.u,, even though both of them may be less than 
0-1. In spite of the great stabilising effect of gravity the flow was evidently turbu- 
lent, for the values of .,, are some hundreds of times as great as the viscosity due to 
molecular agitation. 
TasLe I.—Data for Schulz’s Grund, Kattegat. C.G.S. units. 
Depth m B ts Pu Usitu | o&/g9B 
Metres. 
| | 
2:5 —10x10-3) 7:5x10-7} O38 | 31 0-09 0-14 
5-0 —17x10 11-2 0-4 °° | 93-1 0-13 | 0-24 
75 —22x10 28-5 0-18 2-7 0-067 0-17 
10-0 —24x 10 60-0 0-05 2-2 0-023 0-098 
12-5 —19x10 | 105-0 0-04 1-9 0-021 0-035 
| 15-0 — 8x10 82-5 0-2 3-8 0-05 0-008 
|. , 16 0x10 46-0 0-74 
| 20-0 +4x10 
TasLe II.—Data for Randersfjord. C.G.S. Units. 
| 
Depth 0 Bo} ts) Ba tealtee | eoB | 
Metres. | | 
1 —0-033 | 0-67x10-5) 06 | 385 0-17 0-17 
2 —0-071 | 135x10-5, 05 2-5 0-20 | 0:38 — 
3 —0-062 | 150x10-5| 04 | 26 O-15 0-26 
1 Except close to a region where « =0. 
ey 
