David L. George

Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation

We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Typically, a simple solver for a system of m conservation laws uses m such discontinuities. We present a four wave solver for use with the the shallow water equations-a system of two equations in one dimension. The solver is based on a decomposition of an augmented solution vector-the depth, momentum as well as momentum flux and bottom surface. By decomposing these four variables into four waves the solver is endowed with several desirable properties simultaneously. This solver is well-balanced: it maintains a large class of steady states by the use of a properly defined steady state wave-a stationary jump discontinuity in the Riemann solution that acts as a source term. The form of this wave is introduced and described in detail. The solver also maintains depth non-negativity and extends naturally to Riemann problems with an initial dry state. These are important properties for applications with steady states and inundation, such as tsunami and flood modeling. Implementing the solver with LeVeques wave propagation algorithm [R.J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys. 131 (1997) 327-335] is also described. Several numerical simulations are shown, including a test problem for tsunami modeling.

Tsunami modelling with adaptively refined finite volume methods

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.

The GeoClaw software for depth-averaged flows with adaptive refinement

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 and dam-break flooding problems. Documentation and download information is available at www.clawpack.org/geoclaw.

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I. physical basis

To simulate debris-flow behaviour from initiation to deposition, we derive a depth-averaged, two-phase model that combines concepts of critical-state soil mechanics, grain-flow mechanics and fluid mechanics. The models 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, vs. 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 ∇⋅vs. Calculation of the dilation rate involves the effects of an elastic compressibility and an inelastic dilatancy angle proportional to m−meq, where meq 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−meq 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.

HIGH-RESOLUTION FINITE VOLUME METHODS FOR THE SHALLOW WATER EQUATIONS WITH BATHYMETRY AND DRY STATES

We give a brief review of the wave-propagation algorithm, a highresolution finite volume method for solving hyperbolic systems of conservation laws. These methods require a Riemann solver to resolve the jump in variables at each cell interface into waves. We present a Riemann solver for the shallow water equations that works robustly with bathymetry and dry states. This method is implemented in clawpack and applied to benchmark problems from the Third International Workshop on LongWave Runup Models, including a two-dimensional simulation of runup during the 1993 tsunami event on Okushiri Island. Comparison is made with wave tank experimental data provided for the workshop. Some preliminary results using adaptive mesh refinement on the 26 December 2004 Sumatra event are also presented.

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

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 models 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 large-scale 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 pore-fluid pressure, which mediates basal Coulomb friction and thereby regulates debris-flow dynamics.

High-Resolution Methods and Adaptive Refinement for Tsunami Propagation and Inundation

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.

New methodology for computing tsunami generation by subaerial landslides: Application to the 2015 Tyndall Glacier landslide, Alaska

Landslide-generated tsunamis pose significant hazards and involve complex, multiphase physics that are challenging to model. We present a new methodology in which our depth-averaged two-phase model D-Claw is used to seamlessly simulate all stages of landslide dynamics as well as tsunami generation, propagation, and inundation. Because the model describes the evolution of solid and fluid volume fractions, it treats both landslides and tsunamis as special cases of a more general class of phenomena. Therefore, the landslide and tsunami can be efficiently simulated as a single-layer continuum with evolving solid-grain concentrations, and with wave generation via direct longitudinal momentum transfer—a dominant physical mechanism that has not been previously addressed in this manner. To test our methodology, we used D-Claw to model a large subaerial landslide and resulting tsunami that occurred on 17 October 2015, in Taan Fjord near the terminus of Tyndall Glacier, Alaska. Modeled shoreline inundation patterns compare well with those observed in satellite imagery.

Debris flow runup on vertical barriers and adverse slopes

Runup of debris flows against obstacles in their paths is a complex process that involves profound flow deceleration and redirection. We investigate the dynamics and predictability of runup by comparing results from large-scale laboratory experiments, four simple analytical models, and a depth-integrated numerical model (D-Claw). The experiments and numerical simulations reveal the important influence of unsteady, multidimensional flow on runup, and the analytical models highlight key aspects of the underlying physics. Runup against a vertical barrier normal to the flow path is dominated by rapid development of a shock, or jump in flow height, associated with abrupt deceleration of the flow front. By contrast, runup on sloping obstacles is initially dominated by a smooth flux of mass and momentum from the flow body to the flow front, which precedes shock development and commonly increases the runup height. D-Claw simulations that account for the emergence of shocks show that predicted runup heights vary systematically with the adverse slope angle and also with the Froude number and degree of liquefaction (or effective basal friction) of incoming flows. They additionally clarify the strengths and limitations of simplified analytical models. Numerical simulations based on a priori knowledge of the evolving dynamics of incoming flows yield quite accurate runup predictions. Less predictive accuracy is attained in ab initio simulations that compute runup based solely on knowledge of static debris properties in a distant debris flow source area. Nevertheless, the paucity of inputs required in ab initio simulations enhances their prospective value in runup forecasting.

Combining InSAR and GPS to Determine Transient Movement and Thickness of a Seasonally Active Low‐Gradient Translational Landslide

The combined application of continuous Global Positioning System data (high temporal resolution) with spaceborne interferometric synthetic aperture radar data (high spatial resolution) can reveal much more about the complexity of large landslide movement than is possible with geodetic measurements tied to only a few specific measurement sites. This approach is applied to an ~4 km reactivated translational landslide in the Columbia River Gorge (Washington State), which moves mainly during the winter rainy season. Results reveal the complex three-dimensional shape of the landslide mass, how onset of sliding relates to cumulative rainfall, how surface velocity during sliding varies with location on the topographically complex landslide surface, and how the ground surface subsides slightly in weeks prior to downslope sliding.