The role of bed-load thickness closure in a two-phase morphodynamical model

In the present paper the mathematical character of the two-phase depth-averaged morphodynamical model proposed by Di Cristo et. al (2014) is discussed. In particular, in this model the bed-load concentration is dynamically evaluated provided that an algebraic expression for bed-load thickness is assumed. To this aim, some adaptations of empirical correlations proposed for uniform equilibrium flow conditions may be considered, modifying the differential structure of the problem. In this paper the model eigenstructure is studied considering thee different expressions of the bed-load thickness, showing that the mathematical character of the model is significantly affected by the adopted relationship. In particular, when relationships accounting for only the bed-load volume for unit bottom area or both the solid phase velocity and the bed-load volume for unit bottom surface are considered the model is hyperbolic. In contrast, a hyperbolicity loss is detected for the expression which accounts for only the liquid velocity. The domains of nonhyperbolicity are individuated in terms of the relevant dimensionless parameters and represented through shaded maps. The results of the analysis aim to guide the users to define the correct number of boundary conditions and to choose the most appropriate numerical solver in simulating fast geomorphic transients.