[Paper Review] Self-Arresting and Runaway Earthquakes:Nucleation, Propagation, Gutenberg-Richter law and Dragon-King Events

Didier Sornette, Xueting Wei+1 more|arXiv (Cornell University)|Feb 22, 2024

earthquake and tectonic studiesEarth and Planetary Sciences3 citations

TL;DR

This paper proposes a statistical thermodynamic framework for earthquake nucleation on homogeneous faults, showing that self-arresting ruptures arise when the critical sliding distance exceeds the local critical radius, leading to a Gutenberg-Richter distribution of seismic moments. It identifies runaway ruptures (subshear/supershear) as dragon-king events, distinct from self-arresting events due to their unique physical origins and different statistical behavior.

ABSTRACT

We develop a dissipation-based framework for earthquake rupture on homogeneous faults that explicitly separates the onset of unstable slip from the conditions required for self-sustained rupture propagation. This distinction explains the coexistence of self-arresting earthquakes and run-away ruptures (subshear and supershear events) observed in numerical simulations and empirical studies. We identify two distinct characteristic fault sizes: a nucleation radius controlling the instability of slip, and in general a larger propagation radius controlling whether an unstable rupture can be energetically sustained. Ruptures initiated above the nucleation scale but below the propagation scale spontaneously arrest. We further derive the Gutenberg-Richter law for self-arresting earthquakes by linking rupture physics to the fractal geometry of faulting. Finally, we interpret run-away ruptures as extreme events generated by an amplifying mechanism, consistent with the dragon-king concept. These results provide a unified physical basis for earthquake initiation, arrest, and seismicity statistics.

Motivation & Objective

• To explain the physical mechanism behind self-arresting ruptures in homogeneous faults, which halt spontaneously without requiring strong barriers.
• To derive the Gutenberg-Richter law for seismic moments from the interplay between fractal fault geometry and critical nucleation physics.
• To classify runaway ruptures (subshear/supershear) as dragon-king events, distinct from self-arresting events in both physical origin and statistical distribution.
• To unify nucleation theory with fracture mechanics by adapting Griffith's theory to frictional sliding, offering a new thermodynamic perspective on earthquake onset and arrest.
• To reconcile the observed power-law distribution of earthquake sizes with the underlying physics of rupture nucleation and energy dissipation mechanisms.

Proposed method

• Adapts classical nucleation theory and Griffith fracture theory to frictional sliding, modeling rupture as a thermodynamic phase transition from a metastable to a ruptured state.
• Introduces a critical sliding distance as the key determinant: if the rupture patch exceeds this size, it becomes unstable and runs away; otherwise, it self-arrests.
• Uses the scaling relation $ M_0 \propto R^3 $ to transform the probability distribution of rupture radii $ R $ into the seismic moment distribution $ p(M_0) \propto M_0^{-1 - (D_f - 1/2)/3} $, where $ D_f $ is the fractal dimension of faulting.
• Derives the Gutenberg-Richter law by assuming $ D_f = 2.5 $, yielding the standard $ b $-value of 1 in magnitude space.
• Distinguishes two rupture classes: self-arresting ruptures (stable, dissipative, governed by friction and seismic radiation) and runaway ruptures (unstable, driven by positive feedback, stopped only by strong barriers).
• Applies the dragon-king theory to classify runaway ruptures as extreme events with unique physical origins, implying their statistical distribution differs from that of smaller, self-arresting events.

Experimental results

Research questions

• RQ1 What physical conditions determine whether a rupture self-arrests or becomes runaway on a homogeneous fault?
• RQ2 How does the fractal dimension of fault networks influence the seismic moment distribution in self-arresting earthquakes?
• RQ3 Can the Gutenberg-Richter law be derived from first principles of nucleation physics and fracture mechanics?
• RQ4 Why do large earthquakes differ in physical and statistical properties from smaller ones, and can this be linked to rupture dynamics?
• RQ5 How do self-arresting ruptures and runaway ruptures differ in their energy dissipation mechanisms and statistical behavior?

Key findings

• Self-arresting ruptures occur when the critical sliding distance is smaller than the local critical radius, leading to spontaneous arrest without requiring strong barriers.
• The seismic moment distribution of self-arresting earthquakes follows the Gutenberg-Richter law with $ b = 1 $, derived from the fractal dimension $ D_f = 2.5 $, linking statistical scaling to nucleation physics.
• Runaway ruptures (subshear and supershear) are identified as dragon-king events—extreme, physically distinct events with unique origins, not part of the standard power-law distribution.
• The distribution of runaway ruptures is expected to differ from self-arresting events due to dependence on rare, strong geometric barriers rather than local nucleation conditions.
• The model explains the prevalence of small earthquakes over large ones by showing that self-arresting ruptures are more probable than runaway events, which require both instability and a strong barrier to halt.
• The theory extends to heterogeneous faults by allowing the friction work $ F_f $, analogous to interfacial energy in nucleation, to vary spatially, enabling generalization beyond uniform fault assumptions.