Sediment Transport in Antidune Flow
Understanding antidunes, their behavior, and the sedimentary structures that they produce requires insight into how sediment is transported in antidune flows. It is remarkable that after nearly 90 years of study, there are no published studies detailing sediment transport in antidune flow.
The following discussion provides an overview of sediment transport in antidune flow. A detailed discussion gets into some hairy equations that I haven't looked for over 10 years, so we will avoid that for now. But the general discussion will show how the distribution of flow velocity and depth along the bed sets up the distribution of bed shear stress. The bed shear stress distribution, in turn, controls the sediment flux and determines the zones of deposition and erosion on the bed.
Starting with a simple configuration of antidunes with an amplitude (a) and wavelenth (λ). Flow over the trough and crest is supercritical, meaning that at both the trough and crest the ratio of flow velocity (U) to the square root of flow depth (d) times gravitational acceleration is greater than one (>1).
Supercritical flows respond to changes in bed elevation differently than subcritical flows. If a supercritical flow encounters positive feature on the bed (like a mound sticking up in the flow), it responds by slowing down and increasing in depth (Middleton and Southard, 1984). Similarly, for a negative feature, the flow becomes faster and shallower.
With antidunes, the crest is a positive feature on the bed, while the trough is a negative feature. The flow responds so that the flow velocity in the trough (UT) is greater than the flow velocity at the crest (UC), while the flow depth in the trough (dT) is less than the flow depth at the crest (dC).
Bed Shear Stress
Sediment transport is driven by bed shear stress. The shear stress (force per unit area) of the flow on the bed is what plucks sand gains off of the bed and entrains them in the flow. If the bed shear stress exceeds a threshold value for a particular grain size and composition, that grain will be lifted from the bed and carried by the flow. If the bed shear stress drops below that threshold value, the entrained grain will drop back to the bed. So the bed shear stress controls whether erosion (removal of grains) or deposition (settling of grains) will occur.
The bed shear stress (τ) is a function of the gradient in the velocity profile of the flow. This gradient is essentially a function of the flow velocity over flow depth (U/d). In antidunes, this gradient is at a maximum in the trough and minimum at the crest since UT/dT is greater than UC/dC.
Since the flow has a mass, there are additional components to the bed shear stress. The flow's inertia causes it to impinge on the bed at trough and separate from the bed at the crest. These effects shift the bed shear stress maximum (τmax) and minimum (τmin) in a downstream direction from the trough and crest as shown in the figure above.
Patterns of Erosion and Deposition in Antidune Flow
The distribution of shear stress along the bed drives sediment transport (Middleton and Southard, 1984), with the sediment discharge in the flow being a function of the bed shear stress. Sediment discharge is greater where bed shear stress is greater, because the higher bed shear stresses can remove more sediment from the bed.
McLean and Smith (1986) indicate that changes in bed elevation with time (whether the sediment is being eroded from or deposited to the bed) is a function of sediment discharge along the bed in a two-dimensional flow. Which in turn is directly related to the bed shear stress. If you move along the bed from a zone of high sediment discharge to low sediment discharge, sediment will accumulate with time. Conversely if you move from a zone of low sediment discharge to high sediment discharge, sediment will be removed from the bed with time.
Because sediment discharge is directly related to bed shear stress, the points of maximum and minimum bed shear stress on the bed define boundary points between zones of increasing and decreasing sediment discharge. Erosion (removal of sediment from the bed over time) occurs where bed shear stress is increasing along the bed. Deposition (addition of sediment to the bed over time) occurs where bed shear is decreasing along the bed.
Now if you look at the patterns of bed shear stress along the bed and the resulting zones of deposition and erosion, you can see why antidunes migrate upstream and why they typically build in amplitude.
Starting at the crest of the upstream antidune, there is a shear stress minimum just downstream of the crest. From this minimum, bed shear stress increases along the downstream side of the bedform to a maximum just downstream of the trough. Erosion occurs along this interval.
Moving on from this bed shear stress maximum, bed shear stress drops along the upstream side of the bedform, as the flow slows down and gets deeper, to the bed shear stress minimum just downstream from the crest. Depostion occurs in this interval of decreasing bed shear stress.
The result is that sediment is added to the upstream side of the bedform and removed from the downstream side. As this happens, the bedform moves upstream. Similarly, since the trough is in a zone of erosion and the crest is in zone of deposition. The bedform will build in amplitude over time.