508 BAGNOLD [chap. 21 



is measured by tlie ratio of mass to overall volume, irrespective of the density 

 of the constituent material. The effective density of marine organic debris, 

 which frecjucntly has hollow or spiny grains, and of clay crystals may be 

 considerably iess than the true density of the solid material composing them. 

 Taking buoyancy into account, tlie dynamic effects of gravity vary as the excess 

 density {ps —p). The true density of most solid mineral grains is approximately 

 that of quartz (2.65 in c.g.s. units), which gives an excess density of 1.65 in 

 water. However, if a marine organic grain with material density of 2.65 has an 

 effective density ps of half this amount (1.32), the effective excess density will 

 be only 0.32, or only one-fifth that of the solid mineral grain. 



Neglecting variations due to grain shape, the general criterion for dynamic 

 similarity of grain behaviour under a combination of gravity and fluid forces 

 may be derived very simply as follows. If the grain diameter is D, the net 

 gravity force acting on it, i.e. the immersed weight, will vary as {ps—p)gD'-^\ 

 and if the fluid stress (fluid force per unit of effective grain area) is /, the fluid 

 force on the grain varies as-Z)2. Whence the fluid stress per unit of immersed 

 grain weight varies as //(ps —p)gD. This dimensionless quantit}^ which will be 

 denoted by d, is useful because it combines the effects of both grain size and 

 grain density. 



For instance, if / is the fluid shear stress exerted on a grain bed, the 

 threshold of grain movement is definable in terms of d. It is found that, under 

 conditions of steady turbulent flow, the threshold value of 6 approaches a 

 universally constant value of approximately 0.06 as {ps —p)gD is increased. 



Again, when a grain is falling freely, the fluid and gravity forces acting on it 

 are equal and opposite ; whence d is unity. Expressing the fluid stress as 

 capw^, where Ca is the drag coefficient, the fall velocity, w, is given at once by 



w = 



(ps -p)gD 



2. Transport during Fall through the Sea 



According to its origin a grain of sea sediment may begin its existence as 

 such at a great distance above the sea bed, or it may be already at or very close 

 to the sea bed, e.g. if introduced as bed load by a river or as moraine material 

 by a polar glacier, or if pre-existing as a marine bed organism. Ultimately, 

 being by definition heavier than the sea-water, all sediment grains must find 

 themselves so close to the sea bed as to be within a zone of water whose move- 

 ment is affected by the presence of the bed boundary. But in the former case 

 this state will be preceded by one in which the grains are falling steadily down- 

 A\'ards through the sea at the fall velocity appropriate to their size and density. 



During tills fall the grains are sus])ended by the water, which exerts an u])- 

 ward force on them equal to their innnersed weight ; and the grains conform 

 without resistance to any motion which the surroiuiding water may have. 

 From a knowledge of the fall velocity the problem of predicting the drift of 

 such grains away from the vertical, from a given point in the sea to that at 



