A striking observation from both Cascadia and Japan is that the tremor associated with slow slip often migrates along strike at speeds close to 10 km/d but updip and downdip at speeds approaching 100 km/h. In this paper I adopt the view that the friction law appropriate for these regions is unknown, and I ask what constraints the observed behavior places on the friction law that must be operating. A simple relation, relying only on kinematics and elasticity, states that for a moving front the ratio of propagation speed to slip speed equals the ratio of the elastic shear modulus to the peak-to-residual stress drop at the front. Thus, larger propagation speeds require some combination of larger slip speeds and smaller peak-to-residual stress drops. As a proof of concept I design a two-state-variable friction law in which the strength drop associated with the main laterally propagating front, moving into a region that has not slipped since the last slow event, is much larger than that at the secondary fronts developing on the active slip surface. Preliminary numerical simulations demonstrate that this law can generate secondary fronts that propagate updip and downdip more than 2 orders of magnitude faster than, and even at some distance from, the main lateral front, as has been observed in Cascadia and Japan.
The recent discovery of episodic slow slip and tremor in subduction zones has exposed the geophysics community to previously unrecognized styles of fault behavior. Occurring just downdip of the seismogenic portion of large faults, this slip is potentially important for evaluating seismic hazards as well. But a major difficulty with using numerical models to investigate episodic slow slip is that we do not know the proper constitutive law. Although some theoretical work has been done on the micromechanics of friction [e.g., Sleep, 2006], existing constitutive laws are primarily empirical and are unable to reproduce the full range of behaviors observed in laboratory experiments. Furthermore, those experiments have explored only a small portion of the pressure, temperature, and pore fluid regimes relevant to deep subduction faults. Recent experiments have even shown that fault “state” can evolve directly with stress [Nagata et al., 2009], upending the decades old notion that state evolution requires the passage of significant time or slip, where “significant” is measured relative to either a characteristic contact age or size.
The premise of this paper is that even if we do not know the proper friction law, we can still use observations of slow slip to assess the characteristics of the laws that must be operating. In particular, in both Cascadia and Japan tremor (and by inference slow slip) migrates along strike at speeds of roughly 10 km/d, but updip and downdip at speeds 2 orders of magnitude faster. A basic relation from elasticity, independent of any particular friction law, states that the ratio of propagation speed to slip speed nearly equals the ratio of the elastic shear modulus to the peak-to-residual stress drop at the propagating front. Motivated originally by observations from Cascadia [Ghosh et al., 2010b], where the dip-parallel migration occurs close enough to the main lateral front to suggest that the peak slip speeds might be similar, I design a two-state-variable friction law, including dilatancy, intended to make the stress drop at the lateral front much larger than that at the secondary fronts moving in the dip direction. Geologically this seems plausible; for example, precipitation of minerals in the months since the previous slow slip event might increase the strength of the main lateral front, which once disrupted by slip might not contribute to strength loss at the secondary fronts piggybacking on top of this.
To validate the underlying concepts, but certainly not the friction law used to illustrate them, I run simplified numerical simulations of slow slip on a 2-D fault. To minimize computation time while varying parameter values I first run cycle simulations on a 1-D fault, and then choose a particular snapshot to impose as initial conditions on the 2-D fault. That 2-D fault is then perturbed behind the main front to initiate the secondary fronts. Parameter values are chosen using analytical rules of thumb to generate slip speeds, propagation speeds, recurrence intervals, and stress drops that are within the ranges inferred from observations. In addition to the relation between propagation speed and slip speed from elasticity, these rules of thumb entail an energy balance that equates the reduction in strain energy during slow slip with the fracture energy at the propagating front.
Mechanism (1) has been observed for halite [Shimamoto, 1986], and while experimental support for such behavior in silicates is quite limited [Moore et al., 1997; Beeler, 2009], this mechanism has some theoretical justification. The dominant contribution to frictional resistance is thought to arise from the shearing of microscopic contact points bridging the sliding surfaces [Dieterich, 1978]. If one thinks of the “state” part of rate- and state-dependent friction as a measure of the true area of contact, and that this area increases with contact age due to time-dependent deformation of the asperities supporting the applied load, then more rapid slip results in on average younger contacts and smaller contact area [Dieterich and Kilgore, 1994]. Steady state velocity-weakening behavior results when this decrease in contact area outweighs the increased stress required to shear the contacts faster (the latter being the “rate” part of rate-and-state friction). As discussed by Nakatani and Scholz , however, there is some finite elastic contact area even at zero age. Once the surface is slipping fast enough, and the contact lifetimes are short enough, that contact area cannot increase significantly beyond its instantaneous elastic value, state evolution is plausibly negligible and the surface becomes steady state velocity strengthening.
Mechanism (2) arises when the rate-and-state friction equations are coupled to slip in an elastic continuum. For velocity-weakening spring block sliders, infinitesimal perturbations to steady sliding lead to instability when the spring stiffness is below a critical value [Ruina, 1983]. In rock, the analog of the spring stiffness is the size of the region of accelerating slip, with larger nucleation zones being less stiff (smaller drop in driving stress for a given slip). By considering the stiffness at the center of a nucleating patch, one can estimate the minimum size of the region capable of undergoing spontaneous acceleration [Rice and Ruina, 1983; Dieterich, 1992; Rice, 1993]. However, the stiffness of a fixed-length nucleation zone is larger near the edges than at the center. As such a region accelerates the edges undergo less slip, smaller reductions in state (less weakening), and so gradually lag behind, causing the stiffness at the center to increase as the effective length of the nucleation zone shrinks. For sufficiently short faults this ultimately quenches the instability. In this way, unlike spring slider systems, faults long (compliant) enough to allow spontaneous acceleration of slip may not be compliant enough for that slip to reach dynamic speeds. In numerical models, faults in this intermediate length range undergo episodic slow slip.
The drawback of this explanation is that the range of fault lengths exhibiting this behavior depends upon the fracture energy implied by the friction law [Rubin and Ampuero, 2005, 2009; Ampuero and Rubin, 2008], and to be within the proper range requires rather severe tuning of the fault length for the common state evolution law that, at least according to existing experiments, seems to get the fracture energy right (the “slip” law). For example, for the range of parameters explored by Rubin , less than a factor of two in length separates “too small for episodic slip” from “too large for slip to remain slow”.
Mechanism (3) has several appealing attributes. Strength changes due to changes in the coefficient of friction are proportional to the effective normal stress (normal stress minus pore pressure), whereas strength changes due to dilatancy are proportional to changes in and not itself. Thus, if dilatancy is independent of [Samuelson et al., 2009], pore pressure reductions associated with dilatancy become more important than rate-and-state effects at sufficiently low . Interpretations of experiments by Marone et al.  and Samuelson et al.  are consistent with dilatancy dominating rate-and-state strength changes at the low values of inferred for slow slip regions [Segall et al., 2010]. Furthermore, the many orders of magnitude difference between plate convergence rates and elastodynamic speeds means that for a wide range of material properties the fault can behave as effectively drained (insignificant pore pressure reduction) at tectonic slip speeds, but relatively undrained (insignificant pore fluid flow) at elastodynamic speeds. Thus, slip can spontaneously accelerate above the plate convergence rate but still be prevented from becoming dynamic.
Recurrence intervals of episodic slow slip in Cascadia are of order 1 year, with peak slip often exceeding 2 cm [Szeliga et al., 2008; Schmidt and Gao, 2010]. In Japan, recurrence intervals are somewhat shorter and slips average closer to 1 cm [Sekine et al., 2010]; in both regions the inverted slip tends to be a modest to a significant fraction of the product of recurrence interval and plate convergence rate.
The average stress drop Δτ in an event can be estimated from
where μ′ is the effective shear modulus (equal to the shear modulus μ for antiplane strain and μ/[1 − ν] for plane strain, where ν is Poisson's ratio), δ is slip, and W is the downdip extent of slip [e.g., Pollard and Segall, 1987]. Equation (1) is simply the statement that stress in an elastic solid equals the elastic modulus times the strain, here of order δ/W. For δ = 2 cm, W = 80 km, and μ′ = 40 GPa, from (1) Δτ = 10 kPa, 2–3 orders of magnitude smaller than for “typical” earthquakes. If most of the slip accumulates over a distance of only a few tens of kilometers behind the propagating front, as could be inferred from recent observations of tremor in Cascadia [Ghosh et al., 2010a], then the stress drop on the actively sliding portion of the fault could be a few times larger. In a compilation of recent events, Schmidt and Gao  give the mean as roughly 30 kPa in Cascadia and Obara  estimates similar values for Japan.
In both Cascadia and Japan, slow slip sometimes propagates over 150 km along strike. The propagation speed, as inferred from tremor migration rates, is remarkably similar in the two regions, ranging from about 2 to 20 km/d and averaging between about 5 and 10 km/d [Obara, 2010; Schmidt and Gao, 2010].
If most of the 2 cm of slip in Cascadia occurs within 20 km of the migrating front, at a migration rate of 10 km/d the average slip speed is ∼10−7 m/s. For understanding propagation speeds the maximum (sustained) slip speed is more relevant but harder to estimate. Both the tremor observations of Ghosh et al. [2010a] and numerical simulations suggest that the slip speed can be strongly peaked at the front, so I view 10−6 m/s as a plausible order of magnitude estimate of the maximum sustained slip speed.
In both Cascadia and Japan, tremor is sometimes triggered by teleseismic surface waves with amplitudes of a few tens of kPa [Miyazawa and Mori, 2006; Rubinstein et al., 2007]. In both regions tremor occurrence and amplitude are modulated by the 12.4 h tidal cycle, which imparts stress changes of only a few kPa [Shelly et al., 2007a; Rubinstein et al., 2008]. Analysis of borehole strainmeter data indicates that slow slip in Cascadia is also modulated, at the tens of percent level, by the 12.4 h tide [Hawthorne and Rubin, 2010]. These observations are consistent with a very small stress drop for these events.
In Cascadia and Japan, bursts of tremor propagate both updip and downdip at speeds of several tens to over 100 km/h, 2 orders of magnitude faster than the along-strike propagation. Where the dip and slip directions differ significantly it is clear that these tremor bursts more closely parallel slip [Ghosh et al., 2010b], but for simplicity I do not always make the distinction here. In Japan this migration has been mapped by locating low-frequency earthquakes (LFEs) embedded within the tremor [Shelly et al., 2007a]. In Cascadia, Ghosh et al. [2010b] used array data to locate an “average” source of radiated energy in overlapping 2 or 5 min intervals. While this does not yield the spatial resolution of the LFE locations, the more continuous temporal coverage makes the association with the main front more clear. An important observation from Cascadia is that while the slip-parallel propagation often appears coincident with the main front, it is just as often located up to 10 km behind (e.g., Figure 1). Judging from the sharpness of both the main front and the tremor streaks themselves, this separation from the main front far exceeds the relative location error. In some cases later episodes of slip-parallel propagation also occur to the SE of earlier episodes, even though the main event is propagating to the NW. Collectively, these observations suggest that most of the rapid slip-parallel propagation reflects complexity of slip behind the main front, a conclusion reached previously by Shelly et al. [2007a]. The alternative, that the slip-parallel migration is just an apparent velocity that arises at the leading edge of the slipping region, either as a dislocation style “kink” [e.g., Gershenzon et al., 2011] or as the main front intersects a line of tremor sources at a small angle [e.g., Ando et al., 2010; Ghosh et al., 2010b], seems harder to justify from these data. In Japan, not enough LFEs were identified to locate the main front unambiguously, but the rapid slip-parallel propagation occurred in narrow bands a few kilometers wide, distributed over a few tens of kilometers along strike. Thus, the rapid migration did not appear to be as closely tied to the main front as was seen in Cascadia.
More recently it has been recognized that Cascadia, and perhaps Japan as well, experience “rapid tremor reversals” [Obara, 2010; Houston et al., 2011], where secondary fronts sometimes propagate tens of kilometers back in the direction from which the main front came, at speeds intermediate between that of the main front and those propagating parallel to slip (roughly 100 km/d). These reversals are not the focus of the present study.
In this paper I treat tremor mainly as proxy for slow slip, as tremor can be more readily located in space and time, but some of its attributes are relevant. Tremor appears to be composed of myriad LFEs, with locations and focal mechanisms consistent with shear slip on the plate interface [Shelly et al., 2007b; Ide et al., 2007b; Brown et al., 2009]. LFEs have seismic moments comparable to typical magnitude 1–1.5 earthquakes, but their apparent durations are up to 10 times longer [Ide et al., 2007a]. Their stress drop is unknown, but if it is as low as 10 kPa the large size this would imply could explain most of their slow nature. However, this would not explain a moment that increases quasi-linearly with time, as has been inferred from their spectra, albeit with considerable uncertainty [Ide et al., 2007a]. It also would not explain the absence of magnitude 2 and larger earthquakes whose duration is 10 times larger than that of their more typical counterparts. In fact, there appears to be a seismologically defined “slow earthquake trend,” with events up to magnitude 4 where moment increases linearly with time, both within individual events and for the trend as a whole [Ide et al., 2007a]. Possible explanations include very elongate rather than circular source regions, and perhaps a heterogeneous fault surface where a substantial amount of slip during the event occurs at less than seismic speeds [Ariyoshi et al., 2009].
3. Designing a Friction Law for Slow Slip
3.1. Propagation Speed
If a front migrates with a quasi steady slip distribution, then from kinematics the maximum slip speed Vmax equals the propagation speed times the maximum slip gradient (e.g., Figure 2, top and middle). In addition, sharp fronts are associated with strength drops that are large compared to the average stress drop over the event (Figure 2, middle). In this case the near-tip slip gradient is dimensionally the peak-to-residual stress drop Δτp−r at the front divided by μ′; this is just the stress-strain relation (1) applied to the near-tip region rather than to the crack as a whole. Putting these two statements together leads to
where Vprop is the propagation speed. The inverse of the coefficient α equals the ratio of the maximum slip gradient to its dimensional estimate Δτp−r/μ′; α depends upon the spatial distribution of the strength reduction behind the tip but is close to 1 for quasi-static elasticity [Ida, 1973; Shibazaki and Shimamoto, 2007; Ampuero and Rubin, 2008].
An appealing feature of (2) is that it relies only upon elasticity, which is well characterized, and not the friction law, which is not. Of course the various terms (excepting μ′) depend upon the friction law, but several of these can be inferred directly from observations. Vprop is known from tremor migration speeds. In numerical simulations α is between 0.4 and 0.75 for a range of friction laws [e.g., Ampuero and Rubin, 2008; Liu and Rubin, 2010]. For a lateral propagation rate Vprop = 10 km/d, μ′ = 30 GPa, and α = 0.5, Δτp−r ranges from 15 to 150 kPa for Vmax from 10−7 to 10−6 m/s. 15 kPa seems low if the average stress drop on the sliding surface is a few times larger than this; for this reason I prefer the Vmax = 10−6 m/s and Δτp−r = 150 kPa estimates.
3.2. Slip Speed
A useful constraint on Vmax comes from the equilibrium crack growth requirement that the fracture energy Gc (units of J m−2) is balanced by the reduction in stored elastic strain energy G that comes from crack growth [Lawn, 1993]. If Vmax is well above the plate convergence rate, such that the time-varying contribution to G from the ongoing plate motion is negligible, then for a quasi-uniform stress drop Δτ on a crack of full-length W,
In numerical simulations Gc can be accurately approximated by considering the fault at the edge of an expanding nucleation zone to undergo an instantaneous velocity increase, much as in a labaoratory “velocity-stepping” experiment [Rubin and Ampuero, 2005]. Dimensionally, Gc is the product of Δτp−r and an effective slip-weakening distance δc for the stress to decay, through state evolution at the new higher slip velocity, to its steady state value. That is,
where τ is shear stress and δ is slip. As [τ(δ) − τ(δc)] and perhaps Δτ depend upon slip speed, equating (3) with (4) provides the constraint on Vmax.
3.3. Building a Friction Law: General Considerations
I wish to derive a set of constitutive laws that gives rise to the following characteristics of slow slip outlined in section 2.2: (1) stress drops of a few tens of kPa, (2) recurrence intervals of order 1 year, (3) maximum sustained slip speeds at the lateral front of order 10–6 m/s, (4) lateral propagation speeds of roughly 10 km/d, and (5) secondary fronts that propagate updip and downdip at speeds up to 100 km/h. It is not my intention to argue that the resulting set of equations is in any sense the “correct” one; this seems impossible. Rather, the goal is to validate through any plausible-seeming equations the rules of thumb outlined in sections 3.1–3.2 and further developed in section 3.5.
To generate episodic fault slip, the fault surface must be steady state velocity weakening at the background slip speed (<10−9 m/s). To generate secondary accelerations behind the main front, the surface (at least if homogeneous) must also be steady state velocity weakening at velocities appropriate for slow slip (10−7–10−6 m/s). An alternate possibility, that the secondary accelerations occur on a heterogeneous but on average velocity-strengthening fault, will be considered in the Discussion but not in the following.
That the rapid slip-parallel propagation in Cascadia often occurs within a few kilometers of the main lateral front suggests that it is worthwhile exploring models where a similar Vmax is seen by both fronts. If this is the case, then large differences in propagation speeds require equally large differences in Δτp−r.
To ensure that most of the moment release occurs at less than seismic slip speeds, one or more stabilizing mechanisms of the sort discussed in section 2.1 must operate. To avoid excessive tuning of the fault length I consider only dilatancy and a cutoff velocity for state evolution. One mechanism must act to prevent instability of the main lateral front while still allowing the growth of secondary fronts on the active slip surface. This seems to preclude a single-state-variable formulation with either a cutoff velocity or velocity-independent dilatancy; both of these would quench any nucleation attempts behind the main front. Thus, I introduce two state variables, one a “standard” Dieterich/Ruina type, θ, and a second, Ψ, suggested by Beeler , that transitions from a value of 1 at low speed to 0 at high speed. Conceptually, Ψ might increase with mineralization of the fault surface in the interval between slow slip events, and decrease with loss of cohesion over a relatively narrow velocity range as slip accelerates. To stabilize the main front during a substantial increase in event size I associate decreases in Ψ with dilatancy, but to allow secondary instabilities there is no dilatancy associated with θ. To limit these secondary instabilities to less than dynamic speeds I introduce a cutoff velocity Vθ for the evolution of θ. Table 1 lists the parameters appearing in this study.
Two alternatives to a two-state-variable formulation might be considered. First, if the slip speed decays rapidly enough behind the main front, secondary velocity excursions could develop that again accelerate to the slip speed of the main front. Because slip speed is higher and “state” smaller behind the main front than ahead of it, Δτp−r for these secondary fronts would be less (see, e.g., equation (19) to follow), leading to a higher propagation speed than at the main front even for the same Vmax. However, because for standard rate-and-state friction Δτp−r depends only logarithmically upon state, achieving a factor of 100 difference in propagation speed in this way seems infeasible. Second, Ampuero  has pointed out that for standard rate-and-state friction, following a given stress perturbation (for example, from a tremor event) the new (perturbed) slip speed is roughly proportional to the background slip speed. Because the appropriate “background” slip speed is orders of magnitude larger behind the main front than ahead of it, slip speeds and hence propagation speeds could be much larger for the secondary fronts than for the main front. In addition, the expected reduction in slip speed with increasing distance behind the main front could explain why “rapid tremor reversals” have a propagation speed intermediate between that of the main front and those close to the main front that propagate parallel to slip. However, with a single-state-variable formulation it is not obvious what would prevent the slip speed at the main front from accelarating to the same limit imposed (by whatever mechanism) on the secondary fronts. Thus, I explore the two-state-variable formulation here.
Table 1. Parameters
a, bθ, bΨ
Background value (subscript)
Shear wave speed
Characteristic slip distances for state evolution
Mechanical energy release rate
Length scale for spontaneous acceleration
Length scale of propagating front
Pore fluid pressure within fault zone
Ambient pore pressure
p − p0
Characteristic pore pressure diffusion time
Steady state value (subscript)
Maximum slip speed at front
Characteristic velocities for state evolution
Plate convergence velocity
Length of velocity-weakening fault
Subscripts referring to main or secondary front
Combined pore and fluid compressibility
Effective critical slip distance
(1 − ν)/μ
Drop in Ψ across front (Ψbg − Ψss)
Effective normal stress σ − p
Ambient effective normal stress
Peak-to-residual stress drop
Average stress drop
3.4. A Candidate Friction Law
A set of constitutive equations consistent with the above considerations is given below. They include only processes that have previously been proposed, but I reiterate that I am presenting them as a “proof of concept” only.
The frictional strength is given by
The strength is proportional to the effective stress σ − p, where σ is the normal stress (assumed constant) and p is the time-variable pore pressure. The first 3 terms within the brackets represent standard Dieterich/Ruina friction with a cutoff velocity Vθ for state evolution [Okubo, 1989]. f′ is a reference value of the friction coefficient. The parameter a determines the magnitude of the “direct effect”, the increase in frictional strength with increasing slip speed at constant state, while bθ and bΨ determine the magnitudes of the “evolution effects” for the state variables θ and Ψ, the decrease in frictional strength with decreasing state at constant slip speed.
The evolution law for θ follows the “slip” form introduced by Ruina :
To the extent that the characteristic slip distance Dθ for the evolution of stress (or ln[θ]) can be thought of asperity size, the steady state value of state θss can be thought of as an asperity age Dθ/V, where V is slip speed. This dictates how the cutoff velocity functions in this formulation: θss decreases as slip speed increases, but for θ small enough that Vθθ/Dθ ≪ 1, further decreases in state do not lead to weakening (θss represents contact age but not area; in the language of section 2.1, the zero-age contact area is accounted for by the “1” in equation (5)).
The evolution law for Ψ is given by either of the equivalent formulations
Equation (7a) shows that Ψss varies from 1 for V ≪ VΨ to 0 for V ≫ VΨ. With Ψ substituting for ln(θ) some parallels with equation (6) can also be seen. Equation (7b) is the form originally written by Beeler , with the first term showing that Ψ decreases exponentially with slip (on the length scale DΨ), and the second that it approaches its limiting value of 1 exponentially with time (on the time scale VΨ/DΨ). Here I choose VΨ to be several times the background slip speed, so that Ψ ∼ 1 ahead of the main lateral front and Ψ ∼ 0 behind. Making the fault dilate with decreasing Ψ stabilizes the main front, but with no dilatancy associated with θ, secondary instabilities can develop provided the fault is still steady state velocity weakening (requiring a < bθ and V ≪ Vθ).
The first term on the right represents diffusive recovery of pore pressure driven by the difference between p and its far-field value p0, and the second represents a source or sink driven by changes in porosity ϕ. The first term implies a linear variation in pressure between p and p0 over a finite width zone with diffusive timescale tf. This formulation is motivated by observations of very fine-grained and low-permeability fault cores surrounded by fractured and much more permeable rock. Even if this condition is met, however, (8) is not accurate for times shorter than tf. It is adopted here for expediency rather than accuracy, in that it approximates fluid diffusion without requiring a one-dimensional fault-normal finite difference grid at every point along the fault, but for purposes of illustration it should be adequate.
Porosity changes are related to changes in Ψ via
For F(Ψ) = 1, the coefficient of proportionality is ε and this equation reduces to that of Segall and Rice  with Ψ substituting for ln(θ). However, for the parameters adopted below, dilatancy associated with reductions in Ψ behind the main front can impede nucleation of the secondary fronts even though Ψ is very low in this region. For this reason I define F(Ψ) such that F ∼ 1 for Ψ ∼ 1 but F ≪ 1 for Ψ ≪ 1 (see section 3.5.2). In other words, most of the dilatancy associated with decreasing Ψ occurs for Ψ near 1, with little occurring for Ψ near 0.
For completeness, when fault slip is coupled to changes in pore pressure there should be a delay between changes in effective stress and changes in strength, with the latter evolving with either slip or time [e.g., Cochard and Rice, 2000]. In the ∼ micron/sec sliding experiments of Linker and Dieterich , this evolution occurred over a time scale of seconds. Provided this evolution or delay is short compared to the duration of large pore pressure reductions (of order 103 s in the simulations to follow), it will not influence the model results significantly and for simplicity I do not include it here. Alternate formulations of the friction laws are also certainly plausible, including hybrid versions that merge the “aging” and “slip” state evolution laws at different slip speeds [Kato and Tullis, 2001], and having one state variable linked to porosity with two distinct mechanisms for healing and closing of pores acting in parallel at different locations within the fault gouge [Sleep, 2010]. Again, my intent here is not to advocate for the particular equations (5)–(9), but simply to present them as a plausible equation set that has the properties required to match the cited observations.
3.5. Estimating Δτp−r and Gc
3.5.1. Main Lateral Front
To choose parameter values it is useful to estimate Δτp−r and Gc by integrating the relevant subset of equations (5)–(9) through a velocity jump to a slip speed Vmax. The main results are summarized below, with details given in Appendix A. Readers not interested in how the adopted equations give rise to the desired Δτp−r, Vmax, and stress drop Δτ can skip to section 3.6.
By design, at the main lateral front we are concerned mostly with the evolution of Ψ and p. Neglecting changes in p, the contributions to Δτp−r and Gc from the reduction in Ψ alone are
where ΔΨ is the reduction in Ψ across the front and the approximation assumes Vmax ≫ VΨ. Across the main front ΔΨ ∼ 1.
The pore pressure reduction is zero at t = 0 (the time of the velocity jump), when there has not yet been any dilatancy, and for t → ∞, when dilatancy is no longer occurring (t ≫ DΨ/Vmax) and the pressure has equilibrated to the far-field value (t ≫ tf). Significant pore pressure reductions require that the time scale for pore pressure equilibration, tf, be longer than the time scale for porosity changes, DΨ/Vmax. In this limit (Vmaxtf/DΨ ≫ 1), the maximum pore suction Δp(t) = p(t) − p0 is estimated to be
Neglecting the small changes in rate-and-state friction, Δτp−r and Gc from dilatancy and membrane diffusion in the limits Vmaxtf/DΨ ≫ 1 and Vmax ≫ VΨ become
The peak suction occurs at a normalized slip distance δ/DΨ of ∼ln(Vmaxtf/DΨ), and then decays over the slip scale Vmaxtf.
The relative contributions of Δp and ΔΨ to Δτp−r and Gc can be determined by comparing equations (13) to (10) and (14) to (11), with the result that
Both of these conditions are met using the parameters chosen below, so it is sufficient to evaluate the contributions to Δτp−r and Gc from the pore pressure reduction when considering the behavior of the laterally propagating front.
Finally, to anticipate Vmax it is useful to estimate the average stress drop Δτ in the slow event. For t ≫ tf the fault is approximately drained well behind the lateral front and the pore pressure change there is zero. If the contribution from the Dieterich/Ruina state variable is also small (bΨ ≫ [bθ − a]ln[Vmax/Vpl]), then from (5) the stress drop is just
Substituting this into equation (3) for G and equating G with GcΔp yields
where I have made use of π/(4f′) ∼ 1 and ΔΨ ∼ 1. In the simulations to follow this simple estimate predicts the slip speed at the main front to within a factor of two or so.
3.5.2. Secondary Fronts
Behind the main front Ψ ≪ 1, so unless further decreases from already low values of Ψ are important, the behavior of the piggybacking fronts is governed by the evolution of θ. From equation (5)
Here Vmax(2) represents the maximum slip speed at the secondary front, and θbg(2) the background value of state ahead of that front but behind the main front (Figure 3). Behind the main front the fault is near steady state, so θbg(2)/Dθ can be approximated as Vbg(2), the slip speed ahead of the secondary front. Then the ratio of Δτp−r at the main front to that at the secondary front is, from (13) and (19),
For the parameters adopted below this ratio exceeds 102; however, the actual ratio is somewhat smaller because for the reasons mentioned in section 4.1equation (12) overestimates the maximum pore suction.
In the simulations below, dilatancy associated with decreasing Ψ is strong enough to impede the growth of secondary fronts even with the small values of Ψ behind the main front (the resulting pore pressure reduction is is given by (13) with ΔΨ being roughly the residual value of Ψ behind the main front, ∼VΨ/[VΨ + V]). For this reason I introduced a version of F given by
(Figure 4a). F is close enough to 1 over most of the domain of Ψ that the scalings in section 3.5.1 remain essentially unchanged, but with F ≪ 1 for Ψ ≪ 1, secondary accelerations become more common.
3.6. Choosing Parameter Values
Parameter values used in the simulations below are listed in Table 2. Where available I have taken these to be close to those inferred from laboratory experiments (e.g., a and bθ near 10−1; Dθ of order 10 μm). Those not so constrained were chosen to generate the desired stress drop, propagation speed, or slip speed, based on the estimates in sections 3.1–3.5, or to minimize the number of grid points needed for the 2-D fault calculations. For a fuller discussion, including an assessment of the accuracy of the dimensional estimates in section 3.5, see Appendix B. The resulting steady state friction is shown in Figure 5.
Table 2. Parameter Values
4 × 10−5 m/s
5 × 10−9 m/s
μ′ (1-D fault)
μ (2-D fault)
ν (2-D fault)
3.7. Model Geometries and Strategy
To keep the numerical simulations tractable, I run multicycle simulations on 1-D faults, and then extract snapshots from those to use as input for models of 2-D faults. Even with this simplification the 2-D simulations can cover only the portion of the 1-D model domain near the main front. After importing the 1-D snapshot the 2-D fault is perturbed locally to initiate a secondary slow event. Although one might think of this perturbation as representing a “tremor event”, the goal here is not to model tremor in any physically meaningful way.
The 1-D fault is driven at the “downdip” edge at the plate velocity Vpl = 10–9 m/s, and is “locked” (V = 10−6Vpl) at the other edge. The “updip” half of the fault, of length W, has the constitutive parameters listed in Table 2. The “downdip” half is velocity-strengthening with a = 0.006 and bΨ = 0, but these details are relatively unimportant to the behavior near the propagating front. When imported into the 2-D fault model this “updip-propagating” front logically becomes the laterally-propagating front, but as the goal here is just to generate a front that is faithful to the underlying equations this seems acceptable. Details of the numerical implementation are given in Appendix C.
4. Numerical Results
4.1. One-dimensional Faults
The maximum slip speed is shown as a function of time for two 1-D fault simulations in Figure 6a. The black curve is for the function F given by equation (21) and Figure 4, while the green shows results for F(Ψ) = 1. In both cases slow slip events occur quasi-annually, with maximum slip speeds close to the anticipated value of 10−6 m/s, but the black curve exhibits occasional secondary excursions to larger slip speeds. Figure 6b shows a zoom of the roughly weeklong event starting near 4.62 years. For most of its duration Vmax ∼ 10−6 m/s, but toward the end higher-velocity excursions appear as narrow spikes. Figure 6c shows snapshots from this event, starting at about 4.625 years and ending after the first velocity spike. The blue curves show the main front propagating to the left from ∼37 to 30 km. At this point the secondary instability develops near 36 km, with the maximum slip speed now limited by Vθ (black snapshots). Because its slip speed is much greater and its peak-to-residual stress drop much smaller (∼20 kPa compared to ∼330 kPa; see Figure 6d), the secondary front quickly overtakes the main front (equation (2)). At this point it decays (cyan curves); to continue propagating at this larger speed would entail a large Gc (equation (14)) that cannot be met by the small stress drop Δτ added by the secondary event. This Δτ is a few kPa, too small to be seen in Figure 6d, whereas Δτ for the main event is 40–50 kPa. A more extended version of this event and its multiple secondary excursions can be seen in Animation S1.
Over the last few kilometers the primary front is propagating at ∼5 km/d (α ∼ 0.7 in equation (2)), well within the observed range. The pore suction reaches 500 kPa at the front and decays over a length scale of Vproptf ∼ 50 m; this dominates the strength drop at the tip. The maximum pore suction is only about 25% of the estimate of ε/β = 2 MPa from equation (12), mostly because the velocity jump at the front is not sufficiently “instantaneous” (some drainage occurs during the accelerating but still slow slip speeds before the arrival of the peak stress). Behind the front both Ψ and θ quickly approach steady state, setting the stage for the secondary velocity excursions.
It is well known that for the “slip” law for state evolution, a spring block slider subjected to a large stress perturbation can go unstable even if the spring stiffness is larger than the critical value appropriate for infinitesimal perturbations [Gu et al., 1984]. However, for the range of perturbations I applied, the secondary excursions on the 2-D faults were not self-sustaining unless the 1-D fault was already close to undergoing spontaneous secondary accelerations on its own. The time of the snapshot used as input for the 2-D fault simulation is shown by the dashed red line in Figure 6b; it occurs shortly before the onset of the secondary excursions in the 1-D fault simulation.
4.2. Two-dimensional Faults
Computational demands require reducing the size of the model domain when moving from a 1-D to a 2-D fault. The thick red line in Figure 6c (1150 grid points) shows where the values of V, Ψ, and θ from the 1-D fault were used as initial conditions in the along-strike (x) dimension for the 2-D fault. The thick green curve (130 points) shows the velocity distribution that is enforced as a boundary condition. There are 2048 grid points in the orthogonal (dip or y) dimension with the same initial values of V(x), Ψ(x), and θ(x). For simulations that are uniform in the y direction, the proximity of the spatial replicates in the x direction (arising from the spectral solution method) promotes growth of secondary instabilities somewhat sooner than in the 1-D case. However, the increased stressing rate due to interaction with these replicates is only ∼1 kPa/h, smaller than the peak stressing rate during the 12.4 h tidal cycle in some regions of Cascadia.
At t = 0 and y = 0 a circular region 500 m in diameter centered 400 m behind the peak slip speed is perturbed by multiplying a and bθ by 3 and V by 8 (500 m equals the background value of h* for the 1-D fault, a crude estimate of the minimum nucleation dimension given in Appendix B). Snapshots of the subsequent velocity evolution are shown in Figure 7. The top and bottom axes are mirror planes and the horizontal axis shows the region where the velocity is free to vary. After accelerating in place (Figure 7a), the perturbation ultimately detaches from its source and propagates parallel to the main front as a slowly decaying pulse (Figures 7b–7d). The maximum slip speed approaches 10−3 m/s within the initial perturbation but is ∼10−5 m/s in the detached front, an order of magnitude larger than Vmax at the main front but still smaller than Vθ. Comparison with the 1-D fault in Figure 6 suggests that Vmax for the secondary event in this 2-D fault example is limited at least in part by its limited x dimension.
The secondary front propagates at an average rate of ∼70 km/h (Figure 7b, inset), within the observed range. The factor of ∼300 higher propagation rate than for the main front is due to a factor of ∼10 increase in Vmax and a factor of ∼30 decrease in Δτp−r (Figure 8). Δτp−r at the secondary front is slightly more than 10 kPa, consistent with the value anticipated from equation (19) with Vmax(2)/Vbg(2) ∼ 10. The propagation speed is consistent with equation (2) with α ∼ 0.7 (Figure 2).
Note that progress of the main front is perturbed very little by passage of the secondary front. This is due in part to the much larger Δτp−r at the main front, which, as intended, would make the main front propagate much more slowly than the secondary front even if their peak slip speeds were the same. But two other factors contribute as well: (1) at any point along its path the secondary velocity excursion is a transient feature (the slip speed behind the secondary front actually drops below its previous value after the front passes), and (2) the maximum slip speed along the secondary front occurs ∼0.75 km behind the main front, not at the intersection of the two. The pulse-like nature of the secondary fronts seems consistent with the LFE locations of Shelly et al. [2007a, Figure 17], which are similarly concentrated at the fronts and not distributed over a broad region behind them (see also Figures 1b and 1d). That Vmax is larger for the secondary front than at the intersection of the two appears to be due to the fact that the pore pressure is still significantly below background (by 0.27 MPa, more than half the peak suction of 0.51 MPa) at the location of the main front Vmax. From the standpoint of a growing instability behind the main front, increasing the slip speed at the larger effective stress near the main front increases the fracture energy there without a corresponding increase in the stress drop driving the front. This is also why the secondary fronts in Figure 6c and Animation S1 decay as they approach the main front.
The secondary front dies out as it propagates to the right because of the decrease in V behind the main front. This leads to an increase in Ψ (healing), more dilatancy associated with the secondary front, and, because the propagation speed is large enough for the fault to behave as undrained even well behind the secondary front, a larger sliding stress (Figure 8b, inset). The extent to which the secondary front is restricted along strike is enhanced here by the choice of F, which is zero for Ψ < 0.015; in Figure 7a this occurs for 1.48 < x < 2.14 km. In principal, however, propagation to the right can be limited by any process that leads to healing with decreasing slip speed behind the main front, implying at least a larger peak stress and perhaps, as in Figure 8b, a larger sliding stress for any secondary fronts propagating in that direction.
Three additional episodes of dip-parallel migration can be seen in Animation S2. When the secondary front in Figure 7 reaches the top boundary, it collides with its mirror image and accelerates before decaying. When the main front reaches 1.3 km, at t ∼ 50 min, a new, spontaneous secondary event develops in the region of the original perturbation (because the increased values of a and bθ persist). This perturbation has an amplitude similar to that of the first, but it grows, rather than decays, as it propagates. After the main front reaches 1.2 km (t ∼ 75 min), a secondary front propagates back to the right, centered along y = 0. As this decays it spawns a third front that propagates parallel to dip, in this case farther (∼3 km) behind the main front, and well within the region where F > 0. The location of this event has been established by the slip accumulated during the previous two. It is noteworthy that the rapid slip-parallel tremor migration observed by Ghosh et al. [2010b] similarly occurred at variable distances from the main front, ranging from essentially at the front to 5–10 km behind (e.g., Figure 1). At the end of the animation a fourth event propagates in the dip direction, but unlike the third this one is clearly guided by the right edge of the model domain.
Because the secondary events described here have a restricted width perpendicular to the propagation direction, they have a quasi-constant moment rate, as has been inferred for events along the “slow earthquake trend” of Ide et al. [2007a]. The event in Figure 7 has a moment less than that of observed events with a comparable duration [Ide et al., 2007a], but most of the difference could be made up by increasing its width by a factor of 2–3. Although its stress drop is quite low (∼4 kPa; Figure 8d), the increase in slip relative to its surroundings is more than 10dθ = 0.2 mm. Using standard relations for earthquakes, 0.2 mm is equivalent to the slip in a circular magnitude 1 LFE with a stress drop of 90 kPa (diameter ∼ 160 m), and 4 times the slip in a magnitude 1 LFE with a stress drop of 11 kPa (diameter ∼ 320 m). Thus, even the small amount of slip in this secondary event is plausibly sufficient to trigger tremor.
5.1. Generating Secondary Velocity Excursions
Equations (5)–(9) produce two generations of slow slip speeds essentially by invoking two state variables, each with its own characteristic velocity. It would be preferable to devise a simpler single-state-variable system that could do the same, but for homogeneous faults this appears to be very difficult. If the main front is stabilized because the entire slow slip region is too small to reach elastodynamic speeds, then the fact that the tremor propagating parallel to slip outlines regions only a few kilometers across (more than 10 times smaller) means that such regions would be too small to undergo spontaneous acceleration (certainly for the slip state evolution law, and almost certainly even for the less defensible aging law). If the main front is stabilized by a cutoff velocity for state evolution, then the fault surface behind that front is strongly velocity strengthening and not prone to further acceleration. And if the main front is stabilized by dilatancy that is proportional to d ln(V), independent of V, then new events cannot nucleate behind the main front because dilatancy becomes much more effective at large slip speeds (e.g., equation (14)).
Whether long-lived secondary fronts could be produced by a simpler set of constitutive laws on a heterogeneous fault remains to be determined. In this context an important question is whether the tremor associated with the secondary fronts is just a passive indicator of an increased slip rate, as has been assumed here, or if it plays an essential role in maintaining that slip. As an example of the latter, if the tremor sources consist of small velocity-weakening patches embedded in a velocity-strengthening background, the secondary event might persist only because the creep fronts propagating from tremor sources into the velocity-strengthening surroundings continually trigger new tremor events in some sort of cascade [Ariyoshi et al., 2009]. Even if this is the case, however, to generate a detectable “event” the average stress over the slipping region should decrease with increased slip speed, or at least increase by less than any increase in applied loads; in the latter case the fault could conceivably be “on average” velocity-strengthening. Tidal loads could facilitate this; at least during the January 2006 event in western Shikoku, the rapid updip and downdip propagation occurred only during a consistent portion of the 12.4 h tidal cycle [Shelly et al., 2007a]. Note that the average stress drops associated with the secondary events in Figure 8 are only ∼4 kPa, whereas that associated with the main event is over 40 kPa. Given typical tidal stresses of a few kPa, it is plausible that the tides could strongly influence the secondary events without influencing the main event.
5.2. Generating Bimodal Propagation Speeds
The governing equations adopted here were designed to ascribe most of the observed difference in propagation speeds to smaller values of Δτp−r at the secondary fronts. That the maximum slip speed in these simulations was ten times larger at the secondary fronts than at the main front also raises the question of whether the observed difference in propagation speeds could be ascribed entirely to differences in Vmax, with negligible contribution from Δτp−r. Even if so, provided the stress drop associated with the secondary events is no larger than that of the main event, and provided it is correct to infer from the tremor distribution that the secondary events are much smaller than the main event, it seems that Gc associated with the secondary fronts must be significantly less (because from equation (3)G is proportional to size). Given the same Δτp−r, for example, a factor of 10 reduction in Gc could be achieved with a factor of 10 reduction in the critical slip distance δc. This also seems difficult to do with a single-state-variable formulation. If, alternatively, Δτ associated with the secondary events is much larger than that associated with the main event, Gc need not be less, but this seems to call into question the whole notion of a “main event” that has an identity independent of an amalgamation of events propagating rapidly parallel to slip.
5.3. Generating Slip-Parallel Propagation
In principal, secondary fronts can be guided parallel to the slip direction by the fault healing that comes with decreasing slip speed behind the main front. In the simulations here this behavior was accentuated by the choice of F. Using parameter values that diminish the role of dilatancy (a = 0.008; bθ = 0.01; bΨ = 0.03; ε/β = 0.6 MPa), I had no difficulty generating two generations of velocity excursions with appropriate slip and propagation speeds using F(Ψ) ≡ 1. However, the healing behind the main front was not rapid enough; the secondary fronts propagated almost as fast back in the direction the main front had come from as parallel to slip. A different choice of the steady state form of or evolution law for Ψ might have achieved slip-parallel propagation with F = 1, but after a few unsuccessful attempts I did not pursue this further.
One observation the model presented here might not explain gracefully is that in western Shikoku the rapid migration parallel to slip occurred at multiple locations up to ∼20 km behind the laterally propagating front [Shelly et al., 2007a]. While secondary fronts propagate parallel to slip at various distances from the main front in Animation S2, reminiscent of the observations from Cascadia (Figure 1), this behavior might be exaggerated by the strictly linear initial front in the simulations. Because the slow slip source region is much longer along strike than along dip, it is reasonable to expect that events larger than the downdip extent propagate mostly along strike. Still, on a homogeneous surface the main front could be expected to have some curvature, and some sort of anisotropy might be required to make the propagation of the secondary fronts strictly slip parallel. Perhaps the fault surface is more continuous in the slip direction than along strike, or has regions of higher hydraulic diffusivity that are elongate in the slip direction. In addition, if tremor is essential for maintaining the secondary events on an otherwise velocity-strengthening surface, the preference for slip-parallel propagation could come from an anisotropic distribution of tremor sources, with those regions being elongate parallel to slip, as has been observed in both Cascadia and Japan [Shelly et al., 2007a; Ide, 2010; Ghosh et al., 2010a].
The next few years are likely to see a steady stream of observations useful for constraining models of slow slip. In the context of the model described here, the fault healing that comes with the decrease in slip speed behind the main front is important for keeping the secondary fronts propagating more rapidly parallel to dip than back from where the main front came. Characterizing this velocity distribution, presumably by assuming it to be approximately proportional to the local tremor rate [Hiramatsu et al., 2008; Aguiar et al., 2009], would be very useful. Also useful would be to map more completely the propagation speed of both the main and the secondary fronts, including the rapid tremor reversals of Houston et al. , especially as a function of the inferred slip speed (or tremor rate) and phase within the tidal cycle. Placing bounds on the total number and moment of LFEs in the secondary events would also be useful for comparing to the area and moment of the secondary events in numerical simulations.
The sharpest images of slow slip currently available are provided by the variable migration patterns of tremor. The purpose of this paper has been to provide a framework for interpreting these patterns in terms of plausible frictional properties of the surface. The basic relation underlying this interpretation is quite simple and independent of any friction law: the ratio of the propagation speed to the maximum slip speed approximately equals the ratio of the elastic shear modulus to the peak-to-residual stress drop at the propagating front. What complicates implementing any numerical example is that the friction law must be capable of limiting the slip speed of the main front while still allowing the secondary fronts to develop. The law presented here does this by superimposing on a velocity-weakening surface a substantial weakening over a limited velocity interval between the plate convergence rate and slow slip speeds. To avoid dynamic instability this weakening is associated with dilatancy and pore pressure reduction. A plausible cause might be disruption of minerals precipitated between consecutive slow slip events.
The numerical simulations reproduce several aspects of episodic slow slip in Cascadia and Japan, including recurrence intervals of order 1 year, stress drops of tens of kPa, average slip speeds of order 10−7 m/s, a main front propagating at ∼5 km/d, and second-generation fronts that propagate orthogonal to this at ∼70 km/h. The factor of 300 larger propagation speed in this case comes from a factor of ∼10 larger Vmax and a factor of ∼30 smaller Δτp−r. Other combinations of Vmax/Δτp−r, even quite different, also seem possible. However, the small size of the secondary velocity excursions, only a few kilometers across as estimated from tremor locations, suggests that their fracture energy (basically the product of Δτp−r and a critical slip distance) must be much smaller than for the main front even if Δτp−r is not. This is also suggestive of a multiple-state-variable rate-and-state friction formulation.
It is noteworthy that in these simulations the stress drop associated with the secondary events is only a few kPa. This should make them susceptible to tidal loading, as was observed by Shelly et al. [2007a] in western Shikoku. Being restricted in their along-strike dimension and propagating parallel to dip automatically gives these events a quasi-constant moment rate, as has been inferred for events along the “slow earthquake” trend of Ide et al. [2007a]. The simulations also generate slip-parallel migration of secondary fronts at variable distances from the main front, as occurs in Cascadia.
From an aesthetic standpoint, it is unfortunate that generating secondary events that are elongate parallel to slip required introducing the ad hoc function F(Ψ) relating decreases in the state variable Ψ to porosity increases. While one could argue that the adopted function is just as likely as F(Ψ) = 1, being forced to adopt it leads one to suspect that the class of laws generating this behavior is in some sense smaller than if F = 1 had sufficed. In practical terms, what this means is that relative to the adopted friction law with F = 1, the operating friction law should have a resistance to velocity increases that increases more rapidly with distance behind the main front, while retaining sufficiently low resistance near that front to permit secondary events to arise. A reasonable alternative is that anisotropy of the fault surface that derives from its slip history plays a supporting role [e.g., Ando et al., 2010].
A different view of the secondary velocity excursions, not explored here, is that they depend critically upon the interaction between asperities on a heterogeneous fault [e.g., Ariyoshi et al., 2009]. This explanation has the appeal that tremor appears to demand that such heterogeneities exist, and perhaps their interaction could lead to the observed behavior under a simpler friction law. Ultimately any proposed constitutive law needs to be tested against a fuller range of observations, including tidal triggering, the intermediate-velocity “rapid tremor reversals” of Houston et al.  and more detailed images of the distribution of tremor locations and amplitudes behind the main front.
Appendix A:: Derivations for Dtp-rand Gc
At the main front, integrating the state evolution law for Ψ (equation (7a) or (7b)) through a step increase from a background state Ψbg to a slip speed Vmax at time t = 0,
For Vmax ≫ VΨ, Ψ evolves over the slip distance DΨ. Neglecting changes in p, inserting (A1) into (5) leads to
Inserting the bΨΨ term from (5) into (4) and integrating (whether to δc = ∞, as is done here, or to just a few times DΨ makes little difference),
With VΨ/Vmax ≪ 1, defining ΔΨ ≡ Ψbg− Ψss leads to equations (10) and (11) in the main text.
A similar strategy can be used to determine the evolution of p. Differentiating (A1) with respect to time, inserting the result into (9) and then (8) and integrating under the assumptions that F(Ψ) = 1 and that p = p0 at t = 0, the pore pressure reduction Δp(t) = p(t) − p0 is
This is equation (49) of Segall et al.  with Vmax + VΨ substituting for V and ΔΨ substituting for ln(Vmaxθbg/Dθ), where θbg is the background value of the state variable θ (the same correspondence exists between equation (A7) below and equation (57) of Segall et al.). Differentiating (A4) with respect to time, the maximum pore pressure reduction occurs at a normalized slip Vmaxt of
and has the value
Using (A4) to evaluate p and inserting only the f′p term from (5) into (4) and integrating, the effective fracture energy from the pore pressure reduction (again neglecting the small changes in the rate-and-state terms) is
For the secondary fronts, integrating (6) through a step increase to a velocity Vmax and substituting the result into (5) leads to
In the limit Vmax ≪ Vθ, substituting (A8) into (4) and integrating leads to
[Ampuero and Rubin, 2008]. I have not found a comparable closed-form solution for the limit Vmax ≫ Vθ, but GcΔθ in this case will be smaller than that given by (A9).
Appendix B:: Choosing Parameter Values
Building on previous models I set 0 = 1 MPa. To minimize Δτp−r at the secondary fronts relative to that at the main front I take a and b to be near the low end of lab values, 0.004 and 0.005. In retrospect, because of the difference between Vmax at the main and secondary fronts this was probably unnecessary. If bΨ is large enough that most of the stress drop for the main event comes from bΨΔΨ, then from (17) Δτ ≈ bΨ0; I choose bΨ = 0.05 so that Δτ ≈ 50 kPa. I take VΨ = 5 × 10−9 m/s, 5 times the plate rate. The cutoff velocity Vθ is 4 × 10−5 m/s, 40 times the target Vmax at the lateral front but far below elastodynamic speeds (roughly 10−3 m/s for these parameters).
Adopting values of 10−4 for the dilatancy parameter ε [Marone et al., 1990; Samuelson et al., 2009] and 0.5 × 10−4 MPa−1 for the compressibility β [Segall et al., 2010] leads to ε/β = 2 MPa. The total porosity increase implied by this ε is quite small; only 0.01% for a drop in Ψ from 1 to 0. From (13) this ε/β implies Δτp−r ≈ 1.2 MPa, nearly an order of magnitude larger than the estimated target of 150 kPa. However, for reasons noted in section 4.1equation (13) overestimates Δτp−r by a factor of nearly 4 in these simulations.
There are few constraints on tf. Generating significant pore suctions at the lateral fronts requires Vmaxtf/DΨ ≫ 1. I choose tf = 103 s (much shorter than the 12.4 h tidal cycle) and DΨ = 100 μm (near the upper end of the lab range for dθ), so that for Vmax = 10−6 m/s, Vmaxtf/DΨ = 10.
For the above parameters and μ′ = 25 GPa, equation (18) suggests that Vmax ∼ 10−6 m/s for W = 20 km, which is of the proper order of magnitude. An estimate of the slip per event from (1) is 4 cm. Interevent slip speeds are low enough that negligible slip accumulates outside the slow events, so for a plate convergence rate of 10−9 m/s the average recurrence interval would be somewhat more than 1 year (but note that the simulations are not strictly periodic). Adding a component of ductile creep to the fault zone in parallel with rate-and-state friction would be expected to increase the recurrence interval without much affecting the slip events themselves, and I neglect it here.
Given the large number of parameters in this model, it is reasonable to ask how finely tuned the results are. To give a semiquantitative sense of this I summarize below the dimensional estimates, obtained from equations (1), (2), (13), and (17)–(19), of the five observables listed in the first paragraph of section 3.3:
where the subscripts (1) and (2) refer to the main and secondary events and for the latter it was assumed that ∼ 0. To the right of each equation is a qualitative assessment of how closely the dimensional estimate matched the 1-D and 2-D simulations. Least accurate was the estimate of Vprop(2), because Vθ is not a robust estimate of Vmax(2) (it is too low for the 1-D simulations and too high for the 2-D simulations). The estimate of Vmax(1) was reasonably accurate despite the fact that f′(ε/β) overestimates Δτp−r by a factor of 3–4, in retrospect because W overestimates the size of the region of significant stress drop.
Equations (B1)–(B5) show that the five target quantities depend upon eight parameters. Of these, four (μ′, W, f′, and Vpl) are reasonably well known from observations or experiment (tens of GPa; tens of km; ∼0.6; 10−9 m/s), so with those constraints there are five equations with effectively four free parameters. However, (B2) is simply the statement that the recurrence interval is of the order of the slip divided by the plate velocity, which is presumably trivially met by many models, so absent this equation there are four equations and four free parameters. For two of those (tf and bθ0) order of magnitude variations lead to order of magnitude changes in the estimated observables, whereas for the other two (bΨ0 and ε/β), order of magnitude variations in some cases lead to 2 order of magnitude changes. Compared to the many orders of magnitude variation in slip speed for a factor of 2 change in fault length for the “tuning the fault length” mechanism for episodic slow slip, this sensitivity is not so extreme. In addition to these four parameters, the derivation behind some of these equations assumed
involving three more parameters (VΨ, DΨ, and a0) that the estimated observables seem only weakly sensitive to. Perhaps the greatest “tuning” involved the introduction of the function F, to reproduce an observation not listed in equations (B1)–(B5) that the secondary excursions propagate much more rapidly parallel to slip than along strike.
Finally, a constraint on Dθ and the ratio a/bθ arises for numerical convenience. To minimize the number of grid points needed for the 2-D fault calculations it is useful to minimize the ratio of the size of the secondary nucleation zones to the grid spacing. These nucleation zones are somewhat larger than the minimum size for spontaneous acceleration h*, given dimensionally by
[Rice, 1993]. Because dilatancy smears out the laterally propagating front somewhat, the smallest features to be resolved are the secondary fronts, of size ∼Lb/ln(Vmax(2)/Vbg(2)), where Lb ≡ μ′Dθ/(bθ) [Ampuero and Rubin, 2008]. Minimizing the ratio h*/Lb thus means minimizing (1 − a/bθ)−1. Here I take a/bθ = 0.8, perhaps not as close to 1 (velocity neutral) as one might expect for the friction transition zone. However, since this value is well within the range for which slip law nucleation zones expand, this compromise should not affect the behavior of the secondary fronts qualitatively. As ln(Vmax(2)/Vbg(2)) is not too large for the secondary fronts, Lb/Δx = Lb/Δy = 20 is adequate to resolve them, where Δx and Δy are the grid spacing in the x and y directions (5 m). Together with the constraint that the width of the secondary events be much smaller than the 2-D model domain (W′), these considerations allow one to write
where δ is slip and cs is the shear wave speed. The first term on the right represents elastic interactions from nonuniform slip, and the second the “radiation damping” approximation to elastodynamics [Rice, 1993]. Slip speeds are generally low enough in these simulations that the radiation damping term is negligible, but it does limit the slip speed within the initial perturbation in the 2-D fault simulations.
For 2-D faults the stresses are given by
where δy is slip in the y (dip) direction, λ is the Lame parameter, and r is the distance from (x, y) to (x′, y′) [Ripperger and Mai, 2004]. For computational efficiency I consider slip to be in the dip direction only. If there is no variation with y, then the elastic interaction contributions to τy(x) in (C2) and τ in (C1) are identical if μ′ in (C1) and μ in (C2) are (as is the case in Table II). 2-D simulations that were as long in the x direction as the 1-D models (but only a few grid points wide for fast computation) proceeded exactly as the 1-D simulations.
Equations (5)–(9) and either (C1) or (C2) are solved using the adaptive time stepping Bulirsch-Stoer routine of Press et al. . A spectral method is used for the elastic stress interactions, which implies periodic replicas of the accelerating region outside the numerical domain. For the 1-D faults these periodic replicas are far enough apart that their influence is negligible. For the 2-D faults they impart stressing rate increases comparable to tidal loads.
I am grateful to Abhijit Ghosh for sharing his catalog of tremor locations for the May 2008 Cascadia slow slip event and for his permission/blessing to publish Figure 1. This paper benefited from discussions with Jean-Paul Ampuero, Abhijit Ghosh, and Jessica Hawthorne. Norm Sleep and Associate Editor Thorsten Becker provided helpful reviews. This research was funded by NSF grant EAR-0911378.