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 Slow slip and tectonic tremor in subduction zones take place at depths where there is abundant evidence for distributed shear over broad zones (∼10–103 m) composed of rocks with marked differences in mechanical properties. Here we model quasi-dynamic rupture along faults composed of material mixtures characterized by different rate-and-state-dependent frictional properties to determine the parameter regime capable of producing slow slip in an idealized subduction zone setting. Keeping other parameters fixed, the relative proportions of velocity-weakening (VW) and velocity-strengthening (VS) materials control the sliding character (stable, slow, or dynamic) along the fault. The stability boundary between slow and dynamic is accurately described by linear analysis of a double spring-slider system with VW and VS blocks. Our results place bounds on the volume fractions of VW material present in heterogeneous geological assemblages that host slow slip and tremor in subduction zones.
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 In subduction zones, there is abundant evidence from laboratory experiments [Hirth and Tullis, 1992; Dimanov and Dresen, 2005; Mehl and Hirth, 2008], seismic data [Eberhart-Phillips and Reyners, 1999; Martin and Rietbrock, 2006; Tsuji et al., 2008], and field observations [Bebout and Barton, 2002; Meneghini et al., 2009] suggesting that, under in situ conditions, shear is distributed over broad zones (∼10–103 m) composed of rocks with marked differences in mechanical properties. Prior to the discovery of slow-slip events, a considerable body of work had documented the structure of the deformation zone beneath seismogenic depths.Shreve and Cloos  developed a model that predicts the fate of subducted sediments and the formation of a heterogeneous mélange along the plate interface. Cloos and Shreve further expanded upon these ideas to discuss how such a “subduction channel” might act as a lubricant and facilitate motion of the down-going plate.Fagereng and Sibson give a concise summary of evidence for a “subduction channel shear zone” (defined here as a shear zone within subducted sediments) in which the style of deformation is determined by the relative abundance and distribution of competent (i.e., brittle or non-compliant) and incompetent (ductile or compliant) material. Efforts to understand the physical interactions that produce aseismic transients (slow slip) must account for the effects of this geological heterogeneity.
 Here, we model the features of such subduction channel shear zones by considering an idealized one-dimensional fault surface along the displacement direction, with spatially variable rate-and-state friction properties. Field observations suggest that two- and three-dimensional variations in the shapes of brittle phacoids have a significant effect on the style of deformation [Fagereng, 2011]. Our model represents a first-order, one-dimensional test of the dynamic implications of geologic heterogeneity in three-dimensional fault systems.
 Our model differs from previous studies by incorporating mixed frictional parameters along the fault surface that are manifest as alternating sections of VW and VS materials in the zone that hosts transient slip events. Linear analysis of a spring-slider system composed of two blocks in parallel, one with VW and one with VS properties, provides an approximate prediction for the sliding stability of the elastic model. Our results demonstrate how different combinations of geological materials produce the full range of sliding styles that are observed along plate-bounding faults.
 The governing equations are solved on a one-dimensional fault segment, referred to simply as “the fault”, with an up-dip boundary corresponding to a seismogenic zone that is locked on the timescale of transient events, and made to slip at the plate convergence velocityvplateat a down-dip boundary corresponding to depths well below (80 km) those of tremor activity and slow slip [e.g.,Rubin, 2008]. Directly down-dip of the locked boundary the fault contains a variable mixture of VW (a/b < 1) and VS (a/b > 1) material. To describe the mixture, we define a parameter ηas the ratio of the combined length of VW material in the fault, to the total length of the mixed region. Strongly VS material extends between the mixed region and the down-dip, forced-slip boundary; this region corresponds to a long transition zone from frictional sliding to distributed deformation below.
 Friction on the fault is described using the single state variable rate-and-state law
where μ0 is the friction coefficient at reference speed v0, θ is a state variable, a and b characterize whether μ evolves to a higher or lower value with sliding rate v, and dc is the characteristic distance over which this evolution takes place according to
Stress balance requires that
where ∂δ/∂ξ represents displacement gradients along the fault, with local coordinate x. The term on the left is derived from plane strain solutions for stress due to displacements on a fault embedded in an elastic medium with shear modulus G and Poisson ratio ν. The first term on the right is the stress due to friction, which is proportional to the effective normal stress σ.The remaining term describes stresses due to inertia, which scale with the ratio of the slip rate to the shear-wave speedvs [Rice, 1993]. Equations (1), (2), and (3) are cast as coupled first order differential equations in v and θ and integrated using ODE solver routines in MATLAB. Additional model details are provided in the auxiliary material.
 We characterize three modes of fault slip [Rubin, 2008]: 1) dynamic sliding, with maximum slip speeds limited by inertia; 2) slow sliding, characterized by repeating transient slip events with slip speeds limited by friction; and 3) stable sliding, with slip speeds approaching a steady-state. Slip speeds and dip-parallel (assuming perfectly dip-slip motion) fault lengthsW are normalized, respectively, by the plate convergence rate vplate, and the critical nucleation length
that must slip to nucleate dynamic events along a homogeneous fault (parameter definitions and nominal values in Table 1). By choosing the ratio of fault length W to h*, the effective normal stress σ on the fault is set by equation (4) and does not exceed 6 MPa for any reported simulations. Parameters are chosen to maintain consistency with previous models [e.g., Rubin, 2008; Liu and Rubin, 2010; Segall et al., 2010] and to reflect the hypothesis that slow slip and tremor are made possible by a combination of temperature dependent frictional behavior and low effective stress.
Table 1. Nominal Parameter Values Used in Model Simulationsa
Ranges are given for those parameters that are varied to explore the different regimes of model behavior.
characteristic slip distance
plate convergence velocity
∼3 cm yr−1
shear wave velocity
3 km s−1
scaled fault length
fraction VW on fault
3. Results and Discussion
 We ran simulations at constant fault length W/h* = 6, for 0.35 < η < 1, keeping frictional parameters that characterize VS material constant and varying the rate sensitivity of VW material by changing the ratio (a/b)vw < 1 (Figure 1) (note: a/b = 1 is velocity neutral; ratios further from unity are more rate sensitive). As (a/b)vw decreases, fault behavior evolves smoothly from stable (not shown) to slow sliding, and switches abruptly from slow to dynamic sliding. With less VW material (lower η), the value of (a/b)vw that marks the transition to dynamic sliding decreases. Whereas previous models [Liu and Rice, 2005; Rubin, 2008; Liu and Rubin, 2010; Segall et al., 2010] have generated similar behavior using faults entirely composed of VW material, our results demonstrate that areas of the plate interface that host tremor and slow slip can also contain sizeable fractions of VS material. The VS material that is required to stabilize slow slip in our model is consistent with expected changes in frictional properties at higher ambient temperatures beneath the seismogenic zone, and field evidence in exhumed fault rocks for ductile deformation [Fagereng and Sibson, 2010; Fagereng, 2011; Fagereng et al., 2011; Bebout and Barton, 2002; Meneghini et al., 2009], which may continue to take place in the time between transient events that this treatment is designed to capture.
 In Cascadia, tremor epicenters correlate with locations of high slip-rate during ETS events [Wech and Creager, 2008; Bartlow et al., 2011]. Such findings support the hypothesis that tremor occurs as locally accelerated slip on heterogeneities within the slipping region. Accordingly, we interpret tremor propagation velocities as corresponding with modeled dip-parallel slip propagation velocitiesvprop. Tremor events in Cascadia propagate along dip at 30–200 km/hr [Ghosh et al., 2010], similar to rates (25–150 km/hr) observed in southwest Japan [Shelly et al., 2007]. These values compare well with vprop for simulations shown in Figure 1, which range from ∼3.6–360 km/hr and depend inversely on η over 0.35 < η < 1 (see auxiliary material).
 Tectonic tremor has been found to consist of numerous low (LFE) and very low (VLFE) frequency earthquakes [Shelly et al., 2006; Wech and Creager, 2007]. Estimating the source parameters of tremor events remains a significant challenge. However, sparse estimates of static stress drop Δτs can be compared to our model results. In southwest Japan, Δτs for tremor events has been estimated at <10 kPa to ∼100 kPa [Obara, 2010]. Also in southwest Japan, more precise determinations of Δτs in the range 0.1–10 kPa have been made for VLFEs in the accretionary wedge [Ito and Obara, 2006]. Similar static stress drops have been inferred for large slow-slip events [Gao et al., 2012]. Our simulations predict static stress drops that are insensitive to η, but increase from ∼0.1–1.8 kPa with decreasing (a/b)vw (see auxiliary material).
 The presence of VS material in the fault produces slow sliding behavior over a larger range of fault lengths than is expected for strictly VW faults. We ran simulations with (a/b)vw = 0.8 for a range of W/h* (Figure 2). For each scaled fault length, as η increases the behavior evolves smoothly from stable to slow sliding, and abruptly from slow slip to dynamic sliding. The results illustrate the persistence of slow sliding events on faults that are much longer than the critical nucleation length (e.g., W/h* = 48). For these calculations, VS material is characterized by frictional parameters that are almost velocity-neutral. Larger values of (a/b)vs would lead to slow sliding behavior for W/h* > 48, corresponding to longer faults, or to smaller values of h* associated with higher effective normal stresses.
4. Double Spring-Slider System
 Validation and insight into the model behavior is gained by considering a simplified spring-slider system consisting of two rigid blocks held in frictional contact with a rigid surface at a constant normal stressσ and connected to each other by a spring with stiffness k2. The first block has (a/b)vw < 1 and base area η, while the second has (a/b)vs > 1 with base area 1 − η. A spring with stiffness k1 connects the VW block to a load point that moves at constant velocity. Elastic compliance is accommodated by the stiffness of the springs, while inelastic deformation is accommodated by the slip displacement of the blocks. Linear stability analysis (see auxiliary material) shows that two overlapping conditions determine the sliding stability (Figure 3). Each condition yields a critical value of k1 that depends on k2. In the limit k2 → ∞, with the characteristic distance for state evolution dc assumed constant, one stability condition vanishes and the remaining one takes the compact form
Recognizing that , the critical length scale for rupture nucleation is
which recovers equation (4) when η → 1. Comparison with h* from equation (4)reveals how VS material effectively increases the critical slip length for instability of a spring-slider system (sincebvs − avs < 0). This underlies the predictions of our elastic model, which demonstrate that longer fault segments are required to slip simultaneously in order to nucleate transient slip events, relative to strictly VW faults. That is, by effectively increasing the critical nucleation length, geologic heterogeneity allows for slow slip to occur over larger sections of fault before slip becomes dynamic.
Figure 4 compares the double slider results against the numerically determined field in which slow slip is observed for the model calculations summarized in Figure 1. Within the context of the double slider system, the value of k2 represents the degree of elastic coupling between the VW and VS materials. When k2 → ∞, the spring connecting the blocks is rigid, so VW and VS materials act as one block and slide at equal velocity. Hence, the stability condition predicted by equation (6) is more restrictive than the behavior expected from the elastic model, which can accommodate significant velocity gradients along the fault. When k2 is finite, the separation of the two blocks varies and the connecting spring stores elastic strain energy so that they slide at different rates – a situation akin to the elastic model. In natural shear zones, the degree of coupling between VW and VS elements scales with the degree of interconnectivity between brittle VW phacoids within a mélange matrix, represented in our model by the VS component. With this interpretation, k2 → ∞ corresponds to larger and/or more phacoids (either would increase the degree of interconnectivity), and decreasing values of k2 correspond with decreasing size and/or fewer phacoids. As k2 decreases, the critical stiffness of k1 decreases, reducing the size of the unstable domain. The elastic model stability corresponds approximately with the minimum critical stiffness. For example, the location of the minimum stiffness point ( , ) from Figure 3 is shown on Figure 4. The stability boundary is found by tracking this point through η–(a/b)vw space for different values of k2 (Figure 4, see auxiliary material for additional details).
 Recent field [Fagereng and Sibson, 2010; Collettini et al., 2011; Fagereng, 2011; Fagereng et al., 2011] and modeling studies [Ando et al., 2010; Daub et al., 2011; Nakata et al., 2011] suggest that fault heterogeneity (e.g., a mixture of VW and VS materials, or similarly, a mixture of brittle and ductile materials) along plate interfaces may be required for the generation of slow sliding transients and tectonic tremor. Matching observed recurrence intervals for tremor along the San Andreas fault at Parkfield, CA with a model based on a single block-spring-slider system, suggests that 40–70% of the frictional contacts at depth must be brittle, the remainder deforming in a ductile manner [Daub et al., 2011]. Additionally, models [Ando et al., 2010; Nakata et al., 2011] that include fault heterogeneity as unstable patches on a stable background fault can reproduce migration speeds (dip- and strike-parallel) and source spectra of LFE/tremor events; including viscous effects [Nakata et al., 2011] produces a transition in rupture behavior from dynamic to slow sliding, similar to the model presented here. Field studies present compelling evidence for these types of fault heterogeneity, notably at the Chrystalls Beach Complex, New Zealand [Fagereng and Sibson, 2010; Fagereng, 2011; Fagereng et al., 2011], where correlations are found between the volume fraction of competent material and inferred mechanisms of deformation (e.g., continuous, discontinuous). Competent volume fractions of ∼0.3–0.85 produce structural markers of mixed continuous-discontinuous sliding behavior that may record LFEs.
 Our results demonstrate that aseismic transients arise naturally along geologically heterogeneous faults, without requiring finely tuned rheological properties. All of our calculations are made assuming σ < 6 MPa, implying that fluid pressures on the fault approach lithostatic values [Shelly et al., 2006; Audet et al., 2009; Matsubara et al., 2009]. We emphasize that further efforts to characterize fault rock properties are necessary to better constrain future modeling efforts. The presence of velocity-strengthening material mixed with velocity-weakening material increases the fault length capable of simultaneous, aseismic sliding. Our numerical results are well described by a double spring-block-slider system, suggesting that further study of such simple models may shed additional light on the physical mechanisms behind aseismic transients on plate boundary faults.
 We thank two anonymous reviewers for helpful and constructive comments. This work was supported by NSF grant EAR-1114380.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.