CHAPTER 24: APPROXIMATE FORMULATION FOR SLOPE AND BANKFULL GEOMETRY OF RIVERS

1. CHAPTER 24: APPROXIMATE FORMULATION FOR SLOPE AND BANKFULL GEOMETRY OF RIVERS In all previous chapters in the e-book with the exception of Chapter 15, rivers have been treated as if they have specified channel widths, and also as if they are independent of their floodplains. Further progress on large-scale morphodynamics calls for the relaxation of these constraints. In this chapter the principles in Chapter 3 and the relations of sediment transport are used to develop an approximate formulation for the prediction of bankfull geometry of streams. No code is presented in this chapter. The formulation is, however, used extensively in subsequent chapters. Small meandering stream and floodplain flowing into the Sangamon Rivers, Illinois

2. BANKFULL GEOMETRY AND FLOODPLAINS Alluvial rivers create their own bankfull cross-sections and floodplains through the interaction of flowing water and sediment erosion and deposition. Rivers establish their bankfull width and depth through the co-evolution of the river channel and the floodplain. The river bed and lower banks are constructed from bed material load. The middle and upper banks are usually constructed predominantly out of wash load, although they usually contain some bed material load as well. As the river avulses and shifts, this material is spread out across the floodplain. The establishment of bankfull depth is equivalent to the construction of a floodplain of similar depth. Minnesota River and floodplain south of the Twin Cities, Minnesota. Flow is from bottom to top. Image from NASA website: https://zulu.ssc.nasa.gov/mrsid/mrsid.pl

3. BANKFULL SHIELDS NUMBER OF RIVERS The estimate bf50* of bankfull Shields number and dimensionless bankfull discharge were introduced and defined in Chapter 3; where D50 refers to a bed surface median size, The diagram below, also introduced in Chapter 3, suggests that rivers evolve toward some relatively narrow range of bankfull Shields number.

4. BANKFULL SHIELDS NUMBER OF RIVERS contd. Within a considerable amount of scatter, the diagram allows the following approximate estimates of bankfull Shields number for gravel-bed and sand-bed streams based on averages (e.g. Parker et al., 1998; Dade and Friend, 1998).

5. BANKFULL SHIELDS NUMBER OF RIVERS contd. The theory behind the evolution of river cross-section toward a quasi-equilibrium bankfull Shields number is outlined in Parker (1978a,b). The theory is, however, incomplete. At this point the numbers below serve as useful empirical results. The considerable scatter in the data is probably due to a) differing fractions of wash load versus bed material load in the various rivers, b) differing amounts and types of floodplain vegetation, which encourages floodplain deposition and c) different hydrologic regimes. Paola et al. (1992) were the first to propose the assumption of constant bankfull Shields number in modeling the morphodynamics of streams. The general form of their analysis is used in the succeeding material. Their results are also applied to basin deposition in a succeeding chapter.

6. SIMPLE THEORY FOR BANKFULL CHARACTERISTICS OF RIVERS • The formulation given here is based on three relations: • a resistance relation describing quasi-normal bankfull flow; • an example sediment transport relation describing transport of bed material load at quasi-normal bankfull flow; • a specified bankfull Shields criterion. • While varying degrees of complexity are possible in the analysis, here the problem is simplified by assuming a constant friction coefficient Cf and a sediment transport relation of generic form (with assumed constant s, t and nt). Where the subscript “bf” denotes bankfull flow, the governing equations are momentum balance bed material transport form* = channel-formative Shields number

7. SIMPLE THEORY FOR BANKFULL CHARACTERISTICS OF RIVERS contd. Let surface median size D50, submerged specific gravity R and friction coefficient Cf be specified. The equations below provide three constraints for five parameters; bankfull discharge Qbf, bankfull volume bed material load Qtbf, bankfull width Bbf, bankfull depth Hbf and bed slope S. Thus if any two of the five (Qbf, Qtbf, Bbf, Hbf and S are specified the other three can be computed. The nondimensionalizations of Chapter 3 are now re-introduced along with a new one for Qtbf;

8. NON-DIMENSIONALIZATION OF GOVERNING EQUATIONS The three governing equations are made dimensionless with the hatted variables; Solving for , S and as functions of and yields the following results;

9. WHAT THE RELATIONS SAY Slope: doubling the water discharge halves the slope; doubling the bed material load doubles the slope. Width: doubling the bed material load doubles the width; doubling the water discharge without changing the bed material load does not change width (but slope drops and depth increases instead). Depth: doubling the water discharge doubles the depth; doubling the bed material load halves the depth (but channel gets wider and steeper).

10. CASE OF SAND-BED RIVERS The Engelund-Hansen (1967) relation can be used for the case of bed-material load of sand-bed rivers. It is obtained by means of the evaluations in which case the relations take the form

11. CASE OF GRAVEL-BED RIVERS In a gravel-bed river the sand load can be treated as wash load, and the volume bed material load Qt can be equated to the volume bed load Qb. The effect of form drag due to bedforms can be ignored as a first approximation. The Meyer-Peter and Müller (1948) relation and its corrected version due to Wong (2003) and Wong and Parker (submitted) both have critical Shields numbers c* that are so high (0.047 and 0.0495, respectively) that the median surface size D50 of a gravel-bed river can barely be expected to move at form* = 0.0487. An alternative form is the Parker (1979) approximation to the Einstein (1950) relation, which takes the form where

12. CASE OF GRAVEL-BED RIVERS contd. Adapting the previous formulation with the transport relation of Parker (1979) leads to the following results; where for form* = 0.0487

13. REFERENCES FOR CHAPTER 24 Dade, W. B. and Friend, P. F., 1998, Grain-size, sediment transport regime, and channel slope in alluvial rivers, Journal of Geology, 106, 661-675. Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service. Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams, Technisk Vorlag, Copenhagen, Denmark. Meyer-Peter, E. and Müller, R., 1948, Formulas for Bed-Load Transport, Proceedings, 2nd Congress, International Association of Hydraulic Research, Stockholm: 39-64. Paola, C., Heller, P. L. and Angevine, C. L., 1992, The large-scale dynamics of grain-size variation in alluvial basins. I: Theory, Basin Research, 4, 73-90. Parker, G., 1978a, Self‑formed rivers with stable banks and mobile bed: Part I, the sand‑silt river, Journal of Fluid Mechanics, 89(1),109‑126. Parker, G., 1978b, Self‑formed rivers with stable banks and mobile bed: Part II, the gravel river, Journal of Fluid Mechanics, 89(1), pp. 127‑148. Parker, G., 1979, Hydraulic geometry of active gravel rivers, Journal of Hydraulic Engineering, 105(9), 1185‑1201. Parker, G., Paola, C., Whipple, K. X. and Mohrig, D., 1998, Alluvial fans formed by channelized fluvial and sheet flow. I: Theory, Journal of Hydraulic Engineering, 124(10), 985-995. Wong, M., 2003, Does the bedload equation of Meyer-Peter and Müller fit its own data?, Proceedings, 30th Congress, International Association of Hydraulic Research, Thessaloniki, J.F.K. Competition Volume: 73-80.

14. REFERENCES FOR CHAPTER 24 contd Wong, M. and Parker, G., submitted, The bedload transport relation of Meyer-Peter and Müller overpredicts by a factor of two, Journal of Hydraulic Engineering, downloadable at http://cee.uiuc.edu/people/parkerg/preprints.htm .