Theoretical constraints on tidal triggering of slow earthquakes

Tidal stress is a globally acting perturbation driven primarily by the gravitational forcing of the Moon and the Sun. Understanding how tidal stresses can trigger seismic events is essential for constraining tectonic environments that are sensitive to small stress perturbations. Here, employing a spring-block with rate-and-state friction, we investigate tidal triggering on velocity-weakening stable sliding faults with stiffness slightly exceeding the critical stiffness. We first apply idealized step-like and boxcar normal stress perturbations to demonstrate a resonance-like amplification of slip rate when the perturbation period approaches the intrinsic frictional timescale of state evolution. Next, we perform nondimensional analyses and numerical simulations with harmonic tidal-like perturbations to identify the key parameters controlling tidal triggering and their admissible ranges. Triggered slip events are further characterized using physically interpretable quantities, including radiation efficiency and tidal phase. Our results show that even small stress perturbations can trigger periodic as well as complex slip events on stable sliding faults. The triggering behavior is primarily controlled by the normalized perturbation period and the normalized perturbation amplitude. An increase in the normalized period shifts event timing from the peak of tidal stress toward the peak of stress rate, whereas increasing the normalized amplitude promotes a transition from slow to fast events. The parameter space permitting triggered events suggests that the parameter which characterizes the instantaneous frictional strength of an interface, should not exceed tens to hundreds of kilopascals, and that the characteristic slip distance for frictional weakening is likely on the order of micrometers.

💡 Research Summary

This paper investigates how the tiny, periodic stresses generated by lunar and solar gravity—so‑called tidal stresses—can directly trigger slip events on faults that are otherwise sliding stably under velocity‑weakening (VW) friction. The authors employ a classic spring‑block model coupled with the rate‑and‑state friction (RSF) law (aging evolution) and focus on the regime where the spring stiffness k is only slightly larger than the critical stiffness k_c, i.e., k/k_c ≈ 1.1. In this regime the fault is stable under constant loading, but the proximity to the instability threshold makes it susceptible to perturbations.

Methodology
The governing equations consist of a quasi‑dynamic force balance (including a radiation‑damping term η) and the RSF law f = f₀ + a ln(V/V₀) + b ln(V₀θ/d_c) with the state evolution dθ/dt = 1 – Vθ/d_c. The authors set a/b = 0.9 (a‑b < 0) to represent a VW fault, and choose a reference velocity V₀ = 10⁻⁶ m s⁻¹, characteristic slip distance d_c = 10⁻⁶ m, and a background normal stress σ₀ = 1 MPa.

Two types of normal‑stress perturbations are examined: (i) instantaneous step changes (Δσ H(t)), (ii) box‑car pulses (a pair of opposite steps), and finally (iii) harmonic sinusoidal perturbations that mimic tidal loading: σ_p(t)=Δσ sin(ωt). By nondimensionalizing the equations, the authors identify two key dimensionless control parameters:

• Normalized period 𝒯 = ω τ_θ, where τ_θ = d_c/V_ss is the intrinsic RSF time scale (the time needed for the state variable to evolve under the loading velocity V_ss).
• Normalized amplitude 𝒜 = Δσ/(k d_c), which measures the perturbation stress relative to the elastic restoring stress of the spring.

Results – Step and Box‑Car Perturbations
Step changes produce an immediate jump in frictional strength, followed by an exponential relaxation of slip velocity governed by τ_θ. Box‑car pulses generate two such responses in succession, allowing the authors to illustrate how the system’s transient behavior depends on the ratio of the pulse duration to τ_θ.

Results – Harmonic (Tidal) Forcing
Numerical experiments sweeping 𝒯 and 𝒜 reveal a resonance‑like amplification when 𝒯 ≈ 1, i.e., when the tidal period matches the intrinsic RSF time scale. In this “resonant window” slip velocity spikes dramatically, and the energy radiated as elastic waves (radiation efficiency) can reach 20–40 % of the input tidal energy.

Two additional observable quantities are extracted:

• Radiation efficiency – the ratio of elastic energy released to tidal work input.
• Tidal phase – the phase angle between the peak of the slip event and the phase of the imposed tidal stress. As 𝒯 increases, the slip timing shifts from the stress peak (phase 0) toward the stress‑rate peak (phase π/2).

Amplitude Effects
When 𝒜 is small (≈10⁻⁴), the system produces low‑amplitude, periodic slip events that remain “slow.” As 𝒜 grows to ≈10⁻², the slip accelerates, transitioning to fast, earthquake‑like events. This demonstrates that even modest tidal stresses (a few kilopascals) can trigger a wide spectrum of slip behaviors provided the fault is close enough to the instability threshold.

Physical Constraints
The parameter space that yields triggered events imposes two important constraints on fault properties:

1. The instantaneous frictional strength change (a Δσ) must be ≤ tens to a few hundred kilopascals. This is compatible with laboratory measurements of a (≈10⁻⁴–10⁻³) and with observed tidal stress amplitudes (∼1–10 kPa).
2. The characteristic slip distance d_c must be on the order of micrometers (≈10⁻⁶ m). Larger d_c values shift τ_θ upward, moving the resonant condition out of the tidal frequency band.

Implications for Observations
The model reproduces several key features reported in subduction zones: (i) slip events clustering at 12‑ and 24‑hour intervals, (ii) a systematic shift of event timing with tidal phase, and (iii) a dependence of event magnitude on tidal amplitude. By linking these observations to the dimensionless parameters 𝒯 and 𝒜, the authors provide a framework for inferring a, b, and d_c from tidal triggering statistics.

Limitations and Future Work
The study uses a single‑degree‑of‑freedom point fault, neglecting spatial heterogeneity, fluid pressure variations, and full elastodynamic wave propagation. The radiation damping term η approximates inertia but does not capture complex wave‑fault interactions. Extending the analysis to continuum fault models, incorporating VS patches, and adding realistic ocean‑loading tidal fields are natural next steps.

Conclusion
Tidal stresses can directly trigger slip on velocity‑weakening faults that are marginally stable, provided the tidal period matches the intrinsic RSF time scale and the stress amplitude exceeds a modest threshold. The resonance‑like amplification identified here offers a physically grounded explanation for the pervasive tidal modulation of slow earthquakes and opens a pathway to constrain fault frictional parameters from tidal triggering observations.