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.

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 also the case for mass movements including extremely rapid flow-type landslides such as debris avalanches (Pudasaini and Hutter, 2007). The dynamics of a landslide are primarily controlled by the flow velocity. Estimation of the flow velocity is key for assessment of landslide hazards, design of protective structures, mitigation measures, and land use planning (Tai et al., 2001; Pudasaini and Hutter, 2007; Johannesson et al., 2009; Christen et al., 2010; Dowling and Santi, 2014; Cui et al., 2015; Faug, 2015; Kattel et al., 2018). Thus, a proper understanding of landslide velocity is a crucial requirement for an appropriate modeling of landslide impact force because the associated hazard is directly and strongly related to the landslide velocity (Huggel et al., 2005; Evans et al., 2009; Dietrich and Krautblatter, 2019). However, the mechanical controls of the evolving velocity, run-out, and impact energy of the landslide have not yet been understood well.

Due to the complex terrain, infrequent occurrence, and very high time and cost demands of field measurements, the available data on landslide dynamics are insufficient. Proper understanding and interpretation of the data obtained from field measurements are often challenging because of the very limited nature of the material properties and the boundary conditions. Additionally, field data are often only available for single locations and determined as static data after events. Dynamic data are rare (de Haas et al., 2020). So, many of the low-resolution measurements are locally or discretely based on points in time and space (Berger et al., 2011; Schürch et al., 2011; McCoy et al., 2012; Theule et al., 2015; Dietrich and Krautblatter, 2019). Therefore, laboratory or field experiments (Iverson et al., 2011; de Haas and van Woerkom, 2016; Lu et al., 2016; Lanzoni et al., 2017; Li et al., 2017; Pilvar et al., 2019; Baselt et al., 2021) and theoretical modeling (Le and Pitman, 2009; Pudasaini, 2012; Pudasaini and Mergili, 2019) remain the major source of knowledge in landslide and debris flow dynamics. Recently, there has been a rapid increase in comprehensive numerical modeling for mass transports (McDougall and Hungr, 2005; Medina et al., 2008; Cascini et al., 2014; Frank et al., 2015; Iverson and Ouyang, 2015; Cuomo et al., 2016; Mergili et al., 2020a, b; Qiao et al., 2019; Liu et al., 2021). However, to a certain degree, numerical simulations are approximations of the physical–mathematical model equations. Their usefulness is often evaluated empirically (Mergili et al., 2020a, b). In contrast, exact analytical solutions (Faug et al., 2010; Pudasaini, 2011) can provide better insights into the complex flow behaviors, mainly the velocity. Moreover, analytical and exact solutions to nonlinear model equations are necessary to elevate the accuracy of numerical solution methods (Chalfen and Niemiec, 1986; Pudasaini, 2011, 2016; Pudasaini et al., 2018). For this reason, here, we are mainly concerned with presenting exact analytical solutions for the newly developed general landslide velocity equation.

Since Voellmy's pioneering work, several analytical models and their solutions have been presented in the literature for mass movements including extremely rapid flow-type landslide processes, avalanches, and debris flows (Voellmy, 1955; Salm, 1966; Perla et al., 1980; McClung, 1983). However, on the one hand, all of these solutions are effectively simplified to the mass point or center of mass motion. None of the existing analytical velocity models consider advection or internal deformation. On the other hand, the parameters involved in these models only represent restricted physics of the landslide material and motion. Nevertheless, a full analytical model that includes a wide range of essential physics of the mass movements incorporating important process of internal deformation and motion is still lacking. This is required for the more accurate description of landslide motion. Moreover, recently presented simple analytical solutions for mass transports considered debris avalanches (Pudasaini, 2011), two-phase flows (Ghosh Hajra et al., 2017, 2018), landslide mobility (Pudasaini and Miller, 2013; Parez and Aharonov, 2015), fluid flows in debris materials (Pudasaini, 2016), mud flow (Di Cristo et al., 2018), granular front down an incline (Saingier et al., 2016), granular monoclinal wave (Razis et al., 2018), and submarine debris flows (Rui and Yin, 2019). However, neither a more general landslide model as we have derived here nor the solution for such a model exists in the literature.

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. The new landslide velocity model and its analytical solutions are more general and constitute the full description for velocities with a wide range of applied forces and the internal deformation. Importantly, the newly developed landslide velocity model covers both the classical Voellmy and inviscid Burgers' equations as special cases; it unifies and extends them further, but it also describes fundamentally novel and broad physical phenomena.

It is a challenge to construct exact analytical solutions even for the simplified problems in mass transport (Pudasaini, 2011, 2016; Di Cristo et al., 2018; Pudasaini et al., 2018). In contrast to the existing models, such as Voellmy-type 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). However, here, we construct various analytical and exact solutions to the new general landslide velocity model by applying different advanced mathematical techniques, including those presented by Nadjafikhah (2009) and Montecinos (2015). We reveal several major novel dynamical aspects associated with the general landslide velocity model and its solutions. We show that a number of important physical phenomena are captured by the new solutions. This includes landslide propagation and stretching, wave generation and breaking, and landslide folding. We also observe that different methods consistently produce similar analytical solutions. This highlights the intrinsic characteristics of the landslide motion described by our new model. 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.

A geometrically two-dimensional motion down a slope is considered. Let

We start with the multi-phase mass flow model (Pudasaini and Mergili, 2019) and include viscous drag (Pudasaini and Fischer, 2020). For simplicity, we assume that the relative velocity between coarse and fine solid particles (

The momentum balance equation (Eq.

Now, with the notation

Equation (

From the structure, Eq. (

Exact analytical solutions to simplified cases of nonlinear debris avalanche model equations provide important insight into the full behavior of the system and are necessary to calibrate numerical simulations. Physically meaningful exact solutions explain the true and entire nature of the problem associated with the model equation (Pudasaini, 2011; Faug, 2015) and should thus be developed, analyzed and properly understood prior to numerical simulations. These exact analytical solutions provide important insights into the full flow behavior of the complex system (Pudasaini and Krautblatter, 2021a) and are often needed to calibrate and validate the numerical solutions (Pudasaini, 2016) as a prerequisite before running numerical simulations based on complex numerical schemes. This is very useful to interpret complicated simulations and/or avoid mistakes associated with numerical simulations.

One of the main purposes of this contribution is to obtain exact analytical velocities for the landslide model (Eq.

For a sufficiently long time and sufficiently long slope, the time-independent steady-state motion can be developed. Then, Eq. (

In situations when the Coulomb friction is dominant and the motion is slow, the viscous drag contribution can be neglected (

The landslide velocity distributions down the slope as a function of position both without and with drag given by Eqs. (

In general, depending on the magnitude of the net driving force (that also includes the Coulomb friction), the viscous drag coefficient and the magnitude of the velocity, either

The major aspect of viscous drag is to bring the velocity (motion) to a terminal velocity (steady, uniform) for a sufficiently long travel distance. This is achieved by the following relation obtained from Eq. (

In what follows, unless otherwise stated, we use the plausibly chosen physical parameters for rapid mass movements: slope angle of about 50

Time evolution of the landslide velocity down the slope with drag given by Eq. (

Assume no or negligible local deformation (e.g.,

Since

Figure

The solutions in Eqs. (

The influence of the model parameters

In Figs.

A landslide can reach its maximum or terminal velocity after a relatively short travel distance or time with a value on the order of 50 m s

Depending on the magnitudes of the involved forces and whether the initial mass was triggered with a small (including zero) velocity or with high velocity, e.g., by strong seismic shaking or when high potential energy is available and is converted quasi-instantaneously into kinetic energy (the situation prevails when the vertical height drop of the detachment area is huge and the slope angle of the terrain is high), Eq. (

We now have two possibilities. First, we can describe

There are explicit models for the interfacial drags between the particles
and the fluid (Pudasaini, 2020) in the multi-phase mixture flow (Pudasaini and Mergili, 2019). However, there is no clear representation of the viscous drag coefficient for a landslide, which is the drag between the landslide and the environment. Often in applications, the drag coefficient (

Furthermore, we note that following the classical method by Voellmy (Voellmy, 1955) and extensions by Salm (1966) and McClung (1983), the velocity models in Eqs. (

We mention that, for two-dimensional cycloidal or parabolic tracks, Gauer (2018) presented analytical velocities for the mass block motions with simple dry Coulomb or constant energy dissipation along the track. For such idealized path geometries he found an important relationship: the maximum front velocity,

For shallow motion the velocity may change locally, but the change in the landslide geometry may be parameterized. In such a situation, the force produced by the free-surface pressure gradient can be estimated. A particular situation is the moving slab for which

The Lagrangian description of a landslide motion is easier. However, the Eulerian description provides a better and more detailed picture
as it also includes the local deformation due to the velocity gradient. So, here we consider the model equation (Eq.

Here, we present the detailed derivation of the analytical solution (Eq.

Utilizing these functions in Theorem 4.1, we finally constructed the exact analytical solution (Eq.

The amazing fact is that the newly constructed general analytical solution (Eq.

Moreover, we mention that Eqs. (

However, we note that

Velocity distribution given by Eq. (

Here, we present some interesting particular exact solutions of Eq. (

Moreover, with the definition of

Interestingly, by directly taking the limit as

Evolution of the velocity field along the slope as given by Eq. (

Time evolution of the velocity field as given by Eq. (

For a properly selected function

For example, with

Furthermore, Fig.

The velocity profiles for a landslide with the mass point motion as given by Eq. (

A crucial aspect of a complex analytical solution is its proper
interpretation. The general solution (Eq.

The new general solution (Eq.

Below, we have constructed a further exact analytical solution to our velocity equation based on the method of Montecinos (2015). Consider the model (Eq.

The solution strategy is as follows: use the definition of

The solution method involves some sophisticated mathematical procedures. However, here we present a compact but quick solution description to our problem. The equivalent ordinary differential equation to the partial differential equation system (Eq.

It is interesting to observe the structure of the solutions given by Eqs. (

The velocity profile down a slope as a function of position for a landslide given by Eqs. (

For the choice of the initial condition

Time evolution of velocity profiles of propagating and stretching landslides down a slope and as functions of position including the internal deformations as given by the general solution in Eqs. (

Any initial condition can be applied to the solution system (Eqs.

Time evolution of the front and rear positions of the landslide as it moves down the slope including the internal deformation given by the general solution in Eqs. (

The stretching (or deformation) of the landslide propagating down the slope depends on the evolution of its front (

In order to better understand the rate of stretching of the landslide, in Fig.

Time stretching of the landslide down the slope including the internal deformation given by the general solution in Eqs. (

Spatial

The dynamics observed in Figs.

It is compelling to see how the solution system (Eqs.

The breaking wave and folding as a landslide propagates down a slope. Panel

Next, we show how the new model (Eq.

Recovering the Burgers' shock formation and breaking of the wave by the solution system (Eqs.

Wave breaking and folding are often-observed important dynamical aspects in mass transport and formation of geological structures. Figure

As the external forcing vanishes, i.e., as

It is important to understand the dynamic control of the viscous drag on the landslide motion. For this, we set

The control of the viscous drag on the dynamics of the landslide. The net driving force is set to zero, i.e.,

Exact analytical solutions of the underlying physical–mathematical models significantly improve our knowledge of the basic mechanism of the problem. On the one hand, such solutions disclose many new and essential physics and may thus find applications in environmental and engineering mass transports down natural slopes or industrial channels. The reduced and problem-specific solutions provide important insights into the full behavior of the complex landslide system, mainly the landslide motion with nonlinear internal deformation together with the external forcing. On the other hand, exact analytical solutions to simplified cases of nonlinear model equations are necessary to calibrate numerical simulations (Chalfen and Niemiec, 1986; Pudasaini, 2011, 2016; Ghosh Hajra et al., 2018). For this reason, this paper is mainly concerned about the development of a new general landslide velocity model and construction of several novel exact analytical solutions for landslide velocity.

Analytical solutions provide the fastest, cheapest, and probably the best solution to a problem as measured from their rigorous nature and representation of the dynamics. Proper knowledge of the landslide velocity is required in accurately determining the dynamics, travel distance, and enormous destructive impact energy carried by the landslide. The velocity of a landslide is associated with its internal deformation (inertia) and the externally applied system forces. The existing influential analytical landslide velocity models do not include many important forces and internal deformation. The classical analytical representation of the landslide velocity appear to be incomplete and restricted from both the physics and the dynamics point of view. No velocity model has been presented yet that simultaneously incorporates inertia and the externally applied system forces that play a crucial role in explaining important aspects of landslide propagation, motion, and deformation.

We have presented the first-ever physics-based, analytically constructed, simple but more general landslide velocity model. There are two main collective model parameters: the net driving force and drag. 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. In this regard, we have presented a novel dynamical model for landslide velocity that precisely explains both the deformation and motion by quantifying the effect of nonlinear advection and the system forces.

Different exact analytical solutions for landslide velocity constructed in this paper independently support each other. These physically meaningful solutions can potentially be applied to calculate the complex nonlinear velocity distribution of the landslide. Our new results reveal that solutions to the more general equation for the landslide motion are widely applicable.
The new landslide velocity model and its advanced exact solutions have made it possible to analytically study complex landslide dynamics, including nonlinear propagation, stretching, wave breaking, and folding. Moreover, these results clearly indicate that proper knowledge of the model parameters

The new model may describe the complex dynamics of many extended physical and engineering problems appearing in nature, science, and technology – connecting different types of complex mass movements and deformations. Specifically, the advantage of the new model equation is that the more general landslide velocity can now be obtained explicitly and analytically, which is very useful in solving relevant engineering and applied problems, and it has enormous application potential.

There are three distinct situations in modeling the landslide motion: (i) the spatial variation of the flow geometry and velocity can be negligible for which the entire landslide effectively moves as a mass point without any local deformation. This refers to the classical Voellmy model. (ii) The geometric deformation of the landslide can be parameterized or neglected; however, the spatial variation of the velocity field may play a crucial role in the landslide motion. In this circumstance, the landslide motion can legitimately be explained by the full form of the new landslide velocity equation (Eq.

The models in Eq. (

Burgers' equation has no external forcing term. The solution domain remains fixed and does not stretch and propagate downslope. So, the initial velocity profile deforms and the wave breaks within the fixed domain. In contrast, our model (Eq.

Within their scopes and structures, many of the analytical solutions constructed in Sects. 3–5 are similar. This effectively implies the physical aspects of our general landslide velocity model (Eq.

Structurally, the solutions presented in Sect. 3 are only partly new, yet they are physically substantially advanced. However, in Sects. 4 and 5 we have presented entirely novel solutions both physically and structurally. From a physical and mathematically point of view, particularly important is the form of the general velocity model (Eq.

Moreover, as viewed from the general structure of the model (Eq.

The new model (Eq.

As the analytically obtained values represent the velocity of natural landslides well, technically, this provides a very important tool for landslide engineers and practitioners in quickly, efficiently, and accurately determining landslide velocity. The general solutions presented here reveal an important fact: accurate information about the mechanical parameters, state of the motion, and initial condition is very important for the proper description of the landslide motion. We have extracted some interesting particular exact solutions from the general solutions. As direct consequences of the new general solutions, some important and nontrivial mathematical identities have been established that replace very complex expressions by straightforward functions.

While existing analytical landslide velocity models cannot deal with the internal deformation and mostly fail to integrate a wide spectrum of externally applied forces, we developed a simple but general analytical model that is capable of including both of these important aspects. In this paper, we (i) derived a general landslide velocity model applicable to different types of landslide motions, (ii) solved it analytically to obtain several exact solutions as a function of space and time for landslide motion, and (iii) highlighted the essence of the new model. The model includes the internal deformation due to nonlinear advection and the external nonlinear forcing consisting of the extensive net driving force and viscous drag. The model describes a dissipative system and involves dynamic interactions between the advection and external forcing that control the landslide deformation and motion. Our model constitutes a unique and new class of nonlinear advective–dissipative system with quadratic external forcing as a function of state variable, containing all system forces. The new equation may describe the dynamical state of many extended physical and engineering problems appearing in nature, science, and technology. There are two crucial novel aspects: first, it extends the classical Voellmy model and additionally explains the dynamics of a locally deforming landslide, providing a better and more detailed picture of the landslide motion. Second, it is a more general formulation but can also be viewed as an extended inviscid, nonhomogeneous, dissipative Burgers' equation by including the nonlinear source term as a quadratic function of the field variable. The source term accommodates the mechanics of an underlying problem through the net driving force and the dissipative viscous drag.

Due to the nonlinear advection and quadratic forcing, the new general landslide velocity model poses a great mathematical challenge to derive explicit analytical solutions. Yet, we constructed several new and general exact analytical solutions in more sophisticated forms. These solutions are strong, recover all the mass point motions in many different ways, and provide a much wider spectrum for the landslide velocity than the classical Voellmy and Burgers' solutions. The major role is played by the nonlinear advection and system forces. The general solutions provide essentially new aspects in our understanding of landslide velocity. We have also presented a new model for the viscous drag as the ratio between one-half of the system force and the relevant kinetic energy.

With the general solution, we revealed that different classes of landslides can be represented by different solutions under the roof of one velocity model. General solutions allowed us to simulate the progression and stretching of the landslide. We disclosed the fact that the shifting and stretching of the velocity field stem from the external forcing and nonlinear advection. After a long time, as drag strongly dominates the system forces, the velocity gradient vanishes, and thus the stretching ceases. Then, the landslide propagates down the slope just at a constant (steady-state) velocity. The general solution system can generate complex breaking waves in advective mass transport and describe the folding process of a landslide. Such phenomena have been presented and described mechanically for the first time. The most fascinating feature is the dynamics of the wave breaking and the emergence of folding. This happens collectively as the solution system simultaneously introduces three important components of the landslide dynamics: downslope propagation and stretching of the domain, velocity uplift, and breaking or folding in the frontal part while stretching in the rear. This physically proves that nonuniform motion is the basic requirement for the development of breaking wave and emergence of the landslide folding. This is a novel understanding. We disclosed the fact that the translation and stretching of the domain, as well as lifting of the velocity field, solely depend on the net driving force. Similarly, the viscous drag fully controls the shock wave generation, wave breaking, and folding, as well as the magnitude of the landslide velocity. Furthermore, the new model can describe the deposition or the halting process of the mass transport. As the external forcing vanishes, the general solution automatically reduces to the classical shock wave generated by the inviscid Burgers' equation. This proves that the inviscid Burgers' equation is a special case of our general model.

The theoretically obtained velocities are close to the often-observed values in natural events including landslides and debris avalanches. This indicates the broad application potential of the new landslide velocity model and its exact analytical solutions in quickly solving engineering and technical problems as well as in accurately estimating the impact force that is very important in delineating hazard zones and for the mitigation of landslide hazards.

The relevant data supporting the findings of this study are available within the paper.

The physical–mathematical models were developed by SPP, who also designed and wrote the paper, interpreted the results, and edited the paper through reviews. MK contributed to the discussions of the results with enhanced descriptions to better fit broader geosciences audiences.

At least one of the (co-)authors is a member of the editorial board of

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Shiva P. Pudasaini acknowledges the financial support provided by the Technical University of Munich with the Visiting Professorship Program and the international research project AlpSenseRely – Alpine remote sensing of climate‐induced natural hazards – from the Bayerisches Staatsministerium für Umwelt und Verbraucherschutz, Munich, Bayern. We thank the reviewers and the associate editor Jens Turowski for their constructive comments and suggestions that helped to substantially improve the paper. We are grateful to Martin Mergili and Ivo Baselt for fruitful discussions. This paper is based on arXiv:2103.10939v1 at

This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.

This paper was edited by Jens Turowski and reviewed by two anonymous referees.