| 24 Jul 2026
Delayed State Evolution in Rate-and-State Earthquake Nucleation: Amplification, Localization, and Scaling
Abstract. Rate-and-state friction assumes instantaneous adaptation of state to slip velocity, but the consequences of this temporal assumption for earthquake nucleation remain poorly constrained. We introduce a constitutive response time tc into ageing-law feedback and isolate its effect in a spatially distributed fault model. Eight simulations use the same Master configuration and differ only in tc = 0–0.12 s. Peak velocity increases by approximately 87 %, from 7.38 × 10⁻³ to 1.38 × 10⁻² m s⁻¹, while finite delays permit local crossing of 10⁻² m s⁻¹. The response separates into a moderate-velocity footprint that remains approximately 0.918 km wide and a delay-sensitive high-velocity core that expands to approximately 0.449 km. The response remains localized, separating constitutive amplification from subsequent large-scale propagation. To our knowledge, this is the first spatially distributed demonstration that explicit finite response time in ageing-law feedback can create a high-velocity inner core within a localized nucleation region. The diagnostic Πd = tc Vmax/Dc relates delay to state-evolution time. The results establish temporal state adaptation as an independent constitutive dimension and provide benchmarks for laboratory calibration and fully dynamic modelling.
General comments
This manuscript examines whet her a finite delay in state evolution can change the way earthquake nucleation develops in space and time. Moving from the authors previous block-based mnodels to a spatially distributed fault model is potentially useful. BUt, I do not think the current results are sufficient to support the main claims about a new constitutive mechanism, a distinct high-velocity nucleation regime, or a general scaling relationship. At present, the study mainly shows that an imposed delay amplifies a strongly forced response in one particular numerical configuration. According to me, several important physical and mathematical issues therefore need to be addressed before the novelty can be properly evaluated. I recommend major revisions.
Specific comments
The Meaning of the proposed delay needs to be explained much more clearly. In Equation 4, both velocity and state are taken from an earlier time, while healing remains instantaneous and the current state is still used to calculate friction. This is one possible mathematical choice, but it is not clear why it is the appropriate description of contact renewal or gouge reorganization.... Classical rate-and-state friction already includes gradual state adjustment through the characteristic slip distance. I think the authors therefore need to explan what additional physical process the new response time represents, why it should be a fixed time rather than a slip-dependent quantity, and whether other reasonable descriptions of delayed adjustment would produce the same behaviour. Without this foundation, the dselay currently appears more phenomenological than constitutive.
The spatial model also needs a stronger physical basis. The effective thickness of the model is set equal to the grid spacing, meaning that the physical mass, acceleration, and damping change with the numerical discretization. This is particularly important because the reported core widths correspond to relatively small numbers of grid cells. The authors should either assign the thickness an independently defined physical value or clearly derive the formulation as a discrete block-chain model. There is also an inconsistency between the stated Mode II rupture configuration and the scalar wave equation used in the manuscript. The elastic formulation cud be derived carefully, including an explanation of why the shear-wave operator and radiation damping are both required.
I am not convinced that the high-velocity core represents a new physiccal regime. The instajntaneous model already reaches a velocity close to the selected high-velocity threshold, and the smallest delay moves it only slightly above that threshold. The apparent transition may therefore reflect a smooth increase passing through a chosen diagnostic value rather than a genuine change in fault dynamics. The authors should use a physically based instability measure, such as acceleration rate, energy release, or a change in the stability of the governing system. I recommend that the authors also clarify whether the reported peak velocities are actual maxima or simply the values reached when the simulations end. If the fault is still accelerating at that point, the reported amplification will depend on the selected twelve-second analysis window.
The relationship between the imposed perturbation and the inferred nucleation scales needs to be separated more carefully. The simulations begin with a prescribed two-kilometre stress perturbation, and the broader accelerated region remains largely confined within this imposed footprint. This makes it difficult to determine how much of the spatial structure is generated by delayed friction and how much is inherited from the initial forcing. The authors should calculate the expected nucleation length for the adopted friction,al and elastic properties and compare it with the imposed perturbation size. It wud also be useful to compare the delayed and instantaneous cases at equivalent stages of evolution, such as the same central velocity or accumulated slip, rather than only at the same time. the sharper delayed profiles could partly reflect comparisons between different stages of the same accelerating process.
The proposed memory-strength scaling is not still independent or predictvie because it contains the maximum velocity, which is itself one of the main model outputs. Plotting ma ximum velocity against a quantity that already includes maximum velocity naturally produces a relationship between the axes. Since the characteristic slip distance and all other model properties remain fixed, the simulations also do not demonstrate a broader scaling law. I sugget the authors to develop a dimensionless measure using only prescribed input quantities and support it with a stability or timescale analysis. More generally, a linear stability analysis of the delayed system would greatly strengthen the manuscript by showing whether the delay changes the growth rate, creates a new unstable mode, or simply amplifies a transient response. This would provide a much firmer technical basis for the novelty than the current threshold-based classification , acc to me.
Technical corrections
Section 3.3 refers to Figures 5 and 6 when the discussion appears to concern Figures 3 and 4. Several sourc.es cited in the Supplement, including Bizzarri and colleagues, Guatteri and colleagues, and Wei and Chen, are missing from the main reference list. The displacement integral described as a moment proxy should also be reported with its units and should not be interpreted as seismic moment without a defined rupture area and shear modulus.