520 



BAGNOLD 



[chap. 21 



and the fluid in tliis case contributes nothing to the appHed power. The avail- 

 able power is therefore given by 



a> = {ps —p)gXh sin ^-u, (11) 



which is the tangential gravity force on the solids times the velocity of their 

 movement. 



Of this power a part {ps —p)gNhw must be expended via the agency of the 

 turbulence in the driven water in maintaining the load of solids at a constant 

 height above the bed. A further part must be expended in maintaining the flow 

 of the current water at the velocity u. The power for this is to«, where to is 



Fig. 4. Initiation, and progress of a self-maintaining turbidity current. 



the fluid resistance exerted at the flow boundary. Neglecting for a moment the 

 extra resistance exerted at the upper flow boundary, we can regard the water 

 current as a river. The stress to can then be expressed in terms of u and h by 

 the semi-empirical relationship 



_ ,-Jto\* lS.2h 



u = 5.751 — 1 logio J. , 



which appears (Francis, 1957) to be a rational generalization consistent both 

 with the Prandtl-Karman theory and with well established practical flow 

 formulae applicable to broad straight flow in open channels. This relationship 

 gives 



'3 I3.2h 



rou - — log 10- 



Di 



(12) 



I where Di is the effective size of the boundary roughness. 



The criterion that the turbidity current shall be self-maintaining is, therefore, 



{p, -p)gNh{u sm ^ -iv) -^ ^ log-^ -^- 



or 



1 p.-p 



u^ p 

 where Ti must always exceed wl^in §. 



gNh log2 H^' (sin 13 -A ^ 0.03, 



(18) 



