Distinguishing run-up height from pressure distribution during avalanche impact on narrow obstacles: mechanisms and semi-empirical prediction from three-dimensional simulations

In many avalanche engineering guidelines, run-up height is assumed to define the vertical extent of significant impact pressure of snow avalanches on narrow obstacles, such as cableway masts or transmission towers. Laboratory experiments indicate that this assumption is invalid in fast, inertia-driven regimes; however, the underlying mechanisms remain poorly understood, limiting the physical basis for improving design practice. To clarify these mechanisms and refine engineering formulations, we conduct a comprehensive numerical investigation using a three-dimensional Material Point Method, varying avalanche velocity and rheological parameters. Our results substantiate that run-up height and the vertical extent of significant impact pressure, which we term pressure height, are distinct quantities governed by different mechanisms. Both include a gravity-driven component controlled by avalanche rheology and independent of flow velocity. In fast flows, however, the two quantities diverge: run-up height gains a dominant inertia-driven component that scales with the square of the flow velocity, while avalanche rheology controls energy dissipation. Pressure height, by contrast, remains velocity-independent. This divergence arises from flow deflection at a dead zone near the obstacle base, where momentum is redirected and impact pressure concentrates. Building on these distinctions, we propose separate semi-empirical formulations for run-up height and pressure height. The run-up formulation follows the structure of the most common avalanche engineering guidelines, but replaces simplified avalanche-type classifications with a rheological parametrization. The pressure height relation adopts the same framework, while reflecting its distinct, velocity-independent governing mechanism.