We present eighteen 3D numerical fluid dynamics experiments of the subduction-transform plate boundary system in southern Alaska, in which we varied the viscosity structure (Newtonian versus composite viscosity), the subducting plate geometry (slabE325 versus slabE115), the yield stress (σy), the plate boundary coupling (ηwk), and the thermal structure (Table 1). Because the approach taken here is to model a regional plate boundary, it was necessary to map regional observations of slab geometry and the plate boundary configuration, as well as the thermal structure of the overriding plate and slab, onto a model grid. A C and C++ code, referred to as SlabGenerator, was written to generate and map this initial configuration onto a 3D model mesh that was locally refined in regions of geometric complexity and where large viscosity contrasts were expected to occur [Jadamec, 2009; Jadamec and Billen, 2010]. The 3D plate boundary configuration and thermal structure were then used as input to the finite element mantle convection code, CitcomCU [Moresi and Solomatov, 1995; Moresi and Gurnis, 1996; Zhong, 2006], as will be described in the sections that follow.
3.1. 3D Model Domain and Driving Forces
 The 3D regional models of the subduction-transform plate boundary system in southern Alaska include an overriding plate (the North American plate), a subducting plate (the Pacific plate), and the underlying mantle (Figure 3). A viscous shear zone separates the Pacific and North American plates. The driving forces in the system are the negative thermal buoyancy of the subducting slab (slab pull) and the positive thermal buoyancy of the Juan de Fuca ridge (ridge push), which lies along the southeast boundary of the Pacific plate within the model domain (Figure 3). The resisting forces are the viscous stresses in the mantle, the plate boundary shear zone, and within the interior of the slab. There are no driving velocities applied anywhere in the model, rather the models predict the flow velocities for the plates and mantle. We note that while the crust and harzburgitic residue of melting making up the lithosphere have a net positive buoyancy, once the crust is converted to eclogite the net compositional buoyancy is zero because the positive buoyancy of the thicker (2.5 times) harzburgitic layer cancels out the effect of the eclogitized crust [Oxburgh and Parmentier, 1977; Hacker et al., 2003]. Therefore, we choose not to include these compositional effects as the imposed shape of the slab essentially accounts for the positive buoyancy of the crust at shallow depths.
Figure 3. 3D model set-up. NAM- North American plate; PAC- Pacific plate; Slab- subducted part of Pacific plate with portion of slab geometry that is varied outlined with short-dashed black line; PBSZ- plate boundary shear zone; SMSZ - southern mesh boundary shear zone; and JdFR - Juan de Fuca Ridge. Cross sections A and B correspond to those in Figure 2.
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 The 3D model domain spans from 185° to 240° longitude, 45°N to 72°N latitude, and 0 to 1500 km in depth (Figure 3). The finite element mesh varies in resolution from 0.04° to 0.255° in the longitudinal direction, 0.0211° to 0.18° in the latitudinal direction, and 2.35 km to 25 km in the radial direction, with the highest resolution centered on the plate boundary corner in south central Alaska. The mesh contains 960 x 648 x 160 elements in the longitudinal, latitudinal, and radial directions, respectively. Models were run using 360 processors on Lonestar, a Dell Linux cluster at the Texas Advanced Computing Center, for approximately 48 hours per job in models with the composite viscosity and for less time in models with the Newtonian only viscosity.
 Reflecting (free-slip) boundary conditions are used on all boundaries. Because the North American plate has a high viscosity, it is essentially fixed horizontally at the model domain sides by the free-slip boundary conditions providing a fixed reference frame for the relative motion of the Pacific plate. In contrast, we use a low viscosity zone to decouple the Pacific plate from the southern edge of the model domain (see below). This allows the Pacific plate to move freely in response to the local driving forces. 2D tests were also used to determine the necessary box depth and width in order to minimize boundary condition effects on the flow in the subduction zone. These are all instantaneous flow simulations designed to explore the present-day balance of forces, lithosphere and mantle structure.
3.2. Subducting Plate Geometry
 We construct two 3D slab shapes, slabE325 and slabE115, to test competing hypotheses for the geometry and continuity of the subducted oceanic lithosphere beneath south central Alaska (Figure 1). We assume that the Wadati-Benioff zone represents the shape of the subducting lithosphere, and use seismic data to constrain the shape and depth extent of the slab (Figure 1; Table 4; and references therein). In models using slabE325, the depth of the slab is everywhere 325 km, and thus the Aleutian and Wrangell slabs are continuous. In contrast, in models using slabE115, east of 212° longitude the slab surface extends to only to 115 km depth. Thus in models using slabE115, the slab is shorter beneath the Wrangell Mountains and barely protrudes beneath the lithosphere of the overriding plate in this region of the model (Figure 1; Table 4).
 The 3D slab surfaces (Figure 1) were generated using a tensioned cubic spline algorithm in GMT [Wessel and Smith, 1991]. The slab surface was then mapped onto the finite element grid using the SlabGenerator code by calculating the perpendicular distances of the finite element mesh points to the slab surface. In this way, the varying strike, dip and depth of the slab are smoothly represented on the model grid. From these perpendicular distances, the initial 3D thermal structure and 3D plate boundary shear zone structure are constructed.
3.3. Subducting Plate Thermal Structure
 A 3D thermal structure constrains the temperature-dependent viscosity and the density anomaly that drives the flow in the geodynamic models. The thermal structure is based on geologic and geophysical observables, for both the subducting and overriding plates, thereby capturing the regional variability in the plate boundary system (Figure 2).
 The 3D thermal structure for the subducting plate, generated by the SlabGenerator code, is constructed using a semi-infinite half-space cooling model in which the thickness of the thermal lithosphere is a function of the plate age [Turcotte and Schubert, 2002]:
where Tm = 1400°C, Ts = 0°C, κ = 1 × 10−6 m2/s, d is depth perpendicular from the slab surface, and t is the plate age. The half-space cooling model is an appropriate approximation of the thermal structure of oceanic lithosphere where the plate age is less than approximately 80 Myr [Turcotte and Schubert, 2002; Hillier and Watts, 2005], which is the case for the majority of the Pacific plate included in our models (Figure 2a) [Muller et al., 1997].
 Along the length of the Aleutian trench, the seafloor age decreases eastward, from approximately 70 Myr at 185° E longitude to approximately 30 Myr in the plate boundary corner (Figure 2a). The seafloor age at the trench is extrapolated onto the subducted part of the Pacific plate so that there is also an eastward decrease in the age of the slab. The eastward decrease of plate age creates an the eastward decrease in thermal and mechanical thickness of the Pacific plate. Note, that we do not include a compositional density anomaly corresponding to the Yakutat block in our models, and therefore, in the plate boundary corner, we assume the age of the seafloor beneath the Yakutat block is the same as that just southwest of the Transition fault (Figure 2).
 To simulate conductive warming of the slab in the mantle, a depth-dependent correction factor based on a length-scale diffusion analysis is applied to d, the distance perpendicular from the slab surface used in equation (1). We adjust the minimum value of d as a function of increasing vertical depth in the model, such that the minimum value for d on the slab surface increases from 0 km for the unsubducted part of the Pacific plate to 15 km on the slab surface at 300 km depth. Before this correction is added, T = 0°C everywhere on the slab surface, even on the parts of the slab surface that are at 300 km depth. After this correction, the minimum temperature on the slab surface smoothly increases with increasing slab depth, simulating the warming that would occur as the slab is immersed within the mantle, with the deeper parts having been immersed longer and hence warmed longer. The slab thermal field is then blended into the ambient mantle thermal field using a sigma-shaped smoothing function.
3.4. Overriding Plate Thermal Structure
 The 3D thermal structure for the overriding North American plate, constructed with SlabGenerator, also assumes a semi-infinite half-space cooling model (equation (1)), but uses effective thermal ages assigned to the continental regions as a proxy to capture the temperature structure within the continental lithosphere (Figures 2a and 2b). We elect this approach rather than using surface heat flow observations to directly invert for the temperature as a function of depth, because there are few coupled surface heat flow and radiogenic heat production measurements in Alaska. Without coupled measurements, the relative contribution of heat from radioactive decay within the crust cannot be quantitatively separated from the contribution of heat from the underlying mantle [Turcotte and Schubert, 2002].
 Thus, using equation (1), we construct a spatially variable depth-dependent thermal structure for the overriding plate, with the overriding plate subdivided into four thermal domains: the Cordilleran region, the magmatic arc, the forearc, and ancestral North America (Figures 2a and 2b). We base these subdivisions on a synthesis of regional geophysical and geologic observations, including the surface heat flow [Blackwell and Richards, 2004], location of Neogene volcanism (Alaska Volcano Observatory) [Plafker et al., 1994], seismic profiles [Fuis et al., 2008], moho temperature estimates, major terrane boundaries [Greninger et al., 1999], and the integrative work characterizing the thermal structure of the lithosphere in western Canada by Currie and Hyndman  and Lewis et al. . For the Cordilleran domain, we assume the relatively warm Cordilleran back arc lithosphere in western Canada [Currie and Hyndman, 2006; Lewis et al., 2003] extends into mainland Alaska (Figures 2a and 2b), and assign this region an effective age of 30 Myr, corresponding to the lithospheric thickness of a continental mobile belt [Blackwell, 1969; Hyndman et al., 2005]. The magmatic arc is assigned an effective age of 10 Myr, giving it the thinnest (warmest) lithosphere with the 1200°C isotherm at approximately 40 km. The forearc and ancestral North America (effective ages of 80 Myr) have the thickest lithosphere reaching 1200°C at approximately 110 km depth (Figure 2b). The resultant 3D thermal structure is shown in Figures 2c–2e.
 To test the effect of the lateral variations in the overriding plate thermal structure, we also construct a depth-dependent thermal structure that uses a single effective age and thus has a laterally uniform thermal field throughout the entire overriding plate. We choose an effective age of 30 Myr, which corresponds to a lithospheric thickness of approximately 60 km at the 1200°C isotherm and 90 km at the 1350°C isotherm (Figure 2b), and is comparable to that of warm continental back arcs [Blackwell, 1969; Hyndman et al., 2005].
3.6. Governing Equations and Numerical Method
 The open source finite element code, CitcomCU [Zhong, 2006], based on CITCOM [Moresi and Solomatov, 1995; Moresi and Gurnis, 1996], is used to solve for the viscous flow in the 3D models of the Alaska plate boundary corner. CitcomCU solves the Navier-Stokes equation for the velocity and pressure, assuming an incompressible fluid with a high Prandlt number, given by the conservation of mass:
and conservation of momentum:
where u, σ, ρo, α, T, To, g, and δrr are the velocity, stress tensor, density, coefficient of thermal expansion, temperature, reference temperature, acceleration due to gravity, and Kronecker delta, respectively [Moresi and Solomatov, 1995; Zhong, 2006]. The models are instantaneous, therefore the energy equation is not solved.
 The constitutive relation is defined by
where P is the dynamic pressure, ηeff is the effective viscosity defined in equation (9), and is the strain rate tensor. The models are defined and solved in spherical coordinates. The equations of motion are non-dimensionalized by the Rayleigh number, Ra, is defined by
where ΔT = To−Tsurf and ρo, α, g, R, ηref, and κ are as defined in Table 2.
Table 2. Dimensionalization Parameters
|Ra||Rayleigh number||2.34 × 109|
|g||acceleration due to gravity, m/s2||9.8|
|To||reference temperature, K||1673|
|Tsurf||temperature on top surface, K||273|
|R||radius of Earth, m||6371 × 103|
|ηref||reference viscosity, Pa⋅s||1 × 1020|
|ρo||reference density, kg/m3||3300|
|κ||thermal diffusivity, m/s2||1 × 10−6|
|α||thermal expansion coefficient, K−1||2.0 × 10−5|
3.7. 3D Model Rheology
 Experimental studies and the observation of seismic anisotropy throughout the upper mantle, suggest that dislocation creep is the dominant deformation mechanism of olivine in the upper mantle [Hirth and Kohlstedt, 2003]. The non-linear relationship between stress and strain rate for deformation by dislocation creep leads to a faster rate of deformation (higher strain rates) for a given stress, which can be expressed as a reduction in the effective viscosity (i.e., ). Previous 3D numerical modeling studies using only Newtonian viscosity (diffusion creep) have shown that imposing a region of low viscosity in the mantle wedge provides a better fit to the geoid and topography [Billen and Gurnis, 2001; Billen et al., 2003; Billen and Gurnis, 2003]. Because mantle flow strain rates are high in the corner of the mantle wedge, the imposed low viscosity region in these previous models may be due to the weakening effect of the dislocation creep mechanism, providing one example of the possible importance of including the non-Newtonian (dislocation creep) viscosity. Therefore following the implementation of Billen and Hirth , in our 3D models of the Alaska subduction zone, we modified CitcomCU to use a composite viscosity in the upper mantle, that includes the effects of both dislocation (ds) and the diffusion (df) creep deformation mechanisms [Jadamec and Billen, 2010].
 For deformation under a fixed stress (driving force) the total strain rate is the sum of the contributions from deformation by the diffusion and dislocation creep mechanisms:
where without the subscripts i-j refers to the second invariant of the strain rate tensor, . Substituting, , the composite viscosity, ηcom, can be defined by
 The viscosity components, ηdf and ηds, are defined assuming the experimentally-determined viscous flow law governing deformation for olivine aggregates [Hirth and Kohlstedt, 2003] such that
where Pl is the lithostatic pressure, R is the universal gas constant, T is non-adiabatic temperature, Tad is the adiabatic temperature (with an imposed gradient of 0.3 K/km), and A, n, d, p, COH, r, E, and V are as defined in Table 3, assuming no melt is present [Hirth and Kohlstedt, 2003].
Table 3. Flow Law Parameters, Assuming Diffusion and Dislocation Creep of Wet Olivinea
|A||pre-exponential factor||1.0||9 × 10−20|
|d||grain size, μm (if A in μm)||10 × 103||–|
|p||grain size exponent||3||–|
|COH||OH concentration, H/106 Si||1000||1000|
|r||exponent for COH term||1||1.2|
|E||activation energy, kJ/mol||335||480|
|V||activation volume, m3/mol||4 × 10−6||11 × 10−6|
 For diffusion creep, the strain rate depends linearly on the stress (n = 1) but non-linearly on the grain size (p = 3) [Hirth and Kohlstedt, 2003]. A grain size of 10 mm is used for the upper mantle giving a background viscosity of 1020 Pa s at 250 km. In contrast, for dislocation creep of olivine, the strain rate dependence is non-linear for dislocation creep of olivine (n = 3.5), but there is no grain-size dependence, giving the same background viscosity at 250 km for a strain rate of 10−15 s−1 [Hirth and Kohlstedt, 2003]. For higher strain rates, the composite viscosity will lead to lower effective viscosity due to most of the deformation being accommodated by the dislocation creep mechanism.
 The lower mantle viscosity is included using the Newtonian flow law for olivine, with a larger effective grain size (70 mm) in order to create a viscosity jump by a factor of 30 from the upper to lower mantle. This simplified viscosity structure for the lower mantle is consistent with the magnitude of the viscosity jump constrained by post-glacial rebound [Mitrovica, 1996] and models of the long wavelength geoid [Hager, 1984], as well as the lack of observations of seismic anisotropy in the lower mantle and the few experimental constraints on the viscous behavior of perovskite.
 Close to the earth's surface and within the cold core of the subducted slab, the viscosity values determined by equation (7) become unrealistically large in that they imply a rock strength much greater than that predicted by laboratory experiments [Kohlstedt et al., 1995]. To allow for plastic yielding where the stresses calculated in the model exceed those predicted from laboratory experiments, the stresses calculated in the model are limited by a depth-dependent yield stress. The yield stress increases linearly with depth, assuming a gradient of 15 MPa per km, from 0.1 MPa at the surface to a maximum value of either 500 MPa or 1000 MPa (Table 1), with the maximum value based on constraints from experimental observations, low-temperature plasticity, and the dynamics in previous models [Kohlstedt et al., 1995; Weidner et al., 2001; Hirth, 2002; Billen and Hirth, 2005, 2007].
 We thus define the effective viscosity, ηeff:
where ηcom, σII, , and σy are the composite viscosity, second invariant of the stress tensor, second invariant of the strain rate tensor, and yield stress, respectively. The effective viscosity is solved for as an additional solution loop in CitcomCU that iterates until the global difference between the velocity field of consecutive solutions is less than some specified value, typically 1% [Billen and Hirth, 2007].The models allow for viscosity variations of up to seven orders of magnitude, with a minimum viscosity cutoff value of 5 × 1017 Pa s and a maximum value of 1 × 1024 Pa s.
 To incorporate the 3D PBSZ defined by SlabGenerator into CitcomCU, we modified CitcomCU to read in a weak zone field, such that the weak regions indicative of the plate boundaries are smoothly blended into the background viscosity, ηeff, using
where ηeff is the effective viscosity as defined in equation (9), and ηo is the reference viscosity equal to 1 × 1020 Pa s [Jadamec, 2009; Jadamec and Billen, 2010]. Awk is the scalar weak zone field, defined using SlabGenerator, assuming a sigma-function with values ranging from 0 to 1 [Jadamec, 2009; Jadamec and Billen, 2010]. Values of 1 correspond to fully weakened regions in the center of the shear zone and values of 0 correspond to unweakened regions. The ηwk value serves as an upper bound on the viscosity in the shear zone and will be overwritten if the viscosity calculated by equation (9) is lower. Thus, the final form of the viscosity, ηf, becomes
 Although we cannot compare the complex 3D numerical models presented in our paper to an analytic solution, previous 2D modeling by Moresi and Solomatov  and Moresi et al.  using CITCOM indicates that for models with large viscosity variations, error can be reduced by limiting the viscosity jump across each element, preferably to a factor of 3 or less. CitcomCU implements the full multigrid method to accelerate convergence [Zhong, 2006], which was found to save on valuable compute time in the high resolution regional models of the Alaska plate boundary containing large viscosity variations [Jadamec, 2009]. In our models, to limit the viscosity jump in the PBSZ, the PBSZ width varies with the resolution of the finite element mesh so that the PBSZ always spans a minimum number of elements. We found that numerical stability could be obtained using 8 elements to span four orders of magnitude of viscosity, i.e., we allow for viscosity jumps of up to a factor of 5 across the elements in the PBSZ. This implementation was tested on a series of 3D models with a simplified subduction zone geometry and we found that viscosity contrasts on this order led to good convergence behavior [Jadamec, 2009].
3.8. 3D Visualization
 The open source software 3DVisualizer [Kreylos et al., 2006; Billen et al., 2008; Jadamec et al., 2008] was used in the Keck Center for Active Visualization in the Earth Sciences (KeckCAVES) for rapid inspection and interactive exploration of the 3D plate boundary geometry and thermal structure output from the SlabGenerator code. In this way, the quality and smoothness of the features mapped onto the model grid, containing over 100 million finite element nodes, could be assessed efficiently. This was especially useful because of the large model size (which amounted to inspecting almost two thousand 2D model slices for each 3D model) and because of the geometrically complex slab and plate boundary configuration.
 3DVisualizer was also used in the KeckCAVES for inspection and exploration of the composite viscosity field, velocity field, and strain rate output from CitcomCU for each viscous flow simulation. Interactive exploration of the model output in a 3D immersive environment allowed for on-the-fly query of streamlines, velocity vectors, and of isosurfaces of scalar variables such as velocity magnitude, viscosity, and pressure, allowing for real-time assessment of hypotheses. Using the interactive 3D virtual reality facility enabled the massive amounts of data (several Gigabytes per model run) to be conceptualized and allowed for identification of key areas to target for more quantitive analysis with MATLAB or other tools like GMT. To preserve the choices and features that were explored and identified, numerous sessions were recorded with a playback feature making the sessions portable to a movie format. 3D model results were also visualized with the desktop version of 3DVisualizer.