Deriving formulas for an unsteady virtual velocity of bedload tracers

摘要

In a flume experiment with steady flow conditions, H. A. Einstein recognised the transport of bedload particles as consisting of steps of rolling, sliding, or saltation with intermittent rest periods, and introduced the concept of an average, virtual' transport velocity. This virtual velocity then has also been derived from tracer studies in the field by dividing the travelled distance of a tracer by the duration of competent flow. As a consequence, the virtual velocity in the field is represented by one single value only, despite the unsteady flow variables. Tracer measurements in a river have not been yet used to express transport velocity as a direct function of these actual variables, and insights from tracer measurements into the processes of sediment transport remain limited. In particular, the unsteady conditions for bedload in the field have impeded the derivation of sediment transport characteristics as determined from laboratory experiments, as well as the transfer of laboratory insights to a field setting. We introduce a method of data regression for the derivation of an unsteady' virtual velocity from repeated surveys of tracer positions. The regression program called graVel (provided as supplementary material) relates the integral of an excess flow variable term to measured travel distances, yielding the most probable threshold value for entrainment and the coefficient of linear and non-linear formulas. An extended regression allows additional fitting of the exponent in non-linear formulas. Application to published tracer data from the Mameyes River, Puerto Rico, shows that the unsteady virtual velocity is more likely governed by non-linear relations to excess Shields stress, similar to bedload transport, than by relations linking the particle velocity linearly to excess shear velocity. Partial agreements with non-dimensional results derived from the larger, non-wadeable Mur River encourage the establishment of a generalised formula for the unsteady vir