Phase-locking in Coupled Non-linear Relaxation Oscillators: an Explanation for Observed Temporal and Spatial Correlation and Anti-correlation of Large Earthquakes

There is mounting paleoseismological evidence that large earthquakes on a given fault network tend to occur in temporal clusters. Examples include the southern San Andreas system in the Imperial Valley (Rockwell et al., in prep, 2003), the Eastern California Shear Zone (Rockwell et al., BSSA, 2000), the Garlock system (Dawson et al, in prep., 2003) and the Los Angeles area (Dolan et al., in prep., 2003). This last study has also found evidence that clusters within the Los Angeles area tend to be anti-correlated with similar clusters in the Eastern California shear zone and on the Garlock fault. This clustering behavior is expected if large earthquakes behave as coupled non-linear relaxation oscillators. As a simplest case, we consider two identical faults which are loaded at constant strain rate and which fail at a prescribed stress threshold. Each thus produces the saw-tooth stress strain curve characteristic of a relaxation oscillator. The faults are non-linear oscillators because we assume the stress-strain curve is non-linear, having the negative curvature typical of laboratory experiments and regional damage mechanics models (Ben-Zion and Lyakhovsky, 2002). The two faults are coupled by symmetric stress transfer, in that we assume each fault either increases or decrease the Coulomb stress on the other by an equal amount. We find that events on the two faults phase-lock either in phase if the Coulomb stress transfer is positive or 180 degrees out of phase if the transfer is negative. This phase-lock is driven by the non-linear stress-strain relation. When a fault is close to failure, the increment of stress transfer causes a larger increment in strain. Since time is linked to strain through the assumption of constant strain rate loading, the time shift of the impending event is larger the nearer a fault is to failure. For a positive stress transfer, this shortens the interval and leads to in-phase locking. For a negative stress transfer, the interval is lengthened and the system evolves to a 180 degree out of phase lock. Preliminary models of simple faults controlled by rate- and state-dependent friction have found that non-linearity introduced by friction has a similar effect.