Slow slip events are regular earthquakes


 Slow slip events usually occur downdip of seismogenic zones in subduction megathrusts and crustal faults, with rupture speeds much slower than earthquakes. The empirical moment-duration scaling relation can help constrain the physical mechanism of slow slip events, yet it is still debated whether this scaling is linear or cubic and a fundamental model unifying slow slip events and earthquakes is still lacking. Here I present numerical simulations that show that slow slip events are regular earthquakes with negligible dynamic-wave effects. A continuum of rupture speeds, from arbitrarily-slow speeds up to the S-wave speed, is primarily controlled by the stress drop and a transition slip rate above which the fault friction transitions from rate-weakening behaviour to rate-strengthening behaviour. This continuum includes tsunami earthquakes, whose rupture speeds are about one-third of the S-wave speed. These numerical simulation results are predicted by the three-dimensional theory of dynamic fracture mechanics of elongated ruptures. This fundamental model unifies slow slip events and earthquakes, reconciles the observed moment-duration scaling relations, and opens new avenues for understanding earthquakes through investigations of the kinematics and dynamics of frequently occurring slow slip events.

Steady rupture propagation with a continuum of speeds can be understood and quantitatively predicted by the 3D theory of dynamic fracture mechanics of elongated ruptures (Figure 1b). A basic condition for steady ruptures is the energy balance condition, G c = G 0 , where G c is the dissipated fracture energy and G 0 is the energy release rate of subshear dip-slip ruptures 41 . In addition, a stability condition is necessary (Methods A5) where the peak slip rate V p is interchangeable with the rupture speed v r due to their monotonically  To explore a universal scaling relation in the global dataset that is consistent with fracture me-153 chanics theory, I calculate the rupture speed and peak slip rate of the SSEs and tsunami earthquakes 154 8 observed globally 36, 44-46, 59 . The rupture speed is estimated by v r = L/T , with an uncertainty of rates in numerical simulations ( Figure S4). For pulse ruptures on elongated faults, the rise time is 158 approximately estimated by τ rise = T W/L. In general, there is an increasing trend between the 159 observed rupture speed and peak slip rate, enveloped by two theoretical predictions assuming con-  The frictional strength, τ , of faults is controlled by a rate-and-state friction law with rateweakening behaviour at low slip rates and rate-strengthening behaviour at high slip rates 72 , which has been used to investigate the rupture propagation of SSEs 15-17, 73 where σ is the effective normal stress, f * and V * are arbitrary reference values, D c is the characteristic slip distance, a and b are nondimensional parameters, V is the slip rate, θ is the state, and V c is a critical slip rate. Rock exhibits rate-weakening frictional behaviour when a − b < 0, and the critical slip rate V c controls the transition from rate-weakening to rate-strengthening 15 . The evolution of state θ is described by the aging law 74 whereθ is the time derivative of θ.
For each single-rupture model, one of the primary parameters that affects the rupture propagation is the initial shear stress τ i , which equals to the frictional strength and is prescribed by the values of initial slip rate V i and state θ i The nondimensional parameters, a/b and W/L b , also affect the rupture propagation 15 , where In this study, I fix the nondimensional ratios of a/b = 0.8 and W/L b = 400, and systematically where λ is a geometrical factor, with λ = 1/π for a deep buried fault 41 . The fracture energy G c depends on the strength evolution on the fault 79 : where where τ i are τ f are the initial shear stress and final shear stress, respectively. Previous rupture simulations of V-shape rate-and-state friction 15 Combining equations 4 and 8 yields the close-form static stress drop Equation 10 well predicts the numerical values of ∆τ in all the simulated steady models ( Figure   S2c). Substituting equation 10 into equation 6 yields the theoretical energy release rate The main feature in equation 11 is that G 0 only depends on the prescribed parameters and is 249 independent of the peak slip rate V p . As only τ i and V c are systematically investigated in this study,

251
G c is an integral function of the fault strength τ (δ) about the slip δ. The numerical simulations show that fault strength governed by V-shape rate-and-state friction has two weakening stages: the first stage accounts for the fast weakening process within the narrow cohesive zone and the second stage accounts for the slow weakening process outside the cohesive zone ( Figure S6).
In the first weakening stage, the strength drop ∆τ p−r and the critical slip-weakening distance d c can be well predicted by previous theoretical equations 15 where V p is the peak slip rate and the factor 3 is an approximation of the non-uniform slip rate within the cohesive zone, which was proposed to be 2 by Hawthorne and Rubin 15 . Thus, the fracture energy within the cohesive zone is estimated as The contribution of fracture energy of the second weakening stage has not explicitly been considered before. Here, I account for this part of the total fracture energy by where D is the final slip, τ r − τ f is the overshooting stress, and τ r is the fault strength at the tail of the cohesive zone For ruptures on long faults with finite width W , the final slip D is linearly proportional to the static stress drop ∆τ , that is 14 Substituting equations 10, 17, 12, and 16 into equation 14 yields the close-form function of the 252 second part of the fracture energy, G c2 . Therefore, the close-form function of total fracture energy 253 17 is given by G c = G c1 + G c2 . As G c depends on τ i , V c , and the undetermined peak slip rate V p , it 254 can be written as G c (V p , τ i , V c ).

255
For steady ruptures, the energy release rate shall be balanced by the dissipated fracture energy: Equation 18 shows that the peak slip rate V p of steady ruptures can be uniquely determined from

258
A4. Relation between peak slip rate and rupture speed A linear relation between peak slip rate and rupture speed for steady SSEs has been proposed by Hawthorne and Rubin 15 where C ≈ 0.5 − 0.55 is an empirical geometrical factor. But this relation does not include the  Weng and Ampuero 41 demonstrated that if the cohesive zone size on elongated faults is much smaller than fault width, L c ≪ W , then the energy release rate has the following form: where A(v r ) = 1/α s , α s = 1 − (v r /v s ) 2 is the Lorentz contraction term and K tip is the stress intensity factor. By removing the strike-slip term 1 − ν and replacing A(v r ) by 1/α s in equation (18) where the correction of a factor of 2 is made to fit the numerical results. where G c and G 0 are the fracture energy and energy release rate, γ, A, and P are known coefficients, and α s = 1 − (v r /v s ) 2 is the Lorentz contraction factor. M (v r ) is nearly constant when v r /v s ≪ 1 and increases to infinity when v r /v s → 1. For a steady rupture, the accelerationv r is zero, thus G c = G 0 , which is the energy balance condition. In addition, the stability of steady ruptures also depends on the sign of dF (G c /G 0 )/dv r . If dF (G c /G 0 )/dv r > 0, a tiny positive/negative perturbation of v r acting on the steady rupture induces a further increase/decrease of v r . Therefore, dF (G c /G 0 )/dv r < 0 is another condition for steady ruptures. Combining this inequality equation Considering the monotonic relation between v r and V p (Method A4), equation 25 can also be written as 20 For runaway/steady ruptures, the energy balance condition of equation 27 is Assuming the fault is steady-state before rupture, that is V i θ i /D c = 1, equations 10 and 12 yield a lengthy expression of ∆τ p−r /bσ that depends on ∆τ /bσ, a/b, and V p /V c . Although the derivation of closed-form ∆τ is complex and lengthy, the dimensional analysis of equation 28 shows that ∆τ /bσ is a function of V p /V c , a/b, and W/L c . For an extreme condition, W/L c ≫ 1, equations 28 and 16 leads to a minimum stress drop Hawthorne and Rubin 15 noted that the minimum stress drop for steady ruptures can be approxi- ≈ 50V c µ/σ, where σ is the effective normal stress, V c is the critical slip rate, and µ is the shear modulus. Therefore, the moment and duration can be normalized         Figure S1.