CHAPTER 18: MOBILE AND STATIC ARMOR IN GRAVEL-BED STREAMS.
MOBILE AND STATIC ARMOR IN GRAVEL-BED STREAMS
Whereas sand-bed rivers often show dunes on the surface of their beds, gravel-bed streams often show a surface armor layer. That is, the surface layer is coarser than the substrate. In addition, the surface layer is usually coarser than the mean annual load of transported gravel (e.g. Lisle, 1995).
The surface of even an equilibrium gravel-bed stream must be coarser than the gravel load because larger material is somewhat harder to move than finer material. The river renders itself able to transport the coarse half of its gravel load at the same rate as its finer half by overrepresenting coarse material on its surface, where it is available for transport.
Bed sediment of the River Wharfe, U.K., showing a pronounced surface armor. Photo courtesy D. Powell.
The principle of mobile-bed armor is explained in Parker and Klingeman (1982) and Parker and Toro-Escobar (2002).
Most gravel-bed streams display a mobile armor. That is, the surface has coarsened to the point necessary to move the grain size distribution of the mean annual gravel load through without bed degradation or aggradation. In the case of extremely high gravel transport rates, no armor is necessary to enable the coarse half of the gravel load to move through at the same rate as the fine half (e.g. Powell et al., 2001). A mobile-bed armor gives way to a static armor as the sediment supply tends toward zero.
Bed sediment of the unarmored Nahal Eshtemoa, a wadi in Israel subject to severe flash floods with intense gravel transport. Photo courtesy D. Powell.
The mechanism of armoring can be explained with e.g. the transport equation of Powell, Reid and Laronne (2001), which can be cast in the following form.
Recall that here s50* denotes the Shields number based on the surface median size Ds50.
The functional form that drives armoring (and its disappearance at sufficiently high flows) is the term [1-(1/i)]4.5 in the above relation.
Now consider the function
plotted on the next slide.
Note that the bedload transport rate is a multiple of K(i), which is a steeply-increasing function of i for values of i that are not much greater than 1 (just above the threshold of function), but becomes nearly horizontal for value of I that are large compared to 1 (far above the threshold of motion.
In the next slide it is shown that this feature of the function biases the bedload to be finer than the surface material (or surface material to be coarser than the bedload) at conditions not far above the threshold of motion. By the same token, at conditions far above the threshold of motion the bedload and surface grain size distributions become nearly identical.
Now consider a mixture of only three grain sizes, D1 = 0.5 Ds50, D2 = Ds50 and D3 = 2 Ds50.
A condition fairly typical of bankfull flows in many perennial gravel-bed streams is characterized by the value s50 = 1.5, i.e. 50% above the threshold of motion for the surface median size. The value s50 = 8, on the other hand, corresponds to a condition far above the threshold of motion for the surface median size. Using the relations
the following values are obtained:
Note that the value of i is largest for the finest grain and smallest for the coarsest grain for both values of s50.
Now the bedload transport equation can be written in the form
When s50 = 1.5, the values of Ki are strongly dependent on grain size Di, such that K1/K3 = 34.6. At such a condition, then, the finer sizes will be overrepresented in the bedload compared to the surface (underrepresented in the surface compared to the bedload). The result is a mobile armor.
When s50 = 8, the values of Ki are weakly dependent on the grain size Di, such that K1/K3 = 1.26. At such a condition, the grain size distributions of the bedload and the surface material will not differ much, and only weak mobile armor is present.
In principle the computation of equilibrium mobile-bed armor is a direct calculation (Parker and Sutherland, 1990). Let the bedload transport rate qT and fractions in the bedload pbi be specified. A knowledge of pbi allows computation of the geometric mean size Dlg and arithmetic standard deviation l of the load. The bedload transport relation of Parker (1990), for example, can be written in the form
where W*( ) denotes a function. After some rearrangement,
Letting i = ln2(Di) and recalling that
and taking the 0th, 1st and 2nd moments of the equation below,
three equations for the three unknowns u*, Dsg and s are obtained;
The solution for u*, Dsg and s is obtained iteratively (e.g. using a Newton-Raphson scheme). Once this is done the surface fractions are obtained directly from the relation
It can be verified from e.g. the Parker (1990) relation that the armor becomes washed out as the Shields number based on the geometric mean size of the sediment feed becomes large:
On the other hand, the mobile-bed armor approaches a constant static armor as
An alternative way to compute armor is with the code of the Excel workbook RTe-bookAgDegNormGravMixPW.xls. Specified water discharge per unit width qw, sediment feed rate qbTf and grain size fractions pbf,i of the feed specify a final equilibrium bed slope S, flow depth H and surface fractions Fi regardless of the initial conditions.
It thus becomes possible to study equilibrium mobile-bed armor by allowing the calculation to run until it converges to equilibrium. In the succeeding calculations the sediment feed rate qbTf ( which eventually becomes equal to the equilibrium sediment transport rate qbT)is varied from 1x10-8 m2/s to 1x10-2 m2/s, while holding the following parameters constant: qw = 6 m2/s, If = 0.05 and L = 20 km. In addition, the size distribution of the sediment feed is held constant as given in the table to the right.
The input parameters for the highest value of sediment feed rate qbTo of 0.01 m2/s are given below. The duration of the calculation is longer for smaller feed rates, because more time is required to approach the final equilibrium.
Lisle, T. E., 1995, Particle size variations between bed load and bed material in natural gravel bed channels. Water Resources Research, 31(4), 1107-1118.
Parker, G. and Klingeman, P., 1982, On why gravel‑bed streams are paved. G. Parker and P. Klingeman, Water Resources Research, 18(5), 1409‑1423.
Parker, G., 1990, Surface-based bedload transport relation for gravel rivers. Journal of
Hydraulic Research, 28(4): 417-436.
Parker, G. and Sutherland, A. J., 1990, Fluvial Armor. Journal of Hydraulic Research, 28(5).
Parker, G. and Toro-Escobar, C. M., 2002, Equal mobility of gravel in streams: the remains of the Water Resources Research, 38(11), 1264, doi:10.1029/2001WR000669.
Powell, D. M., Reid, I. and Laronne, J. B., 2001, Evolution of bedload grain-size distribution with increasing flow strength and the effect of flow duration on the caliber of bedload sediment yield in ephemeral gravel-bed rivers, Water Resources Research, 37(5), 1463-1474.