It has been shown that the onset of frictional instability is characterized by a transition from stable, quasi-static rupture growth to unstable, inertially-controlled high-speed rupture. In particular, slow rupture fronts propagating at a steady speed Vslow of the order of 5% of the S-wave speed have been observed prior to the onset of dynamic rupture in recent fault-friction laboratory experiments. However, the precise mechanism governing this Vslow stage is unknown. Here we reproduce this phenomenon in numerical simulations of earthquake sequences that incorporate laboratory-derived rate-and-state friction laws. Our simulations show that the Vslow stage originates from a stress concentration inherited from the coalescence of interseismic slow creep fronts. Its occurrence is limited to a narrow range of the parameter space but is found in simulations with two commonly-used state-variable evolution laws in the rate-and-state formulation. The sensitivity of the speed Vslow to the model parameters suggests that the propagation speed Vslow reported in laboratory experiments may also be sensitive to parameters of friction and stress conditions. Our results imply that time and space dimensions associated with the propagation of Vslow on natural faults can be as much as a few seconds and several hundred meters, respectively. Hence the detection of such preseismic signals may be possible with near-field high-resolution observations.
 The onset of frictional instability is a key mechanism governing the nucleation of crustal earthquakes and landslides. It has been demonstrated both in laboratory experiments [e.g., Dieterich, 1979; Ruina, 1983; Dieterich and Kilgore, 1996; Ohnaka and Shen, 1999; Nielsen et al., 2010] and numerical simulations [e.g., Okubo, 1989; Rice and Ben-Zion, 1996; Lapusta et al., 2000; Rubin and Ampuero, 2005] that the onset of frictional instability is characterized by a transition from stable, quasi-static rupture growth to unstable, inertially-controlled high-speed rupture. While these studies have advanced our understanding of the transitional behavior, direct comparisons between numerical models and laboratory observations remain quite challenging due to difficulties in accurately monitoring the transitional behavior in laboratory friction experiments and in reproducing the laboratory observations in numerical simulations that incorporate appropriate friction laws.
 Several laboratory studies attempted to understand the behavior of the quasi-static to dynamic transition of stick-slip motion on experimental faults (i.e., pre-cut interfaces). Ohnaka and Shen  reported an initial, quasi-static phase in which the rupture grows at a slow and steady speed, followed by a rupture acceleration phase up to dynamic speeds. In friction experiments on polymethyl-methacrylate (PMMA), Rubinstein et al.  observed a slow detachment front propagating at 5% of the S-wave speed (Vs) after the passage of a supershear rupture front. More recently, Nielsen et al.  reported steady rupture fronts systematically propagating at about 5% of Vs on experimental faults where a stick-slip instability was spontaneously nucleated under slow loading. However, under different loading conditions, Ben-David et al.  found variability of rupture speeds ranging from a few percent of Vs to P-wave speed (Vp) depending on the ratio of local shear to normal stresses, suggesting that the speed of slow steady fronts (∼5% of Vs) observed in Rubinstein et al.  and Nielsen et al.  is not universal. Hence the mechanism of a slow steady propagation front remains elusive.
 In this study, we reproduce the occurrence of slow steady propagation fronts preceding fast dynamic rupture in numerical simulations of spontaneous earthquake sequences, consistent with the laboratory findings of Nielsen et al. . We refer to this phenomenon as a ‘Vslow stage’ and to its propagation speed as ‘Vslow’. We discuss the mechanism of the Vslow stage, the dependence of Vslow on model parameters, and its potential occurrence on natural faults.
2. Model Set-Up
 We conduct 2-D simulations of spontaneous earthquake sequences on a 1-D fault subjected to slow, tectonic loading [Kaneko et al., 2010]. We consider an antiplane (Mode III) configuration in which purely dip-slip motion is assumed. The simulations resolve all stages of the seismic cycle: the aseismic nucleation process, the subsequent dynamic rupture event, the postseismic slip, and the interseismic quasi-static deformation.
 The fault resistance to sliding is described by laboratory-derived rate and state friction laws [Dieterich, 1979; Rice and Ruina, 1983; Ruina, 1983]. For time-independent effective normal stress , the shear strength τ on the fault is expressed as
where a and b are rate and state constitutive parameters, is slip velocity, f0 is a reference friction coefficient corresponding to a reference slip velocity 0, θ is a state variable, and L is the characteristic slip for state evolution [e.g., Dieterich, 1979; Ruina, 1983; Rice and Ruina, 1983]. We consider the two most classical state variable evolution laws:
[Dieterich, 1979; Ruina, 1983]. Recently, Ampuero and Rubin  rekindled the discussion of which state evolution laws are more appropriate to use in earthquake modeling. Bayart et al.  showed that the slip law provides a better match to velocity-jump experiments. On the other hand, Beeler et al.  found, during slide-hold-slide experiments using a servo-control system, that a frictional surface heals with time, rather than with slip rate, suggesting that the aging law is a more appropriate representation.
 The parameter combination a − b < 0 corresponds to steady-state velocity-weakening friction and can lead to unstable slip, whereas a − b > 0 corresponds to steady-state velocity-strengthening and leads to stable sliding. Throughout this article, we omit the words “steady-state” and simply refer to velocity weakening or strengthening.
 Under slow tectonic loading, a frictional instability (i.e., an earthquake) is able to develop only if the velocity-weakening region of the fault exceeds the nucleation size h* [Rice and Ruina, 1983; Rubin and Ampuero, 2005]. Two theoretical estimates of the earthquake nucleation size for 2-D problems are given by
where μ is shear modulus for mode III. The estimate h*RR was derived from the linear stability analysis of steady sliding by Rice and Ruina , while h*RA was obtained for a/b ≳ 0.5 by Rubin and Ampuero  on the basis of energy balance for a quasi-statically expanding crack governed by the aging law (2). Rubin and Ampuero  gave formulae for half of the nucleation size but we use full sizes.
 Our simulated fault is 300-cm long. Slip evolution is computed based on the assumed friction law on the 150-cm long central portion. The slip rate Load = 5 cm/year is prescribed on the two 75-cm long outer portions of the fault. The central portion is divided into three segments: a 75-cm long velocity-weakening segment surrounded by two 37.5-cm long velocity-strengthening segments (Figure 1a). The velocity-strengthening segments mimic viscous silicon patches placed at each end of the fault in some of the experiments of Nielsen et al. , creating similar loading conditions. Note that, in the study of Nielsen et al. , the Vslow stage was observed for smooth fault surfaces with or without viscous patches (S. Nielsen, written communication, 2009), but was not observed for rough surfaces with viscous patches.
Table 1 gives a range of model parameters considered in this study. Since values of rate-and-state parameters a, b, and L are not known for the laboratory specimen of Nielsen et al. , we do not attempt to exactly reproduce the spatial and temporal scales of the Vslow stage observed in their laboratory experiments. Instead, we set the spatial scale and nucleation sizes to be roughly equal to those in the laboratory experiments by adjusting the value of L. The values of other model parameters are chosen to represent plausible stress and friction values on natural faults in seismogenic conditions rather than those in the laboratory experiments.
Table 1. A Range of Parameters Used in This Study
The indicated values of b are valid for the VW region.
Figure 1 shows one of the simulation examples in which slow steady fronts (Vslow) propagate at about 5% of Vs systematically during the nucleation processes of seismic events. Motivated by the fact that a/b of many velocity-weakening materials in laboratory experiments is closer to 1 than 0, we set a/b = 0.93. In this example (Figure 1b) we adopt the slip law (3). The blue lines show the continuous slow sliding of the velocity-strengthening segments, which creates a stress concentration at its tip and penetrates into the velocity-weakening segment (Figure 1b). In due time, seismic rupture nucleates and propagates bilaterally (its progression is shown by red lines in Figure 1b). After a seismic event, the velocity-strengthening segments experience postseismic sliding due to the transferred stress. The interseismic period between two successive events is 5 hours.
Figure 1c shows a close-up look at the onset of one of the seismic events. The rupture fronts, the positions of which are defined here by the peak values of shear stress, begin at a localized point, expand quasi-statically, then steadily propagate at about 5% of Vs before accelerating to much faster speeds. The behavior of the Vslow fronts (Figure 1c) is similar to that in the laboratory experiments of Nielsen et al.  (Figure 1d) in that the speed Vslow is about 5% of Vs and remains about the same for all events in a given simulation (except for the first few that are affected by initial conditions).
 To understand the mechanism of the slow steady fronts, we look at the evolution of slip rates , shear stress, and the quantity θ/L before and during the occurrence of the Vslow stage (Figure 2). During the interseismic period, two creep fronts emanate from the rheological boundaries (x ≈ 45, 105 cm) and then propagate inward. Their eventual coalescence creates favorable conditions for earthquake nucleation near the center of the fault (Figures 2a, 2c, and 2e). Behind the two creep fronts, the condition is near steady-state (θ/L ≈ 1), and there is no evolution of state variable θ according to (3). The Vslow stage originates from a stress concentration inherited from the coalescence of the creep fronts (Figures 2b, 2d, and 2f). During the Vslow stage, the peak slip velocities remain roughly constant in time (∼0.05 m/s in this example). The expansion of rupture takes the form of a bilateral crack (Figure 2b) as opposed to a unilateral pulse reported by Ampuero and Rubin  under different loading conditions.
 The relation between propagation speed Vslow and peak slip rate max is consistent with a theoretical relation given by equation (53) of Ampuero and Rubin :
where θi is the value of the state variable prior to the arrival of the rupture front. For instance, for the case shown in Figure 2, ln(maxθi/L) ≈ 11–16 and max ∼ 0.05 m/s, which yields Vprop/Vs = 0.04–0.06. This is consistent with the speed Vslow obtained in the simulation (Figure 1c).
 Setting max = 2aσVs/μ in (6), the slip velocity at which the effect of radiation damping is comparable to the direct effect of rate-and-state friction, Perfettini and Ampuero  proposed that a typical rupture speed at the onset of elastodynamic effects was Vprop ∼ 0.05Vs. To assess the relevance of elastodynamics, Figure 2 shows results obtained by first simulating three earthquake cycles (Figure 1b), then continuing with a quasi-static simulation (by turning off the inertial effects). We find that the Vslow stage occurs despite the quasi-static assumption, suggesting that the speed Vslow does not explicitly depend on Vs. A subtle effect of elastodynamics is however not discarded because the arrest of the last dynamic event sets the background conditions for the creep front propagation.
 A fracture mechanics argument based on an idealized model provides insight on how the stress concentration induced by the coalescence of the creep fronts gives rise to a period of a steady propagation speed. Let us consider the quasi-static growth of a mode III crack of length 2ℓ under a stress field that consists of a uniform background value (and hence constant stress drop ) plus a highly concentrated force F at the center of the crack. The balance between the energy release rate at the crack tip, G, and the fracture energy, Gc, provides a relation between crack speed Vprop and crack half-length ℓ. On the one hand, the static energy release rate G, given by [e.g., Ampuero et al., 2006]
reaches a minimum when ℓ = ℓmin = 2F/(π). On the other hand, the effective fracture energy of rate-and-state friction behaves as Gc ≈ bLln(max/Load)n, where n = 1 for the slip law and n = 2 for the aging law [Ampuero and Rubin, 2008]. Equation (6) implies that max, and hence Gc, is an increasing function of Vprop. Since G(ℓ) = Gc(Vprop), the crack propagation speed is almost steady when the energy release rate G(ℓ) is almost constant. This happens when the crack length is close to 2ℓmin. This analysis shows that a quasi-statically expanding crack subjected to a stress concentration can lead to a period of a steady propagation speed. The previous analysis also provides an estimate of the propagation speed, which depends on the product FΔτ. However, deriving an adequate estimate of F is not straightforward, and the quantitative comparison of the idealized model to the numerical simulation results remains a subject of future work.
4. Dependence of the Speed of Steady Propagation Fronts on Model Parameters
 To identify the parameters controlling Vslow, we perform a number of earthquake-sequence simulations with different sets of model parameters indicated in Table 1. First, we vary the size of the velocity-weakening segment Dvw while other parameters are held fixed. We normalize Dvw by h*RR/2, a length that agrees well with our simulated nucleation sizes. We find that by increasing Dvw/h* the speed Vslow increases (Figure 3a), the duration of the Vslow stage decreases dramatically, and its spatial extent remains about the same (Figure 3b).
 Different speeds of Vslow fronts (Figure 3a) are related to different amplitudes of slow creep fronts during the interseismic period. For a larger value of Dvw/h*, the slip rate of slow creep fronts is larger (e.g., ∼10−8 vs. ∼10−9 m/s in Figure 3c) and, upon their coalescence, the resulting peak slip rate (max) during the Vslow stage also becomes larger (Figure 3c). From (6), the speed Vslow is strongly correlated to the peak slip rate (Figure 3b). Hence the coalescence of larger-amplitude creep fronts emerging during the interseismic period leads eventually to a faster propagation speed during the Vslow stage.
 We also vary the value of the rate-and-state parameter b in the velocity-weakening segment. We find that Vslow decreases with increasing a/b (Figure 3d). This behavior can be attributed to the dependence of Vslow on Dvw/h* shown in Figure 3a, which indicates that a smaller nucleation size h* leads to a faster Vslow; theoretical estimates (4) and (5) and numerical simulations show that h* generally decreases as b increases. When a/b becomes even smaller (a/b < 0.88), the nucleation occurs near one of the rheological transitions and takes the form of a unilateral pulse (Figures 4a and 4d). Since the nucleation size becomes smaller and the fault becomes more unstable with larger b − a, one of the slow creep fronts can self-accelerate towards earthquake nucleation, without coalescence (Figure 4d). Its peak slip rate increases monotonically with time, and so does its propagation speed as predicted by (6) (Figures 4a and 4d). Hence there is no steady Vslow stage when the nucleation occurs near one of the rheological transitions (Figures 4a and 4d). This behavior was also observed by Nielsen et al.  in experiments with rough surfaces and viscous patches. We find that this style of nucleation is the most common in the parameter range shown in Table 1. This is probably why a Vslow stage was not reported in previous theoretical studies.
 Since the assumed loading rate Load is orders of magnitudes slower than that in the laboratory experiments, we further explore the dependence of Vslow on Load. Figure 3e shows that the speed Vslow increases with the loading rate. Kaneko and Lapusta  demonstrated how increased loading conditions could change the nucleation process and, in particular, cause order-of-magnitude smaller nucleation sizes. Since a larger loading rate leads to a smaller nucleation size, the quantity Dvw/h* increases with the loading rate, resulting in larger Vslow in our simulations (Figure 3e).
 We also find the occurrence of a Vslow stage in simulations with the aging law (2) (Figure 4c). The dependence of the properties of the Vslow stage on model parameters is qualitatively similar to the case of the slip law, although the parameter ranges in which a Vslow stage occurs are different. Under the same rate and state parameters, Vslow ∼ 0.05Vs when Dvw/h* ∼ 24 for the slip law and Dvw/h* ∼ 3.4 for the aging law, respectively (Figures 4a and 4b). For both laws a Vslow stage exists when nucleation proceeds by coalescence of two slow creeping fronts (Figures 4c and 4f), but not if nucleation occurs near a rheological transition (Figures 4b and 4e).
 We have found that in rupture models nucleated by the coalescence of aseismic slip fronts, dynamic rupture is preceded by a stage of slow, steady rupture propagation. Its rupture speed Vslow is not universal, but depends on frictional and loading parameters (Figures 3 and 4). This suggests that the value Vslow ∼ 0.05Vs reported in the laboratory experiments of Nielsen et al.  may also be sensitive to parameters of friction and stress conditions. Future work may be directed towards determining the values of the rate-and-state parameters of the laboratory specimen to allow quantitative comparison between laboratory observations and numerical simulations.
 Slow propagation fronts have been observed in laboratory experiments under a variety of loading conditions [e.g., Rubinstein et al., 2004; Nielsen et al., 2010; Ben-David et al., 2010]. In this study, the occurrence of a Vslow stage results from coalescence of slow creep fronts emanating from rheological boundaries, and hence loading conditions play an important role in generating a Vslow stage. We have not yet explored scenarios where the fault is characterized by velocity-weakening conditions only (i.e., no rheological transitions). Whether the Vslow stage would occur under such scenarios remains a subject of future work.
 Earthquake nucleation is relevant to earthquake prediction because nucleation determines the origin time and hypocenter of seismic rupture and may result in detectable precursors. Laboratory rock-sliding experiments showed that values of characteristic slip L range from ∼1 to ∼500 μm, depending on the fault roughness and gouge width [Marone, 1998]. Space dimension in numerical models can be non-dimensionalized, for example, by x' = x/L such that the results of numerical simulations can be scaled with L. The results shown in Figures 1c and 4c combined with the upper bound of the laboratory value L = 500 μm suggest that the space and time dimensions associated with hypothetical Vslow fronts propagating at 5% of Vs on natural faults correspond to ∼100 m and ∼0.6 s ( = 100 m divided by 0.05Vs) for the cases with the slip law, respectively, and ∼500 m and ∼3 s ( = 500 m divided by 0.05Vs) for the cases with the aging law, respectively. This implies that inferences of such preseismic steady rupture propagation before the break-out of a seismic event may be possible with near-field high-resolution observations.
 The authors thank Stefan Nielsen for sharing the results of his laboratory experiments. The reviews by Maria Elina Belardinelli and an anonymous reviewer helped us improve the manuscript. This study was supported by NSF (grant EAR-1015698) and SCEC (funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agreement 02HQAG0008). This is SCEC contribution number 1511.
 The Editor thanks Maria Elina Belardinelli and an anonymous reviewer for their assistance in evaluating this paper.