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Supplementary Information for manuscript:
Extreme localized exhumation at syntaxes initiated by subduction geometry
Rebecca Bendick1,2, Todd A. Ehlers2
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Department of Geosciences, University of Montana, Missoula, MT 59802, USA.
Department of Geosciences, University of Tübingen, 72074 Tübingen, Germany.
Extended description of methods
The numerical simulations used in this study are built with the COMSOL multiphysics
finite element solver including the Arbitrary Lagrangian Eulerian (ALE) formulation
(Fullsack, 1995) to accommodate large deformation and erosion on the upper boundary
of the model domain. The specifics of the model setup follow:
ALE
All of the model domain boundaries are fixed in space. Mass may be advected through
the top boundary only; once mass passes through the z=zmax surface (the Earth’s
surface), it disappears from the computation. Therefore, the top temperature and
velocity boundary conditions are always applied on z=zmax (100 km), Total mass is
conserved such that the sum of the flux out of the upper surface and the outflow over the
rigid indenter equals the inflow at x=0 (Fig. S1).
Fluid mechanics
The velocity field is calculated for a viscous fluid on the model domain using the NavierStokes equation where the inertial terms are assumed to equal zero (laminar flow),
𝜕𝒖
𝜌 𝜕𝑡 = ∇ ∙ [−𝜌 + 𝜇(∇𝒖 + (∇𝒖)𝑻 )] + 𝑭 with 𝜌∇ ∙ 𝒖 = 0. In this expression,  is the fluid
density, u is the velocity vector,  is the viscosity, and F is the body force tensor. The
upper boundary at z=zmax is free, so that fluid can move through z=zmax (100 km) to form
topography (or be “eroded” using the ALE formulation). The inflow condition at x=0 is a
constant normal velocity of 20 mm/yr. The outflow condition is a free boundary, where
again fluid can move out of the model domain. The y=0 and y=ymax (800 km) boundaries
have free boundary-parallel slip, and the basal boundary, including the top surface of the
indenter, has a fixed velocity boundary condition set at the inflow velocity to approximate
net convergence between the indenter and the overriding fluid (Fig. S1). The viscosity of
the fluid in the model domain is either constant (in the constant viscosity simulations,
Table 1 in manuscript), or determined by the temperature solution for the domain,
1−𝑛
1 −1
𝑄
through 𝜇 = 2 𝐴 ⁄𝑛 𝜀̇ ⁄𝑛 𝑒𝑥𝑝 (𝑛𝑅𝑇), with varying values for n used in the sensitivity
testing (Table 1 in text, and Fig. S2). In this expression,  is the viscosity, A is a
conditioning coefficient, n is a viscosity exponent, 𝜀̇ is the strain rate, Q is the activation
energy, R is the ideal gas constant, and T is the temperature.
Temperature
The temperature field is calculated for a viscous fluid including both advection and
𝜕𝑇
diffusion with the advection-diffusion equation, 𝜌𝐶𝑝 𝜕𝑡 + 𝜌𝐶𝑝 𝒖 ∙ ∇𝑇 = ∇ ∙ (𝑘∇𝑇) + 𝑄. In
this expression,  is the fluid density, 𝐶𝑝 is the heat capacity, T is the temperature and t
Bendick and Ehlers Supplemental Material
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the time, u is the velocity vector, k is the thermal diffusivity, and Q is internal heat
production. The initial temperature condition on the model domain is a 1D steady state
continental geotherm for the boundary conditions specified below, no heat advection,
−𝑧
and with internal heat production given by 𝑄 = 𝑆0 𝑒𝑥𝑝 ( ). This temperature relation is
ℎ𝑟
also imposed on the inflow boundary (x=0) for the duration of the model run (e.g. Fig.
S3). The outflow condition is free in the temperature problem. The y=0 and y=ymax
boundaries are insulated, so the net lateral heat flux equals zero. The basal boundary
condition is set at a fixed temperature of 980ºC; the surface boundary condition at a
fixed temperature of 30ºC.
The temperature on the top of the rigid indenter (Fig. S1) at different depths is given by
the solution (Molnar and England, 1990) for the steady state limit on a slipping
𝑇
discontinuity, 𝑇 ′ =
. In this expression, T’ is the steady state temperature
1+𝑏√(𝑧𝑓 𝑉𝑠𝑖𝑛𝛿)/𝜅
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on the “fault”, when T is the standard geothermal temperature for the given depth, b is a
dimensionless correction factor, 𝑧𝑓 is the depth to the discontinuity, V is the slip rate  is
the component of the dip of the fault in the thrust direction, and  is the thermal
diffusivity.
The initial mesh and the model conditions are also summarized in Figure S1. The range
of parameter values used in the sensitivity testing are given in Table 1 in the text and
shown in manuscript Figures 2 and 3a.
Thermochronometer Age calculation:
Apparent cooling ages are predicted for rocks exhumed to the surface for comparison to
observed zircon fission track (ZFT) ages. The temperature history of exhumed rocks is
determined using transient particle tracking. A discrete set of 10,000 particles was
released into the model domain on the x = 0 plane (Fig. 2a, Fig. S5). The timedependent coupled velocity-temperature solutions were then recorded as particles move
through the domain. The resulting time series for particle temperature, pressure, and
position were exported from COMSOL and used in the TERRA cooling age prediction
software (Ehlers et al., 2005). Cooling rate dependent ZFT ages were calculated for
samples exposed at the surface using the effective closure temperature concept for
natural radiation damaged zircon (Brandon et al., 1998; Ehlers et al., 2005).
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Supplemental Figures
Figure S1: Model setup including mesh geometry and boundary conditions for the
baseline model. (Top) Top view, (Bottom) View including basal boundary conditions.
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Figure S2: Temperature-dependent viscosity laws. All of the viscosity relations except
the constant viscosity case have the Arrhenius form; only the viscosity exponent varies.
Larger values of n approach constant viscosity. In all cases, the average effective
viscosity, given by the integral of the viscosity curve, is 1021 Pa s.
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Figure S3: The temperature distribution in the model domain after 10 Ma of collision.
Note the upward advection of geotherms at the indenter front, resulting in locally steep
temperature gradients, hence young cooling ages for particles passing near the indenter
front.
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Figure S4: Surface velocity contours for an additional “fast” model, where the
convergence rate is doubled from the baseline model to 40 mm/yr. The maximum
vertical velocity is 9 mm/yr at the center of the bull’s eye.
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Figure S5: (a) Temperature-time plot for selected particles released in the model
domain. Note that it takes approximately 15 Ma for particles to cross the modeled
volume from the inflow point; all of the vertical advection occurs within a subsequent 1015 Ma. (b) Depth-temperature plot for selected particles released in the model domain.
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Supplemental Material References
Brandon MT, Roden-Tice MK, Garver J, 1998, Late Cenozoic exhumation of the
Cascadia accretionary wedge in the Olympic Mountains, northwest Washington
State: GSA Bulletin, v. 110, p. 985-1009.
Ehlers, T.A., Chaudhri, T., Kumar, S., Fuller, C.W., Willett, S.D., Ketcham, R., Brandon,
M.T., Belton, D., Kohn, B., Gleadow, A.J., Dunai, T., Fu, F., 2005, Computational
tools for low-temperature thermochronometer interpretation: Reviews in
Mineralogy and Geochemistry, v. 58, p. 589-622.
Fullsack, P., 1995, An arbitrary Lagrangian-Eulerian formulation for creeping flows and
its application in tectonic models: Geophys. J. Int., v. 120,p. 1-23.
Molnar, P., and England, P., 1990, Temperatures, heat flux, and frictional stress near
major thrust faults: J. Geophys. Res., v. 95, p. 4833-4856.
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