Another Chapter in THE SEARCH FOR THE HOLY GRAIL:

1. Another Chapter in THE SEARCH FOR THE HOLY GRAIL: A MECHANISTIC BASIS FOR HYDRAULIC RELATIONS FOR BANKFULL FLOW IN GRAVEL-BED RIVERS Gary Parker With help from François Metivier and John Pitlick

2. What is the physical basis relations for bankfull geometry of gravel-bed streams?

3. Where do the following relations come from? • Bankfull Depth Hbf ~ (Qbf)0.4 • Bankfull Width Bbf ~ (Qbf)0.5 • Bed Slope S ~ (Qbf)-0.3 • where Qbf = bankfull discharge

4. THE GOAL: A Mechanistic Description of the Rules Governing Hydraulic Relations at Bankfull Flow in Alluvial Gravel-bed Rivers The Parameters: Qbf = bankfull discharge (m3/s) QbT,bf = volume bedload transport rate at bankfull discharge (m3/s) Bbf = bankfull width (m) Hbf = bankfull depth (m) S = bed slope (1) D = surface geometric mean or median grain size (m) g = gravitational acceleration (m/s2) R = submerged specific gravity of sediment ~ 1.65 (1) The Forms Sought:

5. DATA SETS • Alberta streams, Canada1 • Britain streams (mostly Wales)2 • Idaho streams, USA3 • Colorado River, USA (reach averages) • 1 Kellerhals, R., Neill, C. R. and Bray, D. I., 1972, Hydraulic and • geomorphic characteristics of rivers in Alberta, River Engineering • and Surface Hydrology Report, Research Council of Alberta, Canada, • No. 72-1. • 2 Charlton, F. G., Brown, P. M. and Benson, R. W., 1978, The • hydraulic geometry of some gravel rivers in Britain, Report INT 180, • Hydraulics Research Station, Wallingford, England, 48 p. • 3 Parker, G., Toro-Escobar, C. M., Ramey, M. and Beck S., 2003, • The effect of floodwater extraction on the morphology • of mountain streams, Journal of Hydraulic Engineering, 129(11), • 2003. • 4 Pitlick, J. and Cress, R., 2002, Downstream changes in the channel of a • large gravel bed river, Water Resources Research 38(10), 1216, • doi:10.1029/2001WR000898, 2002.

6. NON-DIMENSIONALIZATION These forms supersede two previous forms, namely which appear in reference 3 of the previous slide. Note:

7. WHAT THE DATA SAY The four independent sets of data form a coherent set!

8. REGRESSION RELATIONS BASED ON THE DATA To a high degree of approximation, Remarkable, no?

9. WHAT DOES THIS MEAN?

10. THE PHYSICAL RELATIONS NECESSARY TO CHARACTERIZE THE PROBLEM • Required: four relations in the four unknowns • Hbf, Bbf, S, QbT,bf. • Resistance relation (Manning-Strickler): • Gravel bedload transport relation (Parker 1979 approximation of Einstein 1950): • Relation for channel-forming Shields number bf* (Parker 1978): and • Relation for gravel yield from basin (not determined solely by channel mechanics).

11. RESISTANCE RELATION Manning-Strickler form: where Ubf = Qbf/(Bbf Hbf) denotes bankfull flow velocity, Here we leave r and nr as parameters to be evaluated.

12. BEDLOAD TRANSPORT RELATION Use Parker (1979) approximation of Einstein (1950) relation applied to bankfull flow:

13. RELATION FOR CHANNEL-FORMING SHIELDS NUMBER Base the form of the relation on Parker (1978):

14. RELATION FOR GRAVEL YIELD FROM BASIN AT BANKFULL FLOW This relations is external to the channel itself, and instead characterizes how the channels in a watershed interact with the unchannelized hillslopes. The necessary relation should be a dimensionless version of the form where nbT must be evaluated.

15. WORKING BACKWARD Rather than working forward from the basic physical relations to the hydraulic relations, let’s work backward and find out what the form the physical relations must be to get the observed hydraulic relations. Recall that

16. RESISTANCE RELATION The desired form is Now using the definition of Cz, the non-dimensionalizations and the relations it is found that But so that

17. RELATION FOR BANKFULL SHIELDS NUMBER By definition Using the relations it is found that This can be rewritten as

18. RELATION FOR GRAVEL TRANSPORT AT BANKFULL FLOW Recall that Now from the last relation of the previous slide, Using the previously-introduced non-dimensionalizations, Thus

19. EVALUATION OF THE CONSTANTS From the regression relations, In addition, for natural sediment it is reasonable to assume In the Parker approximation of the Einstein relation, The data of the four sets indicate an average value of bf* of 0.04870, or thus

20. THE RESULTING RELATIONS

21. TEST OF RELATION FOR Cz using all four data sets

22. TEST OF RELATION FOR bf* using all four data seta

23. FINAL RESULTS If we assume mechanistic relations of the following form: resistance bedload transport channel-forming Shields number sediment yield relation then we obtain the results The first three of these correspond precisely to the data!

24. Test against the original data set

25. Test against the original data set

26. Test against the original data set

27. Test against the original data set

28. Test against four new data sets

29. Test against four new data sets

30. Test against four new data sets

31. Test against three new data sets

32. BRITAIN II STREAMS: ROLE OF BANK STRENGTH Class 1 has least vegetation, Class 4 has most vegetation

33. RELATION BETWEEN VEGETATION DENSITY AND BANK STRENGTH, BRITAIN II STREAMS

34. HOW WOULD VARIED BANK STRENGTH (r), SEDIMENT SUPPLY (Y) AND RESISTANCE (r) AFFECT HYDRAULIC GEOMETRY?

35. VARIATION IN r (BANK STRENGTH)

36. VARIATION IN y (GRAVEL SUPPLY)

37. VARIATION IN r (CHANNEL RESISTANCE)

38. QUESTIONS?