On evolution of bed material waves in alluvial rivers

Abstract

Field and laboratory measurements have shown distinct characteristics of bed sediment waves under differing conditions, whilst their theoretical interpretation has emerged to be equivocal. This note aims to clarify the interpretation of evolution of bed material waves. The complete set of governing equations for the flow-sediment-morphology system is deduced to demonstrate its universally hyperbolic nature, irrespective of the sediment transport functions implemented to close the equations. The hyperbolic nature can admit not only attenuating bed material waves, but also shock-like waves that are not unusual in the real world. It is suggested that the theory of dispersion/diffusion is not universally appropriate for evolution of bed material waves.

More on the evolution of bed material waves in alluvial rivers

“…That is, the extra terms in Cao and Carling (2003) are negligible in most rivers. The extra terms in Equations 5 and 2 are not universally negligible.…”

“…The formulations of Cao and Carling (2003) for the conservation of total flow and sediment mass (equations 1 and 3 in their paper, which also appear in Cao et al (2002) as equations 1 and 3, corresponding to Equations 1 and 2 below) contain additional terms that have not been included in the work of most previous authors.…”

“…The last term on the left-hand side of Equation 1 and the second term on the left-hand side of Equation 2 are the terms added by Cao and Carling (2003); they argue for the significance of these terms in morphodynamic modelling. In this section we suggest that Equation 1 may not be correct, and then provide our proposal for a replacement.…”

“…(In the present contribution the terms 'sediment wave' and 'sediment pulse' are used interchangeably.) This recent short communication (Cao and Carling, 2003) served three primary purposes: (a) to offer additional terms in the governing equations of river morphodynamics; (b) to demonstrate that the governing equations are universally hyperbolic; and (c) to point out limitations of the recent work of Lisle et al (2001). Here we suggest that one of Cao and Carling's governing equations may not be correct, and the additional terms are likely to be negligible in most applications of interest, whether field or experimental.…”

Kinematic Wave Theory for Transient Bed Sediment Waves in Alluvial Rivers

“…͑8͒-͑10͔͒, the mathematical formulation has been closed by relating the sediment flux in the movable bed layer to the flow variables ͑sediment transport function͒ as q s = u h where , , and are parameters ͑Mahmood 1975; Ribberink and Van Der Sande 1985;Vreugdenhil and de Vries 1973͒. In the case of the simple dynamic wave ͓Eqs. ͑5͒-͑7͔͒ or the full diffusion wave approach, the formulation is, in addition to the sediment transport function, closed by relating the suspended sediment concentration in the water flow layer to the flow variables as c = ␦u h , where parameters ␦, , and are functions of water flow and sediment characteristics ͑Ching and Cheng 1964;de Vries 1975;Pianese 1994;Cao and Carling 2003͒. For example, if Velikanov's approach ͑Ching and Cheng 1964͒ is employed for relating the suspended sediment concentration to the flow variables, i.e., c = ͑u 3 / gv f h͒, where = coefficient of sediment transport capacity; g = gravitational acceleration ͑L / T 2 ͒; v f = average fall velocity of sediments ͑L / T͒; ␦ = / ͑gv f ͒; =3; and = −1.…”