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