The Landslide Velocity

Abstract

Proper knowledge of velocity is required in accurately determining the enormous destructive energy carried by a landslide. We present the first, simple and physics-based general analytical landslide velocity model that simultaneously incorporates the internal deformation (non-linear advection) and externally applied forces, consisting of the net driving force and the viscous resistant. From the physical point of view, the model stands as a novel class of non-linear advective − dissipative system where classical Voellmy and inviscid Burgers' equation are specifications of this general model. We show that the non-linear advection and external forcing fundamentally regulate the state of motion and deformation, which substantially enhances our understanding of the velocity of a coherently deforming landslide. Since analytical solutions provide the fastest, the most costeffective and the best rigorous answer to the problem, we construct several new and general exact analytical solutions. These solutions cover the wider spectrum of landslide velocity and directly reduce to the mass point motion. New solutions bridge the existing gap between the negligibly deforming and geometrically massively deforming landslides through their internal deformations. This provides a novel, rapid and consistent method for efficient coupling of different types of mass transports. The mechanism of landslide advection, stretching and approaching to the steady-state has been explained. We reveal the fact that shifting, up-lifting and stretching of the velocity field stem from the forcing and non-linear advection. The intrinsic mechanism of our solution describes the fascinating breaking wave and emergence of landslide folding. This happens collectively as the solution system simultaneously introduces downslope propagation of the domain, velocity up-lift and non-linear advection. We disclose the fact that the domain translation and stretching solely depends on the net driving force, and along with advection, the viscous drag fully controls the shock wave generation, wave breaking, folding, and also the velocity magnitude. This demonstrates that landslide dynamics are architectured by advection and reigned by the system forcing. The analytically obtained velocities are close to observed values in natural events. These solutions constitute a new foundation of landslide velocity in solving technical problems. This provides the practitioners with the key information in instantly and accurately estimating the impact force that is very important in delineating hazard zones and for the mitigation of landslide hazards.

The Mechanics of Landslide Mobility with Erosion

“…However, we mention, that the exact solution of ( 18) can also be obtained in Eulerian form, but is very demanding mathematically. Pudasaini and Krautblatter 67 have presented various exact analytical solutions for landslide velocity, however, without considering the erosion effects. Here, we focus on the velocity of an erosive landslide considering the case where there is an increase in the landslide velocity due to the erosion-induced entrainment, i.e., for P M > 0 corresponding to λ b > 1/2.…”

Helmoltz problem for the Riccati equation from an analogous Friedmann equation

“…where c 0,1 are constants, which describes many physical systems including, e.g., falling raindrops [21,22] or charges in constant electric field, avalanches [15], debris slides [16], geomagnetic fields [17], and box models of ocean basins [18]; the Riccati equation is also related to the Schrödinger, the Ermakov-Pinney, and other equations of fundamental physics [19,20]. A second order equation is naturally associated with Eq.…”