Text S1: Numerical Method Details
(Auxiliary material on “Comprehensive model of short- and long-term slow slip events
in the Shikoku region of Japan, incorporating a realistic plate configuration” by
Takanori Matsuzawa, Bunichiro Shibazaki, Kazushige Obara, and Hitoshi Hirose,
Geophysical Research Letters, 2013, doi:10.1002/grl.51006)
We employ a numerical method similar to our previous studies [e.g., Matsuzawa et
al., 2010]. In our numerical simulations, a rate- and state-dependent friction law
(RS-law) with cut-off velocities [Okubo, 1989] is adopted to reproduce SSEs [Shibazaki
and Shimamoto, 2007]. The frictional stress  is given as

 v

 v1

 1  b ln  2  1 ,
v

 dc

   e   0  a ln 

(1)
where  e is the effective normal stress given by the difference between the normal
stress and the pore pressure. a , b , and  0 are frictional constitutive parameters in
the RS-law. v and  are a slip velocity and a state variable, respectively. v1 and v2
are cutoff velocities for the direct and evolution effects, respectively. d c is a critical
displacement scaling factor for evolution effects. For simplicity, the ratio between
effective normal stress and critical displacement is kept constant in our model, as
assumed in our previous study [Matsuzawa et al., 2010].  e d c  1.6  10 3 (MPa/m) is
used in all results shown in our paper. As we adopt the aging law [Ruina, 1983] in our
simulation, the temporal evolution of the state variable is given by
d
v
1
.
dt
dc
(2)
In our model, the RS-law defines the frictional stress on the subducting plate
interface. The interface of the Philippine Sea plate is modeled using 93,144 triangular
elements. To make numerically stable simulations, the cell size should be smaller than a
critical nucleation size. Rice [1993] shows that the critical nucleation size is given by
h* 
2G d c
,
 b  a  e
(3)
where G is defined as rigidity ( G ) for antiplane strain and G 1   for plane strain.
If fault healing is unimportant, Rubin and Ampuero [2005] show that the half-length of
the nucleation patch size is
Lv 
1.3774G d c
.
b e
(4)
In our models, the minimum values of h* and Lv are 2.28 km and 1.95 km,
respectively. All triangular elements with negative a  b values are completely
covered by a rectangle 1.73 km long and 1.82 km wide. This means that the area of the
triangular mesh is less than half the rectangle’s area, and smaller than either h* or Lv .
Temporal evolution of the slip velocity and state variable on the plate interface are
calculated based on the above friction law and elastic response between triangular
elements. The frictional parameters of each element are described in Fig. 1 and in the
body of the paper. The time derivative of shear stress on the i -th element is given by
d i
G dvi
  k ij Vpl  v j  
dt
2 dt
j
(5).
V pl , and  are the loading velocity at the plate interface and S-wave velocity,
respectively. k ij is an elastostatic kernel that gives the stress change at the collocation
point of the i -th element. The elastostatic kernel for a triangular element is calculated
assuming a semi-infinite elastic medium [Stuart et al., 1997]. Equations (1), (2), and (5)
are solved by the Runge-Kutta method with adaptive step-size control [Press et al.,
1996].
Reference of text01
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