Rouse Suspension Number

Hunter Rouse's 1937 analysis of suspended sediment asked a simple question with a two-line answer: sediment falls at its settling velocity $w_s$, and turbulence mixes it upward at a rate set by $\kappa u_*$. The ratio $P = w_s/(\kappa u_*)$ says which wins. Small $P$ means turbulence dominates and the grain is carried through the whole depth; large $P$ means gravity dominates and the grain stays near the bed.

The conventional dividing lines are $P < 0.8$ for full suspension, roughly 0.8 to 1.2 for suspension with a concentration gradient, 1.2 to 2.5 for suspension confined near the bed, and above about 2.5 for bed load — rolling, sliding and saltating in a layer a few grain diameters thick. Those boundaries are conventions drawn through a continuous transition, and different texts place them a few tenths apart. There is no discontinuity in nature at 2.5.

The number matters because the two regimes behave nothing alike. Suspended material moves at very nearly the water's own speed, so it travels a long way — the wash load in a river typically bears no relation to the bed material beneath it, because it came from somewhere far upstream and is going somewhere far downstream. Bed load creeps along at a small fraction of the flow speed, and it is the material that builds bars, migrates dunes and shapes the channel itself. A reservoir traps almost all the bed load and only some of the suspended load. A sediment budget that does not distinguish the two is not a budget.

Rouse's fuller result is the concentration profile that carries his name: concentration falls with height above the bed as a power law whose exponent is exactly $P$. At $P = 0.5$ the profile is nearly uniform; at $P = 2$ almost everything is in the bottom tenth of the depth. This has an immediate practical consequence for measurement — a bottle dipped near the surface will underread the depth-averaged concentration by a large and unknown factor whenever $P$ is above about 1, which is why depth-integrating samplers exist and why comparing sediment records taken by different methods is treacherous.

Two cautions on the inputs. The settling velocity $w_s$ is only given by Stokes' law for fine material — above roughly 100 µm the particle Reynolds number leaves the Stokes regime and a Stokes calculation overpredicts the fall velocity, badly for sand. Use a settling relation valid for the size you have. And von Kármán's $\kappa$, a genuine constant of turbulence at 0.40 to 0.41 in clear water, is measured to drop in heavily sediment-laden flow, sometimes to 0.3 or below: the suspended load damps the very turbulence that carries it. That is a real effect and a fascinating one, but it is a diagnostic, not a knob to turn until the answer looks nicer.