Sediment Entrainment and Channel Lifetimes in Braided Streams

Braided streams are typically found on steep slopes (high energy environments) with an abundance of sediment. Their channels are constantly choking up with sediments and shifting. This gives rise to the concept of the lifetime of a channel, i.e., how long water continues flowing in it. The channel lifetime is known to be proportional to the square root of its length¹. This vignette summarizes research that suggests that the origin of this relationship is in the mechanics of particle transport. We show that transport distances relate to particle sizes. Then, if the channel's length is proportional to how far sediment is transported in it, this argument succeeds.

The process of sediment entrainment (lifting into the flow) initiates fluvial (water) and aeolian (wind) transport. The wide range of beautiful patterns of ripples and dunes, the dimensions and slopes of channels, the development of Sahara Desert dust storms, the sorting of sediments, as well as the dynamics of braided streams, derive from sediment transport processes. Understanding of entrainment began with Bagnold, an English engineer interested in the saltation (bouncing transport, Figure 1) of sand grains in Northern Africa. Bagnold's description of sediment entrainment derived from basic fluid dynamics and Newtonian mechanics of particles on the surface².

Consider Bagnold's argument. Turbulent flow near the earth's surface (in a stream, or the atmosphere) is described by a vertical velocity profile (Fig. 2), in which the mean velocity, <v> increases logarithmically with height,

<v> = v*ln(y/y₀)

Here, v* is called the friction velocity, while y₀ is a parameter understood as the bed roughness. The bed stress, a force per unit area which can be exerted on a particle to lift it into the flow, is given by (1/2)ρv*², where ρ is the density of the fluid. The area presented by a spherical particle of diameter d is proportional to d², and the lifting force, F_lift, is then proportional to (1/2)d²ρv*². If F_lift is greater than or equal to the effective weight, F_g, of the particle, itself proportional to (ρ_s-ρ)gd³, the particle is entrained (Fig. 3). Here g is the acceleration due to gravity. An important result obtained from equating these two forces is that the volume d³ of the largest particle entrained in a given flow is proportional to v*⁶.

Physics usually offers more than one way to understand a phenomenon and an alternate method based on energy^3,4 rather than forces allows new interpretations. Consider this energy perspective. If the energy a particle absorbs from the fluid exceeds the gravitational potential necessary to lift it over its nearest neighbor, the particle is entrained into the flow. The advantage here is that one can easily address the issue of what happens to the entrained particle if the force is greater than necessary for entrainment. The excess kinetic energy may be converted to gravitational potential energy, and knowledge of that potential energy (how high the particle can be pushed) and an understanding of the process of settling in turbulent flow allows predictions of the total distance downstream a particle is carried before it falls back to the bed or ground.

The following considers, for simplicity, a streambed with equal sized particles. A particle of volume d³ can absorb an energy (1/2)ρv*²d³ from turbulent flow³. If this energy exceeds (ρ_s-ρ)gd³(d), the particle is lifted off the bed (or ground). Equating the two energies leads to the identical result for the maximum particle size entrained found above. But now the height, h, a given particle can reach is found by setting its initial excess kinetic energy equal to its final potential energy⁴:

(1/2)ρv*²d³ = (ρ_s-ρ)gd³(d+h)

and solving for h. The distance that the particle is transported before it settles is approximately the product of v*t, where t is the time the particle takes to settle back to the bed. In turbulent flow t has been given as



Using this result and the frequency, v*/d, such a particle is likely to be entrained, it is possible to calculate an effective rate of transport of particles as a function of their diameter⁴. The combination of variables (v*²/gd) can be put into the following form, (v*²/gH) (H/d), where H is the depth of the stream and v*²/gH is the square of the (unitless) Froude number, F_r. F_r² is the ratio of stream kinetic to potential energy. When F_r=1, these two energies are equal and the flow is termed critical. The Froude number is also relevant to wave propagation in a stream. When F_r < 1 (F_r > 1), surface wave propagation is downstream (upstream), but at critical flow standing wave patterns (Fig. 4) develop^5,6. Constant F_r with constant g implies the existence of a constant ratio of length to time squared in the remaining factors, e.g., v* and H.

It seems no coincidence that the relationship of channel length to channel lifetime squared¹ is the same ratio of units required by constant F_r. In fact, if the length of a channel tends to be proportional to the typical transport distance of the particles found in it, the above formula for the distance of transport of particles in turbulent flow can be used to show that the lifetime squared of a channel is proportional to its length⁷.

1. Foufoula-Georgiou, E., Sapozhnikov, V., 2001, Scale invariance in the morphology and evolution of braided rivers, Math. Geol. 33; 273-291.
   
2. Bagnold, R. A., 1966, An approach to the sediment transport problem from general physics. U.S. Geological Survey Professional Paper 429-I. 37 pp.
   
3. Hunt, A. G., 1999, A probabilistic treatment of fluvial entrainment of cohesionless particles, Journal of Geophysical Research.104 15409-15413.
   
4. Hunt, A. G., and Papanicolaou, T., 2003, Tests of predicted downstream transport of clasts in turbulent flow, Advances in Water Resources, 26, 1205-1211, 2003.
   
5. Tinkler, K. J., 1997, Indirect velocity measurements from standing waves in rockbed rivers, J. Hydraul. Eng. 123; 918-921.
   
6. Grant, G. E., 1997, Critical flow constrains flow hydraulics in mobile-bed streams: a new hypothesis. Water Resour. Res. 33; 349-358.
   
7. Hunt, A. G., G. E. Grant, and V. K. Gupta, 2005, Spatio-temporal scaling of braided streams, Journal of Hydrology, doi:10.1016/j.jhydrol.2005.02.034