the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 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.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-4365', Anonymous Referee #1, 25 Aug 2026
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AC1: 'Reply on RC1', Srdjan Kostic, 27 Sep 2026
Response to Reviewer 1
We thank Reviewer 1 for the detailed technical review. We accept the recommendation for major revision and have addressed each point through additional numerical, analytical, and sensitivity analyses. The additional analyses make explicit the physical quantities used throughout the study: amplification is evaluated as an increase in near-critical modal and nonlinear growth tendency, localization is assessed relative to the intrinsic nucleation stability length and by equal-stage spatial profiles, and scaling is expressed through the input-only ratio χ = tcVs/Dc. We performed independent time-step and grid convergence tests, linear-stability analysis, reduced-layer-thickness sensitivity tests, perturbation-size tests, equal-stage comparisons, and alternative temporal-response tests. The corresponding changes in text are highlighted in yellow.
Reviewer comment: This manuscript examines whether 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.
Response: We accept this suggestion. To make the physical basis of the study explicit, we performed the requested convergence, stability, perturbation-scale, and equal-stage analyses. These analyses distinguish three complementary quantities. Amplification is quantified as the delay-induced increase in near-critical modal and nonlinear growth tendency, rather than by the crossing of a single velocity threshold. Localization is evaluated relative to the intrinsic critical wavelength and by profiles compared at matched center velocity and matched accumulated slip. Scaling is expressed through the prescribed-state ratio χ = tcVs/Dc. Temporal refinement also shows that finite-window peak velocity is not the sensitive diagnostic in the subcritical 2 km Master case: converged Vmax values for tc = 0 and 0.12 s differ by only about −0.01 to −0.02%. The main evidence is therefore based on characteristic roots, the critical wavelength Lc, post-transient logarithmic growth rates, perturbation-size sensitivity, and equal-stage spatial structure.
Reviewer comment: 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.
Response: We accept this suggestion. We have expanded the physical interpretation of tc and tested several alternative response architectures. The parameter tc is treated as a phenomenological coarse-grained response time for an unresolved internal reorganization process whose adjustment is controlled primarily by elapsed time; it is distinct from the slip-controlled renewal scale Dc and is not presented as a uniquely calibrated contact-scale constant. Equation (4) is written as retarded ageing-law feedback g[V(t−tc), θ(t−tc)]; because the healing contribution is the constant 1, delaying the arguments of g is mathematically equivalent to delaying the complete right-hand side. A causal exponential time relaxation, tc dm/dt = Vθ − m, produces essentially the same linear stability shift as the discrete delay: at tc = 0.12 s, Lc changes from 2.300323 km to 2.293780 km, compared with 2.293771 km for Eq. (4), and the nonlinear near-critical turnaround differs by only about 2 ms. A matched slip-controlled relaxation behaves differently and remains close to the classical trajectory. Velocity-only and state-only decompositions further show that the destabilizing contribution is associated primarily with retarded state adjustment rather than with a generic delay applied to any variable.
Reviewer comment: 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.
Response: We accept this suggestion. We have provided an explicit mechanical derivation and separated the reduced-layer thickness from the numerical grid. Equation (6) is derived as the continuum limit of a reduced 1-D scalar shear layer / nearest-neighbour block chain. For a cell of length Δx, m = ρHΔx and Ks = μH/Δx, giving utt = cs²uxx + [τload − τf − η(V − Vinit)]/(ρH). The formulation is described consistently as a reduced scalar shear-layer model. H is defined independently of Δx, with H = 20 m used as the reference reduced-layer thickness and H = 5-160 m explored in sensitivity tests. At fixed H, the N = 2048 solution differs from N = 4096 by about 0.00036% in Vmax and about 0.03% in the continuous RMS width. We also state explicitly that cs²uxx represents elastic redistribution along the modeled layer, whereas η(V − Vinit) represents local radiation loss to the unresolved surrounding medium.
Reviewer comment: 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.
Response: We accept this suggestion. We therefore evaluate the dynamics with physically based instability measures in addition to the descriptive velocity threshold. The 10−2 m s−1 level is treated only as a diagnostic velocity level, not as a stability boundary. In the 12 s finite-window runs, the reported values are maxima within that analysis window and the trajectories are still accelerating at its endpoint. In the converged 2 km Master calculation, the true peak occurs during the initial direct-effect transient at about 2.5-2.6 ms, with Vmax ≈ 4.744 × 10−4 m s−1, after which the center velocity decreases through 16 s. We therefore use d ln V/dt, characteristic-root growth rates, post-transient turnaround time, and a mechanical-energy balance as the primary dynamical diagnostics. The numerical energy balance closes to an RMS relative error of approximately 2.4 × 10−4. Near the intrinsic stability boundary, these measures show a genuine delay-dependent increase in growth tendency even though the subcritical Master-case Vmax is nearly unchanged after convergence.
Reviewer comment: 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.
Response: We accept this suggestion. We quantified the relationship between the imposed forcing scale and the intrinsic stability scale and added the requested equal-stage comparisons. For H = 20 m and the input-derived reference speed VΔ = 4.7463 × 10−4 m s−1, the instantaneous critical wavelength is Lc = 2.300323 km, so the 2 km Master perturbation is subcritical, corresponding to 0.869 Lc. At tc = 0.12 s, Lc decreases to 2.293771 km. A finite-amplitude perturbation-size sweep gives a grid-converged end-member shift of the 8-16 s crossover diameter of about −4.17 m, consistent in sign and order of magnitude with the −6.55 m linear-stability shift. For a near-critical 2.542 km perturbation, tc = 0.12 s advances the post-transient velocity turnaround by approximately 0.07 s. At equal accumulated center slip, the normalized spatial profiles nearly collapse; at equal center velocity, the differences are sub-percent. We therefore characterize localization through the relation between imposed perturbation size, intrinsic stability length, and matched-stage spatial structure, while common-time sharpening is treated as sensitive to evolutionary phase.
Reviewer comment: 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.
Response: We accept this suggestion. We use the input-only timescale ratio χ = tcVs/Dc for prescribed steady sliding and χΔ = tcVΔ/Dc for the Master forcing, where VΔ is calculated entirely from prescribed stress and friction inputs. The quantity Πd = tcVmax/Dc is retained only as a post-simulation descriptive coordinate and is not used as an independent predictor. We also performed the requested linear-stability analysis. Linearization gives [λ² + Aλ + cs²k²][λ + ω exp(−λtc)] − Bλω exp(−λtc) = 0. At VΔ and H = 20 m, tc = 0.12 s shifts Lc by −0.285%. A 2.270 km mode changes from Re(λ) = −3.14 × 10−5 s−1 at tc = 0 to +3.19 × 10−5 s−1 at tc = 0.12 s, with the stability crossing near tc = 0.0594 s. Thus finite response time can change the sign of a near-critical growth rate, while long already-unstable modes need not be amplified. The fractional stability-boundary shift grows systematically with χ, providing the requested input-based scaling and timescale interpretation.
Reviewer comment: 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.
Response: We accept this suggestion. We corrected the Section 3.3 figure cross-references and reconciled the Supplement citations with the main reference list, including the cited works by Bizzarri and colleagues, Guatteri and colleagues, and Wei and Chen. We also report ∫u dx explicitly as an integrated-displacement proxy with units of m². It is not interpreted as seismic moment because the reduced model does not prescribe the physical rupture area and complete geometric information required for that conversion.
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AC1: 'Reply on RC1', Srdjan Kostic, 27 Sep 2026
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RC2: 'Comment on egusphere-2026-4365', Anonymous Referee #2, 21 Sep 2026
General comments
This manuscript proposes a modified state-variable evolution model for the classic rate-and-state friction (RSF) aging law that accounts for a delayed state response. The study shows that incorporating a delay time tc into the state variable and slip rate in the classic aging law modifies both the temporal and spatial evolution of slip during the nucleation stage, including the maximum slip rate and the emergence and extent of a high-velocity core. The authors further introduce a new dimensionless parameter, Πd, that relates the delay parameter tc to the spatiotemporal evolution of the high-velocity nucleation core. The manuscript has novelty and potential contribution to the field by proposing a simple constitutive model that captures a delayed update of the state variable. However, several important concerns need to be addressed before publication.
My main concern is that the physical, observational, and theoretical basis for the motivation and interpretation of the study is not yet sufficiently developed. In particular, it is not clear why a finite constitutive response time needs to be introduced, or how this modification helps reproduce important observations or theoretical predictions. The authors state that the instantaneous update of the state variable is “convenient and often appropriate, yet it is not equivalent to demonstrating that the physical processes represented by the state are instantaneous”, while an in-depth review of previous work discussing when this approximation is expected to be appropriate is missing.
A related issue is the emphasis on the development of a high-velocity core within the nucleation zone. Given the lack of target observations or theoretical predictions supporting the presence or importance of such a core, it is not yet clear why this behavior should be considered a central feature that makes the proposed formulation distinctively important.
In addition, the current Discussion section reads more like an extension of the Introduction, with continued emphasis on justifying the proposed model rather than discussing its physical implications and possible links to observations. For example, the physical meaning of tc remains unclear. Because the classic RSF formulation is largely empirical, I am not sure how introducing an additional parameter without a clear physical meaning, and without support from a specific microphysical framework, would improve the physical relevance of the friction law. Delay parameters or thermal dependences, as introduced by Ikari et al. (2016), Ende et al. (2018), and Barbot (2022), aim to address this type of problem. A more focused discussion of what tc represents physically, and how it may be constrained or tested, would substantially strengthen the manuscript.
The authors outline several useful follow-up directions in the Conclusions section (L423-L443). However, in my view, most of these points should be addressed in the present study rather than deferred to future work. In particular, additional parameter exploration seems essential. For example, the a/b value used in this study appears to be ~0.67. As highlighted by Rubin and Ampuero (2005), a/b values can affect the style of nucleation. Testing different shapes of the imposed stress perturbation would also help clarify the robustness of the reported behavior.
Another concern is the limited set of references. The subjects of RSF and earthquake nucleation have been extensively studied over the past few decades, yet the manuscript cites fewer than 40 references, with substantial repetition in the Introduction and Discussion sections. Expanding the reference list would better acknowledge existing work and would help place the proposed model in a clearer scientific context. It may also help strengthen the motivation and interpretation of the study.
Therefore, I suggest major revision, or possibly reject-and-resubmit, to allow the authors enough time to substantially revise the manuscript.
Specific comments
- The usage of the term “reduced-order” needs caution. It could be associated with surrogate models, which is not the case here. I suggest being more specific about what the authors intend to convey when using this term.
- L273 reads as if this is an observational feature in current seismic/geodetic instrumentation. I suggest rephrasing this sentence.
- What is the rationale for choosing the simulation time T = 12 s? In Figure 4, V seems to keep increasing over time. This makes me wonder whether, if the simulation were run for longer than 12 s, the internal core in the instantaneous response model (tc = 0) might also grow above the dynamic threshold.
- I am also not sure whether “simple” is a critical virtue for proposing a new state evolution law. Numerically, the aging law is already a well-adopted framework for explaining existing RSF experiment results and is reasonably efficient computationally.
- I do not understand L117-118, where the authors state that the model accounts for “inertia” while using radiation damping. Radiation damping approximates inertial effects while ignoring the actual term in the elastodynamic equation for computational efficiency. This part needs clarification or revision.
- Is there an explanation for the sharp transition of the slip rate from the background to the accelerated nucleating zone at x = +/- 0.5 km, as shown in Figures 3 and 5? Since the stress perturbation is imposed over x = +/- 1 km, I would expect a somewhat smooth connection from the background slip rate to the accelerated slip rate within the x = +/- 0.5-1 km range, similar to, for example, Figure 1 in Rubin and Ampuero (2005).
- How did the authors determine the time step size? I see that the authors tested two values, but the manuscript does not explain how they chose them.
- I am concerned about the forced nucleation setup. The manuscript should more clearly explain how the imposed stress perturbation affects the resulting nucleation process, and whether the reported high-velocity core is controlled by the proposed delay formulation or by details of the imposed perturbation.
Technical corrections
- A number of terms are not defined before being used. A non-exhaustive list of such examples is:
- The concept of “velocity-weakening” in L29 and “velocity-strengthening” in L127 is not yet introduced.
- The definition of tc in L79 is not yet introduced in the main text. It is introduced in the abstract, but not in the main text.
- The definition of Ldyn and its reference should be added for Figures 6 and 7.
- Figure 1 is not properly rendered. The colorbar is blocking the main figure. Information about contour colors should be added to the figure caption, not in the master configuration box embedded in the figure.
- In all figures, please remove the master set configuration. This makes the figures look busy and is unnecessary since the model setup does not vary across the study. Referring to Table S1 in the caption should be enough.
- L112-113 is hard to understand. Adding relevant equations and/or illustrations would help.
- Figure 5 would be more informative if it focused only on the nucleation zone (x = +/- 0.5 km), with associated colorbar limits. The current color range from yellow to bright green makes it hard to see the velocity pattern within the nucleation zone. The caption also needs to explain what the red contour indicates.
- Figure 7 needs a clearer caption explaining why Ldyn/Lnuc is shown as a shaded area rather than a curve.
- L423-L443 does not belong in the Conclusions section. It would be more suitable for the Discussion section.
Citation: https://doi.org/10.5194/egusphere-2026-4365-RC2 -
AC2: 'Reply on RC2', Srdjan Kostic, 27 Sep 2026
Response to Reviewer 2
We thank Reviewer 2 for the detailed and constructive review. The comments motivated additional physical framing, frictional-parameter and forcing-shape sensitivity tests, an explicit spatial-transition diagnostic, a reorganized Discussion, and a cleaned figure set. Corresponding changes in the manuscript are highlighted un blue.
Reviewer comment: This manuscript proposes a modified state-variable evolution model for the classic rate-and-state friction (RSF) aging law that accounts for a delayed state response. The study shows that incorporating a delay time tc into the state variable and slip rate in the classic aging law modifies both the temporal and spatial evolution of slip during the nucleation stage, including the maximum slip rate and the emergence and extent of a high-velocity core. The authors further introduce a new dimensionless parameter, Πd, that relates the delay parameter tc to the spatiotemporal evolution of the high-velocity nucleation core. The manuscript has novelty and potential contribution to the field by proposing a simple constitutive model that captures a delayed update of the state variable. However, several important concerns need to be addressed before publication.
Response: We accept this suggestion. We have substantially expanded the physical, theoretical, numerical, and robustness basis of the study. The revised analysis now combines time-step and grid convergence, an explicit linear-stability calculation, near-critical finite-amplitude tests, equal-stage comparisons, alternative temporal architectures, a/b sensitivity, and stress-perturbation-shape sensitivity. The central dynamical quantities are the critical nucleation/stability length, modal and nonlinear growth tendency, and response timing; threshold-defined high-velocity contours are retained only as descriptive diagnostics.
Reviewer comment: My main concern is that the physical, observational, and theoretical basis for the motivation and interpretation of the study is not yet sufficiently developed. In particular, it is not clear why a finite constitutive response time needs to be introduced, or how this modification helps reproduce important observations or theoretical predictions. The authors state that the instantaneous update of the state variable is “convenient and often appropriate, yet it is not equivalent to demonstrating that the physical processes represented by the state are instantaneous”, while an in-depth review of previous work discussing when this approximation is expected to be appropriate is missing.
Response: We accept this suggestion. We now describe the conventional ageing law as a local-in-time (Markovian) closure: the state-evolution rate at time t is evaluated from the current slip velocity and state, although the state variable itself evolves over a finite time and slip history. The revised Introduction places this assumption in the broader RSF literature, including contact-age/area, gouge-structure, thermally activated, and microphysical interpretations of state. We define tc narrowly as a phenomenological coarse-grained response time of an unresolved internal reorganization process, distinct from Dc. The input ratio χ = tc Vs/Dc provides an explicit criterion: when χ ≪ 1 the local-in-time closure is the natural leading-order approximation. For the Master input-derived state, χΔ is approximately 5.7 × 10−3 at tc = 0.12 s, consistent with the small converged effect in that configuration.
Reviewer comment: A related issue is the emphasis on the development of a high-velocity core within the nucleation zone. Given the lack of target observations or theoretical predictions supporting the presence or importance of such a core, it is not yet clear why this behavior should be considered a central feature that makes the proposed formulation distinctively important.
Response: We accept this suggestion. We clarify that the high-velocity inner contour is not treated as an established seismic or geodetic observation that the model is intended to reproduce. The physically relevant targets are instead the intrinsic nucleation/stability length, characteristic-root growth rate, acceleration/turnaround timing, and the phase lag of state adjustment. The 10−2 m/s contour is retained only as a descriptive diagnostic in the Supplement. Equal-stage comparisons further show that the normalized spatial profiles nearly collapse at matched accumulated slip, so the revised localization analysis is based on the intrinsic stability scale and evolutionary stage rather than on a presumed observational core.
Reviewer comment: In addition, the current Discussion section reads more like an extension of the Introduction, with continued emphasis on justifying the proposed model rather than discussing its physical implications and possible links to observations. For example, the physical meaning of tc remains unclear. Because the classic RSF formulation is largely empirical, I am not sure how introducing an additional parameter without a clear physical meaning, and without support from a specific microphysical framework, would improve the physical relevance of the friction law. Delay parameters or thermal dependences, as introduced by Ikari et al. (2016), Ende et al. (2018), and Barbot (2022), aim to address this type of problem. A more focused discussion of what tc represents physically, and how it may be constrained or tested, would substantially strengthen the manuscript.
Response: We accept this suggestion. The Discussion has been reorganized into physical interpretation and observable targets, robustness to frictional properties and forcing geometry, and testability/limitations. We distinguish tc from Dc and discuss physical interpretations of state and time-dependent frictional strengthening using a broader literature set, including Nakatani (2001), Carpenter et al. (2016), Ikari et al. (2016), van den Ende et al. (2018), and Barbot (2022). We also test the physical architecture directly: a causal exponential time-relaxation model reproduces the discrete-delay stability shift and near-critical retiming, whereas a matched slip-controlled relaxation does not. This provides a concrete experimental target for large velocity-step, slide-hold-slide, or spatially resolved nucleation tests that distinguish elapsed-time from slip-controlled response.
Reviewer comment: The authors outline several useful follow-up directions in the Conclusions section (L423-L443). However, in my view, most of these points should be addressed in the present study rather than deferred to future work. In particular, additional parameter exploration seems essential. For example, the a/b value used in this study appears to be ≈0.67. As highlighted by Rubin and Ampuero (2005), a/b values can affect the style of nucleation. Testing different shapes of the imposed stress perturbation would also help clarify the robustness of the reported behavior.
Response: We accept this suggestion. We moved the forward-looking material from Conclusions into the Discussion and carried out two of the requested parameter studies in the present revision. First, a/b is varied from 0.40 to 0.90 while the input-derived response speed is held fixed. The classical critical wavelength changes from about 1.70 to 4.44 km, confirming the strong control of a/b on nucleation style, whereas the fractional tc = 0.12 s stability shift remains approximately −0.285 % throughout the sweep. Second, we compare raised-cosine, Gaussian, compact-quartic, and super-Gaussian stress perturbations normalized to the same peak and integrated stress increment. The absolute nonlinear crossover varies strongly with forcing shape, but the finite-response shift retains the same sign for all four profiles. These tests are summarized in the main text and documented in the Supplement.
Reviewer comment: Another concern is the limited set of references. The subjects of RSF and earthquake nucleation have been extensively studied over the past few decades, yet the manuscript cites fewer than 40 references, with substantial repetition in the Introduction and Discussion sections. Expanding the reference list would better acknowledge existing work and would help place the proposed model in a clearer scientific context. It may also help strengthen the motivation and interpretation of the study.
Response: We accept this suggestion. The reference list and literature framing have been expanded substantially, especially for physical interpretations of state, laboratory frictional healing, microphysical and temperature-dependent formulations, extensions of state evolution, and nucleation sensitivity to the state law. New references include Nakatani (2001), Baumberger and Caroli (2006), Linker and Dieterich (1992), Nagata et al. (2012), Bhattacharya et al. (2015), Carpenter et al. (2016), and the already cited microphysical studies are integrated more directly into the motivation and Discussion.
Reviewer comment: Therefore, I suggest major revision, or possibly reject-and-resubmit, to allow the authors enough time to substantially revise the manuscript.
Response: We accept this suggestion. The manuscript has been substantially revised and is accompanied by expanded Supplementary Material and reproducibility files. The revision adds the requested physical framing, parameter and forcing robustness tests, extended-time calculations, convergence tests, and a reorganized Discussion and figure set.
Specific comments
Reviewer comment: The usage of the term “reduced-order” needs caution. It could be associated with surrogate models, which is not the case here. I suggest being more specific about what the authors intend to convey when using this term.
Response: We accept this suggestion. We removed the term “reduced-order” where it could imply a surrogate or projection-based model. The mechanics are now described specifically as an effective one-dimensional scalar shear-layer model, or equivalently as the continuum limit of a nearest-neighbour block chain. H is described as an effective-layer thickness parameter.
Reviewer comment: L273 reads as if this is an observational feature in current seismic/geodetic instrumentation. I suggest rephrasing this sentence.
Response: We accept this suggestion. The relevant sentence has been rephrased so that no high-velocity inner core is presented as an established feature of current seismic or geodetic observations. The revised text identifies theoretical and experimentally testable targets instead: nucleation length, growth rate, acceleration timing, and state-response lag.
Reviewer comment: What is the rationale for choosing the simulation time T = 12 s? In Figure 4, V seems to keep increasing over time. This makes me wonder whether, if the simulation were run for longer than 12 s, the internal core in the instantaneous response model (tc = 0) might also grow above the dynamic threshold.
Response: We accept this suggestion. The 0-12 s interval is now described only as a common finite-window used for the original response-time sweep. It is not treated as a physical termination criterion. We added converged extended calculations through 16 s, using a 120 km domain where necessary to exclude periodic return. In the converged 2 km Master case the true velocity maximum occurs during the initial direct-effect transient, and the center velocity subsequently decreases through 16 s for both tc = 0 and 0.12 s. Thus the instantaneous case does not cross the 10−2 m/s descriptive threshold when the calculation is extended.
Reviewer comment: I am also not sure whether “simple” is a critical virtue for proposing a new state evolution law. Numerically, the aging law is already a well-adopted framework for explaining existing RSF experiment results and is reasonably efficient computationally.
Response: We accept this suggestion. Simplicity is no longer presented as a physical or numerical virtue of the proposed state evolution. The motivation is instead the isolation of a distinct time-controlled response hypothesis and the ability to test its stability consequences and its robustness to alternative causal temporal architectures.
Reviewer comment: I do not understand L117-118, where the authors state that the model accounts for “inertia” while using radiation damping. Radiation damping approximates inertial effects while ignoring the actual term in the elastodynamic equation for computational efficiency. This part needs clarification or revision.
Response: We accept this suggestion. The mechanical formulation has been rewritten and derived explicitly. Distributed inertia is retained through the acceleration term ρH utt in the scalar momentum balance. The separate η(V−Vinit) term is described as a local radiation-loss closure representing energy carried from the effective layer into unresolved surrounding material. We explicitly note that this is not the quasi-dynamic construction in which radiation damping substitutes for the inertial term, and that the model is not a full elastodynamic boundary formulation.
Reviewer comment: Is there an explanation for the sharp transition of the slip rate from the background to the accelerated nucleating zone at x = ± 0.5 km, as shown in Figures 3 and 5? Since the stress perturbation is imposed over x = ± 1 km, I would expect a somewhat smooth connection from the background slip rate to the accelerated slip rate within the x = ± 0.5-1 km range, similar to, for example, Figure 1 in Rubin and Ampuero (2005).
Response: We accept this suggestion. We performed a dedicated spatial-transition diagnostic. The feature near x = ± 0.5 km is not a frictional boundary: the raised-cosine stress perturbation extends smoothly to ± 1 km and has half its peak amplitude at ± 0.5 km. Because the RSF direct effect is logarithmic in velocity, this smooth stress taper maps into a multi-order-of-magnitude velocity contrast. In the converged calculation, V at 12 s decreases continuously from approximately 1.73 × 10−8 m/s at 0.5 km to 8.5 × 10−10 at 0.6 km, 5.7 × 10−11 at 0.7 km, and approaches the 10−12 m/s background only near 1 km. The tc = 0 and 0.12 s outer tails are essentially identical. We therefore clarify that the visually sharp transition is not a physical discontinuity or a delay-created boundary.
Reviewer comment: How did the authors determine the time step size? I see that the authors tested two values, but the manuscript does not explain how they chose them.
Response: We accept this suggestion. The Methods now state that the reported reference time step Δt = 1.25 × 10−4 s was selected from a temporal-refinement sequence rather than chosen a priori. It is checked at 6.25 × 10−5 s, and independent semi-implicit and deeply refined explicit implementations converge to the same solution. The principal near-critical timing and architecture results are also unchanged under time-step halving.
Reviewer comment: I am concerned about the forced nucleation setup. The manuscript should more clearly explain how the imposed stress perturbation affects the resulting nucleation process, and whether the reported high-velocity core is controlled by the proposed delay formulation or by details of the imposed perturbation.
Response: We accept this suggestion. The revised manuscript now separates the imposed forcing scale from the intrinsic stability scale. At H = 20 m the 2 km Master perturbation is only 0.869 Lc and is subcritical in the converged model. We added a perturbation-size sweep around Lc, equal-stage profile comparisons, and a new forcing-shape sensitivity test. The shape test shows that forcing geometry strongly controls the absolute nonlinear crossover, while the smaller tc-dependent shift retains the same sign across four matched shapes. We therefore state explicitly that the imposed perturbation is a first-order control on the absolute nucleation trajectory and that the robust constitutive effect is the smaller near-critical stability/timing shift.
Technical corrections
Reviewer comment: A number of terms are not defined before being used. A non-exhaustive list of such examples is:
Response: We accept this suggestion. Definitions have been moved forward and the revised manuscript and Supplement now define the requested quantities at first use.
Reviewer comment: The concept of “velocity-weakening” in L29 and “velocity-strengthening” in L127 is not yet introduced.
Response: We accept this suggestion. The paper now states at first use that velocity weakening denotes b > a and velocity strengthening denotes a > b; the same definitions are repeated in the constitutive section.
Reviewer comment: The definition of tc in L79 is not yet introduced in the main text. It is introduced in the abstract, but not in the main text.
Response: We accept this suggestion. The main text now explicitly introduces tc as the additional constitutive response time in the Introduction before it is used in the research questions or scaling discussion, and Section 2.2 gives its formal constitutive definition.
Reviewer comment: The definition of Ldyn and its reference should be added for Figures 6 and 7.
Response: We accept this suggestion. In the retained finite-window diagnostic figure, now Supplementary Figure S17, Ldyn is explicitly defined as the width over which the finite-window maximum exceeds V = 10−2 m/s and Lnuc = 2 km as the imposed perturbation diameter. The caption also states that these are descriptive threshold quantities rather than intrinsic stability lengths.
Reviewer comment: Figure 1 is not properly rendered. The colorbar is blocking the main figure. Information about contour colors should be added to the figure caption, not in the master configuration box embedded in the figure.
Response: We accept this suggestion. The main-text figure set has been reorganized, and the former space-time Figure 1 is retained only as Supplementary Figure S11 for continuity. Its layout has been cleaned so the colorbar no longer interferes with the panels, the embedded Master-configuration box has been removed, and the caption now states explicitly that cyan contours denote 10−3 m/s and green contours denote 10−2 m/s. The common configuration is referenced through Table S1.
Reviewer comment: In all figures, please remove the master set configuration. This makes the figures look busy and is unnecessary since the model setup does not vary across the study. Referring to Table S1 in the caption should be enough.
Response: We accept this suggestion. The repeated Master-configuration boxes have been removed from the retained finite-window figures. Figure captions now refer to Table S1 for the common setup.
Reviewer comment: L112-113 is hard to understand. Adding relevant equations and/or illustrations would help.
Response: We accept this suggestion. The mechanical section now gives the explicit cell mass mi = ρHΔx and neighbour stiffness Ks = μH/Δx and derives the scalar momentum equation. The text separately identifies along-layer elastic redistribution, explicit inertia, and the local radiation-loss closure.
Reviewer comment: Figure 5 would be more informative if it focused only on the nucleation zone (x = ± 0.5 km), with associated colorbar limits. The current color range from yellow to bright green makes it hard to see the velocity pattern within the nucleation zone. The caption also needs to explain what the red contour indicates.
Response: We accept this suggestion. The former Figure 5 is now Supplementary Figure S15 and has been replotted over x = ± 0.5 km with a restricted color scale appropriate to the accelerated nucleation zone. The contour line is explicitly identified as Vmax = 10−2 m/s, and the common configuration is referenced to Table S1.
Reviewer comment: Figure 7 needs a clearer caption explaining why Ldyn/Lnuc is shown as a shaded area rather than a curve.
Response: We accept this suggestion. The former Figure 7 is now Supplementary Figure S17. Its caption defines Ldyn/Lnuc and explains that the filled region is only a visual fill from zero to the ratio, not an uncertainty band or a separate data quantity.
Reviewer comment: L423-L443 does not belong in the Conclusions section. It would be more suitable for the Discussion section.
Response: We accept this suggestion. The forward-looking material has been moved from Conclusions to the reorganized Discussion, where it is presented under testability and remaining limitations. The Conclusions are now restricted to the results established in the present study.
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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.