A general analytical model for superelevation in landslide

Superelevation is an often observed phenomenon in landslide and debris flow down a complex three-dimensional topography. The degree of superelevation is controlled by the geometry of the channel, the material involved and also the flow dynamics. Empirical methods are usually applied to estimate superelevation. However, those models are incomplete and lack important aspects of the channel geometry, material properties and flow dynamics, instigating serious errors in estimating flow velocities that could only be controlled empirically by introducing ad hoc correction factors. Here, we present new and complete analytical models for superelevation and superelevation velocities down a general topography providing a fully dynamical method. New models formally include essential forces that play an important role in the flow dynamics, namely gravitational forces, topographic- and hydraulic-pressure gradients and Coulomb friction. We discuss the importance of geometry in inducing superelevation and that one directional channel without twist cannot produce superelevation. With the new models we can, in principle, exactly obtain the flow velocities of deformable landslide, dynamic impact pressures and the explicit description of deposition. We have formally provided two alternative analytical representations for superelevation: geometrical and dynamical definitions of superelevation, which is a new concept. We proved that for superelevation to take place, the transversal velocity must have a gradient across the channel. We have analytically constructed a new non-dimensional superelevation number. Superelevation velocity appears to be a non-linear function of the superelevation number. We can now explicitly quantify the superelevation intensity in landslide motion. It has several implications. We proved that superelevation is higher for fluid-saturated debris flows than for dry granular flows. New superelevation models have been validated against a laboratory granular flow down a multi-dimensionally curved channel, and a natural debris flow event in Chamoson, Valais, Switzerland. Our theoretical superelevation velocities appear to be very close to the velocity measured in laboratory and in the field, which however, are largely overestimated by the empirical models. We further validated the model by constructing an exact analytical solution and by applying it to describe superelevation-induced propagation and deposition of the natural debris flow event. New simulations