Physical basis for quasi-universal relationships describing bankfull hydraulic geometry of sand-bed rivers

Empirical data indicate that hydraulic geometry relationships for single-thread sand-bed rivers (i.e., rivers with median bed-material size between 0.062 and 0.502mm) can be delineated such that bankfull width, bankfull depth, and channel slope are related in consistent ways to bankfull discharge. Such relationships ought to be the external expression of physical relationships intrinsic to sand-bed river dynamics. In this study, a back-calculation is performed to identify parameters (exponents and coefficients) for three relationships taken to be intrinsic to sand-bed rivers: (1)2a generalized Manning-Strickler resistance relationship; (2)2a relationship for channel-forming Shields number; and (3)2a relationship for sand yield at bankfull flow. To back-calculate parameters for the physical relationships, first the hydraulic geometry relationships are expressed in suitable dimensionless form. Second, the physical relationships are expressed with coefficients and exponents that are analytically related to parameters in the hydraulic geometry relationships. Third, parameters from the hydraulic geometry relationships are used to calculate parameters for the physical relationships. The analysis yields the following results for the sand-bed rivers: (1)2no physical basis exists for using an exponent of 1/6 in the resistance relationship; (2)2channel-forming Shields number decreases with particle Reynolds number, and thus grain size, in a consistent way; and (3)2sand concentration at bankfull flow must decline with increasing bankfull discharge. Although each of these relationships could have been established independently on its own, in this study they have been obtained as the only conclusions consistent with the observed hydraulic geometry relationships and the proposed physical framework. The analysis also yields a useful, dimensionally homogeneous predictive relationship for bankfull discharge as a function of bankfull width, bankfull depth, bed slope, and bed-material median grain size.