An analytical solution to river profile concavity and downstream fining

Abstract:

We present an analytical solution to the steady state upward-concave bed profile, as well as downstream fining, for a river dominated by gravel and sand. The model is based on (i) the conservation equation of streamwise momentum of the flow, (ii) the conservation equation of the mass of each grain size fraction in the surface layer of the bed (the Hirano equation) yet including sediment abrasion, and (iii) the conservation equation of total sediment mass (the Exner equation). The model includes downstream fining induced by abrasion as well as by grainsize-selective transport of gravel and sand. In order to arrive at an analytical solution, the model is deliberately kept simple through assumptions such as a constant width and no tributaries. The model is then reduced to the case of steady state conditions, which means that all time derivatives in the equations are set to zero. We find a solution to the steady state streamwise profile of both the bed slope and the volume fraction of gravel in the surface layer of the bed. Like existing empirical predictors, the analytical solution is of an exponential type. The main variables that affect the solution are the total load at the upstream boundary, the gravel fraction in this upstream load, the abrasion coefficient, the grain sizes of the sediment, and the water discharge at the upstream boundary. The below image shows an example of how the gravel fraction in the upstream load affects the solution to the longitudinal profiles of, respectively, bed slope (S), the gravel fraction at the bed surface (F_g), and the mean grain size of the sediment at the bed surface (D_gs). We can see how an increase in the gravel fraction in the upstream load results in a larger overall slope and an increase in profile concavity. It also induces an increase of the gravel fraction at the bed surface.