Topography curvature effects in thin-layer models for gravity-driven flows without bed erosion

Depth-averaged thin-layer models are commonly used to model rapid gravity-driven flows such as debris flows or debris avalanches. However, the formal derivation of thin-layer equations for general topographies is not straightforward. The curvature of the topography results in a force that keeps the velocity tangent to the topography. Another curvature term appears in the bottom friction force when frictional rheologies are used. In this work, we present the main lines of the mathematical derivation for these curvature terms that are proportional to the square velocity. Then, with the SHALTOP numerical model, we quan- tify their influence on flow dynamics and deposits over synthetic and real topographies. This is done by comparing simulations in which curvature terms are exact, disregarded or approximated. With the Coulomb rheology, for slopes θ = 10° and for friction coefficients below μ = tan(5°), neglecting the curvature force increases travel times by up to 10% and 30%, for synthetic and real topographies respectively. When the curvature in the friction force is neglected, the travel distance may be increased by several hundred meters on real topographies, whatever the topography slopes and friction coefficients. We observe similar effects on a synthetic channel with slope θ = 25° and μ = tan(15°), with a 50% increase of the kinetic energy. Finally, approximations of curvature in the friction force can break the non-invariance of the equations and decelerate the flow. With the Voellmy rheology, these discrepancies are less significant. Curvature effects can thus have significant impact for model calibration and for overflows prediction, both being critical for hazard assessment.