The GeoClaw software for depth-averaged flows with adaptive refinement

Marsha J. Berger, David L. George, Randall J. LeVeque, and Kyle Mandli

Courant Institute of Mathematical Sciences, NYU,Cascades Volcano Observatory, U.S. Geological Survey, 1300 SE Cardinal Ct. #100, Vancouver, WA 98683

Department of Applied Mathematics, University of Washington, Box 352420, Seattle, WA 98195-2420

Abstract

Many geophysical flow or wave propagation problems can be modeled with two-dimensional depth-averagedequations, of which the shallow water equations are the simplest example. We describe the GeoClaw soft-ware that has been designed to solve problems of this nature, consisting of open source Fortran programstogether with Python tools for the user interface and flow visualization. This software uses high-resolutionshock-capturing finite volume methods on logically rectangular grids, including latitude–longitude grids onthe sphere. Dry states are handled automatically to model inundation. The code incorporates adaptivemesh refinement to allow the efficient solution of large-scale geophysical problems. Examples are given illus-trating its use for modeling tsunamis, dam break problems, and storm surge. Documentation and downloadinformation is available at www.clawpack.org/geoclaw.

1. Introduction

Many geophysical flow or wave propagation prob-lems take place over very large spatial domains, forwhich detailed three-dimensional modeling of thefluid dynamics is not an efficient option. Fortu-nately, two-dimensional depth-averaged equationssuch as the shallow water equations often providemodels that are sufficiently accurate for many appli-cations. Even with two-dimensional models, how-ever, it is often necessary to use adaptive mesh re-finement (AMR) techniques in order to concentrategrid cells in regions of interest, and to follow suchregions as the flow evolves. This is often the onlyefficient way to obtain results that have sufficientspatial resolution where needed without undue re-finement elsewhere, such as regions the flow or wavehas not yet reached or points distant from the studyarea.

We will briefly describe and illustrate the useof GeoClaw, an open source research code thatuses high-resolution finite volume methods togetherwith adaptive mesh refinement to tackle geophys-ical flow problems of this nature. In particular,this code has recently been used together with theshallow water equations to model tsunamis, dambreaks, and storm surges. In Section 7 we give abrief illustration of each. For other geophysical flow

problems it may be necessary to replace the shallowwater equations by a different set of depth-averagedequations. For example, in modeling landslides, de-bris flows, or lahars, it is necessary to incorporateterms modeling internal stress or pore pressure (e.g.[14, 39]). The software is written in a manner thatallows such extensions. Other depth-averaged mod-els are currently being investigated using GeoClaw,but will not be described in this paper.

GeoClaw is based on the Clawpack software andis incorporated as a part of the general Clawpackdistribution [30]. Clawpack (Conservation LawsPackage) is an open source software package thathas been under development since 1994 and iswidely used for both teaching and research pur-poses. It is designed to solve hyperbolic systems ofpartial differential equations (PDEs) in one, two,and three space dimensions. This class of PDEsgenerally models wave propagation or fluid trans-port, and a wide variety of physical problems giverise to mathematical models of hyperbolic form, in-cluding for example compressible gas dynamics, lin-ear and nonlinear acoustics, and elastic wave propa-gation. The theory of nonlinear hyperbolic systemsand a variety of applications are described in [27],which also describes in detail the high-resolutionfinite volume methods implemented in Clawpack.

Preprint submitted to Elsevier August 1, 2010

Nearly all of the examples given in this text areavailable as working examples via the Clawpackwebsite.

Clawpack is written in a formulation that al-lows the user to specify the system of equationsbeing solved by providing a “Riemann solver” asdescribed in Section 3. The software incorporatesa general form of AMR as reviewed briefly in Sec-tion 4, in a manner that is easy to apply to manyhyperbolic problems. However, there are severaldifficulties that arise when solving depth-averagedequations over realistic topography or bathymetrythat required some substantial modifications to thegeneral approach taken in Clawpack, and the Geo-Claw variant of the code provides an implementa-tion specific to such problems.

In particular, this code addresses the followingissues:

• The flow takes place over topography orbathymetry that may be specified via multi-ple data sets covering overlapping regions atdifferent resolutions. (Henceforth we will gen-erally use the term topography to refer also tobathymetry.)

• Some problems can be tackled on purely Carte-sian grids, but many applications require usinglongitude-latitude grids on the earth’s surface.

• The flow is of bounded extent; the depth goesto zero at the margins and the “wet-dry inter-face” is a moving boundary that must be cap-tured as part of the flow. This is handled byallowing the fluid depth to be zero in some gridcells (“dry cells”). Cells can change dynami-cally between wet and dry to model evolvingflows or inundation, and AMR can be used toprovide sufficient resolution of the shoreline ormargin.

• There often exist nontrivial steady states (suchas an ocean at rest) that should be maintainedexactly. Often the desired flow or wave prop-agation is a small perturbation of this steadystate, as in tsunami or storm surge modeling.This requires the use of a “well-balanced” nu-merical method as discussed in Section 3.

These issues and the algorithms in GeoClaw arediscussed in more detail elsewhere [6, 17, 18, 20, 32]and here we give only a brief summary of some keyaspects of the numerial algorithms (in Section 3)and the AMR procedure (in Section 4).

The computational core of GeoClaw is written inFortran, but a user interface written in Python isprovided to simplify the setup of a single run, or ofa series of runs as is often required for parameterstudies. Python and Matlab plotting tools are alsoprovided for viewing the results in various forms,either on the dynamically changing set of adaptivegrids or on a set of fixed grids, or in other forms suchas gauge plots of depth vs. time at fixed spatial lo-cations. Some of these software tools are describedbriefly in Section 6, and more details can be foundin the on-line documentation [31].

2. Depth-averaged mathematical models

The simplest depth-averaged set of fluid equa-tions in two lateral space dimensions are the shallowwater equations

ht + (hu)x + (hv)y = 0,

(hu)t + (hu2 +12gh2)x + (huv)y = −ghBx −Du,

(hv)t + (huv)x + (hv2 +12gh2)y = −ghBy −Dv,

(1)

where u(x, y, t) and v(x, y, t) are the depth-averaged velocities in the two horizontal directions,B(x, y, t) is the topography or bathymetry, andD = D(h, u, v) is the drag coefficient. Coriolisterms can also be added to the momentum equa-tions. These equations have the form

qt + f(q)x + g(q)y = ψ(q, x, y) (2)

where q = (h, hu, hv) is the vector consisting of thedepth and momentum of the fluid. In the absenceof bathymetry (B ≡ constant, so Bx = By = 0)and drag (D ≡ 0), the source terms would be zero(ψ ≡ 0) and these equations would express the con-servation of mass and horizontal momentum. Weuse conservative finite volume methods that in gen-eral conserve mass to machine precision (since thereis no source term in the mass equation) and wouldalso conserve momentum in the absence of sourceterms. This is true even when AMR is applied ex-cept perhaps in cells that intersect the coastline, asdiscussed further in Section 4.

Note that for an ocean at rest, in which h(x, t) +B(x, y) ≡ 0 (sea level) in any wet cells, the topog-raphy source terms exactly cancel the derivativesof the hydrostatic pressure 1

2gh2. Maintaining this

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balance numerically is critical and is discussed inSection 3. The drag term could have many forms;for the experiments reported here we use

D =gM2

√(u2 + v2)h5/3

(3)

where M is the Manning coefficient, which we taketo be 0.025. (Typical values for the Manning coef-ficient for a given substrate are empirically based.See [11] for a description and examples of valuesused in various applications.)

For tsunami modeling, the equations (1) are gen-erally considered to provide a very good model, ashas been confirmed in comparisons done by manygroups (e.g. [48, 25, 33, 41]) Most tsunamis are gen-erated by motion of the sea floor, due to an earth-quake or submarine landslide, which sets the entirewater column in motion. The wave length is gener-ally very long compared to the depth of the ocean,and under these conditions shallow water equationsare appropriate. In Section 7 we illustrate the useof GeoClaw for tsunami modeling.

For other applications it is less clear that theclassical shallow water equations are sufficient. Forshallow flow on steep terrain, such as following adam break for example, vertical acceleration termsmay need to be added to improve the model. How-ever, the simple equations (1) are often still used formany practical problems and can give fairly accu-rate results. In Section 7.3 we display some resultsfrom [19].

The shallow water equations are also often usedto model storm surge, e.g. [40, 47]. We presentsome preliminary results in Section 7.4 illustratinghow the GeoClaw software can be applied in thiscontext.

3. Numerical methods

The algorithms used in GeoClaw are describedin detail elsewhere and here we only give a briefsummary with pointers to other sources for furtherreading. Clawpack provides a general implemen-tation of “wave-propagation algorithms”, a class ofhigh-resolution finite volume methods in which eachgrid cell is viewed as a volume over which cell aver-ages of the solution variables q are computed. Logi-cally rectangular grids are used and Qn

ij denotes thecell average in cell (i, j) at time tn. In each timestep the cell averages are updated by waves propa-gating into the grid cell from each cell edge. These

are Godunov-type methods in which the waves arecomputed by solving a “Riemann problem” at eachcell edge that consists of the one-dimensional shal-low water equation in the direction normal to thecell edge, with piecewise constant data determinedby the cell averages and bathymetry values on eachside of the interface. The topography source termsin the momentum equations are incorporated intothe Riemann problem as described in [18]. Thef-wave formulation of the wave-propagation algo-rithms is used, which was originally presented forthe shallow water equations in [4]. In this approachit is the difference in the flux normal to the cell in-terface that is split into propagating waves (ratherthan the jump in Q), but only after modifying themomentum components of the flux difference bythe topography source terms. This is done in sucha way that the methods are well-balanced for theocean at rest: if the initial data is in equilibriumwith zero velocities and a flat surface (h + B =constant) then the modified flux difference is thezero vector, leading to zero-strength f-waves and nochange to the solution. This approach to well bal-ancing is discussed in more detail in [4], [29], andSection 17.14 of [27].

The f-waves modify the cell averages on each sideof the interface. We also solve a “transverse Rie-mann problem” in which the waves moving normalto the cell edge are split in the transverse directionand modify the cell averages in adjacent rows ofgrid cells. This improves stability and accuracy ofthe method and this general approach is discussedin more detail in [26] and Chapters 20–21 of [27], forexample. The f-waves are also used to define cor-rection terms modeling second derivatives normalto the interface that, together with the transverseterms, make the method second order accurate forsmooth solutions. Before calculating these terms,however, wave limiters are applied to the f-waves toreduce their amplitude in regions where the solutionvaries rapidly. This results in a “high-resolutionmethod” that avoids nonphysical oscillations in re-gions where the solution is rapidly varying. Thismethodology has been well developed in the contextof shock-capturing for general nonlinear hyperbolicsystems of equations and leads to a robust method.

Developing a Riemann solver that works robustlyin the presence of dry states is particularly challeng-ing — it must handle the case where one state inthe Riemann problem is already dry as well as sit-uations where a cell dries out as a wave recedes,and must work robustly when the topography has

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arbitrary jumps from one cell to the next. Detailsof the solver we use are given in [19].

The topography source terms (those involving Bx

and By on the right hand side of (1)) are incorpo-rated in the Riemann solver, in order to obtain awell-balanced method and to handle the dry stateproblem. The drag source terms are handled viaa fractional step approach: after each time step ofthe hyperbolic problem a time step is taken in whichthe momenta are adjusted due to the drag terms.Coriolis terms can also be incorporated in the samemanner and this is included as an option in Geo-Claw, though for tsunami modeling at least thisappears to be a negligible effect, both in or own ex-periments and elsewhere in the literature, e.g., [25].

In general Clawpack allows the solution of hy-perbolic problems on any logically rectangular grid,with an arbitrary mapping function specified thatmaps points in computational space (a rectangu-lar grid with uniform spacing) to the physical do-main. In two space dimensions the grid cells arealways assumed to be quadrilaterals with linear celledges joining vertices that are obtained by apply-ing the mapping function to the rectangular grid ofvertices in the computational domain. The wave-propagation algorithms with transverse Riemannsolvers work very robustly, even on highly distortedgrids or with non-smooth mapping functions. Un-like many approaches to mapped grids, we do notincorporate metric terms that depend on deriva-tives of the mapping function into the differentialequations. Instead, we always solve Riemann prob-lems for the original set of equations in the di-rection orthogonal to each cell edge. The lengthsof the edges and the area of the quadrilateral cellthen come into the formulas for updating cell aver-ages. The transverse wave propagation also takesaccount of the fact that adjacent cell edges are notnecessarily orthogonal to one another. For prop-agation on the earth, we use distances and areasas measured on a sphere, although in principle thiscould be replaced by a geoid or other surface. Cur-rently in GeoClaw the domain choices are limitedto the sphere with longitude-latitude coordinatesor purely Cartesian domains, primarily because themore general routines for integrating topographydata sets (see Section 5) over a general quadrilateralhave not yet been developed. Longitude-latitudecoordinates are suitable for modeling tsunamis orstorm surge on the earth, where the domain of inter-est is bounded away from the poles. In recent workwe have also explored another class of quadrilateral

grids on the sphere and present some results usingAMR for a tsunami-type problem over the wholesphere in [5], for synthetic bathymetry where therequired integrals could be more easily computed.

4. Adaptive mesh refinement

In this section we describe the patch-based meshrefinement that is used to span the orders of magni-tude needed for example to go from ocean scale tocoastline resolution in a tsunami model. Multiplelevels of patches can be used until a sufficiently fineresolution is reached. We also give an overview ofthe numerical algorithms needed to initialize andremove fine grid patches, and present the organi-zation of the time stepping procedures on the gridhierarchy.

The time step on the refined patches is chosen sothat stability of the explicit finite volume methodis maintained. This generally requires refining intime by the same factor as in space. For example,if the level 2 grids are refined in both x and y by afactor of 4 relative to level 1, then four time stepson all level 2 grids must be taken for each time stepon level 1. The code is organized so that the timestep is first taken on level 1, which covers the entiredomain. Then four time steps are taken on eachof the level 2 grids. In each time step it is neces-sary to fill in “ghost cell” values around the edgesof each level 2 grid in order to provide boundaryconditions for the time step. For each ghost cell,the value is either taken from a neighboring gridat the same level, if one exists, or otherwise is ob-tained by space-time interpolation from the valueson the underlying coarse grid, which has alreadybeen advanced in time.

This same procedure is used recursively at all lev-els: after each time step on the level 2 grids, therewill be typically be several time steps taken for alllevel 3 grids and so on. This AMR procedure is de-scribed in more detail in [8] and has been success-fully used for many years in Clawpack for problemssuch as shock wave propagation where dozens ofgrid patches are used to track shock waves obliqueto the grid.

The application to tsunami modeling promptedthe addition of a new feature to the code: the capa-bility of specifying anisotropic refinement in time,in which the time step on fine grids is refined by adifferent factor than the refinement in space. Thisis crucial for problems where very fine grids are usedonly near the coast of an ocean, since in the shallow

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water equations the wave speed is given by√gh. In

an ocean with a maximum depth of 4000m, say, thisgives a wave speed of 200m/s (and in some regionsthe maximum ocean depth is much greater). If afine grid level covers only portions of the continen-tal shelf with a maximum depth of 100m, say, thenthe maximum wave speed on this level is roughly32m/s. Hence the refinement factor in time couldbe up to 6 times smaller than the spatial refinementfactor on this level. Since the bulk of the compu-tational work is often on the finest grids, this canmake a substantial difference in computing time.

Grids are refined by flagging cells where the reso-lution is insufficient and then clustering the flaggedcells into refinement patches. Flagging is done ei-ther using an error estimate (Richardson extrapola-tion), by examining gradients of the solution (whichwill detect where the largest waves are), or for thetsunami application simply by flagging cells wherethe surface elevation is perturbed from sea levelabove a specified level. In addition, it is possibleto specify rectangular regions where a certain levelof resolution is required. This can be used to in-sure that particular portions of a coastline are al-ways refined to a resolution of meters whereas thedeep ocean uses a mesh width of many kilometers.These routines are all part of the GeoClaw software,controlled by user-specified tolerances along withoptional user-specified regions in space and timewhere a minimum and maximum allowable level re-finement is specified. Regardless of which method(or combination of methods) is used to control therefinement criteria, the result is a set of flagged cellsneeding to be covered by finer grid(s).

Using heuristics from the pattern recognition lit-erature [9], these flagged cells are clustered intogrid patches that are efficient, in the sense thatthe grids do not contain too many unflagged cells(which would be wasteful), and there are not toomany of them (since there is boundary overheadassociated with each fine grid patch). The gridsalso should obey a proper nesting criterion — a gridpatch at level 4 should be surrounded by a level 3patch, and not border directly on level 2 patches.Figures 3 and 4 show several frames from a 5 levelcomputation. In this example the grids were refinedby a factor of 4 in going from each level to the next.

Every few timesteps the features in the solutionneeding refinement will have moved, and the gridpatches should move too. The grids do not actu-ally move; rather, at discrete times new grid patchesare created and their solution is interpolated from

the finest previously existing grids, which are thenremoved. This interpolation step must be donecarefully. For example, a constant sea level shouldbe maintained even in the presence of variablebathymetry so that no waves are generated solelyfrom grid refinement. This is accomplished by in-terpolating the surface elevation h + B for coarsergrids and then computing the depth h in the finecells by subtracting the fine cell value of topogra-phy B. This maintains conservation of mass pro-vided that the fine and coarse topography are con-sistent, in the sense that the topography value usedin a coarse cell should be the average of the val-ues in all fine cells that cover this cell. This is ac-complished by computing exact integrals of a singlepiecewise bilinear representation of the topography,as described further in the next section.

Similarly when a grid is removed, the coarsegrid solution underneath it should be the volume-weighted average of the fine grid cells it containedso that mass is not lost or gained. Most of the timethese numerical procedures are straightforward, butthere are difficulties associated with the wet-dry in-terfaces. A coarse cell that covers a shoreline regionwill be either wet or dry, depending on the level ofthe averaged topography. Suppose it is dry, for ex-ample. When the cell is refined then typically someof the fine cells will have to have nonzero depthin order to represent the shoreline and to maintainthe constant sea level required before a wave arrives.Hence it is essential that water be introduced (massincreased) in this situation. These subtleties will bedescribed more fully in [6].

5. Topography data sets

To use GeoClaw the user must provide one ormore files that specify the topography for the ter-rain on which the flow evolves. Each topographydata set specifies the z coordinate (relative to sealevel, for example) at a set of points on a rectangu-lar grid (a longitude–latitude grid if this coordinatesystem is being used). Several different formats areallowed (see the documentation [31]).

Appropriate data sets for many regions of theearth are available online, for example from theNational Geophysical Data Center (NGDC) [37].A few data sets for test problems are available inthe GeoClaw topographic database [16], and morewill be added in the future. The test problems inGeoClaw include Python scripts to automatically

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download this data as needed. Some example in-stead use synthetic data (for example the tsunamimodel presented in Section 7.1) and again a Pythonscript is provided to create this when needed.

Some applications also require a dataset that de-scribes the motion of the topography relative to aninitial topography, for example if seafloor motionresulting from an earthquake or submarine land-slide is used to generate a tsunami. In this case oneor more files must also be provided to specify therelative displacement at one or more times.

For the examples provided in GeoClaw, the com-mand “make topo” in an applications directory willdownload or create the required data, both for thebackground topography and for any relative dis-placements.

Often more than one topography file is used atdifferent resolutions. For example, in a tsunamisimulation a large region of the ocean may be mod-eled, for which a fairly coarse resolution such asthe 10-minute or 4-minute ETOPO2 data availablefrom NGDC is sufficient. These have resolutions ofroughly 18.5km or 7.5km respectively in each di-rection near the equator. Since the wavelength oftsunami waves is typically 10s to 100s of km thisis sufficient for modeling propagation across theocean. However, it is not sufficient for modelinginundation of specific regions along the coast, andso this data must generally be supplemented withone or topography files at much higher resolutionover small regions.

In GeoClaw, an arbitrary number of topographyfiles can be provided for a single run and at eachpoint in space the topography will be determinedfrom the dataset covering this point at the finestresolution. The user should be aware, however, thatthis means there will generally be discontinuitiesin the effective topography along the boundaries offine scale datasets, and some preprocessing of thedata may be necessary.

We use finite volume methods as described in Sec-tion 3 and so the first element of the vector Qn

ij ingrid cell (i, j), the value hn

ij , represents the cell aver-age of the fluid depth. We also need a cell averageof the topography in each grid cell, a value Bij .Topography data sets generally give the pointwisevalue of z on a grid of spatial locations. To con-vert this into cell averages, we construct a piece-wise bilinear function that interpolates the point-wise values and then compute the exact integral ofthis interpolating function over a grid cell to ob-tain Bij . This is easy to do if there is a single

best-resolution data set in the region around a cell,but if a grid cell covers an area where two or moredifferent data sets must be sampled then compu-tation of the integral is more difficult. This oftenhappens in realistic tsunami simulations. Fine gridtopography for a small region of the coast may lieentirely within one grid cell on the coarsest compu-tational grids, for example. We have implementedthis quadrature in full generality to guarantee thatthe topography values used are consistent betweendifferent adaptive mesh refinement levels.

6. Software tools and user interface

More details about many of the tools mentionedin this section can be found in the on-line GeoClawdocumentation [31].

Early versions of the Clawpack software were en-tirely in Fortran 77 with a set of Matlab plottingroutines for visualizing the results. Some special-ized versions of the Matlab plotting routines werecreated for dealing with topographic data sets andare available in GeoClaw. Recently, however, themain development of plotting tools for Clawpackhas shifted from Matlab to Python for a numberof reasons. Many users of Clawpack do not haveaccess to Matlab and it is desirable to have anopen source alternative. Moreover, in the past fewyears substantial improvements have been made inPython plotting packages that provide quality thatequals or exceeds that of Matlab graphics. For two-dimensional plots of the type shown in this paper,we use the matplotlib module [34].

For three-dimensional surface plots we arecurrently investigating several options, includ-ing Mayavi [35], which is included (along withmatplotlib) in the Enthought Python Distribution[15], and VisIt, an open source visualization pack-age being developed at Lawrence Livermore Na-tional Laboratory [45]. VisIt provides more func-tionality for large scale visualization problems andis designed to work well with AMR data and dis-tributed memory supercomputers. Development of3D plotting tools for GeoClaw and Clawpack moregenerally is an on-going project.

We have developed a Python plotting modulethat allows the user to easily specify a set of plots tobe produced for each frame of a simulation. WhenAMR is being used it is necessary to loop over allgrids and combine the solution on each grid into asingle plot. It is often desirable to combine severalplots in a single figure. For example, we may want

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to do a pseudo-color plot of the water surface el-evation using one color map while the topographyin dry regions is also plotted with a different colormap. Contour lines of bathymetry may be addedto this along with indications of locations of tidegauges, resulting in a plot such as the ones shownin Section 7. The logic of looping over the gridsis handled by the plotting module and the inter-face provides a mechanism for specifying a varietyof different plots or combinations of plots on a sin-gle axis without the user needing to deal with theAMR data structures. Other useful tools such ascodes for dealing with the topographic data setsand colormaps appropriate for these problems arealso included.

There are also several ways the user can viewplots coming from a simulation. There is an inter-active module Iplotclaw for stepping through theframes of a simulation and producing the plots onthe screen, facilitating data exploration via zoom-ing in on features of interest, for example. Alterna-tively, it is easy to generate a set of hardcopy filesin formats such as png or jpg, one for each figureat each time frame, together with a set of webpagesdesigned to easily browse through the collection ofplots. Webpages are automatically created to loopthrough all frames of each figure, creating an ani-mation that is often extremely useful in developinga better understanding of the time-evolution of theflow. This set of webpages also simplifies the pro-cess of archiving past experiments for later viewing,or for sharing sharing simulations with others. Anexample can be viewed on the webpages that ac-company this paper [7]. Many of these tools havebeen developed with the aim of encouraging users toadopt the paradigm of reproducible research in com-putational science. The approach we have takenwith Clawpack is discussed in more detail in [28].

7. Applications

We briefly describe four applications of GeoClaw.The first is a new synthetic tsunami test problem.The second example models the 27 February 2010earthquake in Chile as an illustration of the use ofreal data sets, and is included in the GeoClaw dis-tribution. The third example is the simulation ofthe Malpasset dam catastrophe of 1959, which hasbeen well studied and often used as a benchmarkproblem. We give a summary of results obtainedwith GeoClaw that are discussed in more detail in

[19]. The final example shows some preliminary re-sults in which GeoClaw is applied to a test problemfor storm surge.

7.1. Synthetic tsunami test

First we present some results obtained using asynthetic data set that has been designed to illus-trate the power of our adaptive refinement approachwith a realistic range of spatial scales, but in a con-text where it is also possible to assess the accuracyof the solution. We start with a radially symmetricocean that has a depth depending only on distancefrom some central point, which we take to be lon-gitude x0 = 0 and latitude y0 = 40◦N. Distance ismeasured as the great circle distance on a sphericalearth by the formula

d(x, y; x0, y0) = 2R arcsin((sin(0.5(y − y0))2

+ cos(y0) cos(y) sin(0.5(x− x0))2)1/2).

where R = 6367.5 × 106m is the average radius ofthe earth. In this formula we assume the longitude-latitude pairs (x, y) and (x0, y0) are in radians toavoid the factors π/180. The bathymetry profile isshown in Figure 1(b), where the horizontal scale isin kilometers and the vertical scale in meters. Thecentral portion of the ocean is flat and is boundedby a continental slope and flat continental shelf, fol-lowed by a linear beach. We use a continuous piece-wise cubic function whose derivative is also contin-uous except at r3, the start of the beach:

B(d) =

z1 if d ≤ r1C(d) if r1 ≤ d ≤ r2z2 if r2 ≤ d ≤ r3z2 + σ(r − r3) if d ≥ r3.

(4)

where the cubic C(d) is given by

C(d) = z1 +(x2 − z1)(d− r1)

r2 − r1

(1− 2

d− r2r2 − r1

)and smoothly connects the ocean floor to the con-tinental shelf. Here r1 = 1.5 × 1500 × 103m is thestart of the continental slope, r2 = 1580 × 103mis the start of the flat continental shelf, and r3 =1640 × 103m is the start of the beach, which hasslope σ = 0.02. We take z1 = −4000m for thedepth of the ocean and z2 = −100m for the depthof the shelf. The initial shoreline is at 1645×103m.

As a smaller scale feature we add an island on thecontinental shelf at one point that can be varied.

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Figure 1: (a) Geometry of the radially symmetric ocean. The outer solid curve is the position of the shoreline, with constantdistance from the center when measured on the surface of a sphere. The dashed line shows the extent of the continental shelf.The boxes labelled Test 1 and Test 2 are regions where an island is located in the tests presented in the following figures. Thesmall circle near the center shows the extent of the hump of water used as initial data. (b) The topography defined by thepiecewise cubic function (4). (c) A zoom view of the topography of the continental shelf along the ray going throug the centerof the island.

Figure 2: Topography in regions near the island for Test 1 and Test 2. The solid contour lines are shoreline (B = 0) and thedashed contour lines are at elevations B = −40,−80,−120,−160m. Note that the continental shelf has a uniform depth of−100m. The level 4 grid is shown onshore. The rectangle around the island shows the level 5 grid in each computation, whichare refined by an additional factor of 4 in each direction from the level 4 grids. The location of four gauges is also shown. Thetime history of the surface at these gauges is shown in Figure 5.

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The island is defined by a piecewise function of dis-tance about a specified center point (x1, y1) (againusing great circle distance). The center point is cho-sen to have distance 1600km from the center of theocean (hence 45km off shore). The island is speci-fied by B2(d) = 120(1− (d/r4)2(1− 2(d− r4)/r4))for d ≤ r4 and zero outside this radius. Here d isthe distance from (x1, y1) and the radius of the is-land is r4 = 30 × 103m. The island rises from thecontinental shelf to a peak height of 20m above sealevel. Figure 1(c) shows how the cross section ismodified along the radial slice that passes throughthe center of the island.

The full bathymetry at any longitude–latitudepoint is thus given by

B(x, y) = B1(d(x, y; x0, y0) +B2(d(x, y; x1, y1))

A Python script that can be used to generatebathymetry files with arbitrary resolution is pro-vided in the directory for this example, which canbe downloaded from [7]. Figure 1(a) shows the en-tire ocean in longitude-latitude coordinates. Thedashed line indicates the extent of the continentalshelf. As initial data we take an ocean at rest andadd a Gaussian hump of water at the center of theocean:

h(x, y, 0) = 20 exp(−0.5× 10−9d(x, y; x0, y0)2)−B(x, y).

The innermost contour on Figure 1(a) shows thecontour where the initial hump has an elevation of2m above sea level, 10% of its peak value.

We have performed several different runs in whichthe central point of the island (x1, y1) always hasthe same distance from (x0, y0) but is located atdifferent angular locations. The solution to theshallow water equations with this set up should beexactly radially symmetric if there were no island.With the island the solution with different island lo-cations should ideally be rotations of one another.This is a good test of the numerical method sincethe grid orientation to the shoreline near the islandvaries greatly depending on the its location. Thetwo nearshore boxed regions in Figure 1(a) indicatethe two test cases considered here. In each case anisland is centered in the 2-degree square box seenin Figure 1(a). See Figure 4 for closeups of theseregions for the two tests.

The longitude-latitude domain [−20, 20]×[20, 60]is covered with a coarse 40 × 40 grid, so the meshwidth is one degree on this level 1 grid. Note

that one degree of latitude is about 111km andone degree longitude varies from 38km at 60◦N to96km at 20◦N. We use 5 levels of mesh refinement,with refinement factor 4 in going from each levelto the next, and hence a total refinement factor of44 = 256 from the coarsest to finest grids. The level5 grid has a mesh width 1/256 ≈ .0039 degrees,or 434m in the latitude direction. (More levels orhigher refinement ratios could be used to refine fur-ther at particular points along the shoreline. Inthe tsunami computation presented in [32], for ex-ample, we used a total factor of 4096 refinementbetween coarsest and finest levels.)

Refinement to level 3 is allowed over the portionof the ocean that is in the direction of the studyarea. Refinement to levels 4 and 5 is only allowedin a small region near the island, as seen in Figure 4.

The Gaussian initial hump spreads out into awave that propagates radially. Figure 3 shows thesea surface elevation at time t = 5000 seconds, fora calculation in which only 3 levels of refinementhave been allowed so far, in the case where the is-land is located in the square indicated as Test 1 inFigure 1(a). The edges of refinement patches aredrawn and the spreading wave is poorly resolvedon the coarse grid, but well resolved in the refinedregions.

Note that there is very little evidence of spuriousreflected waves at the refinement boundaries in thisfigure. This is true in general with the AMR ap-proach used in Clawpack. Moreover, for problemswhere the computational domain does not coverthe full ocean (such as in Figure 6), it is impor-tant that the method does not generate spuriousnumerical reflections at these outflow boundaries.The Godunov-type wave-propagation algorithms doa very good job of providing non-reflecting bound-ary conditions simply by using constant extrapola-tion into ghost cells at the domain boundaries: thevalues in interior cells adjacent to the boundariesare copied into ghost cells. Then solving the Rie-mann problems at the interfaces along the bound-aries results in zero-strength waves propagating intothe domain, and hence no apparent reflection of theout-going waves.

Note also that the calculation is done on the sur-face of the sphere and so the wave spreads as acircle on the sphere, but in longitude-latitude spacethe wave front is not circular. The wave appears tobe on track to reach all points at the shore simula-teously, as should happen, and this is confirmed inFigures 4 and 5 which show nearly identical time

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Figure 3: Computed surface elevation for Test 1 at two different times. Left: at time t = 5000 seconds, at which point at mostthree refinement levels are allowed, and only in part of the domain. Right: at time t = 10000, when 5 levels are allowed butonly near the island. See Figure 4 for a zoom of the region around the island.

histories at two locations near the shore.The top row of Figure 4 shows three later times

for an island located in the box labelled Test 1 inFigure 1(a). The bottom row of this figure showsthe same three times for a second test run, in whichthe island was located in the box labelled Test 2 inFigure 1(a).

Figure 5 shows tide gauge data from the com-putation at four gauge locations on the radial linefrom the center of the ocean that passes throughthe center of the island. Gauges 1 and 2 are dis-tance 1570km and 1590km from the center, on theseaward side of the island, while Gauges 3 and 4 aredistance 1610km and 1630km from the center of theocean, on the lee side. Gauges 1 and 4 are in regionsthat are never refined beyond level 4, while Gauges2 and 3 are with the region refined to level 5. Eachfigure shows two curves for each gauge, one fromTest 1 (solid lines) and one from Test 2 (dashedlines). In principle these should lie on top of eachother and in fact the agreement is quite good.

Each of these calculations (for Test 1 and Test 2)required roughly 18 minutes of one processor on a2.26GHz MacBook Pro laptop computer. Approxi-mately 55 million grid cells were advanced in timeover the entire computation, of which roughly 14Mwere on level 3, 28M on level 4, and 11M on level5. Levels 1 and 2 combined accounted for less than1M.

7.2. The 27 February 2010 tsunamiThe GeoClaw code has been used to model sev-

eral historical tsunamis using bathymetry and to-

pography data sets obtained from NDGC [37] andother sources. Some simulations of the 26 Decem-ber 2004 tsunami in the Indian Ocean followingthe Sumatra-Andaman earthquake are presented in[20, 32] and several other studies are under way tobe published elsewhere. See [31] for links to someanimations.

Here we present some results for the 27 February2010 Chile event. This example is included in Geo-Claw as a sample to illustrate the use of topographydata sets. In this case we use 10-minute ETOPO2topography from NGDC [37]. The seafloor motionis generated using the Okada model [38], whichtranslates earthquake parameters taken from [43]into seafloor deformation, using a general Pythonfunction implementing the Okada model that is in-cluded in GeoClaw.

Figure 6 shows four frames from a 3 level simula-tion. Level 1 has cell size 2 degrees and refinementfactors of 2 and 6 are used, so the finest grid hascell size of 10′ and matches the topography data.The dot labelled 32412 shows the position of DARTbuoy 32412 (Deep-ocean Assessment and Reportingof Tsunamis [12]), which collected data during theevent. Figure 7 shows this data together with mod-eling results from the MOST model used at NOAA(Method of Splitting Tsunamis [36, 42]). We havesuperimposed the GeoClaw results on the NOAAplot. The results agree fairly well in view of thecoarseness of this calculation. Moreover, our com-putation was done with a simplified source modelwhereas the MOST calculation was done using a

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Figure 4: Top row: Surface elevation at two times for Test 1. Bottom row: Same times for Test 2. At time t = 10000 secondsthe wave is approaching the shore. At time t = 12000 the wave has reflected off the shore and is outgoing. In all cases thesame contour levels are shown: the solid contours are at elevations of 0.4m and 0.8m above sea level, the dashed contours areat the same elevations below sea level. The inner rectangle is the interface between level 4 and level 5 grids. The level 4 gridis shown onshore. The level 5 grid is finer by a factor of 4 in each direction.

Figure 5: Comparison of gauge output from Test 1 and Test 2, showing the surface elevation as a function of time for thegauges shown in Figure 2. In each case the solid blue curve is from Test 1 and the dashed red curve is from Test 2.

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linear combination of 7 fault models chosen by asource inversion procedure to yield a good matchto the DART data by the MOST model.

This computation ran in about 4 minutes on aMacBook Pro laptop, advancing 25.8M grid cells.By contrast, a 360× 360 uniform grid computationon this domain (at the resolution of the finest AMRgrid) takes about 16 minutes of computer time, ad-vancing 121.8M grid cells. The results computed onthe uniform grid agree very well with the AMR so-lution. The results at DART buoy 32412 are nearlyidentical as seen in Figure 7. The advantage ofAMR is clear, even for this problem where we arenot zooming in on regions of the coast to modelinundation.

The uniform grid calculation exhibits a largernumber of “cells advanced per second of compu-tation” (126K vs. 107K), due to the overhead ofadaptive refinement. This overhead is greater inGeoClaw than normally with Clawpack because ofthe need to recompute the topography in each gridcell each regridding time. In this calculation we re-grid every 3 time steps to insure waves do not leaverefinement patches between regridding. In the fu-ture we hope to improve the efficiency of regriddingthe topography.

7.3. Riverine and overland floodingThe shallow water equations are often used to

model riverine or overland flooding problems, suchas those due to dam/levee breaches (e.g. [21, 44, 2]).For flooding problems in rugged mountainous ter-rain, where rapidly varying contours in the topogra-phy create highly irregular domains, adaptive meshrefinement can be a valuable tool because the op-timal grid resolution is highly spatially and tem-porally dependent yet unpredictable before doingthe computation. A common approach for model-ing floods in complicated topographic regions is touse static irregular meshes that are fit to the to-pography in some fashion (e.g. [44]). However, byusing adaptive mesh refinement, uniform rectangu-lar grids can be used for such problems, resolvingthe flood on an evolving patchwork of finer gridsthat advance with the flood waves through topogra-phy. This makes it much simpler to set up a generalproblem.

We have tested GeoClaw in this context by mod-eling the historic Malpasset dam-break flood, whichoccurred in southern France in 1959 (see [19]).This thin-arch dam failed suddenly and explosively,sending a roughly 60 m deep flood wave into the

winding ravine below, eventually inundating theReyran River Valley. This disaster has served asa valuable test case for code validation due to theextensive field data, such as high water marks, col-lected after the event. For this problem we used a6.47 km east–west (x-direction) by 16.58 km north–south (y-direction), rectangular grid. The coars-est level 1 grid was 16 by 40 grid cells respectively(∆x × ∆y ≈ 404.4m × 414.5m) . Level 2, level 3and level 4 grids were then used to refine the flowingwater, with refinement ratios of 8, 4 and 4, yield-ing ≈ 3m × 3m grid cells on the finest level. Somesnapshots from the simulation are shown in Fig-ure 8. Comparision of the maximum surface eleva-tion computed by GeoClaw was compared to othercodes and empirical field and model data, shown inFigure 9. A detailed explanation of this test prob-lem and comparison can be found in [19].

We are currently developing depth-averagedmodels for two-phase flows consisting of granular-fluid mixtures, applicable to debris-flow floods andvolcanic lahars or mudslides (e.g. [23, 13]). Theseflows often occur in rugged mountainous regions,and present many of the same difficulties as over-land flooding in terms of domain geometry.

7.4. Storm surgeStorm surges can lead to potentially catastrophic

flooding of coastal areas during hurricanes or othertropical storms. Although the physics of these flowsare not fully understood, current modeling effortsuse depth-averaged techniques in order to modellarge domains efficiently (e.g. [24, 46, 47]). Adap-tive refinement strategies lend themselves to theresolution of important features in these flows, inorder to capture the critically important coastalregions where topographical effects such as leveesmust be resolved on the order of meters [40]. Sub-stantial efforts have been made to predict surge inthe case of real storms such as Hurricane Katrinain the region around New Orleans [47].

The primary difference between tsunamis andstorm surge is the manner in which the waves aregenerated. In particular, wind stress on the sur-face needs to be handled carefully since it is theprimary mode of momentum transfer to the water.Since this is a surface effect rather than a motion ofthe seafloor, it is less clear that shallow water equa-tions are a sufficient model. However, most currentoperational models of surge use these equations.

We are in the preliminary stages of modelingstorm surges using shallow water equations in Geo-

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Figure 6: Four frames from a 3 level simulation of the 27 February 2010 Chile event. The location of DART buoy 32412 is alsoindicated, for which the time history is shown in Figure 7. Note the reflection from the Gallapagos near the equator at 6 hours.

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Figure 7: Time history at DART buoy 32412. The thick blue and green line are from GeoClaw computations using AMR anda uniform fine grid respectively.

Claw. Here we present one simple test problem,using idealized bathymetry and an empirical formfor a storm’s wind and pressure fields introducedin [22]. The domain is a large (700 × 600km) butshallow (100m) sea, flat over most of the domainand bounded by a linear beach. The profile anddomain is shown in Figure 10. The storm trav-els in the x-direction as indicated in figure 10(b)at constant velocity 5 m/s with a ramp up timeof 12 hours. The parameters used for this stormare taken from [22], where they were determinedfrom a fit of the 1974 Hurricane Tracy. Wind forc-ing is handled as a source term in the momentumcomponents, implemented in a fractional step pro-cedure in the same manner as bottom friction. Weuse piecewise-continuous wind friction coefficientscalculated as in [46]. The response to the stormis shown in Figure 11, where gauge data is shownalong with current contours. In Figure 11(c) wecan see the storm is shedding vortices, which de-tach themselves from the main flow at certain in-tervals and remain persistent. GeoClaw identifiesthese structures even after the storm has left thedomain and continues to construct grids to followthem adaptively as the simulation reaches its con-clusion in Figure 11(d).

In this calculation three grid levels are used. The

coarse grid has a 5 km resolution and refinement bya factor of 2 is used in each additional level.

8. Conclusions and future plans

The GeoClaw software project was initiated withthe TsunamiClaw code developed by one of the au-thors in his 2006 PhD thesis [17], and has undergoneseveral more years of development and testing, pri-marily on tsunami simulation. The version recentlyreleased with Clawpack 4.5 (in July, 2010) is fairlyrobust and stable, but will continue to be developedand improved in the future. We are also workingon incorporating OpenMP into the code to take ad-vantage of multi-core shared memory computers.

A number of on-going projects by the authorsmake use of this software. In particular we arestudying extensions to debris flows and landslides.We are also developing multi-layer versions of thecode, both to model tsunamis generated by sub-marine landslides and as a possible improvementover single-layer shallow water equations for mod-eling storm surge due to surface forcing. Multi-layer shallow water equations introduce a numberof new mathematical and numerical challenges (seee.g. [1, 3, 10]).

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Figure 8: A GeoClaw simulation of the Malpasset dam-break flood from [19]. Initial and later times are shown from left toright. A very coarse level 1 grid is used where the flood has yet to arrive. The reservoir in the northeast corner (upper right)is resolved on a level 2 grid. Level 3 and 4 grids surround and evolve with the flood as it winds down into the Reyran RiverValley. Individual level 3 and 4 grids are outlined; their grid lines are omitted for clarity.

Figure 9: Comparison of the maximum water elevation produced by a GeoClaw simulation from [19] with those presentedby other authors [44, 21] using static topography-fit irregular meshes. Simulation results are compared with field data forhigh-water marks collected after the flood at 17 surveyed points (left), and with a physical scale-model experiment, where themaximum water level was recorded by electronic gauges at 12 locations (right). Figures are reprinted from [19] (permissionpending).

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(a) (b)

Figure 10: (a) Profile view of the bathymetry used. (b) The computational domain with the storm track marked and gaugelocations used in Figure 11. The diamonds represent locations of the storm at the corresponding times. Note that at timet = 0 the storm is at coordinates (x, y) = (0, 0) and has ramped up to full strength; it was started at the left boundary at timet = −12.

Extensions to handle more general quadrilateralgrids are also of interest, either to use other gridson the sphere, as mentioned at the end of Section 3,or for problems such as river flows or dam breakswhere it would be advantageous to define a coarsequadrilateral grid that follows a long river valleyand then use adaptive refinement to better resolvethe actual flow domain.

We use the Subversion version control softwareand the Trac interface as a development wiki andfor its ticket system for bug tracking. These can befound via the Clawpack webpage [30]. Animationsand some GeoClaw code to accompany the simula-tions presented in this paper are available at [7].

Acknowledgments. This research was sup-ported in part by DOE grant DE-FG02-88ER25053,AFOSR grant FA9550-06-1-0203, NSF Grant DMS-0914942, ONR Grant N00014-09-1-0649, and theFounders Term Professorship in Applied Mathe-matics at the University of Washington.

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