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: open (until 28 Sep 2026)
- RC1: 'Comment on egusphere-2026-4365', Anonymous Referee #1, 25 Aug 2026 reply
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RC2: 'Comment on egusphere-2026-4365', Anonymous Referee #2, 21 Sep 2026
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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
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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.