JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, B02304, doi:10.1029/2011JB008563, 2012
The role of rheology and slab shape on rapid mantle flow:
Three-dimensional numerical models of the Alaska slab edge
M. A. Jadamec1,2 and M. I. Billen1
Received 31 May 2011; revised 6 December 2011; accepted 8 December 2011; published 9 February 2012.
[1] Away from subduction zones, the surface motion of oceanic plates is well
correlated with mantle flow direction, as inferred from seismic anisotropy. However,
this correlation breaks down near subduction zones, where shear wave splitting studies
suggest the mantle flow direction is spatially variable and commonly non-parallel to
plate motions. This implies local decoupling of mantle flow from surface plate motions,
yet the magnitude of this decoupling is poorly constrained. We use 3D numerical
models of the eastern Alaska subduction-transform plate boundary system to further
explore this decoupling, in terms of both direction and magnitude. Specifically, we
investigate the role of the slab geometry and rheology on the mantle flow velocity at a
slab edge. The subducting plate geometry is based on Wadati-Benioff zone seismicity
and tomography, and the 3D thermal structure for both the subducting and overriding
plates, is constrained by geologic and geophysical observations. In models using the
composite viscosity, a laterally variable mantle viscosity emerges as a consequence of
the lateral variations in the mantle flow and strain rate. Spatially variable mantle
velocity magnitudes are predicted, with localized fast velocities (greater than 80 cm/yr)
close to the slab where the negative buoyancy of the slab drives the flow. The same
models produce surface plate motions of less than 10 cm/yr, comparable to observed
plate motions. These results show a power law rheology, i.e., one that includes the
effects of the dislocation creep deformation mechanism, can explain both observations
of seismic anisotropy and decoupling of mantle flow from surface motion.
Citation: Jadamec, M. A., and M. I. Billen (2012), The role of rheology and slab shape on rapid mantle flow: Three-dimensional
numerical models of the Alaska slab edge, J. Geophys. Res., 117, B02304, doi:10.1029/2011JB008563.
1. Introduction
[2] Plate boundaries are inherently three-dimensional (3D)
tectonic features with variations in geometry and physical
properties along their length [Jarrard, 1986; Gudmundsson
and Sambridge, 1998; Bird, 2003; Lallemand et al., 2005;
Sdrolias and Muller, 2006; Syracuse and Abers, 2006;
Rychert et al., 2008]. For example, subduction zones have
significant changes in slab dip along strike, including flat
slab segments, as in the eastern Alaska subduction zone, the
Peru-Chile Trench, the Cascadia subduction zone, and the
southwestern Japan Trench [Gudmundsson and Sambridge,
1998; Gutscher et al., 2000; Ratchkovski and Hansen,
2002; Lallemand et al., 2005; Syracuse and Abers, 2006;
Tassara et al., 2006]. In addition, many plate boundaries are
not simply end-member subduction zones, transform faults,
or spreading centers in isolation, but rather are hybrid systems. For example, plate boundary corners with slab edges
1
Geology Department, University of California, Davis, California,
USA.
2
School of Mathematical Sciences and School of Geosciences, Monash
University, Clayton, Victoria, Australia.
Copyright 2012 by the American Geophysical Union.
0148-0227/12/2011JB008563
formed by the intersection of a subduction zone with a
transform fault occur in eastern Alaska, the northern KurilKamchatka subduction zone, the southernmost Mariana
trench near the Yap trench, the northern Tonga trench, and
the north and south boundaries of the Antilles and the Scotia
subduction zones [Gudmundsson and Sambridge, 1998;
Bird, 2003]. Subduction-transform intersections also occur
in non-corner geometries, as in the termination of the Kermadec Trench against the Alpine fault in New Zealand and
the north and south terminations of the Cascadian subduction zone [Gudmundsson and Sambridge, 1998; Bird, 2003].
The three-dimensionality in both surface and subsurface
structure can influence the relative motion of the mantle with
respect to the surface plates. Thus, in many cases a twodimensional (2D) approach to studying plate boundary processes can miss critical aspects of the dynamics resulting
purely from the geometry.
[3] Away from subduction zones, the surface motion of
oceanic plates is well correlated with mantle flow direction,
as inferred from seismic anisotropy [Kreemer, 2009; Conrad
et al., 2007; Becker et al., 2003]. However, observations of
seismic anisotropy from numerous subduction zones display
seismic fast directions that are non-parallel to plate motions,
e.g., the Peru-Chile, Kamchatka, Tonga, Marianas, South
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Sandwich, Cascadia, Costa Rica-Nicaragua, and eastern
Alaska subduction zones [Russo and Silver, 1994; Peyton
et al., 2001; Smith et al., 2001; Pozgay et al., 2007; Muller
et al., 2008; Zandt and Humphreys, 2008; Long and Silver,
2008; Abt et al., 2009; Christensen and Abers, 2010]. If
the seismic fast directions are tracking mantle flow [Savage,
1999; Kaminiski and Ribe, 2002; Kneller and van Keken,
2007; Long and Silver, 2009; Jadamec and Billen, 2010],
this implies that local decoupling between the plates and
underlying mantle may be a common feature in subduction
zones. This also indicates differential motion, at least in
terms of direction, of the mantle close to subduction zones
versus that farther away from them.
[4] Several 3D semi-analytic, numerical, and laboratory
experiments have investigated mantle flow dynamics in
subduction zones [Zhong and Gurnis, 1996; Zhong et al.,
1998; Buttles and Olson, 1998; Hall et al., 2000; Kincaid
and Griffiths, 2003; Schellart, 2004; Funiciello et al., 2006;
Piromallo et al., 2006; Royden and Husson, 2006; Schellart
et al., 2007; Kneller and van Keken, 2008; Giuseppe et al.,
2008; Jadamec and Billen, 2010]. For example, 3D subduction studies, commonly in the context of slab rollback, indicate a toroidal component of mantle flow dominates near the
slab edge, where material is transported from beneath the slab
around the slab edge and into the mantle wedge [Kincaid and
Griffiths, 2003; Schellart, 2004; Funiciello et al., 2006;
Piromallo et al., 2006; Stegman et al., 2006]. This component of flow cannot be captured in 2D models of subduction
[Tovish et al., 1978; Garfunkel et al., 1986]. In addition,
numerical models investigating the effect of variable slab dip
and curvature on flow in the mantle wedge show that the
variable geometry can induce pressure gradients that drive
trench parallel flow [Kneller and van Keken, 2007, 2008].
Thus, both observations of seismic anisotropy and 3D fluid
dynamics simulations indicate a difference between the
direction of mantle flow and the surface plate motion, indicating local decoupling of the surface plates from the mantle
in subduction zones [Jadamec and Billen, 2010].
[5] In this paper, we use 3D instantaneous regional models
of the eastern Alaska subduction-transform plate boundary
system to further explore this decoupling of mantle flow
from surface plate motion, in terms of both direction and
magnitude. The 3D models of the subduction-transform
plate boundary system in southern Alaska use a subducting
plate geometry based on Wadati-Benioff zone seismicity and
a 3D thermal structure for both the subducting and overriding plates, constrained by geologic and geophysical observations. No kinematic boundary condition is prescribed to
the subducting plate, rather the negative buoyancy of the
slab drives the flow. We employ a composite viscosity,
which includes both the diffusion and dislocation creep
mechanisms (i.e., Newtonian and non-Newtonian viscosity).
[6] This study builds on the 3D numerical models
presented by Jadamec and Billen [2010] that investigated
the effect of the slab shape and Newtonian versus nonNewtonian rheology on rapid mantle flow and compared
the results to observations of seismic anisotropy from
Christensen and Abers [2010]. In this paper, we explore a
larger range in model parameters and specifically address
how these affect the velocity magnitude in the mantle
surrounding the slab, the overall pattern of mantle flow
around the slab edge, as well as implications of the non-
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Newtonian rheology as a mechanism for plate-mantle
decoupling in subduction zones. In addition, we discuss
implications specific to Alaskan tectonics. The approach
taken for models presented in this paper, as well as those of
Jadamec and Billen [2010], differs from that in previous 3D
numerical models of subduction in that previous models
[Moresi and Gurnis, 1996; Zhong and Gurnis, 1996; Billen
et al., 2003; Billen and Gurnis, 2003; Piromallo et al.,
2006; Stegman et al., 2006; Schellart et al., 2007; Kneller
and van Keken, 2007; Giuseppe et al., 2008; Kneller and
van Keken, 2008] either use a simplified plate geometry,
prescribe a velocity boundary condition to the subducting
plate, do not include an overriding plate, or use a Newtonian
rheology in the mantle.
2. Alaskan Subduction-Transform Boundary
[7] The Aleutian-Alaska subduction zone accommodates
convergence between the northernmost Pacific plate and the
Alaskan part of the North American plate (Figure 1). The
subduction zone length is greater than 3000 km, and along
this length the sense of curvature changes from convex to
the south in the central Aleutians to convex to the north in
south central Alaska [Page et al., 1989; Gudmundsson and
Sambridge, 1998; Ratchkovski and Hansen, 2002; Bird,
2003]. Offshore south central Alaska, the observed Pacific
plate motion with respect to North America is N16°W at
5.2 cm/yr [DeMets and Dixon, 1999]. The easternmost part
of the subduction zone forms a plate boundary corner where
the Aleutian trench terminates against the northwest trending Fairweather-Queen Charlotte transform fault (Figure 1)
[Bird, 2003]. In this plate boundary corner, the dip of the
subducting plate shallows to form a flat slab [Ratchkovski
and Hansen, 2002], and the Yakutat block, a buoyant
oceanic plateau, is subducting beneath the North American
plate [Bruns, 1985; Pavlis et al., 2004; Ferris et al., 2003]
(Figure 1).
[8] East of approximately 212° longitude, where the subduction zone approaches the transform boundary, there is an
abrupt decrease in seismicity at depths greater than 25 km
(Figure 1). Despite the decrease in seismicity, several seismic studies suggest there is a steeply dipping slab, the
Wrangell slab, in this region of the plate boundary corner
beneath the active adakitic Wrangell Volcanoes (Figures 1
and 2) [Stephens et al., 1984; Skulski et al.,1991; Preece
and Hart, 2004]. There is debate over the geometry of the
Wrangell slab as well as its continuity with the main Aleutian slab to the west [Perez and Jacob, 1980; Stephens et al.,
1984; Page et al., 1989; Zhao et al., 1995; Ratchkovski and
Hansen, 2002; Eberhart-Phillips et al., 2006; Fuis et al.,
2008]. Therefore, we construct two slab geometries, one
with and one without the Wrangell slab, to test the influence
of the slab geometry on the mantle flow in this plate
boundary corner (Figure 1).
3. Methodology
[9] 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
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Figure 2. Temperature structure for 3D geodynamic model. (a) Seafloor ages for Pacific plate and thermal domains in upper plate used in half-space cooling model. Seafloor ages from Muller et al. [1997]. See
text for constraints on upper plate age assignment. (b) Temperature profiles for domains in upper plate:
I - forearc, II - magmatic arc, III - Cordillera, IV - Ancestral North America. Note temperature profiles
for domains I and IV overlap because the two regions have the same effective age. (c) Radial slice
through temperature at 26 km depth, which is the same for models using slabE325 and slabE115. Radial
slice at 152 km depth for models with (d) slabE325 and (e) slabE115. (f) Cross section through flat slab
region (AA′), which is the same for models using slabE325 and slabE115. Cross section through the Wrangell
Volcanics (BB′) for models using (g) slabE325 and (h) slabE115. Note, Figures 2b–2h are shown for subset of
model domain.
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
[13] 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 WadatiBenioff 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).
[14] The 3D slab surfaces (Figure 1) were generated using
a tensioned cubic spline algorithm in GMT [Wessel and
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Table 1. Summary of Model Parameters and Resultsa
Model
A1n
A2n
A3n
A4n
A1c
A2c
A3c
A4c
B1n
B2n
B3n
B4n
B5n
B1c
B2c
B3c
B4c
B5c
Slab
325
325
325
325
325
325
325
325
115
115
115
115
115
115
115
115
115
115
Top
var
var
var
var
var
var
var
var
var
var
var
var
unif
var
var
var
var
unif
Rheology
sy Max. (MPa)
Newtonian
Newtonian
Newtonian
Newtonian
Composite
Composite
Composite
Composite
Newtonian
Newtonian
Newtonian
Newtonian
Newtonian
Composite
Composite
Composite
Composite
Composite
500
500
1000
1000
500
500
1000
1000
500
500
1000
1000
500
500
500
1000
1000
500
PBSZ (Pa s)
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Vpacsurf (cm/yr, Azimuth)
20
2.11,
0.76,
2.10,
0.74,
5.28,
2.58,
5.17,
2.47,
1.92,
0.69,
1.92,
0.68,
0.74,
4.40,
2.04,
4.40,
2.02,
2.10,
& 10
& 1021
& 1020
& 1021
& 1020
& 1021
& 1020
& 1021
& 1020
& 1021
& 1020
& 1021
& 1021
& 1020
& 1021
& 1020
& 1021
& 1021
338.91°
328.64°
338.39°
327.29°
345.92°
345.89°
342.93°
342.73°
332.63°
320.38°
332.11°
318.90°
318.45°
339.59°
339.45°
336.57°
336.18°
340.39°
Vop (cm/yr, Azimuth)
Vwedge (cm/yr, Azimuth)
0.23, 263.09°
0.18, 285.15°
0.23, 261.17°
0.18, 282.83°
0.06, 280.92°
0.06, 290.08°
0.06, 275.49°
0.06, 283.39°
0.24, 264.99°
0.19, 279.15°
0.24, 263.61°
0.19, 277.28°
0.26, 264.21°
0.07, 276.41°
0.07, 284.45°
0.07, 273.33°
0.06, 279.12°
0.08, 273.39°
2.56, 185.80°
2.43, 183.03°
2.25, 187.02°
2.07, 183.67°
31.68, 189.84°
28.52, 189.51°
15.40, 180.79°
12.12, 179.42°
3.15, 204.87°
2.86, 203.13°
2.88, 207.91°
2.52, 206.48°
3.16, 203.42°
65.78, 225.59°
79.79, 226.71°
40.82, 225.39°
32.32, 225.24°
94.76, 225.81°
a
Top is temperature structure of overriding plate, either (var) laterally variable effective age or (unif) laterally uniform effective age with both using an
age-dependent temperature formulation. Vpacsurf is the Pacific plate surface velocity offshore south central Alaska at 217.3° longitude, 57.0° latitude. Vop
is the surface velocity in the overriding plate at 210.5° longitude, 64.5° latitude. Vwedge is the velocity in mantle wedge at 210.5° longitude, 64.5° latitude,
and 100 km depth. At this location in the mantle wedge, horizontal velocities dominate with vertical velocities less than 3% of the horizontal velocity
magnitude. Vertical velocities increase with proximity to the pivoting slab tip.
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
[15] A 3D thermal structure constrains the temperaturedependent 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).
[16] 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]:
T ¼ ðTm " Ts Þerf
"
#
d
pffiffiffiffi þ Ts
2 kt
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).
[18] 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
ð1Þ
where Tm = 1400°C, Ts = 0°C, k = 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].
[17] 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
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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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
[19] 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].
[20] 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 [2006] and Lewis et al. [2003]. 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.
[21] To test the effect of the lateral variations in the
overriding plate thermal structure, we also construct a depthdependent 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
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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].
3.5. Plate Boundary Shear Zone
[22] In numerical models of subduction the plate interface
is typically represented either as a discrete fault surface
[Kincaid and Sacks, 1997; Zhong et al., 1998; Billen et al.,
2003] or a narrow low viscosity layer [Gurnis and Hager,
1988; Kukacka and Matyska, 2004; Billen and Hirth,
2007]. We model the plate interface as a narrow 3D low
viscosity layer in which the imposed reduction in viscosity
represents a number of processes including the shear heating, variation in grain size, and the presence of water and
sediments. We use the SlabGenerator code to define the
plate interface, referred to as the plate boundary shear zone
(PBSZ), that is at least 40 km wide and extends to 100 km in
depth [Jadamec, 2009; Jadamec and Billen, 2010]. The
rheological implementation of the narrow 3D low viscosity
layer (PBSZ) is described in section 3.7. The orientation of
the PBSZ is spatially variable following the strike and dip of
the subducting plate and is continuous with the verticallyoriented Fairweather-Queen Charlotte transform boundary
(Figures 1 and 3). The PBSZ location is fixed in space,
which is appropriate for these models of the present-day
deformation [Sdrolias and Muller, 2006]. We also include a
narrow vertical low viscosity layer located along the southern edge of the Pacific plate in our model to decouple the
Pacific plate from the southern model boundary. This low
viscosity layer is implemented by the same method as for the
PBSZ. No low viscosity layer is imposed on the Juan de
Fuca ridge, because the localized warm temperatures at the
spreading center decouple the plates in that region. In addition, no weak zone is imposed on the western-most boundary as this was found to have little affect on the model
results.
3.6. Governing Equations and Numerical Method
[23] 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:
r⋅u ¼ 0
ð2Þ
and conservation of momentum:
r⋅s þ ro aðT " To Þgd rr ¼ 0
ð3Þ
where u, s, ro, a, T, To, g, and drr 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.
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Table 2. Dimensionalization Parameters
Parameter
Description
Value
Ra
g
To
Tsurf
R
href
ro
k
a
Rayleigh number
acceleration due to gravity, m/s2
reference temperature, K
temperature on top surface, K
radius of Earth, m
reference viscosity, Pa⋅s
reference density, kg/m3
thermal diffusivity, m/s2
thermal expansion coefficient, K"1
2.34 & 109
9.8
1673
273
6371 & 103
1 & 1020
3300
1 & 10"6
2.0 & 10"5
where ɛ̇ without the subscripts i-j refers to the second
invariant of the strain rate tensor, ɛ̇II . Substituting, s ¼
2hcom ɛ̇, the composite viscosity, hcom, can be defined by
hcom ¼
hdf ;ds ¼
ð4Þ
where P is the dynamic pressure, heff is the effective viscosity defined in equation (9), and ɛ̇ij 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
Ra ¼
ro agDTR3
khref
ð5Þ
where DT = To"Tsurf and ro, a, g, R, href, and k are as
defined in Table 2.
3.7. 3D Model Rheology
[25] 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
_ Previous 3D numerical modeling studies
(i.e., heff ¼ s=2ɛ).
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 nonNewtonian (dislocation creep) viscosity. Therefore following the implementation of Billen and Hirth [2007], 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].
[26] 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:
ɛ̇com ¼ ɛ̇df þ ɛ̇ds
ð6Þ
ð7Þ
[27] The viscosity components, hdf and hds, are defined
assuming the experimentally-determined viscous flow law
governing deformation for olivine aggregates [Hirth and
Kohlstedt, 2003] such that
[24] The constitutive relation is defined by
sij ¼ "Pd ij þ heffɛ̇ij
hdf hds
hdf þ hds
"
dp
r
ACOH
#1n
$
%
E þ Pl V
1"n
ɛ̇ n exp
nRðT þ Tad Þ
ð8Þ
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].
[28] 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.
[29] 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.
[30] 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
Table 3. Flow Law Parameters, Assuming Diffusion and Dislocation Creep of Wet Olivinea
Variable
Description
Creepdf
Creepds
A
n
d
p
COH
r
E
V
pre-exponential factor
stress exponent
grain size, mm (if A in mm)
grain size exponent
OH concentration, H/106 Si
exponent for COH term
activation energy, kJ/mol
activation volume, m3/mol
1.0
1
10 & 103
3
1000
1
335
4 & 10"6
9 & 10"20
3.5
–
–
1000
1.2
480
11 & 10"6
a
Values from Hirth and Kohlstedt [2003].
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Table 4. Seismic Constraints on Slab Deptha
Study
Data Type
daleut (km)
dakp (km)
dscak (km)
dwr (km)
Qi et al. [2007]
Eberhart-Phillips et al. [2006]
Zhao et al. [1995]
Kissling and Lahr [1991]
Fuis et al. [2008]
Ferris et al. [2003]
Ratchkovski and Hansen [2002]
Page et al. [1989]
Engdahl and Gubbins [1987]
Stephens et al. [1984]
Boyd and Creager [1991]
Gudmundsson and Sambridge [1998]
3D teleseismic tomography
3D tomography
3D tomography
3D tomography
Seismic reflection, refraction earthquake hypocentral locations
Teleseismic receiver function analysis
Earthquake hypocentral locations
Seismic reflection, refraction earthquake hypocentral locations
Simultaneous time travel inversion
Earthquake hypocentral locations
Local seismicity and teleseismic residual sphere analysis
RUM seismic model
–
–
–
–
–
–
–
–
400
–
600
300
>400
>200
>190
–
240
–
210
–
–
–
–
250
300–400
160–180
165
120–150
175
150
165–185
–
–
–
–
150
90
50–60
>90
–
100
–
–
>100
–
85
–
–
a
Abbreviations, daleut, dakp, dscak, and dwr, refer to the depth of the subducting plate beneath the Aleutians, the Alaska Peninsula, south central Alaska, and
the Wrangell Mountains. Data suggest a decrease in slab depth from west to east. Note, there is a range in depth resolution for studies listed in the table.
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].
[31] We thus define the effective viscosity, heff:
&
'
heff ¼ hcom ; if sII < sy
¼
(" #
)
sy
; if sII > sy
ɛ̇II
ð9Þ
ð10Þ
where hcom, sII, ɛ̇II , and sy 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.
[32] 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, heff, using
hwk ¼ ho 10ð1"Awk Þlog10 ðheff =ho Þ
ð11Þ
where heff is the effective viscosity as defined in equation (9),
and ho 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 hwk 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, hf, becomes
hf ¼ minðheff ; hwk Þ
ð12Þ
[33] Although we cannot compare the complex 3D
numerical models presented in our paper to an analytic
solution, previous 2D modeling by Moresi and Solomatov
[1995] and Moresi et al. [1996] 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
[34] The increasing incorporation of high performance
computing and massive data sets into scientific research has
led to the need for high fidelity tools to analyze and interpret
the information [Erlebacher et al., 2001; Kreylos et al.,
2006; Chen et al., 2008; Kellogg et al., 2008]. Immersive
3D visualization facilities provide one approach to fill this
gap in the workflow, especially when working with complex
spatially varying data and nonlinear model behavior [e.g.,
Kreylos et al., 2006; Billen et al., 2008].
[35] 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
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Figure 4. Flow field, viscosity and strain rate for models using (left) slab325 with composite viscosity,
(middle) slab115 with composite viscosity, and (right) slabE115 with Newtonian viscosity. Viscosity isosurface and velocity slices for (a, d) model A1c, (b, e) model B1c, and (c, f) model B1n. Cross section of viscosity and second invariant of the strain rate tensor for (g, j) model A1c, (h, k) model B1c, and (i, l) model
B1n. Profiles at 64.5°N (location of vertical line in Figures 4g–4l) for (m) models A1c-A4c, (n) models
B1c-B5c, and (o) models B1n-B5n. Figures show subset of model domain.
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.
[36] 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
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10° westward so that it is focused around the most northern
part of the slab. This leads to a stronger component of westward directed trench parallel flow beneath the magmatic gap
in south central Alaska for models with slabE115.
[40] Figures 4d–4f show the poloidal component of flow,
where the sinking of the slab draws material into the mantle
wedge toward the slab. Beneath the Pacific plate, a broader
region of poloidal flow is induced in the mantle where the
surface part of the Pacific plate is pulled into the subduction
zone. Cross sections AA′ through the poloidal flow in the
mantle wedge show velocity vectors that are steeper than the
slab dip (Figures 4d–4f), indicating steepening of the slab
dip with time, or retrograde slab motion [Garfunkel et al.,
1986] if the models were to run forward in time. However,
in cross section BB′ for models with slabE115, there is
essentially no poloidal flow component in the mantle wedge
above the Wrangell slab (Figures 4e and 4f). This is because
slabE115 is too short in this region of the model to induce the
poloidal motion of mantle material.
[41] The toroidal and poloidal flow in the mantle around
the subduction zone are not independent. Figures 4a–4c and
Figures 5a–5c show there is motion along strike associated
with the poloidal flow and motion in the vertical direction
associated with the toroidal flow. For example, in all models,
the poloidal flow in cross section AA′ contains a westward
trench parallel velocity component in the mantle wedge,
suggesting the flow of material away from the slab nose
toward the arcuate central Aleutian subduction zone (Figures
4d–4f). The trench parallel flow component around the
northernmost part of the slab is stronger in models with
slabE115, because the center of the toroidal flow is more
westward, i.e., at 212° rather than 222° longitude.
[42] In addition, in all models, the toroidal flow around the
slab edge, whether it is centered at 212° or 222° longitude, is
associated with an upward component of flow (Figures 4a–
4c and Figure 5a–5c). This upward flow component occurs
where the toroidal flow emerges from beneath the slab, i.e.,
on the eastern side of the counterclockwise flow pattern in
Alaska. The magnitude of the upward flow varies depending
on slab shape, slab strength, and mantle rheology, and can
have values greater than 10 cm/yr (Figures 5a–5c). Cross
section BB′ shows the upward component of flow at 100 km
depth for models that use slabE325 with the composite viscosity (Figure 5d), slabE115 with the composite viscosity
(Figure 5e), and slabE115 with the Newtonian viscosity
(Figure 5f). Note the upward component of flow associated
with the slab edge forms a 3D lenticular feature, and thus its
magnitude varies laterally as well as with depth, with the
maximum vertical component occurring at approximately
200 km depth.
[43] The upward component of mantle flow associated
with the toroidal flow at the slab edge combines with the
upward component of the poloidal flow in the mantle wedge
above the slab. The vertical motion of mantle material near
slab edges may play a role in forming melts from both
decompression melting and melting of the slab edge (i.e.,
adakites), as this part of the subduction zone is continuously
exposed to upward advected warm mantle material. The
implications for the adakitic Wrangell Mountains in eastern
Alaska, which are situated at approximately 216° longitude,
are discussed in section 5.4.
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4.2. 3D Variable Viscosity and Mantle Flow
[44] Cross sections through the viscosity structure and
strain rate field are shown for the three representative models: Model A1c (Figures 4g and 4j), Model B1c (Figures 4h
and 4k), and Model B1n (Figures 4i and 4l). Figure 4 also
shows vertical profiles through the viscosity and strain rate
for all models using slabE325 and the composite viscosity
(Figure 4m), slabE115 and the composite viscosity (Figure 4n),
and slabE115 and the Newtonian viscosity (Figure 4o).
[45] Models using the composite viscosity formulation
have higher strain rates and lower mantle viscosities than
models using the Newtonian viscosity. In these models using
the composite viscosity, a laterally variable viscosity emerges in the mantle as a consequence of the lateral variations in
the flow (Figures 4g, 4h, 4j, 4k, 4m, and 4n). Viscosities in
the mantle wedge can be lower than 1018 Pa s, where strain
rates are high, on the order of 10"12 s"1. The weakest viscosities in the mantle wedge occur as elliptical regions
immediately beneath the overriding plate and immediately
above the slab, again correlating to where strain rates are
high ( Figures 4g, 4h, 4j, 4k, 4m, and 4n). Note that although
shown in cross section only, these low viscosity (high strain
rate) regions form three-dimensional ellipsoids in the mantle
wedge that follow the strike of the slab.
[46] For models using slabE115 and the composite viscosity, the high strain rate and low viscosity region in the mantle
wedge along cross section AA′ is broader than for models
using slabE325 (Figure 4h versus Figure 4g). This is because
the velocity gradients and hence strain rates are higher where
both toroidal and polodial flow occur leading to the lower
viscosity, as shown in cross section AA′ for model B1c.
Models that use the composite viscosity and either slab
shape also have high strain rates and low viscosity beneath
and subparallel to the downgoing Pacific plate and slab
(Figures 4g, 4h, 4j, and 4k).
[47] In the Newtonian models, viscosities in the mantle
wedge are higher (1019 to 1020 Pa s) and strain rates are
lower (10"13 to 10"14 s"1). In these models, the mantle
viscosity is laterally uniform, except for temperaturedependent variations, because there is no power law relation
between the stress and strain rate. Therefore, the viscosity is
not reduced in regions of high strain rate as was the case for
models using the composite viscosity (Figure 4).
[48] The weakening effect of the strain rate dependent
viscosity in models using the composite viscosity formulation leads to viscosity contrasts that can be up to seven
orders of magnitude between the mantle and lithosphere.
However, away from the driving force of the slab, strain
rates decrease, and mantle viscosities increase to on the
order of 1020 Pa s. The localized regions of low viscosity and
high strain rate around the slab and beneath the plates significantly reduces the coupling between the lithosphere and
the mantle near the subduction zone.
[49] It is the magnitude of the driving forces that ultimately limits the amount of weakening that occurs in the
models with the composite viscosity formulation, with the
greater amount of weakening in the mantle close to where
the sinking slab is driving the flow. In this way, using this
viscosity formulation redistributes how the stresses are
accommodated in the model by weakening in some regions
(the mantle) while strengthening in others (the lithosphere).
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Figure 7. Predicted Pacific plate motion vectors offshore south central Alaska for Newtonian models
using (a) slabE325 and (b) slabE115 and for composite viscosity models using (c) slabE325 and (d) slabE115.
Observed NUVEL–1A Pacific plate motion vector, assuming North America fixed, shown with thick gray
solid line [DeMets and Dixon, 1999].
leading to faster mantle flow velocities (Figure 6), as seen by
Billen and Hirth [2005], Jadamec and Billen [2010], and
Stadler et al. [2010]. For example, mantle wedge velocities
in models using the composite viscosity can be greater than
ten times that in models using the Newtonian viscosity
(66.8 cm/yr versus 6.5 cm/yr in maximum horizontal
velocity magnitude for models B1c versus B1n at 100 km
depth) (Figure 6). This implies greater decoupling of mantle
flow from surface plate motions for models with the composite viscosity. Note that for both types of upper mantle
rheology, surface plate motions are less than 6 cm/yr (see
section 4.4).
[54] Figures 4–6 illustrate how the velocities are sensitive
to the rheology, including both to the slab strength and the
surrounding mantle viscosity. For models that use the composite rheology, high strain rates located where the slab
bends into the mantle, lead to viscous yielding within the
slab and deformation by dislocation creep on the underside
of the slab, resulting in a narrower and weaker slab core
(Figures 4g, 4h, 4j, and 4k). For example, whereas the slab
core remains on the order of 1024 Pa s in models with the
Newtonian viscosity (Figure 4i), in models with the composite rheology, the viscosity of the slab is reduced to on the
order of 1022 Pa s in the slab hinge (Figures 4g and 4h). As a
result, less weight of the slab can be supported by the slab
strength, allowing for the slab to pivot faster and generate
faster velocities in the mantle (Figures 4–6).
[55] When varying the yield strength, models with a
weaker slab (yield strength of 500 MPa) generate mantle
velocities that are almost twice as fast as the equivalent
models with a stronger slab (yield strength of 1000 MPa)
(black lines versus gray lines in Figures 6d–6f). Again,
models with the lower yield strength allow less of the weight
of the slab to be supported by the slab strength, thus leading
to a slab that is sinking faster and thereby generates more
vigorous mantle flow.
[56] Mantle wedge velocities also vary depending on
whether toroidal and poloidal flow are both occurring. For
example, where both poloidal and toroidal flow are acting
together flow velocities in the mantle are stronger (Figures 5
and 6). This can be seen for example, in comparison of
Model B1c (slabE115) and A1c (slabE325) along cross sections AA′ in Figure 6. The velocities are higher for Model
B1c in which the flow from the toroidal and poloidal cells
combine in the mantle wedge.
[57] In addition, models that use the uniform overriding
plate thermal structure and the composite viscosity have
faster mantle velocities, because the overriding plate is relatively thinner and thus the shallow mantle is warmer,
allowing for lower viscosities and less resistance to flow
(Figure 6e, models B2 versus B5).
4.4. Predicted Surface Motions
[58] The predicted Pacific plate velocity vectors offshore
south central Alaska are shown in Figure 7. For models that
use slabE325 and the Newtonian viscosity, the predicted
velocity of the Pacific plate ranges from 327.3° at 0.74 cm/yr
to 338.9° at 2.11 cm/yr (Figure 7a). For models that use
slabE115 and the Newtonian viscosity, the predicted plate
motion ranges from 318.9° at 0.68 cm/yr to 332.6° at
1.92 cm/yr (Figure 7b). Thus, models that use slabE325 have
a more northerly and faster Pacific plate velocity than the
same models using slabE115. This is because models using
slabE325 have more slab pull in the eastern part of the slab
where it extends to325 km beneath the Wrangell Mountains
(Figure 1). This pattern is also seen in models using the
composite viscosity, where the predicted Pacific plate
velocity ranges from 343° at 2.47 cm/yr to 346° at 5.28 cm/yr
for models using slabE325 (Figure 7c) and from 336° at
2.02 cm/yr to 340° at 4.4 cm/yr for models using slabE115
(Figure 7d).
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[59] Comparison of models using the composite viscosity
with those using the Newtonian viscosity show that, on
average, models using the composite viscosity move in a
more northerly direction and about 2.5 times faster
(Figures 7a and 7b versus Figures 7c and 7d). This is because
the strain rate dependence in models using the composite
viscosity leads to a viscosity reduction in the mantle surrounding the slab, reducing the viscous support of the slab by
the mantle. In addition, high strain rates along the plate
boundary in models using the composite viscosity lead to a
lower viscosity and broader plate boundary shear zone,
allowing less resistance along the plate boundary interface
and consequently faster surface plate motions. Imposing a
higher viscosity of 1021 Pa s versus 1020 Pa s along the plate
interface decreases the surface plate motion by approximately a factor of two, regardless of whether the composite or
Newtonian viscosity is used (Figure 7, dashed versus solid
lines). There is little difference in predicted plate motion for
models that use the uniform overriding plate thermal structure versus those that use the variable overriding plate thermal structure (Figure 7, models B2 versus B5).
[60] Comparison of the predicted Pacific plate velocity to
the observed NUVEL–1A Pacific plate velocity offshore
Alaska [DeMets and Dixon, 1999] is shown in Figure 7.
There is good agreement between the predicted and
observed Pacific plate motion offshore south central Alaska
for models using the composite viscosity and plate boundary
viscosity of 1020 Pa s, regardless of slab shape (Figure 7).
However, before considering how the model parameters
affect the match between the observed and predicted surface
plate motion, it is important to realize that the entire Pacific
plate is not included in the models because doing so would
exceed the computational resources available. Instead, the
model domain was designed such that the bounds of the
Pacific plate included in the models are approximately representative of those of the actual Pacific plate, that is, a
subduction zone to the north (Aleutian-Alaska trench),
transform and spreading center to the east (Queen CharlotteFairweather transform and Juan de Fuca ridge), and a
spreading center to the south (vertical weak zone at the
southern boundary approximates the orientation of the eastwest trending Antarctic spreading center). The main exception is the western boundary, where the Pacific plate has
westward dipping subduction zones that are not included in
the models.
[61] In order for a plate to remain rigid and move in
response to slab-pull and ridge-push forces on multiple
boundaries, the orientations of the individual plate boundaries must be roughly oriented such that the local driving
force is mostly consistent with the overall plate motion. In
the case of southern Alaska, plate reconstructions indicate
this plate boundary configuration has been stationary for the
last 40 Myr [Sdrolias and Muller, 2006] implying that the
local driving forces and orientation of the plate boundary are
dynamically consistent with the overall Pacific plate motion.
Therefore, if the other model parameters are appropriate
(e.g., mantle rheology), then the section of the Pacific plate
and Aleutian slab included in our models should result in
plate motions that are close to Pacific plate motion.
[62] Thus, although we do not expect the models to achieve
an exact match with observed Pacific plate velocities, that
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there is a good fit in the same models that produce rapid flow
in the mantle indicates that rapid mantle flow is not incompatible with surface plate motions. In other words, the same
3D geodynamic models of the Alaska subduction-transform
system that predict velocity magnitudes of greater than
80 cm/yr in the mantle close to the subduction zone, also
predict surface plate velocities comparable to observed plate
motions.
5. Discussion
[63] The results from the 3D models of the Alaska subduction-transform system demonstrate the ranges in mantle
velocities that can be produced assuming an experimentally
determined flow law for olivine and a realistic slab geometry
and density structure. The results show larger velocity ranges as well as sharper velocity gradients in models using the
composite viscosity formulation. The maximum mantle
velocity predicted by the 3D models of the Alaska subduction-transform system ranges from less than 10 cm/yr in the
Newtonian models to over 80 cm/yr in the composite viscosity models. The maximum velocity that occurs in the
mantle wedge, around the slab edge, and underneath the slab
depends on several factors: the (1) slab density anomaly, i.e.,
the driving force; (2) slab strength, which depends in part on
the yield strength; (3) mantle viscosity; (4) plate boundary
coupling, i.e., viscosity along the plate interface; and (5) the
proximity of the mantle to the slab edge.
5.1. Spatially Variable Slab and Mantle Viscosity
[64] The consequence of using an experimentally derived
Newtonian and composite viscosity formulation in the 3D
numerical models of the Alaska subduction-transform system is that the slab strength depends on the thermal structure,
and on the strain rate for the composite viscosity formulation
[Billen and Hirth, 2007; Jadamec and Billen, 2010].
Therefore, a strong slab core emerges from the rheological
flow law [Jadamec and Billen, 2010]. The depth dependent
yield stress places a limit on the maximum strength of the
slab determined from the experimentally derived flow law.
[65] In models with the composite rheology, the slab hinge
area becomes locally weakened, with viscosities reduced
from 1024 Pa s to on the order of 1022 Pa s (Figures 4g and
4h). This gives a viscosity difference of three to four orders
of magnitude between the slab hinge and the mantle surrounding the slab, and of up to six orders of magnitude
between the stronger parts of the slab and the weakest part of
the mantle wedge (Figures 4g and 4h). In models using the
Newtonian viscosity, the slab is not weakened in the slab
hinge and maintains a viscosity on the order of 1024 Pa s in
the core, with a viscosity difference between the slab and
mantle on the order of four orders of magnitude (Figure 4i).
The viscosity ratios in our models are in the upper range of
those suggested from previous 2D and 3D models of subduction, where the viscosity contrasts between the slab and
the mantle range from two to six orders of magnitude
[Piromallo et al., 2006; Stegman et al., 2006; Royden and
Husson, 2006; Billen and Hirth, 2007; Schellart et al.,
2007; Kneller and van Keken, 2008; Schellart, 2008;
Giuseppe et al., 2008; Jadamec and Billen, 2010; Ribe,
2010].
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[66] The models of the Alaska subduction-transform subduction zone that use the composite viscosity formulation
exhibit a localized lower viscosity in the mantle surrounding
the slab, and therefore locally reduced viscous support of the
slab. This local reduction of the mantle viscosity does not
occur in Newtonian models, such as those of Piromallo et al.
[2006], Stegman et al. [2006], and Schellart et al. [2007].
Thus, although 3D Newtonian models of short slabs undergoing slab rollback or slab steepening do show faster
velocities close to the slab [Kincaid and Griffiths, 2003;
Schellart, 2004; Funiciello et al., 2006; Piromallo et al.,
2006; Funiciello et al., 2006; Stegman et al., 2006;
Schellart et al., 2007], the maximum velocities and the
velocity gradients are larger for models using the composite
rheology because of the local reduction in mantle viscosity
[Jadamec and Billen, 2010; Stadler et al., 2010]. This is
consistent with a non-Newtonian viscosity behavior where a
power law dependence between stress and strain rate is
characterized by larger velocity gradients than in a Newtonian viscosity [Turcotte and Schubert, 2002].
[67] In order to match surface observables, several previous models have called upon mantle velocities in the mantle
wedge and back-arc that are faster than plate motions,
although the mechanism for the lower viscosities is typically
attributed to the presence of water or to higher temperatures,
rather than from the weakening effects of a strain rate
dependent viscosity [Hyndman et al., 2005; Currie and
Hyndman, 2006]. Although we use a thermal model with a
warm back-arc lithosphere, as do Blackwell [1969] and
Hyndman et al. [2005], the fast velocities in the mantle
predicted by our models are not a result of the warm, thin
lithosphere imposed for our overriding plate, because the
thermal structure is the same for the instantaneous Newtonian and composite viscosity models. The low viscosity
region in the mantle wedge that emerges from the strain rate
dependent viscosity suggests that in some subduction zones,
additional water or heat [Billen and Gurnis, 2001; Hyndman
et al., 2005; Currie and Hyndman, 2006] may not be
required to reduce the viscosity and that the strain rate
dependent rheology may be as important of a contributor to
reducing the viscosity in the mantle wedge.
5.2. Short Slabs and Fast Velocities
[68] The deepest part of the slab in the 3D models of the
Alaska subduction-transform system extends to approximately 325 km based on seismic constraints (section 3.2).
Therefore, although the model depth extends to 1500 km,
and the models include an increase in mantle viscosity for
the lower mantle, the slab is not supported by the lower
mantle. Thus, the slab in our model is considered a short
slab, and how the model parameters affect the slab dynamics
should be considered in the context of that for short slabs in
this early phase of subduction, as identified by Schellart
[2004], Funiciello et al. [2006], Piromallo et al. [2006],
Billen and Hirth [2007], Giuseppe et al. [2008], and
Jadamec and Billen [2010]. Within this transient state, as a
slab descends into the mantle it does not maintain its initial
dip indefinitely, rather the slab will steepen due to the slab
pull force [Spence, 1987; Conrad and Hager, 1999;
Lallemand et al., 2008; Ribe, 2010]. This pivoting can be
accentuated by the presence of a slab edge, where return
flow can increase the rate of steepening, as indicated by the
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occurrence of steeper slab dips near slab edges [Lallemand
et al., 2005; Schellart et al., 2007].
[69] Viscous yielding and deformation by dislocation
creep which weaken the slab hinge allowing the slab to bend
more easily also reduce the amount of slab pull that can be
directly transmitted to the subducting plate. This effect is
included in all the models and contributes to why the surface
part of the subducting plate moves slower than the slab is
sinking. Some amount of yielding is required to allow the
slab to pivot and sink into the mantle, but the reduction in
slab strength in the hinge limits the amount of slab pull force
that can be directly transmitted to the surface plate.
[70] We expect that it is slabs in the transient state, not
supported by the higher viscosity mantle below the transition
zone, that are more likely to induce localized rapid mantle
flow. Thus, other subduction zones with short slabs where
rapid mantle flow may be expected include central America,
Cascadia, the lesser Antilles, Scotia, and Vanuatu. Seismic
and geochemical studies suggest this may be the case
beneath Costa Rica-Nicaragua. In this region, local S wave
and teleseismic SK(K)S wave anisotropy measurements
contain trench parallel fast axes in the shallow and deeper
parts of the mantle wedge, indicative of trench parallel flow
[Abt et al., 2009, 2010]. Along the volcanic front there is a
systematic northwestward increase in 143/144Nd and decrease
in 208/204Pb suggesting northwestward flow of mantle
[Hoernle et al., 2008]. Using the isotopic ages and with
assumptions about the transport distance, velocity bounds of
6.3–19.0 cm/yr have be placed on the trench parallel component of mantle flow in the mantle wedge [Hoernle et al.,
2008], which is either slightly slower than or more than
twice as fast as the surface motion vector for the Cocos plate.
We also point out that localized rapid flow may not be
limited to subduction zones with slab edges, as previous
studies that include a non-Newtonian rheology predict along
strike flow of up to 50 cm/yr in a low viscosity channel
within the mantle wedge [Conder and Wiens, 2007] as well
as mantle plume ascent velocities on the order of 100 cm/yr
[Larsen et al., 1999].
5.3. Seismic Anisotropy and Plate-Mantle Decoupling
[71] Beneath south-central Alaska teleseismic shear wave
splitting observations show east-west and northeast-southwest trending (trench-parallel) fast directions in the mantle
wedge north of the slab [Christensen and Abers, 2010]. The
seismic fast directions rotate to northwest-southeast (trenchperpendicular) to the south of the 100 km Aleutian slab
contour beneath south central Alaska and are northeastsouthwest (trench perpendicular) to the southeast of the slab
nose [Christensen and Abers, 2010]. In order to use the
seismic anisotropy observations of Christensen and Abers
[2010] as a constraint on the mantle flow field predicted by
the 3D Alaska subduction-transform models, Jadamec and
Billen [2010] calculated infinite strain axes (ISAs) from the
predicted mantle flow field, according to the method of
Conrad et al. [2007], and compared the ISAs to the observations of seismic anisotropy.
[72] Jadamec and Billen [2010] found that models using
slabE115 and the composite rheology provided the best fit
between the predicted ISA orientations and the seismic fast
directions from the observed anisotropy. None of the models
with slabE325, where the toroidal flow is located further to
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the east, were able to match the observed pattern of anisotropy. Models with slabE115, but that used the Newtonian
viscosity, displayed less vigorous flow including a smaller
toroidal flow component and lead to a poor fit to the
anisotropy. Thus, the pattern of observed seismic anisotropy
in this region of south central Alaska is consistent with rapid
toroidal flow around the slabE115 edge. This implies decoupling of the mantle from surface plate motion in terms of
both direction and speed within the Alaska subductiontransform system.
[73] Shear wave splitting studies that probe the character
of the mantle in many other subduction zones find seismic
fast directions that are nonparallel to the direction of subducting plate motion [Russo and Silver, 1994; Fischer et al.,
1998; Hall et al., 2000; Smith et al., 2001; Kneller and van
Keken, 2007; Pozgay et al., 2007; Hoernle et al., 2008;
Long and Silver, 2008; Abt et al., 2010]. In addition, shear
wave splitting measurements from the subduction-transform
junctures in northern Kamchatka and southern Cascadia also
display a curved pattern of the seismic fast directions, suggestive of toroidal flow in the uppermost mantle around a
slab edge [Peyton et al., 2001; Zandt and Humphreys,
2008]. This suggests that partial decoupling of the surface
plate motion from the underlying mantle flow field, at least
in terms of direction, may be common in subduction zones.
Moreover, in some subduction zones the mantle may flow at
rates significantly faster than the surface plate motion
[Conder and Wiens, 2007; Hoernle et al., 2008; Jadamec
and Billen, 2010; Stadler et al., 2010], implying decoupling in speed as well.
[74] A power law rheology, i.e., one that includes the
effects of the dislocation creep deformation mechanism, can
explain both observations of seismic anisotropy and the
decoupling of mantle flow from surface motion [Jadamec
and Billen, 2010]. Dislocation creep of olivine in the upper
mantle leads to lattice preferred orientation (LPO) in olivine
which results in anisotropy in seismic waves [Savage, 1999;
Kaminiski and Ribe, 2002; Karato et al., 2008; Long and
Silver, 2009]. In addition, dislocation creep of olivine leads
to a lowering of the viscosity in regions of high strain rate in
the mantle, which in turn facilitates the decoupling of the
mantle from the surface plates [Billen and Hirth, 2005;
Jadamec and Billen, 2010]. This is because the power law
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 [Hirth and Kohlstedt,
2003; Karato et al., 2008]. This reduction can occur in
regions of variable velocity, or high strain rate, such as in the
mantle wedge and near a slab edge [Billen and Hirth, 2005,
2007; Jadamec and Billen, 2010].
5.4. Implications for Volcanics Above a Slab Edge
[75] In the 3D models of the Alaska subduction-transform
system, where the slab edge is deep enough to induce
toroidal flow, the models indicate there is associated mantle
upwelling within approximately 500 km outward of the slab
edge. This is consistent with mantle flow patterns predicted
by other 3D subduction models where there is a slab edge
and slab steepening [Funiciello et al., 2006; Piromallo et al.,
2006; Stegman et al., 2006; Jadamec and Billen, 2010;
Schellart, 2010b]. Thus, in these tectonic environments,
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warm mantle is expected to be transported from underneath
the slab into the mantle wedge in an upward and arcuate
pattern around the edge of the slab, which could lead to
decompression melting within several hundred kilometers
outward of the slab edge as well as contribute to melting of
the slab edge [Yogodzinski et al., 2001]. In addition, because
toroidal flow is associated with rapid mantle velocities
[Kincaid and Griffiths, 2003], which are significantly faster
for a composite viscosity [Jadamec and Billen, 2010;
Stadler et al., 2010], this may contribute to the preservation
of primitive magmas that can be brought to the surface
[Durance-Sie, 2009; McLean, 2010; P. M. J. Durance and
M. A. Jadamec, Magmagenesis within the Hunter Ridge Rift
Zone resolved from olivine-hosted melt inclusions and
geochemical modelling with insights from geodynamic
models, submitted to Australian Journal of Earth Sciences,
2012].
[76] Adakitic volcanics have been identified at several
subduction-transform plate boundaries, including the eastern
Alaska subduction-transform boundary, the KamchatkaAleutian plate boundary corner, the Cascadia-San Andreas
fault juncture, in southern New Zealand where there is a
subduction-transform transition, and in the New Hebrides
trench-Hunter fracture zone region [Skulski et al., 1991;
Peyton et al., 2001; Yogodzinski et al., 2001; Durance-Sie,
2009]. However, the position of the volcanics with respect
to the slab edge and upwelling in the mantle flow field has
only recently been tested in 3D geodynamic models
[Jadamec and Billen, 2010; Schellart, 2010b; McLean,
2010; Durance and Jadamec, submited manuscript, 2012].
[77] The Wrangell volcanics in Alaska, characterized by
adakitic geochemical signatures indicative of melting of the
slab edge, are located east of and physically separate from
the Aleutian-Alaska magmatic arc (Figure 2) [Skulski et al.,
1991; Preece and Hart, 2004]. In our 3D models that use
slab115, the Wrangells are located above the upward stream
in the counterclockwise quasi-toroidal flow associated with
the deeper slab edge. In models using slabE325, the Wrangells are located above the center line of the return flow from
the slab edge. We suggest the Wrangell volcanics in Alaska
may be due in part to the upwelling associated with the
deeper edge of slabE115 at approximately 212° to 215° longitude. These models do not investigate the link between
melt migration and solid state flow of the mantle, which is an
important and complex process especially in 3D, but beyond
the scope of this paper.
5.5. Slab and Trench Geometry in Alaska
[78] The toroidal flow of the mantle around the eastern
Alaska slab edge and steepening of the slab dip predicted by
the 3D models of the Alaska subduction-transform system
are suggestive of slab rollback [Garfunkel et al., 1986;
Kincaid and Griffiths, 2003; Schellart, 2004; Funiciello et
al., 2006; Stegman et al., 2006] and thereby retreat of the
trench in eastern Alaska. 3D laboratory experiments of free
subduction indicate that during trench retreat, the trench
geometry evolves into an arcuate shape with the concave
side directed away from subducting plate [Schellart, 2004;
Funiciello et al., 2006; Schellart, 2010a]. Along the greater
than 3000 km length of the Aleutian-Alaska subduction
zone, the sense of curvature of the plate boundary changes
concavity [Page et al., 1989; Gudmundsson and Sambridge,
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1998; Ratchkovski and Hansen, 2002; Bird, 2003]. Although
in the central Aleutians the trench is concave to the north,
i.e., away from the subducting plate consistent with that
expected for trench retreat, the trench in eastern Alaska is
concave to the south, i.e., toward the subducting plate
inconsistent with that expected for trench retreat.
[79] A comparison of global plate motion models
[Schellart et al., 2008] indicates that although the trench in
the central Aleutians may be retreating, the trench in eastern
Alaska shows little to no trench retreat, despite that it terminates into the Queen Charlotte-Fairweather transform
fault forming a slab edge. In addition, plate reconstructions
indicate this portion of the plate boundary has been stationary for at least 10 Myr [Sdrolias and Muller, 2006]. These
observations are unexpected, as along many plate boundaries the rate of trench retreat tends to be greatest near a slab
edge, and 3D numerical models indicate that the magnitude
of trench retreat along a subduction zone is greatest closest
to a slab edge [Stegman et al., 2006; Schellart et al., 2007,
2008]. Therefore, other features of the subduction zone may
control the shape of the trench in eastern Alaska or the
subduction zone may be in a transient state.
[80] One such feature is the Yakutat terrane, the oceanic
plateau that is actively colliding with North America in the
plate boundary corner of southern Alaska [Lahr and Plafker,
1980; Bruns, 1983; Fletcher and Freymueller, 1999; Pavlis
et al., 2004; Meigs et al., 2008] and may be subducted to a
depth of 150 km [Ferris et al., 2003]. 3D laboratory and
numerical models of the subduction of an oceanic plateau
oriented at a high angle to a trench indicate that the shape of
the trench becomes arcuate with the concave side toward the
subducting plate [Martinod et al., 2005; Mason et al., 2010].
This is opposite to that predicted for slab rollback and consistent with the trench shape in eastern Alaska. At depth,
these 3D laboratory and numerical models predict that after a
plateau is subducted, the uppermost part of the slab shallows
in dip becoming a flat slab in the vicinity of the oceanic
plateau with the flanks of the slab dipping more steeply and
away from the central shallow slab [Martinod et al., 2005;
Mason et al., 2010]. This geometry is consistent with that
observed in the subducted plate beneath south central
Alaska,where the flat slab is flanked by the more steeply
dipping Aleutian portion of the slab to the west and the
steeply dipping, but shorter, Wrangell slab to the east
[Stephens et al., 1984; Page et al., 1989; Gudmundsson and
Sambridge, 1998; Ratchkovski and Hansen, 2002; Fuis et
al., 2008]. Thus, the toroidal flow around the slab edge in
south-central Alaska may not lead to the formation of a
retreating trench, because the positive buoyancy of the
Yakutat terrane may prevent sinking of this upper portion of
the slab and thus inhibit trench retreat. We point out that,
although the geometry of the subducting plate in our models
is based on Wadati-Benioff zone seismicity, which we
expect is due in part to the subducted Yakutat terrane, the
models do not include an additional compositional buoyancy
force representative of this plateau.
[81] Alaska has long-lived subduction history [Wallace
and Engebretson, 1984; Lonsdale, 1988; Madsen et al.,
2006; Sdrolias and Muller, 2006; Qi et al., 2007] suggesting that a slab could be present to the 660 km discontinuity.
However, the seismicity and seismic tomography indicate
that, although the slab beneath the Aleutians may reach the
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transition zone, the slab beneath south central Alaska likely
does not extend deeper than 300 km, and that the slab beneath
easternmost Alaska is likely even shorter (see Table 4 and
references therein). The lack of a slab continuous to the
transition zone beneath south central Alaska could result
from previous detachment of the deeper part of the subducted
plate due to the subduction of the Yakutat terrane or due to a
weakness in the subducting plate at depth because of subduction of the failed Kula-Farallon spreading center [Madsen
et al., 2006; Qi et al., 2007]. Investigating the potential
detachment of a slab with depth over the course of the subduction history of southern Alaska is beyond the scope of this
paper, but would be interesting to investigate using future
time-dependent modeling studies and would likely be more
tractable with future codes using adaptive mesh refinement
[e.g., Stadler et al., 2010].
[82] The seismicity, seismic tomography, and the agreement between the models using slabE115 and the seismic
anisotropy from the northern mantle wedge, suggest slabE115
is representative of the actual slab shape beneath mainland
Alaska, and that slabE325 is not (Table 4 and Jadamec and
Billen [2010]). The numerical models show that the relative depth of the slab tip and base of the overriding lithosphere influences whether there will be flow induced in the
mantle. Thus, for models using slabE115, although this slab
shape has two slab edges, one at approximately 212° longitude (beneath south central Alaska) and the other at
223° longitude (near the northern end of the Queen
Charlotte-Fairweather fault), only the deeper slab edge at
212° induces toroidal flow. The lack of induced flow near
the northern end of the Queen Charlotte-Fairweather fault
implies that there is little strain induced in the mantle wedge
here and therefore we would not expect a well developed
LPO in the mantle wedge in the vicinity of the easternmost
Wrangell slab. If, however, the actual slab shape were twotiered as in slabE115 but with the slab beneath the Wrangells
extending to 175 km, rather than to 115 km, we would
expect the tiered shape to produce two regions of toroidal
flow in the mantle, one centered at 212° longitude and the
second centered farther to the east at 223° longitude. In this
case, one would expect a well developed LPO, and thus
organized pattern in the seismic anisotropy, in the mantle
wedge near both 212° and 223° longitude. Additional seismic anisotropy measurements near the northern end of the
Queen Charlotte-Fairweather fault would be useful to better
constrain the depth of the shorter slab segment beneath the
Wrangell mountains.
6. Conclusions
[83] We have constructed 3D instantaneous regional
models of the eastern Alaska subduction-transform plate
boundary system to investigate the role of slab geometry and
rheology on the decoupling of mantle flow from surface
plate motion, in terms of both direction and magnitude, at
subduction zones. The 3D models use a subducting plate
geometry based on Wadati-Benioff zone seismicity and
seismic tomography and a 3D thermal structure for both the
subducting and overriding plates, constrained by geologic
and geophysical observations. The models test the effect of a
Netwonian viscosity and a composite viscosity, which
includes both the diffusion and dislocation creep
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mechanisms (i.e., Newtonian and non-Newtonian viscosity).
The models in this paper differ from previous 3D numerical
models of subduction in that previous models either use a
simplified plate geometry, prescribe a velocity boundary
condition to the subducting plate, do not include an overriding plate, or use a Newtonian rheology in the mantle.
However, our models are limited in that they are instantaneous, examining the spatial distribution of the flow rather
than the time-dependent evolution.
[84] Models using the composite viscosity formulation
have higher strain rates and lower mantle viscosities than
models using the Newtonian viscosity. In models using the
composite viscosity, a laterally variable mantle viscosity
emerges as a consequence of the lateral variations in the
mantle flow and strain rate. Spatially variable mantle velocity
magnitudes are predicted, with localized fast velocities
(greater than 80 cm/yr) close to the slab where the negative
buoyancy of the slab drives the flow. The same models produce surface plate motions of less than 10 cm/yr, comparable
to observed plate motions. We expect that it is short slabs
undergoing slab steepening, not supported by the higher
viscosity mantle below the transition zone, that are more
likely to induce the localized rapid mantle flow. These results
show that a power law rheology, i.e., one that includes the
effects of the dislocation creep deformation mechanism, can
explain both observations of seismic anisotropy and the
decoupling of mantle flow from surface motion.
[85] The incorporation of realistic slab geometries into 3D
models of subduction leads to a complex pattern of poloidal
and toroidal flow in the mantle, with the spatial positions of
the mantle flow components sensitive to the slab geometry.
We conclude that slabE115 is representative of the actual slab
shape beneath south central Alaska. The vertical motion of
mantle material near slab edges may play a role in forming
melts from both decompression melting and melting of the
slab edge (i.e., adakites), as this part of the subduction zone
is continuously exposed to upward advected warm mantle
material. The Wrangell volcanics may be due in part to the
upwelling associated with the deeper edge of slabE115.
[86] Acknowledgments. We thank C. Matyska, W. P. Schellart, and
an anonymous reviewer for thoughtful reviews of the manuscript. We thank
D. Turcotte, L. Kellogg, S. Roeske, D. Eberhart-Phillips, T. Taylor, and L.
Moresi for thoughtful discussions. We thank Oliver Kreylos and the UC
Davis Keck Center for Active Visualization in the Earth Sciences (KeckCAVES) for enabling the incorporation 3D immersive data visualization
into this research. We thank the Computational Infrastructure for Geodynamics (CIG) for the CitcomCU source code. Figures were made with
GMT and 3D Visualizer. High resolution models were run on the TeraGrid
site, Lonestar, at the Texas Advanced Computing Center (TACC) through
TG-EAR080015N. This work was supported by National Science Foundation grants EAR-0537995 and EAR-1049545.
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