Andrew Simon USDA-ARS National Sedimentation Laboratory, Oxford, MS andrew.simon@arsda

1. Equilibrium, Shear Stress, Stream Power and Trends of Vertical Adjustment Andrew Simon USDA-ARS National Sedimentation Laboratory, Oxford, MS andrew.simon@ars.usda.gov

2. Non-Cohesive versus Cohesive Materials • Non-cohesive: sands and gravels etc. • Resistance is due solely to particle size, weight, shape and “hiding”. • Cohesive: silts and clays • Resistance is derived from electro-chemical inter-particle forces under zero normal stress

3. Shields Diagram Cohesive Materials Denotes uncertainty

4. Shields Diagram by Particle Diameter Excludes cohesives

5. Heterogeneous Beds ks = 3* D84

6. Erosion of Cohesives by Hydraulic Shear National Sedimentation Laboratory Need for a means to determine critical shear stress (tc) and the erodibility coefficient (k) in-situ for soils and sediments.

7. Erosion Rate is a Function of Erodibility and Excess Shear Stress e = k (to- tc) e = erosion rate (m/s) k = erodibility coefficient (m3/N-s) to =boundary shear stress (Pa) tc =critical shear stress (Pa) (to-tc) = excess shear stress Critical shear stress is the stress required to initiate erosion. Obtained from jet-test device

8. Impinging Jet Applies Shear Stress to Bed National Sedimentation Laboratory Jet Nozzle

9. Impinging Jet Applies Shear Stress to Bed National Sedimentation Laboratory Jet Nozzle As scour hole depth increases, shear stress decreases.

10. From Relation between Shear Stress and Erosion We Calculate tc and e National Sedimentation Laboratory tc Erosion Depth, cm (cm3/Pa/sec) k Time

11. General Relation for Erodibility and Critical Shear Stress Erodibility, m3/N-s k = 0.1 tc -0.5 Where; tc = critical shear stress (Pa), x, y = empirical constants

12. Revised Erodibility Relation

13. Distributions: Critical Shear Stress

14. Distributions: Erodibility Coefficient

15. Mapping Critical Shear Stress: Yalobusha River Basin, Mississippi National Sedimentation Laboratory

16. Idealized Adjustment TrendsFor a given discharge (Q) National Sedimentation Laboratory t gVS Se n tc d

17. Adjustment: Boundary Shear Stress

18. Adjustment: Increasing Resistance

19. Adjustment: Increasing Resistance

20. Adjustment: (Excess Shear Stress)Degrading Reach

21. Boundary Shear Stress: Range of Flows Shearstress, in N/m2

22. Adjustment: Excess Shear StressDegrading Reach Excess shear stress

23. Adjustment: (Excess Shear Stress)Aggrading Reach

24. Adjustment of Force and Resistance

25. Results of AdjustmentDecreasing Sediment Loads with Time Toutle River System

26. Experimental Results

27. Total and Unit Stream Power • W = g w y V S = g Q S • W = total stream power per unit length of channel • g = specific weight of water • w = water-surface width • y = hydraulic depth • v = mean flow velocity • Q = water discharge • S = energy slope • ww = W / (g w y) = V S • where ww = stream power per unit weight of water

28. Adjustment: Unit Stream Power

29. Flow Energy • Total Mechanical Energy H = z + y + (a v2/ 2 g) where H = total mechanical energy (head) z = mean channel-bed elevation (datum head) a = coefficient for non-uniform distribution velocity y = hydraulic depth (pressure head) g = acceleration of gravity • Head Loss over a reach due to Friction hf = [z1 + y1 + (a1 v12/ 2g)]- [z2 + y2 + (a2 v22/ 2g)] • Head, Relative to channel bed Es = y + (av2/ 2g) =y + [a Q2 / (2 g w2 y2)]

30. Adjustment: Total Mechanical Energy As a working hypothesis we assume that a fluvial system has been disturbed in a manner such that the energy available to the system (potential and kinetic) has been increased. We further assume that with time, the system will adjust such that the energy at a point (head) and the energy dissipated over a reach (head loss), is decreased. Now, for a given discharge, consider how different fluvial processes will change (increase or decrease) the different variables in the energy equations.

31. Adjustment: Energy Dissipation Minimization of energy dissipation

32. Trends of Vertical Adjustment and Determining Equilibrium

33. Determining Equilibrium Recall definition A stream in equilibrium is one in which over a period of years, slope is adjusted such that there is no net aggradation or degradation on the channel bed (or widening or narrowing) OR There is a balance between energy conditions at the reach in question with energy and materials being delivered from upstream

34. Causes of Channel Incision

35. Trends of Incision: Channelization

36. Trends of Incision: Below Dams

37. Bed-level Trends Along a Reach

38. Bed-level Trends Along a Reach

39. Empirical Functions to Describe Incision E = a t b E = elevation of the channel bed a = coefficient; approximately, the pre-disturbance elevation t = time (years), since year before start of adjustment b = dimensionless exponent indicating rate of change on the bed (+) for aggradation, (-) for degradation E/ Eo = a +b e-kt E = elevation of the channel bed Eo = initial elevation of the channel bed a = dimensionless coefficient, = the dimensionless elevation a > 1 = aggradation, a < 1 = degradation b = dimensionless coefficient, = total change of elevation b > 0 = degradation, b < 0 = aggradation k = coefficient indicating decreasing rate of change on the bed

40. Empirical Model of Bed-level Response

41. Comparison of the Two Bed-level Functions

42. A Natural Disturbance (Toutle River System)

43. Bed-Level Response

44. Bed Response: Toutle River System Upstream disturbance, addition of potential energy, sub-alpine environment

45. Comparison with Coastal Plain Adjustment Downstream disturbance, increase in gradient, coastal plain environment

46. Model of Long-Term Bed Adjustment