We perform 3D quasi-dynamic modeling of the cycle of a megathrust earthquake in the offshore Tohoku region, Japan, using a rate- and state-dependent friction law with two state variables that exhibits strong velocity weakening at high slip velocities. We set several asperities where velocity weakening occurs at low to intermediate slip velocities. Outside of the asperities, velocity strengthening occurs at low to intermediate slip velocities. At high slip velocities, strong velocity weakening with large displacements occurs both within and outside of the asperities. The rupture of asperities occurs at intervals of several tens of years, whereas megathrust events occur at much longer intervals (several hundred years). Megathrust slips occur even in regions where velocity strengthening occurs at low to intermediate slip velocities, but where velocity weakening is dominant at high slip velocities. The proposed model explains that megathrust earthquakes occur in the same subduction zone as large thrust earthquakes.
 The 2011 off the Pacific Coast of Tohoku Earthquake (Mw 9), Japan, occurred on 11 March 2011 in the subduction zone along the Japan Trench. The results of tsunami waveform inversion [Fujii et al., 2011] indicate that large slips occurred near the trench off Miyagi and off Fukushima, with a maximum slip amount of ∼48 m. Ito et al.  estimated magnitude of the slip along the main fault to be 80 m near the trench. These results indicate that a significant stress drop occurred in the shallow part of the fault zone. Hasegawa et al.  estimated the ratio of the mainshock stress drop to the background deviatoric stress Δτ/τ to be 0.9–0.95 by using the observed rotation of the maximum compressive stress axis. Their results suggest that the background deviatoric stress was completely released and that the frictional strength decreased significantly.
 The region off Miyagi contains the asperities that generated earthquakes of Mw 7.1–7.5 [Yamanaka and Kikuchi, 2004]. Satake et al.  estimated the magnitude of the 869 Jogan earthquake (which caused a giant tsunami) to have been M 8.4, based on analyses of tsunami deposits in Miyagi. Therefore, in the region offshore from Miyagi, large thrust earthquakes occur at certain asperities with relatively short recurrence intervals, and megathrust earthquakes occur with longer recurrence intervals. These observations give rise to the questions of how large slips occur in the shallower part of the fault zone and how large thrust earthquakes (Mw 7–7.5) and megathrust earthquakes (Mw 9) can be generated at overlapping sites within the same subduction zone.
Hori and Miyazaki  succeeded in modeling the generation cycle of Mw 9 earthquakes in an area of repeating Mw 7–7.5 earthquakes, assuming that the Mw 9 source area is an area of large fracture energy and that the Mw 7–7.5 asperities are areas of smaller fracture energy with a nucleation size smaller than the asperity size. Noda and Lapusta  performed 3D simulations of earthquake sequences with evolving temperature and pore pressure due to shear heating and found that the region of more efficient thermal pressurization produces larger slip, resulting in large events with a large interseismic period.
 Recent experimental studies using fault materials have reported that a dramatic drop in the friction coefficient occurs at high slip velocities [e.g., Di Toro et al., 2011]. Tsutsumi et al.  examined the frictional properties of clay-rich fault materials under water-saturated conditions and found that velocity weakening or velocity strengthening occurs at intermediate slip velocities and that dramatic velocity weakening occurs at high slip velocities. These results suggest that large coseismic slips may occur in the shallow subduction zone, where subducted sediments are found.
 In this study, we propose a rate- and state-dependent friction law with two state variables that exhibits weak velocity weakening or velocity strengthening with a small critical displacement at low–intermediate velocities, but strong velocity weakening with a large critical displacement at high slip velocities. We use this friction law for quasi-dynamic 3D modeling of the cycle of a great Tohoku-oki earthquake.
2. Constitutive Law With Two State Variables
 In general, the value of the friction coefficient depends on the slip velocity and on state variables [e.g., Dieterich, 1981]. Recent experimental results suggest that strong velocity weakening occurs at high slip velocities for both cohesive and non-cohesive fault materials [e.g., Di Toro et al., 2011]. The critical displacement observed at high slip velocities appears to be much greater than that observed at low slip velocities, suggesting that the friction law requires a state variable with a larger critical displacement in order to model large-slip events. Tsutsumi et al.  reported that the frictional behavior of clay-rich fault materials under water-saturated conditions shows velocity-weakening or velocity-strengthening behavior with an evolution of a state variable with a small critical displacement at intermediate slip velocities. For higher slip velocities, the friction coefficient shows a marked decrease with increasing slip velocity [Tsutsumi et al., 2011; Ujiie and Tsutsumi, 2010].
 To model the friction behavior observed by Tsutsumi et al. , we formulate a rate- and state-dependent friction law with two state variables that exhibits velocity weakening or strengthening with an evolution of a state variable with a small critical displacement at low–intermediate velocities, but strong velocity weakening with an evolution of a state variable with a large critical displacement at high slip velocities. Frictional resistance τ can be written as
where v is the instantaneous sliding velocity; Θ1 and Θ2 are state variables that characterize the evolving state of the sliding surfaces; σneff is the effective normal stress, defined as the difference between the lithostatic pressure and the pore fluid pressure Pf. We represent the friction coefficient μ as follows:
where μ* is the base friction; a, b1, and b2 are empirical parameters; v0 is the cut-off velocity for the direct effects in the second term; v1 is the higher cut-off velocity for the evolution effect in the third term; v2 is the lower cut-off velocity for the evolution effect in the fourth term; Dc1 and Dc2 are critical displacements scaling the evolution of the state variables; and α is the empirical parameter that determines the effect of the cut-off. In the Dieterich–Ruina friction law, an evolution law for the state variable can be written as
 The steady-state friction can be written as
 The rate dependence of the steady-state friction can be written as
 The fourth term in equation (2) represents the effect of velocity weakening at high slip velocities. This term becomes effective when the slip velocity v is greater than the cut-off velocity v2. α is positive, and decreasing α results in decreasing effect of the cut-off (see the auxiliary material).
Figure 1 shows the frictional behavior of the proposed constitutive law. The constitutive law parameters are given empirically, based on experimental results (see the auxiliary material). In case I (red dashed line in Figure 1a), velocity weakening occurs at low slip velocities, and strong velocity weakening occurs at a high slip velocities. In case II (solid blue line in Figure 1a), velocity strengthening occurs at low slip velocities and strong velocity weakening occurs at high slip velocities. If α is small and less than 1, the effect of the cut-off becomes small. In Figure 1a we take α to be 0.5 so that the corner slip velocity of steady state friction is smaller than the cut-off velocity v2. Figure 1b shows the dependence of friction on fault slip when slip velocity changes from low to high. In the case where slip velocity changes from 10−9 to 10−3 m/s, a decrease in friction with short displacement becomes dominant. When the slip velocity is changed from low values to 1 m/s, we find a significant stress drop with a large critical displacement. This behavior is similar to that indicated by the experimental results reported by Tsutsumi et al. , whereby a significant stress drop occurs with a large displacement. In the present constitutive law, the displacement required for stress to drop becomes larger with decreasing initial value of the slip velocity.
 In this model, we introduce a cut-off velocity v2 to the fourth term in equation (2). The state variable Θ2 evolves following equation (4) at low slip velocities without a cut-off velocity. Another possible way to model a strong velocity-weakening process that becomes effective when the slip velocity v is greater than the cut-off velocity v2 is to introduce a cut-off velocity to the evolution of the state variable, such that dΘ2/dt = 1 − Θ2(v + v2)/Dc2 [Beeler, 2009]. In the present study, however, we adopt the form of the constitutive law described by equations (2)–(4) because it is less computationally expensive.
3. Numerical Simulation
 We consider a quasi-dynamic analysis that assumes a thrust fault in a 3D elastic half-space. We use a curved plate boundary as shown in Figure 2. For simplicity, we set a free surface at the depth of the trench which is assumed to be 8 km below the sea surface. The curved plate interface is divided into triangular elements. For simplicity we assume the direction of plate convergence to be EW. The long-term average slip velocity vector of the i-th element has EW and vertical components and is parallel to the surface of a triangular element. For all elements, the magnitude of the long-term average relative slip velocity is fixed at the rate of plate convergent, vpl = 8 cm/year. The shear stress τi on the i-element is accumulated by the delay of the fault slip relative to the long-term average slip vplt, following a quasi-dynamic equation for tectonic loading [e.g., Rice, 1993] (see the auxiliary material). To solve the coupled equations of the constitutive law (1)–(4) and the equation of tectonic loading (S1) in the auxiliary material, we use the fifth-order Runge–Kutta method with adaptive step-size control [Press et al., 1992].
 We set eight different asperities, as shown in Figure 2, corresponding to the rupture areas of past earthquakes obtained by Yamanaka and Kikuchi [2003, 2004], Murotani , and Central Disaster Prevention Council (http://www.bousai.go.jp/jishin/nihonkaikou/houkoku/sankou1.pdf) (see the auxiliary material). In these asperities, velocity weakening occurs at low to intermediate slip velocities. Outside of the asperities, however, velocity strengthening occurs at low slip velocities but velocity weakening occurs at high slip velocities. The values of the constitutive law parameters a, b1, b2, Dc1 and Dc2 are given in Table 1. a − b1 is negative in the eight asperities and positive outside of the asperities, whereas a − b1 − b2 is negative in all regions shallower than 50 km. Hasegawa et al.  estimated the absolute shear stress of the source region of the Tohoku-oki earthquake to be around 21–22 MPa. Therefore, the effective normal stress is estimated to be 35–36.7 MPa when friction coefficient is 0.6. In the present study the value of σneff increases from 0 to 36.6 MPa with increasing depth from 8 km (the depth of the trench) to 13 km and then gradually increases to 39.74 MPa at a depth of 18 km. The values of the cut-off velocities v0, v1 and v2 are 1.0, 0.1 and 0.001 m/s, respectively. α is taken to be 0.5. The minimum nucleation length scale Lb = GDc/bσneff [Rubin and Ampuero, 2005] in the model is calculated to be 3.1 km for b = b1 and Dc = Dc1, and 18.9 km for b = b2 and Dc = Dc2. In the present calculation, the largest node interval is 2.2 km, which is smaller than the nucleation length scale. The values of a, b1, and Dc1 are determined empirically to reproduce M 7.5 earthquakes with recurrence intervals of several tens of years and a fault slip of 2–3 m. The values of b2 and Dc2 are determined to reproduce megathurst earthquakes with a recurrence interval of 1000 years.
Table 1. Constitutive Law Parameters for Each Region
A southern part of the source area overlaps with the asperity SRK.
a changes from 0.0064 to 0.0256 linearly with depth.
Figure 3 shows temporal changes in slip at points P1–7 defined in Figure 2 (see the auxiliary material for temporal changes in slip at points PS1–2). Megathrust slips occur at elapsed times of 1156.01 and 2042.66 years. At point P1, within a large asperity near the trench (SRK), the fault is locked during the interval between megathrust events. At the time of megathrust events, very large slips of ∼70 m occur at this point. P2 is located in the northern part of the SRK asperity. Two large slips of 5.1 and 13.5 m occur at 1700.42 and 1884.21 years, respectively, during the period between megathrust events. P3 and P4 are located in the Miyagi asperities, P5 and P6 in the Fukushima asperity, and P7 in the Ibaraki asperity. No significant slip events occur within the 150 -year period after megathrust events; however, after 150 years has passed, slips of 2–4 m occur at intervals of several tens of years.
Figure 4 shows the slip velocity distribution for typical events on the curved plate interface. Many events with magnitudes of around M w 7.5 (Figures 4h–4k) occur at the Ibaraki Fukushima, and Miyagi asperities. These events do not grow to larger events because velocity strengthening occurs at low to intermediate velocities in the surrounding region. At 1700.42 and 1884.21 years, large events with Mw 7.9 and 8.3 (Figures 4b–4g) occur in the northern part of the SRK asperity. These events do not extend to the entire region of the SRK asperity because the fault is strongly locked near the center of the SRK asperity (arrow in Figure 4). At 2042.66 years, a megathrust event (Mw 9.1) is initiated near the strongly locked portion of the SRK asperity (Figures 4m–4q). The slip and slip velocity become sufficiently high that velocity weakening occurs at high slip velocities with large displacement. The rupture propagates to the Miyagi asperities and then propagates to the Fukushima and Ibaraki asperities. During this event, coseismic fault slips occur in the regions outside of the asperities because the frictional property changes from velocity strengthening at low to intermediate velocities to velocity weakening at high slip velocities. During this event, large coseismic slip of ∼ 70m occurs in the strongly locked region near the trench (Figure 5).
 In the present model, the occurrence of megathrust events is controlled by the rupture of a strongly locked portion of the SRK asperity. Fault slips are largest around this strongly locked portion because a − b1 − b2 is large and is located near the center of the asperity and the free surface. Furthermore, small asperities (MYG1 and MYG2) exist at the deeper extension of the strongly locked portion. The recurrence intervals and sizes of megathrust events are controlled by the value of a − b1 − b2 and by the size of the SRK asperity. In the present simulation, we use larger values of a, b1, b2, and Dc for the SRK asperity to reproduce longer recurrence intervals. By taking a larger value of Dc, the duration of shallow slip becomes longer. In the present case, megathrust slip propagates throughout almost the entire region. The state after a megathrust slip is almost the same in each cycle; consequently, the system evolves in a similar manner in each cycle. The site of rupture initiation of megathrust events changes in each cycle, but is located near the edge of the SRK asperity, from where the rupture propagates to the strongly locked portion.
 We performed quasi-dynamic 3D modeling of earthquake cycles by considering eight asperities in the region offshore from Miyagi, Fukushima, and Ibaraki in Japan, where velocity weakening occurs at low to intermediate slip velocities. Outside of the asperities, velocity strengthening occurs at low to intermediate slip velocities. At high slip velocities, strong velocity weakening with large displacements occurs both within and outside of the asperities. The modeling results show that ruptures with slips of 2–4 m occur at intervals of several tens of years at the Miyagi, Fukushima and Ibaraki asperities, located close to land. Ruptures of these asperities do not extend to the surrounding region, because of velocity strengthening at low to intermediate slip velocities. Megathrust slips initiated at the Sanriku asperity and propagated to other asperities. During megathrust events, the slip and slip velocity become very large, meaning that in the surrounding region the frictional property changes from velocity strengthening at low slip velocities to velocity weakening at high slip velocities. The results explain why large thrust earthquakes (slips of 2–4 m) and megathrust earthquakes (maximum slips of ∼70 m) occur within the same subduction zone.
 The 1611 (M8.1) and 1896 (M8.5) tsunamigenic earthquakes occurred in the Sanriku region near the Japan trench. The model produced M7.9 and M8.3 Sanriku events at the northern part of the Sanriku asperity, although these events show a relatively high slip velocity. By changing the constitutive law parameters (e.g., by increasing the critical displacement), it may be possible to reproduce the Sanriku slow Tsunami earthquakes. This study proposed one possible form of a friction law that shows strong velocity weakening at high slip velocities. In a future study, it would be important to investigate the frictional properties at high slip velocities, based on friction experiments using rocks with the same properties as those within the northeastern Japan subduction zone.
 This research was supported by MEXT KAKENHI (21107007). We are grateful to two anonymous reviewers for their critical comments and to T. Shimamoto and Y. Fujii for valuable discussions. For this study, we have used the computer systems of the Earthquake Information Center of the Earthquake Research Institute, the University of Tokyo.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.