To simulate debris-flow behaviour from initiation to deposition, we derive a depth-averaged, twophase model that combines concepts of critical-state soil mechanics, grain-flow mechanics and fluid mechanics. The model's balance equations describe coupled evolution of the solid volume fraction, m, basal pore-fluid pressure, flow thickness and two components of flow velocity. Basal friction is evaluated using a generalized Coulomb rule, and fluid motion is evaluated in a frame of reference that translates with the velocity of the granular phase, v s . Source terms in each of the depth-averaged balance equations account for the influence of the granular dilation rate, defined as the depth integral of ∇ · v s . Calculation of the dilation rate involves the effects of an elastic compressibility and an inelastic dilatancy angle proportional to m − m eq , where m eq is the value of m in equilibrium with the ambient stress state and flow rate. Normalization of the model equations shows that predicted debris-flow behaviour depends principally on the initial value of m − m eq and on the ratio of two fundamental timescales. One of these timescales governs downslope debris-flow motion, and the other governs pore-pressure relaxation that modifies Coulomb friction and regulates evolution of m. A companion paper presents a suite of model predictions and tests.

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Numerical modelling of transoceanic tsunami propagation, together with the detailed modelling of inundation of small-scale coastal regions, poses a number of algorithmic challenges. The depth-averaged shallow water equations can be used to reduce this to a time-dependent problem in two space dimensions, but even so it is crucial to use adaptive mesh refinement in order to efficiently handle the vast differences in spatial scales. This must be done in a ‘wellbalanced’ manner that accurately captures very small perturbations to the steady state of the ocean at rest. Inundation can be modelled by allowing cells to dynamically change from dry to wet, but this must also be done carefully near refinement boundaries. We discuss these issues in the context of Riemann-solver-based finite volume methods for tsunami modelling. Several examples are presented using the GeoClaw software, and sample codes are available to accompany the paper. The techniques discussed also apply to a variety of other geophysical flows.

Many geophysical flow or wave propagation problems can be modeled with two-dimensional depth-averaged equations, of which the shallow water equations are the simplest example. We describe the GeoClaw software that has been designed to solve problems of this nature, consisting of open source Fortran programs together with Python tools for the user interface and flow visualization. This software uses high-resolution shock-capturing finite volume methods on logically rectangular grids, including latitude--longitude grids on the sphere. Dry states are handled automatically to model inundation. The code incorporates adaptive mesh refinement to allow the efficient solution of large-scale geophysical problems. Examples are given illustrating its use for modeling tsunamis, dam break problems, and storm surge. Documentation and download information is available at www.clawpack.org/geoclawComment: 18 pages, 11 figures, Animations and source code for some examples at http://www.clawpack.org/links/awr10 Significantly modified from original posting to incorporate suggestions of referee

Summary. We describe the extension of high resolution finite volume methods and adaptive refinement for the shallow water equations in the context of tsunami modeling. Godunov-type methods have been used extensively for modeling the shallow water equations in many contexts, however, tsunami modeling presents some unique challenges that must be overcome. We describe some of the specific difficulties associated with tsunami modeling, and summarize some numerical approaches that we have used to overcome these challenges. For instance, we have developed a well-balanced Riemann solver that is appropriate in the deep ocean regime as well as robust in near-shore and dry regions. Additionally, we have extended adaptive refinement algorithms to this application. We briefly describe some of the modifications necessary for using these adaptive methods for tsunami modeling.

We evaluate a new depth-averaged mathematical model that is designed to simulate all stages of debris-flow motion, from initiation to deposition. A companion paper shows how the model's five governing equations describe simultaneous evolution of flow thickness, solid volume fraction, basal pore-fluid pressure and two components of flow momentum. Each equation contains a source term that represents the influence of state-dependent granular dilatancy. Here, we recapitulate the equations and analyse their eigenstructure to show that they form a hyperbolic system with desirable stability properties.To solve the equations, we use a shock-capturing numerical scheme with adaptive mesh refinement, implemented in an open-source software package we call D-Claw. As tests of D-Claw, we compare model output with results from two sets of largescale debris-flow experiments. One set focuses on flow initiation from landslides triggered by rising pore-water pressures, and the other focuses on downstream flow dynamics, runout and deposition. D-Claw performs well in predicting evolution of flow speeds, thicknesses and basal pore-fluid pressures measured in each type of experiment. Computational results illustrate the critical role of dilatancy in linking coevolution of the solid volume fraction and porefluid pressure, which mediates basal Coulomb friction and thereby regulates debris-flow dynamics.

Some landslides move slowly or intermittently downslope, but others liquefy during the early stages of motion, leading to runaway acceleration and high-speed runout across low-relief terrain. Mechanisms responsible for this disparate behaviour are represented in a two-phase, depth-integrated, landslide dynamics model that melds principles from soil mechanics, granular mechanics and fluid mechanics. The model assumes that gradually increasing pore-water pressure causes slope failure to nucleate at the weakest point on a basal slip surface in a statically balanced mass. Failure then spreads to adjacent regions as a result of momentum exchange. Liquefaction is contingent on pore-pressure feedback that depends on the initial soil state. The importance of this feedback is illustrated by using the model to study the dynamics of a disastrous landslide that occurred near Oso, Washington, USA, on 22 March 2014. Alternative simulations of the event reveal the pronounced effects of a landslide mobility bifurcation that occurs if the initial void ratio of water-saturated soil equals the lithostatic, critical-state void ratio. They also show that the tendency for bifurcation increases as the soil permeability decreases. The bifurcation implies that it can be difficult to discriminate conditions that favour slow landsliding from those that favour liquefaction and long runout.