Abstract

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 (nonlinear advection) and externally applied forces, consisting of the net driving force and the viscous resistant. From the physical point of view, the model represents a novel class of nonlinear advective–dissipative system, where classical Voellmy and inviscid Burgers' equations are specifications of this general model. We show that the nonlinear 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, most cost-effective, and 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 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 the steady state has been explained. We reveal the fact that shifting, uplifting, and stretching of the velocity field stem from the forcing and nonlinear 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 uplift, and nonlinear advection. We disclose the fact that the domain translation and stretching solely depend 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 practitioners with key information for instantly and accurately estimating the impact force that is very important in delineating hazard zones and for the mitigation of landslide hazards.

Highlights

  • There are three methods to investigate and solve a scientific problem: laboratory or field data, numerical simulations of governing complex physical–mathematical model equations, or exact analytical solutions of simplified model equations

  • Eq (5) is a fundamental nonlinear partial differential equation, or a nonlinear transport equation with a source, where the source is the external physical forcing. Such an equation explains the nonlinear advection with source term that contains the physics of the underlying problem through the parameters α and β. The form of this equation is very important as it may describe the dynamical state of many extended physical and engineering problems appearing in nature, science, and technology, including viscous–fluid flow, traffic flow, shock theory, gas dynamics, landslides, and avalanches (Burgers, 1948; Hopf, 1950; Cole, 1951; Nadjafikhah, 2009; Pudasaini, 2011; Montecinos, 2015)

  • By rigorous derivations of the exact analytical solutions, we showed that incorporation of the nonlinear advection and external forcing is essential for the physically correct description of the landslide velocity

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Summary

Introduction

There are three methods to investigate and solve a scientific problem: laboratory or field data, numerical simulations of governing complex physical–mathematical model equations, or exact analytical solutions of simplified model equations This is the case for mass movements including extremely rapid flow-type landslides such as debris avalanches (Pudasaini and Hutter, 2007). This paper presents a novel nonlinear advective– dissipative transport equation with a quadratic source term representing the system forcing, containing the physical and mechanical parameters as a function of the state variable (the velocity) and their exact analytical solutions describing the landslide motion. In contrast to the existing models, such as Voellmytype and Burgers-type, the great complexity in solving the new landslide velocity model analytically derives from the simultaneous presence of the internal deformation (nonlinear advection, inertia) and the quadratic source representing externally applied forces (in terms of velocity, including physical parameters). As exact analytical solutions disclose many new and essential physics, the solutions derived in this paper may find applications in environmental, engineering, and industrial mass transport down slopes and channels

Mass and momentum balance equations
The landslide velocity equation
A novel physical–mathematical system
The landslide velocity: simple solutions
Steady-state motion
Negligible viscous drag
Viscous drag included
A mass point motion
The dynamics controlled by the physical and mechanical parameters
The velocity magnitudes
Accelerating and decelerating motions
Velocity described by the space of physical parameters
A model for viscous drag
Landslide motion down the entire slope
The landslide velocity: general solution – I
Derivation of the solution to the general model equation
Recovering the mass point motion
Some particular exact solutions
Some explicit expressions for u in Eq (19)
Description of the general velocity
A fundamentally new understanding
The landslide velocity: general solution – II
A particular solution
Time marching general solution
Landslide stretching
Describing the dynamics
Rolling out the initial velocity
Breaking wave and folding
Recovering Burgers’ model
5.10 The viscous drag effect
Discussion
Advantages of the new model and its solutions
Compatibility, reliability, and generality of the solutions
Implications
Summary
Full Text
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