Physics of the Earth and Planetary Interiors 171 (2008) 187–197

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Physics of the Earth and Planetary Interiors

journa l homepage: www.e lsev ier .com/ locate /pepi

A community benchmark for subduction zone modeling

Peter E. van Kekena,∗, Claire Currieb, Scott D. Kingc, Mark D. Behnd, Amandine Cagnioncleh,Jiangheng Hee, Richard F. Katz f,g, Shu-Chuan Lin i, E. Marc Parmentierh, Marc Spiegelmanf, Kelin Wange

a Geological Sciences, University of Michigan, 1100 North University Avenue, Ann Arbor, MI 48109, USAb Department of Physics, University of Alberta, 11322-89 Avenue, Edmonton, AB, T6G 2G7, Canadac Department of Geosciences, Virginia Tech, 4044 Derring Hall, Blacksburg, VA 24061, USAd Department of Geology and Geophysics, Woods Hole Oceanographic Institution, Clark 260B, MS22, Woods Hole, MA 02543, USAe Pacific Geoscience Centre, Geological Survey of Canada, Sidney, BC V8L 4B2, Canadaf Department of Earth and Environmental Sciences, Lamont-Doherty Earth Observatory, P.O. Box 1000, Palisades, NY 10964, USAg Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 0WA, UKh Geological Sciences, Brown University, 324 Brook Street, Providence, RI 02912, USAi Department of Geosciences, National Taiwan University, P.O. Box 13-318, Taipei 106, Taiwan

a r t i c l e i n f o

Article history:Received 31 October 2007Received in revised form 15 April 2008Accepted 25 April 2008

Keywords:

a b s t r a c t

We have developed a suite of benchmarks to facilitate the comparison of numerical models for the dynam-ics and thermal structure of subduction zones. The benchmark cases are based on a thermomechanicalapproach in which the slab is prescribed kinematically and the wedge flow is computed dynamically. Wepropose various cases to investigate the influence of boundary conditions and rheology on wedge flowand resulting thermal structure. A comparison between the codes suggest that accurate modeling of the

Subduction zonesThermal structureGN

thermal field requires a good implementation of the velocity discontinuity along the seismogenic zoneand high resolution in the thermal boundary layers. A minor modification to the boundary conditions of

cessa

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eodynamicsumerical modeling

the wedge flow is also necornerflow model.

. Introduction

Subduction zones form the location of explosive arc volcanism,arge underthrusting earthquakes and deep earthquakes in Benioffones. The generation of arc volcanism is generally attributedo the release of volatiles from the subducting plate into theot overlying wedge and subsequent lowering of the peridotiteolidus. Direct melting of subducted sediments and pressure-elease melting in the wedge can also assist in the formation ofagmas. Intermediate-depth seismicity within the slab is gener-

lly attributed to dehydration-embrittlement due to the release ofater once the subducted oceanic crust and mantle go through keyhase changes (Kirby et al., 1996). The up- and down-dip limits ofhe megathrust seismogenic zone are at least partially controlled

∗ Corresponding author. Tel.: +1 734 764 1497.E-mail addresses: keken@umich.edu (P.E. van Keken), ccurrie@phys.ualberta.ca

C. Currie), sdk@vt.edu (S.D. King), mbehn@whoi.edu (M.D. Behn),mandine cagnioncle@brown.edu (A. Cagnioncle), JHe@NRCan.gc.ca (J. He),fk22@cam.ac.uk (R.F. Katz), skylin0@ntu.edu.tw (S.-C. Lin),m parmentier@brown.edu (E.M. Parmentier), mspieg@ldeo.columbia.eduM. Spiegelman), KWang@NRCan.gc.ca (K. Wang).

URL: http://www.geo.lsa.umich.edu/ keken/subduction/benchmarkP.E. van Keken).

iwwtatbamsEoff

031-9201/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.pepi.2008.04.015

ry to avoid a pressure singularity that exists in analytical solutions of the

© 2008 Elsevier B.V. All rights reserved.

y the thermal structure of subduction zones (Hyndman and Wang,995). Temperature and pressure form therefore fundamental con-rols on the processes leading to volcanism and intermediate-deptheismicity (Kerrick and Connolly, 2001). Forward models that usenformation about the subduction zone geometry, convergenceate, and age of the incoming plate can be useful in constraininghe temperature distribution and allow for quantitative tests ofypotheses about the dynamics and structure of subduction zones.

It is generally understood that viscous coupling between theubducting slab and overriding mantle causes a downward flown the wedge which leads to a reduced pressure in the tip of the

edge. This in turn induces flow of the upper portions of theedge (Fig. 1a). This cornerflow in the wedge advects hot man-

le into the region above the slab and below the overriding platend therefore creates the environment for the ‘subduction fac-ory’. Many thermal subduction zone models, past and present, areased on this cornerflow scenario, although different approachesre used to approximate the flow in the wedge. Semi-analyticalodel approaches include those that couple separate parts of the

ubduction zone that can be described analytically (Davies, 1999;ngland and Wilkins, 2004) or use a simplified parameterizationf the thermal effects of the wedge with a finite-difference schemeor the slab (Minear and Toksoz, 1970). In some numerical modelsor thermal structure an analytical cornerflow model for isoviscous

188 P.E. van Keken et al. / Physics of the Earth and Planetary Interiors 171 (2008) 187–197

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dftadcGeeiaatr

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ig. 1. (a) Cartoon of the cornerflow model for subduction zone dynamics (from Vaantle wedge below a rigid overriding plate. (c) Boundary conditions for Stokes an

heology (Batchelor, 1967; McKenzie, 1969) was used to prescribehe flow in the wedge (e.g. Peacock and Wang, 2001). More recentlyt has become common to determine the flow in the wedge self-onsistently by solving the Stokes equations for viscous flow (e.g.urukawa, 1993; Van Keken et al., 2002) or to use fully dynamicodels of subduction zones (e.g. Billen and Hirth, 2007).While fully dynamic models are needed to develop an

nderstanding of the dynamics of subduction zones and the self-onsistent generation of plate tectonics in mantle convectionGurnis and Hager, 1988; King and Hager, 1994; King and Ita, 1995;incaid and Sacks, 1997; Chen and King, 1998; Zhong et al., 1998;an Hunen et al., 2002; Billen and Hirth, 2007), models that pre-cribe the kinematics of the slab are simpler and better suitedn some cases. Such models are particularly appropriate for thetudy of the Earth’s subduction zones where the slab geometry isell described by Benioff zone seismicity and local seismic studies

nd those where the relative plate motion can be extracted fromlobal tectonic models (e.g. Rondenay et al., 2008). This thermo-echanical approach should yield a reasonable prediction of the

emperatures in the slab, which then can be linked to the meta-orphic processes leading to slab dehydration, arc volcanism and

eismogenesis. High accuracy of the thermal models is neededince thermodynamic, field-based and experimental predictions ofhe relevant metamorphic phase changes predict relatively narrowressure–temperature ranges for the key phase changes (Schmidtnd Poli, 1998; Connolly and Kerrick, 2002; Hacker et al., 2003;orneris and Holloway, 2004; Abers et al., 2006).

In recent years it has become clear that the temperature-ependence of the rheology of mantle silicates causes strongocusing of the flow in the wedge. This focusing increases theemperature at the top of the slab, results in narrow thermalnd rheological boundary layers, and enhances the effects ofecompression melting and importance of buoyancy (or secondaryonvection) in the mantle wedge (Furukawa, 1993; Billen andurnis, 2001; Van Keken et al., 2002; Conder et al., 2002; Kelement al., 2003; Gerya and Yuen, 2003; Cagnioncle et al., 2007; Morgant al., 2007). Since the targeted accuracy for use of these modelsn petrological studies is on the order of tens of degrees there isstrong need for methods that can solve the governing equations

ccurately. Achieving such accuracy is made more challenging dueo the strong boundary layers that develop with realistic mantleheology.

As a community of modelers that is interested in the thermaltructure and dynamics of subduction zones we have developed aenchmark that allows us to compare numerical approaches. Theenchmark suite was originally proposed at the 2002 MARGINSorkshop on subduction zone modeling (held at the University of

2

p

n (2003)). (b) Benchmark geometry of a kinematic slab driving flow in the viscousperature equations.

ichigan in Ann Arbor, MI). We compile here the contributionsrom seven different groups with independent codes. The casesre based on a simple 2D Cartesian geometry with a kinemati-ally described slab that dips at 45◦ below an overriding plate ofonstant thickness (Fig. 1b). In the first set of models we explorehe consequences of an isoviscous wedge, starting with the Batch-lor analytical solution (Batchelor, 1967), followed by a numericalolution of the wedge flow. This set of models includes a test ofhe type of boundary conditions that are imposed at the inflownd outflow boundaries of the wedge (Fig. 1c). In the second set ofodels we explore the effects of more realistic temperature- and

tress-dependent rheology in the mantle wedge.An artefact of the mathematical simplification we employ a

iscontinuity in the velocity boundary conditions for the Stokesquations in the wedge tip. This causes an (integrable) singular-ty in pressure but as discussed below leads to some numericalifficulties when the Stokes flow is solved numerically. In naturehe transition from fault contact at the seismogenic zone to theull viscous coupling between slab and mantle wedge is finite andan be approximated by improved and more detailed rheologicalescriptions and freely evolving slabs (Sobolev and Babeyko, 2005;orczyk et al., 2007; Gerya et al., 2008).

The main goal in performing this benchmark is to demon-trate the ability of numerical simulations to give an accurateolution to the problem as posed. It should be noted, however,hat the models described here do not account for many factorshat would modify the flow field in the wedge. The modificationould have impacts on thermal structure that are significantly largerhan differences between simulation results from different codes.hese factors include the dynamical effects of mineralogical phasehanges including the likely serpentization of the wedge cornerHyndman and Peacock, 2003; Gorczyk et al., 2007), melting and

elt transport (Katz et al., 2007; Cagnioncle et al., 2007), and three-imensional effects (Kneller et al., 2007; Behn et al., 2007).

In the remainder of this paper we will first describe the bench-ark cases in detail. This is followed by a description of the

ontributing codes and a full comparison of the benchmark results.e hope that this comparison will spark a broader comparison with

ther codes and that this paper will have a similar impact on theommunity as previous benchmarks for mantle convection model-ng (Blankenbach et al., 1989; Travis et al., 1990; Busse et al., 1993;an Keken et al., 1997).

. Description of the benchmark cases

The benchmark geometry is the same for all cases and is dis-layed in Fig. 1b. The domain is arbitrarily chosen to be 660 km

h and Planetary Interiors 171 (2008) 187–197 189

w(wpicpwabnbas

2

tardcmCf

∇a

∇wd

�

wt

�

is

�

wt

�

wt

�

ie

�

Table 1Nomenclature and reference values

Quantity Symbol Reference value and/orSI units

Velocity �v m/sDynamic viscosity � �0 = 1021 Pa sStress tensor � PaStrain rate tensor � 1/sDynamic pressure P PaDensity � 3300 kg/m3

Temperature T T0 = 1573 K = 1300 ◦CThermal conductivity k 3 W/mKHeat capacity cp 1250 J/kg KThermal diffusivity � � = k/�cp =

0.7272 × 10−6 m2/sActivation energy (diffusion creep) Ediff 335 kJ/molActivation energy (dislocation creep) Edisl 540 kJ/molPowerlaw exponent for dislocation creep n 3.5Pre-exponential constant (diffusion creep) Adiff 1.32043 × 109 Pa sPMG

w

�

wlbviursmaimu

�

�

2

dtrTblpbc

T

P.E. van Keken et al. / Physics of the Eart

ide × 600 km deep. The origin of the coordinate system �x =x, y) = (x1, x2) is at the top left. The inflow boundaries (at bothedge and trench sides) and top of the model have prescribed tem-erature. The wedge is assumed to be an incompressible fluid that

s driven only by the kinematic forcing of the slab. The wedge isonfined by the top of the slab and the base of the rigid overridinglate (located at a depth of 50 km). The boundary conditions for theedge are no-slip below the overriding plate and constant velocity

long the top of the slab. The velocity boundary conditions for theoundaries of the wedge are either provided by the Batchelor cor-erflow solution (cases 1a and 1b) or based on free inflow/outflowoundaries. The velocity field is discontinuous between the slabnd the overriding plate which effectively mimics the fault repre-entative of the seismogenic zone in subduction zones.

.1. Governing equations

The flow in the wedge is assumed to be driven exclusively byhe kinematic velocity of the slab and we ignore thermal buoy-ncy in the wedge. This simplifies the governing equations andeflects that the main source of buoyancy in a subduction zone isue to the excess density of the slab. Secondary forms of buoyancyan become important for low viscosity zones in the wedge whichay be caused by hydration or melting (Billen and Gurnis, 2001;

agnioncle et al., 2007). The velocity and pressure in the wedge isound by solving the conservation of mass

· �v = 0 (1)

nd the conservation of momentum

· � − ∇P = 0 (2)

here �v is velocity, � is the deviatoric stress tensor, and P theynamic pressure. The deviatoric stress tensor is given by

= 2�� (3)

here � is the effective dynamic viscosity and the components ofhe strain rate tensor � are given by

˙ ij = 12

[∂vi

∂xj+ ∂vj

∂xi

](4)

All benchmark cases are assumed to be steady-state and wegnore heat production so that the temperature is found from theteady-state heat advection–diffusion equation

cp(�v · ∇)T = ∇ · (k∇T) (5)

here � is density, cp the specific heat, T the temperature, and k thehermal conductivity.

The effective shear viscosity � follows from (3):

= �

2�(6)

here the second invariants of the strain-rate and deviatoric stressensor are defined by

˙ =

⎡⎣1

2

∑ij

�ij �ij

⎤⎦

1/2

, � =

⎡⎣1

2

∑ij

�ij�ij

⎤⎦

(1/2)

(7)

A general simplified equation for the viscosity of olivine deform-

ng by diffusion creep assuming zero activation volume (ignoringffects of hydration and grain-size dependence) is

diff(T) = Adiff exp(

Ediff

RT

)(8)

w(tt

re-exponential constant (dislocation creep) Adisl 28968.6 Pa s1/n

aximum viscosity �max 1026 Pa sas constant R 8.3145 J/mol K

hile the viscosity for deformation by dislocation creep is given by

disl(T, �) = Adisl exp(

Edisl

nRT

)�(1−n)/n (9)

here A is a prefactor, E the activation energy, and n is the power-aw exponent (e.g. Karato and Wu., 1993). Since the flow is driveny a fixed velocity boundary condition the range of the dynamiciscosity influences the solution, particularly near the return flown the tip of the wedge. A large dynamic range in viscosity leads tonrealistic high stress and unrealistic deformation in high viscosityegions. This could be avoided by inclusion of physically realisticelf-limiting processes, such as a transition to higher stress flowechanisms or the inclusion of viscous dissipation (e.g. Kneller et

l., 2007; Wada et al., 2008). For the ease of the numerical compar-son it is simpler and sufficient to truncate the viscosity at a fixed

aximum �max. For the benchmark cases described below we willse the modified viscosity laws:

diff,effective =(

1�diff

+ 1�max

)−1(10)

disl,effective =(

1�disl

+ 1�max

)−1(11)

A full list of symbols with reference values is provided in Table 1.

.2. Case 1a: analytical cornerflow model

The velocity in the slab is constant and dips at 45◦ with aimensional speed of 5 cm/yr. The wedge flow is prescribed byhe analytical expression for cornerflow (Batchelor, 1967). This caseequires only the solution of the heat advection–diffusion Eq. (5).he top boundary condition is Ts = T(y = 0) = 273 K. At the inflowoundary of the wedge temperature is fixed at T0 = 1573 K and a

inear geotherm is used at the left hand boundary of the overridinglate from 0 to 50 km depth. The temperature at the slab inflowoundary is described by an error-function solution for half-spaceooling for 50 Myr

(x = 0, y) = Ts + (T0 − Ts)erf(

y

2√

�t50

)(12)

here t50 is the age in seconds and � is the thermal diffusivityTable 1). At the slab and wedge outflow boundaries we prescribehe natural boundary condition (zero curvature) for the heat equa-ion.

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90 P.E. van Keken et al. / Physics of the Eart

.3. Case 1b: dynamical flow in isoviscous wedge I

This case is the same as 1a, except that the solution for the wedgeow is determined by solving Eqs. (1) and (2). We use essentialoundary conditions for velocity where the velocity componentsre prescribed by the Batchelor solution. This allows us to eval-ate the quality of the Stokes solution without inducing changesue to modifications of the boundary conditions. One important

ssue is the discontinuity in the velocity boundary conditions at theedge tip (located at 50 km depth). The discontinuity causes a well-

nown pressure singularity that is analytically resolvable, but mayause numerical problems for some numerical algorithms and/oror computational grids that are not sufficiently dense (Van Kekent al., 2002). We will illustrate this in the results section below.

In order to avoid the numerical difficulties we allow the option ofhanging the boundary condition so that the velocity at the bottomf the wedge increases linearly from �v = 0 at position of the corneroint �xc = (50, 50) to �v = �v0 a short distance down stream from theorner point. Formally, this change is from

�(�xs) = �v0 for |�xs| ≥ |�xc| (13)

o

�(�xs) = |�xs − �xc||�xfc − �xc|

�v0 for |�xc| ≤ |�xs| ≤ |�xfc| (14)

�(�xs) = �v0 for |�xs| ≥ |�xfc| (15)

here �xs is a coordinate along the slab–wedge interface and �xfcs the position down-slab where full coupling between slab and

edge is achieved. This should be a short distance D from the corneroint so that

�fc = �xc +(√

2D

2,

√2D

2

)(16)

The velocity ramp is on the wedge side only and this short seg-ent should be treated as an extension of the velocity discontinuity

rom �x = (0, 0 km) to (50, 50 km). This transition can be thought of ashealing fault, where the tangential velocity across the slab–wedge

nterface changes from discontinuous to continuous.

.4. Case 1c: dynamical flow in isoviscous wedge II

This case is the same as 1b, except that natural boundary condi-ions for stress at the inflow and outflow boundaries of the wedgere prescribed. This implies that both the normal and tangentialomponents of the total stress � − PI are zero where I is the identityensor.

.5. Case 2a: dynamical flow with diffusion creep

This case is the same as 1c, except that the viscosity of the wedges given by Eq. (10). See Table 1 for the rheological parameters.

.6. Case 2b: dynamical flow with dislocation creep

This case is the same as 1c, except that the viscosity of the wedges given by Eq. (11). See Table 1 for the rheological parameters.

. Contributing codes

We identify the contributed results of the benchmarks by a shortdentifier (such as VT for Virginia Tech) and define those identifierselow in bold. The finite difference methods use uniform meshesith the same grid resolution in x and y. The finite element methods

sri

v

Planetary Interiors 171 (2008) 187–197

enerally employ grid refinement in the boundary layers. All con-ributors (except Currie, He and Wang) report values on multiplerids with varying number of nodal points.

Mark Behn (WHOI) uses Comsol 3.2b (http://www.comsol.com)ith quadratic triangular elements for the heat equation

nd P2–P1 elements for the Stokes equations. The UMFPACKhttp://www.cise.ufl.edu/research/sparse/umfpack) unsymmetric

ultifrontal method is used for direct LU factorization. For cases 2and 2b a damped Newton method is used starting from an initialolution from the constant viscosity calculation. Grid refinement issed in the boundary layers.

Amandine Cagnioncle and Marc Parmentier (Brown) use aixed finite element and finite volume approach. The heat conser-

ation equation is discretized with a finite volume approximation.dvective fluxes at volume faces are derived by upwinding withsubsequent correction that minimizes the numerical diffusion

hat results from pure upwinding (Smolarkiewicz and Margolin,998). This method is most easily formulated by splitting thedvection and diffusion operators over a time-step. The methods fully explicit and is therefore well suited to multiprocessorpplications. As one of a number of ‘flux corrected transport’ormulations, this offers an efficient alternative to particle tracersn problems where small numerical diffusion is required. Inases where comparisons have been made of advection withoutiffusion, this method appears to work as well as tracer particlesAlley and Parmentier, 1998). Viscous flow solutions are obtainedsing a standard penalty function formulation on a grid of linearour-node elements with nodes at the centers of finite volumelements in the advection–diffusion formulation. Finite volumeace-centered velocities are calculated by linear interpolation ofhe finite element nodal velocities. The advection operator in thenite volume formulation assumes divergence-free velocity fields;o that any numerically non-zero divergence will appear as aource term. To correct for non-divergence-free staggered veloc-ties, we add a small isotropic velocity field in each finite volumehereby making the staggered, face-centered velocities exactlyivergence free. For well-resolved velocity fields, for example onealculated with the penalty formulation, such effects are small. Forhe cornerflow solution used here, which has a discontinuity inelocity and a resulting singularity in pressure, the departure fromivergence-free conditions can be locally large. The magnitude ofhe divergence correction can be monitored as an indicator of theesolution of the velocity fields.

Claire Currie, Jiangheng He and Kelin Wang (PGC) use the finitelement code PGCtherm. Nine-node isoparametric elements aresed for temperature and velocity, with compatible four-node ele-ents for pressure. Galerkin-least squares (Hughes et al., 1989)

s used to stabilize the solution of the heat equation for advec-ion dominated flows. The Stokes and heat equations are solvedimultaneously and a non-linear iteration is performed using aulti-corrector fixed point algorithm. The mesh used to produce

he results in this paper has 11 k elements (46 k nodes). The left-ost portion of the wedge of 1 m horizontal width is removed

o ease mesh construction. The wedge ‘tip’ in the PGC model isherefore a vertical line of 1 m height. Near this tip a special gridefinement is used that results in very small elements (ranging from× 10−8 to 70 m) in the 1 km region near the tip. The maximumlement size is 20 km. The mesh is generated with numerous meshines radiating from the wedge tip. The PGC group was unable toonstruct a mesh with low resolution near the wedge tip with rea-

onable element aspect ratio. For this reason PGC contributed onlyesults obtained on a single mesh with the highest mesh resolutionn the tip.

Richard Katz and Marc Spiegelman (LDEO) use a uniform finiteolume discretization with staggered velocity components and

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P.E. van Keken et al. / Physics of the Eart

ressure. A non-linear system of equations for velocity and pres-ure is built and solved using a Newton–Krylov–Schwarz parallelterative method. The temperature solution is based on a secondrder Fromm upwind scheme (Katz et al., 2007). The code employshe Portable Extensible Toolkit for Scientific Computing (Balay etl., 2004http://www.mcs.anl.gov/petsc).

Scott King (VT) uses a grid of quadrilateral elements withtreamline upwind Petrov–Galerkin (SUPG), which is a form of thealerkin-least-squares approach (Hughes et al., 1989) for the heatquation. A Q1–P0 quadrilateral element is used for the Stokesquation. A penalty function method is used to eliminate pressurerom the discretized set of equations for the Stokes equation whichan then be solved by efficient Cholesky factorization (Hughes et al.,979). Picard iteration is used to speed up convergence to a steadyolution. The finest mesh employed contains 360 × 180 elementsith 1 km grid spacing near the tip of the wedge. The finite element

pproach used here forms also the basis for the free convection codeonMan (King et al., 1990).

Shu-Chuan Lin (NTU) employs a finite volume discretizationith a cell-centered collocated grid using a primitive variable for-ulation. The energy equation is solved with a power-law schemehich is a combination of a central-difference scheme and anpwind scheme with variable weighting (dependent on the localeclet number). The Stokes equations are solved with SIMPLEPatankar, 1980).

Peter van Keken (UM) uses a finite element approach based onhe Sepran software (Cuvelier et al., 1986)(dutita0.twi.tudelft.nl/epran/sepran.html). Linear triangles with SUPG (using the doublesymptotic approach) are used for the heat equation. The Stokesquations are either solved with a penalty function method on2–P1 triangular elements or with the integrated method usinginear Taylor–Hood elements that directly solves for velocity andressure after renumbering of the unknowns in the discretizedet of equations. These methods yield similar accuracy for equallyense meshes. The results obtained with the integrated method arehown here. Picard iteration between Stokes and heat equations issed to arrive at a steady solution for the non-linear cases 2a and

b. Meshes with variable grid resolution are used with a typicalaximum element size of 10 km and a resolution down to 100 m

n the boundary layers. The methodology used here is the same ashat used in Van Keken et al. (2002), Van Keken (2003) and Aberst al. (2006).

ntp

ig. 2. (a) Temperature prediction for case 1a (provided by UM). The bold lines indicate thhe model.

Planetary Interiors 171 (2008) 187–197 191

. Results

We will focus primarily on the temperature structure at theop of the slab and in the mantle wedge in the tip of the corner-ow region. The thermal structure in this area is most relevant forhe physical and chemical processes leading to arc volcanism andeismicity. The accurate solution of the equations is also most dif-cult here due to the formation of the temperature and rheologicaloundary layers. Since the solution to the temperature equation fol-

ows directly from the velocity solution, we find that comparing theemperature fields also provides us with a critical test of the accu-acy of the solution of the Stokes equations. We provide one directomparison of the dynamical solution by focusing on pressure andressure gradients in the wedge. To compare model results eachroup contributed the temperature field as discreted values Tij onn equidistant grid with 6 km spacing, which is a 1 1 1 × 1 0 1 matrixtored row-wise starting in the top left corner. From this grid weave extracted the following measurements for direct comparison:1) the temperature T11,11 which is at coordinates (60, 60 km) andust down-stream from the corner point. This provides thereforene of the most critical tests of accuracy of the numerical codes;2) the L2 norm of the slab–wedge interface temperature betweenand 210 km depth defined by

|Tslab|| =

√√√√√36∑i=1

T2ii

36(17)

nd (3) the L2 norm of the temperature in the triangular part of theip of the wedge, between 54 and 120 km depth:

|Twedge|| =

√√√√√√21∑

i=10

i∑j=10

T2ij

78(18)

We compare these between codes as a function of grid resolutionear the corner point in the sections below. We show all resultshat have been contributed to the benchmark comparison. Somearticipants did not contribute to all cases.

e top of the slab and base of the overriding plate. (b) Close up of the top left part of

192 P.E. van Keken et al. / Physics of the Earth and Planetary Interiors 171 (2008) 187–197

F es 1at long ta he aveS he wea

4

itt

TmwtCtiU

o

asat

w(mDssp

ig. 3. Predictions for selected thermal properties for the isoviscous benchmark cashe slab–wedge interface at 60 km depth (frames a, d, g), the average temperature average temperature in a triangular portion of the wedge (Eq. (18))(frames c, f, i). Tince the mesh from PGC has a special treatment of the corner the resolution near trbitrarily high resolution.

.1. Thermal structure: isoviscous cases

The temperature field for the isoviscous case obtained by UMs shown in Fig. 2a with a close-up on the wedge thermal struc-ure in Fig. 2b. The comparison of the three scalar measurements ofhermal structure is shown in Fig. 3 for the three isoviscous cases.

There are a number of general trends visible in this comparison.he largest spread is observed for the single temperature measure-ents (frames a, d, and g). The numerical values tend to convergeell and generally overlap within a small temperature variation at

he extrapolated highest resolution. All codes agree to within 20 ◦

for grid resolution less than 1 km. The convergence trends for

he equidistant grid codes based on finite volume approximationss somewhat poorer than that of the finite element methods (PGC,M, VT, and WHOI).

Case 1a (frames a–c) is a test of the solution of the heat equationnly. Three of the finite element codes tend to overlap with high

awwci

(frames a–c), 1b (d–f) and 1c (g–i). The quantities represent a spot measurement athe slab–wedge interface from 0 to 210 km depth (Eq. (17)) (frames b, e, h), and therages are computed with an L2 norm from an equidistant grid with 6 km spacing.

dge tip is not defined as in the other codes and we plot their result in each graph at

ccuracy (0.1 ◦C) at the highest resolution (see Table 2). A minorystematic offset, that is somewhat stronger in frames b and c,ppears for NTU and Brown whereas LDEO trends closely towardshe values predicted by PGC, UM, and WHOI.

We have also modeled the thermal structure for this caseith the finite difference approximation of Minear and Toksoz

1970), which has been used extensively in studies of the ther-al structure of slabs (e.g. Toksoz et al., 1971, 1973; Sleep, 1973;e Jonge et al., 1994). This method does not provide an explicit

olution of the wedge flow. Rather, it couples grid size and time-tep and explicitly advects temperatures from a set of higher gridoints to those just below it followed by a diffusional step with

n alternating direction implicit finite difference solver. In theidely available code there is an option to mimic induced flowithout explicitly solving the Stokes equations. We find that this

ode strongly underpredicts the temperature at the slab–wedgenterface (e.g. T11,11 is 275 ◦C compared to 388 ◦C in the present

P.E. van Keken et al. / Physics of the Earth and Planetary Interiors 171 (2008) 187–197 193

Table 2Selected thermal quantities for isoviscous cases 1a–1c

Case Code T11,11 ‖Tslab‖ ‖Twedge‖1a Brown 393.51 520.14 866.52

LDEO 396.63 506.43 855.58NTU 388.87 507.43 852.99PGC 388.21 503.69 854.34UM 388.24 503.77 825.89VT 379.87 502.26 852.05WHOI 388.26 503.75 854.37

1b Brown 391.83 493.76 842.01LDEO 387.15 500.86 852.80NTU 391.42 511.27 853.16PGC 388.21 503.69 854.34UM 388.22 503.65 854.12VT 389.82 504.63 853.04WHOI 389.08 504.50 856.08

1c LDEO 397.55 505.70 850.50NTU 391.57 511.09 852.43PGC 387.78 503.10 852.97UM 387.84 503.13 852.92VT 389.39 503.04 851.68WHOI 388.73 504.03 854.99

T11,11 is the temperature at (60, 60); ‖Tslab‖ is the L2 norm of the slab temperaturefs◦

pk

wflilfUWbfDolrew

Table 3Selected thermal quantities for cases 2a–2b

Case Code T11,11 ‖Tslab‖ ‖Twedge‖2a LDEO

NTU 570.30 614.09 1007.31PGC 580.52 606.94 1002.85UM 580.66 607.11 1003.20VT 577.59 607.52 1002.15WHOI 581.30 607.26 1003.35

2b LDEO 550.17 593.48 994.11NTU 551.60 608.85 984.08PGC 582.65 604.51 998.71UM 583.36 605.11 1000.01

S

vlnuF1fis

antt

a

4

dbwThis is numerically more challenging and explains the slightly

F(

rom 0 to 210 km depth (17); and ‖Twedge‖ is the L2 norm of the temperature in thelab and wedge tip in a triangular area near the wedge tip (18). All quantities are inC. Values at highest resolution are listed for each code.

rediction. The temperature difference increases to 165 ◦C at 210m depth).

Case 1b (frames d–f) show the results for the isoviscous caseith a solution of the Stokes equation that should yield the sameow pattern as that used in case 1a. Except for PGC, the contribut-

ng codes used a small ramp in the velocity boundary condition ofength D to mitigate the effect of the pressure singularity. The needor this velocity smoothing is apparent when comparing results forM with a 2 km ramp (grey circles) to that without (open circles).hile the results overlap at very high resolution the convergence

ehavior is poor and differences in excess of 60 ◦C are obtainedor a grid resolution of less than 4 km. The length of the ramp

is not prescribed. We have found that this is partly dependentn grid resolution near the corner point (as the ramp should be

inear preferably over at least a few grid elements) and that theamp length should not be too high (since then the lower velocityffects become apparent downstream). With one approach (UM)e have verified that the results shown here are not sensitive to

piat

ig. 4. (a) Temperature prediction for case 2a with olivine diffusion creep in the mantle wc) Close up of the model with olivine dislocation creep in the mantle wedge.

VT 574.84 603.80 995.24WHOI 583.11 604.96 1000.05

ee Table 2 for definitions.

ariations in ramp length between 1 and 5 km, provided the reso-ution near the corner point is sufficiently high. For PGC it was notecessary to use a ramp because of the extremely small elementssed near the wedge tip and the high order of the shape functions.or most codes the results trend closely to those obtained in casea, except for VT, that interestingly now trends closely to the othernite element methods, and Brown that shows a slightly differentystematic offset.

The trends for case 1c (in which case open boundary conditionsre used instead of imposing those of the Batchelor solution) areearly identical to those of case 1b, which strongly suggests thathe influence of change in boundary conditions is minimal on thehermal structure near the wedge tip.

Table 2 provides the values predicted by the contributing codest the highest resolution.

.2. Thermal structure: variable rheology

The inclusion of temperature- (or temperature- and stress-ependent) rheology causes significantly thinner thermaloundary layers and higher temperatures near the tip of theedge (Fig. 4; temperature field shown as obtained by UM).

oorer convergence of various codes compared to that of thesoviscous cases (Fig. 5, Table 3). Nevertheless, the agreementt high resolution between various codes is quite good (e.g. lesshan 1 ◦C difference for T11,11 between PGC, UM and WHOI). It is

edge (provided by UM). (b) Close up of the top left part of the model, as in Fig. 2b.

194 P.E. van Keken et al. / Physics of the Earth and Planetary Interiors 171 (2008) 187–197

2b w

id2tl1

4

Fig. 5. As in Fig. 3, but now for cases 2a and

nteresting to note that the temperature values are not significantlyifferent for the cases that simulate diffusion creep in olivine (case

a) and those for dislocation creep (case 2b). This indicates thathe change in activation energies is compensated for by the non-inear rheology as has been previously suggested (Christensen,984).

roe

ith variable rheology in the mantle wedge.

.3. Pressure predictions

One critical test of the numerical codes is their ability to accu-ately predict pressure and pressure gradients. Due to the differentrder of the differentiation of pressure and velocity in the Stokesquation it is often necessary to approximate pressure with a lower

P.E. van Keken et al. / Physics of the Earth and

Table 4Dynamic pressure P and pressure gradients Px and Py in the wedge at two arbitrarypoints for case 1b

Code P1 P2 Px,1 Py,1 Px,2 Py,2

Analytical −929.46 − 7.34 55.97 12.71 0.02935 0.02748LDEO − 916 − 7.3 54.5 11.9 0.0293 0.0275NTU − 911.9 2.2 55.63 16.02 0.02935 0.02751PGC − 929.53 − 7.35 51.16 11.45 0.03047 0.02758UM − 930.51 − 7.70 56.24 12.51 0.0294 0.0275VT − 927.8 − 7.11WHOI − 930.26 − 7.35 56.86 14.45 0.0293 0.0277

P(M

ocuaew

c(ppwaeglatefpTtf“dTT

pml

5

bodahT�xeiScodgi

idvfiatt

Bbdpi

test whether the finite volume methods have more difficulty with

F

oint 1 corresponds to indices (12, 10) and �x = (66, 54 km); point 2 corresponds to100, 60) and �x = (594, 354 km). Pressure is listed in MPa and pressure gradients in

Pa/km.

rder interpolation function than that used for the velocity and itan be questioned whether low order methods (such as those thatse linear interpolation for velocity) can accurately predict pressurend pressure gradients. Accurate solution of pressure gradients isssential for models that couple flow of magma or hydrous fluidsith the solid state flow (Katz et al., 2007; Cagnioncle et al., 2007).

We set pressure in the inflow point at the overriding plate withoordinates �x = (660, 600 km) equal to zero to normalize pressuresince the solution of the Stokes equations strictly only providesressure differences and not absolute pressure). Table 4 shows theressure and pressure gradients for case 1b in two arbitrary pointsith coordinates (66, 54 km) and (594, 354 km). Fig. 6 providescomparison of the analytical pressure (obtained on the 6 km

quidistant grid) with those predicted by VT and UM on the samerid. These represent a point in the wedge close to the pressureow near the tip of the wedge and a point near the outflow bound-ry. These can be directly compared to the analytical values fromhe Batchelor solution. In general we find good, and for some codesxcellent, agreement between numerical and analytical predictionsor pressure. Logically, the differences are somewhat larger for theressure gradients but the differences are significantly below 10%.hese results suggest that codes with low order pressure interpola-ion functions (UM, VT) can still provide highly accurate predictionsor pressure. Because the finite element models are based on the

weak form” of the partial differential equations their pressure gra-ient is by definition discontinuous across elemental boundaries.he most accurate gradient value is obtained in the Gaussian points.he position of the two points listed in Table 4 relative to the Gauss

aaaW

ig. 6. Comparison of predictions for pressure by UM and VT (frame a) with the analytica

Planetary Interiors 171 (2008) 187–197 195

oints depends on the specific mesh used. The results are thereforeore strongly dependent on grid size than for quantities that are

inear or quadratic in space.

. Discussion

The results presented here are part of a multi-year comparisonetween existing and newly adapted methods for the modelingf the thermal structure of subduction zones. We find in generalecent agreement between all codes, and in particular excellentgreement between a number of the finite element methods. Weave identified a number of reasons for the observed differences.hese include the presence of a velocity discontinuity between= (0, 0 km) and (50, 50 km) which makes the solution of the heatquation more challenging, the existence of the pressure singular-ty caused by the discontinuous velocity boundary condition in thetokes equation, and the 45◦ dip of the slab which is more diffi-ult to conform to with finite volume methods. Also, in the casef finite element methods that are based on quadrilaterals withomains that have rectangular layouts (like VT) the elements mayet non-optimal aspect ratios which will contribute to the difficultyn achieving high accuracy.

The velocity discontinuity between the slab and overriding platen the heat equation requires some attention. It may cause principleifficulties with methods that are highly accurate only for smoothelocity fields (like the Smolarkiewicz scheme used by Brown). Thenite volume methods benefit from a special treatment of the nodeslong this discontinuity. For the finite element methods the discon-inuity can be accurately represented by proper integration of theerms in the element stiffness matrix for the discrete heat equation.

We have found that the pressure singularity inherent in theatchelor analytical solution causes numerical difficulties (Fig. 3d),ut that these effects can be efficiently mitigated by removing theiscontinuity in the velocity boundary conditions at the corneroint (50, 50). This is most efficiently and accurately done by includ-

ng a small ramp in the slab–wedge boundary (Eqs. (13)–(15)).We have compared a model with a vertically dipping slab to

slab (and velocity discontinuity at shallow depths) that cutscross the computational grid. A full set of results is availablet http://www.geo.lsa.umich.edu/∼keken/subduction/benchmark.e have summarized aspects of this special case in Fig. 7. The accu-

l values (b). Contours at every 10 MPa are based on the 6 km equidistant grid.

196 P.E. van Keken et al. / Physics of the Earth and

Fvm

rt9l1mm

A

FENdMWh

R

A

A

B

B

B

B

B

B

B

C

C

C

C

C

C

D

D

E

F

F

G

G

G

G

H

H

H

H

H

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K

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ig. 7. Limited comparison of the slab temperature at 60 km depth for a case with aertically dipping slab. The convergence behavior and accuracy of the finite volumeethods are greatly improved compared to that of a 45◦ dipping slab (Fig. 3).

acy and convergence trends are markedly improved. For example,he difference between LDEO and UM was reduced to 1 ◦C from◦C at 0.5 km resolution and the results are within 10◦ for a reso-

ution of 2–3 km (compared to 40◦ at the same resolution for casea). This strongly suggest that the conformability of the finite ele-ent methods provides an attractive aspect for subduction zoneodeling.

cknowledgments

This research was partly supported by the National Scienceoundation (NSF) under grants EAR-0652707 to Mark Behn andAR-0215534 to van Keken. We thank the participants of theSF-sponsored 2002 MARGINS workshop on the modeling of theynamics and thermal structure of subduction zones at Ann Arbor,I (organized by Peter van Keken, Scott King, and Simon Peacock).e thank Jeroen van Hunen and Taras Gerya for constructive and

elpful reviews.

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