1. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 1 CHAPTER 18: 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.

2. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 2 MOBILE AND STATIC ARMOR 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.

3. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 3 MECHANISM OF MOBILE ARMOR 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 D s50. 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.

4. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 4 MECHANISM OF MOBILE ARMOR contd. 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.

5. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 5 MECHANISM OF MOBILE ARMOR contd. Now consider a mixture of only three grain sizes, D 1 = 0.5 D s50, D 2 = D s50 and D 3 = 2 D s50. 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:

6. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 6 MECHANISM OF MOBILE ARMOR contd. 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 K i are strongly dependent on grain size D i, such that K 1 /K 3 = 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 K i are weakly dependent on the grain size D i, such that K 1 /K 3 = 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.

7. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 7 COMPUTATION OF MOBILE AND STATIC ARMOR In principle the computation of equilibrium mobile-bed armor is a direct calculation (Parker and Sutherland, 1990). Let the bedload transport rate q T and fractions in the bedload p bi be specified. A knowledge of p bi allows computation of the geometric mean size D lg 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,

8. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 8 COMPUTATION OF MOBILE AND STATIC ARMOR contd. Letting  i = ln 2 (D i ) and recalling that and taking the 0 th, 1 st and 2 nd moments of the equation below, three equations for the three unknowns u *, D sg and  s are obtained;

9. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 9 COMPUTATION OF MOBILE AND STATIC ARMOR contd. The solution for u *, D sg 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

10. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 10 ALTERNATIVE COMPUTATION OF MOBILE AND STATIC ARMOR 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 q bTf ( which eventually becomes equal to the equilibrium sediment transport rate q bT ) is varied from 1x10 -8 m 2 /s to 1x10 -2 m 2 /s, while holding the following parameters constant: q w = 6 m 2 /s, I f = 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. An alternative way to compute armor is with the code of the Excel workbook RTe- bookAgDegNormGravMixPW.xls. Specified water discharge per unit width q w, sediment feed rate q bTf and grain size fractions p bf,i of the feed specify a final equilibrium bed slope S, flow depth H and surface fractions F i regardless of the initial conditions.

11. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 11 ALTERNATIVE COMPUTATION OF MOBILE AND STATIC ARMOR contd. The input parameters for the highest value of sediment feed rate q bTo of 0.01 m 2 /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.

12. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 12 q bTo = 1x10 -2 m 2 /s After 120 years q bTf = 1x10 -2 m 2 /s After 120 years

13. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 13 q bTo = 3x10 -3 m 2 /s After 240 years q bTf = 3x10 -3 m 2 /s After 240 years

14. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 14 q bTo = 1x10 -3 m 2 /s After 240 years q bTf = 1x10 -3 m 2 /s After 240 years

15. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 15 q bTo = 3x10 -4 m 2 /s After 480 years q bTf = 3x10 -4 m 2 /s After 480 years

16. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 16 q bTo = 1x10 -4 m 2 /s After 480 years q bTf = 1x10 -4 m 2 /s After 480 years

17. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 17 q bTo = 1x10 -5 m 2 /s After 960 years q bTf = 1x10 -5 m 2 /s After 960 years

18. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 18 q bTo = 1x10 -6 m 2 /s After 7680 years q bTf = 1x10 -6 m 2 /s After 7680 years

19. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 19 q bTo = 1x10 -8 m 2 /s After 15360 years q bTf = 1x10 -8 m 2 /s After 15360 years

20. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 20 q bTo = 1x10 -2 m 2 /s After 120 years q bTf = 1x10 -2 m 2 /s After 120 years

21. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 21 q bTo = 3x10 -3 m 2 /s After 240 years q bTf = 3x10 -3 m 2 /s After 240 years

22. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 22 q bTo = 1x10 -3 m 2 /s After 240 years q bTf = 1x10 -3 m 2 /s After 240 years

23. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 23 q bTo = 3x10 -4 m 2 /s After 480 years q bTf = 3x10 -4 m 2 /s After 480 years

24. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 24 q bTo = 1x10 -4 m 2 /s After 480 years q bTf = 1x10 -4 m 2 /s After 480 years

25. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 25 q bTo = 1x10 -5 m 2 /s After 960 years q bTf = 1x10 -5 m 2 /s After 960 years

26. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 26 q bTo = 1x10 -6 m 2 /s After 7680 years q bTf = 1x10 -6 m 2 /s After 7680 years

27. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 27 q bTo = 1x10 -8 m 2 /s After 15360 years q bTf = 1x10 -8 m 2 /s After 15360 years

28. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 28 q bTf

29. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 29 q bTf

30. 1D SEDIMENT TRANSPORT MORPHODYNAMICS with applications to RIVERS AND TURBIDITY CURRENTS © Gary Parker November, 2004 30 REFERENCES FOR CHAPTER 18 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.