Figure 3. Fault friction and hanging wall velocity structure
setup in each scenario. Left panels: friction setup. Middle panels:
hanging wall velocity setup. Right panels: footwall velocity setup.
Color coding in the middle and right panels: the red curve indicatingVp , blue curve indicatingVs , and black dashed curve indicating density.
2.4 Rupture nucleation and resolution
In the dynamic rupture simulation, a region on the fault with
velocity-weakening property is a necessary condition for nucleation. To
initiate an instability, this velocity-weakening zone must be larger
than a critical nucleation patch size h* , which
is determined by the energy balance of a quasi-statically expanding
crack (Lapusta et al., 2000; Rice, 1993; Rubin & Ampuero, 2005). Here,
we use an estimation for 3D modeling according to Chen and Lapusta
(2009) and Lapusta and Liu (2009):
\begin{equation}
h^{*}=\frac{\pi}{2}\frac{\mu^{*}bD_{c}}{(b-a)^{2}\sigma}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3),\nonumber \\
\end{equation}where \(\mu^{*}\) is \(\mu\) for a mode III crack and\(\mu/\left(1-\nu\right)\) for a mode II crack, \(\mu\) is the shear
modulus, and \(\nu\) is the Poisson’s ratio. In our simulations, we
assign a nucleation patch at 110 km along dip with radius of 4 km and
slip rate of 0.01 m/s to artificially initiate a rupture event.
During dynamic rupture process, shear stress and slip rate change
dramatically in the cohesive zone at the rupture front, which requires a
certain number of elements to resolve these features (Day et al., 2005).
The spatial resolution of the cohesive zone is thus critical for
simulating dynamic rupture propagation (Day et al., 2005), which
constrains the element size of the model (e.g., Duan & Day, 2008). The
size of the cohesive zone, Λ0, at rupture speed\(v_{R}\) = 0+ under the RSF law follows
\begin{equation}
\Lambda_{0}=C_{1}\frac{\mu^{*}L}{\text{bσ}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4),\nonumber \\
\end{equation}where C1 is a constant of 9\(\pi\)/32 (Lapusta &
Liu, 2009). For our FEM scheme, it is found that\(\Lambda_{0}/\Delta x\) of 2.4 with an element size \(\Delta x\) of 200
m can well resolve the cohesive zone (Meng et al., 2022). Taken the
parameters choices in Section 2.3 , we set the model parameters
considering equations (3) and (4).
Another consideration for resolution is time step. For dynamic rupture
and seismic wave propagation, the time step (dt ) isα d/Vp , where α is a constant between 0 and
1 and d is the minimum element size (e.g., Liu et al., 2021). Given d =
Δz = 52m, Vp = 6.7 km/ s, and α = 0.26, we set
dt = 0.002 s.
2.5 Rupture scenarios
Figure 3 shows the fault friction and hanging wall material property
setup for five dynamic rupture scenarios. The dynamic rupture scenarios
are all nucleated at 110 km. All scenarios incorporate a downdip
transition from velocity-weakening behavior at 120-km downdip (depth of
~30 km) to velocity strengthening behavior at 150-km
downdip (depth of ~40 km) (white area between the fault
patch and the dashed line in Figure 1). To account for the effects of
updip transition from velocity-strengthening behavior near the trench to
velocity-weakening behavior downdip, we set the along-dip transition
distance L , together with a conditionally stable layer with an
along-dip distance d (Figure 2a; left panels in Figure 3). For
the depth-varying rigidity, we incorporate a multi-layered non-uniform
velocity structure of the upper plate following Sallares & Ranero
(2019) (middle and right panels in Figure 3). Scenario 1 is a reference
scenario that assumes homogeneous velocity structure (Figures 3b and 3c)
and homogeneous friction with L of 0 and d of 0 (Figure
3a). Scenario 2, on the other hand, is a most realistic setup that
includes depth-varying velocity structure in the hanging wall (Figure
3e) and a two-layer velocity structure footwall (Figure 3f) and
depth-varying friction with L of 40 km and d of 30 km
(Figure 3d) among the scenarios. Scenarios 3 and 4 aim to quantify the
effects of friction, both with homogeneous velocity structure in the
both walls (Figures 3h, 3i, 3k, 3l) and depth varying friction withL of 40 km, but different d values of 0 km and 30 km
(Figures 3g and 3j), respectively. Finally, Scenario 5 aims to quantify
the effects of depth-varying rigidity, with depth-varying velocity
structure in the hanging wall (Figure 3n) and a two-layer velocity
structure footwall (Figure 3o) and homogeneous friction (both Land d equal 0) (Figure 3m).
3 Results
We present the simulation results of stress drop, total slip, rupture
time contours, and rupture velocity on the fault plane in each Scenario.
We also analyze the frequency contents of slip rate at selected on-fault
stations to examine seismic radiation. We first compare the most
realistic model (Scenario 2) and the reference model (Scenario 1) to
examine combined effects of the two factors, namely fault friction and
upper-plate rigidity. Then we examine other models and compare them with
Scenario 1 and/or Scenario 2 to determine roles of the two factors in
depth-dependent rupture characteristics.
3.1 Combined effects of depth-varying friction and rigidity on rupture
characteristics
We examine the results of Scenarios 1 and 2 for the combined effects of
heterogeneous friction and rigidity (Figure 4). The stress change
distribution and slip distribution are significantly different between
Scenario 1 (homogeneous rigidity and homogeneous friction) and Scenario
2 (heterogeneous rigidity and heterogeneous friction with L of 40
km and d of 30 km). The stress change distribution on the fault
patch in Scenario 1, except for the boundary surrounded by velocity
strengthening area (Figure 1), is negative (i.e., stress drop) from
downdip toward the trench (Figure 4a). Correspondingly, slip reaches the
trench in Scenario 1 with a maximum slip of ~20 m
occurring at the trench. On the other hand, in Scenario 2, the updip
transition from velocity-weakening behavior downdip to
velocity-strengthening behavior updip (L of 40 km) diminishes
slip at shallow depth, though free surface effects cause some obvious
slip at the trench (Figure 4g). The maximum slip of 8m occurs at depth
in this scenario. Correspondingly, stress drop (blue) mainly occurs at
depth, while stress increases (red) at shallow depth (Figure 4f).
Because both heterogeneous fault friction and heterogeneous wall rock
properties are included in Scenario 2, we will examine contributions
from each of the two factors to the above features in the slip and
stress change distributions by other comparative scenarios in the
sections below.
Rupture times are direct outputs from our dynamic rupture models. At the
rupture time, a fault node reaches a slip rate of 0.01 m/s as the first
time during the simulation. Rupture velocities are calculated from
rupture times, following a method proposed by (Bizzarri and Das, 2012).
In Scenario 1, the rupture time contour (Figure 4c) and the rupture
velocity distribution (Figure 4d and 4e) both indicate that the rupture
generally accelerates toward the trench from the nucleation patch on a
subduction plane with a uniform velocity-weakening friction property
embedded in a uniform medium. In Scenario 2, the rupture accelerates
within the velocity-weakening patch from the nucleation patch but slows
down when it propagates into the conditionally stable part (d of 30 km)
and the updip transition patch (L of 40km) (Figure 4h, 4i, and 4j), in
particular along the depth profile at along-strike-distance of 50km
(black curve in Figure 4j), except near the trench. Supershear rupture
occurs near the trench in both scenarios due to effects of free surface
and shallow-dipping fault geometry. The slow rupture velocity at shallow
depth in Scenario 2 may be attributed to combined effects of updip
transition in friction and low-Vp and
-Vs at shallow depth. The other comparative
scenarios will help clarify and quantify their roles in slower rupture
velocity (and thus longer duration) at shallow depth in Scenario 2 than
that in Scenario 1.