State-variable friction for the Burridge-Knopoff model

This work shows the relationship of the state variable rock-friction law proposed by Dieterich to the Carlson and Langer friction law commonly used in the Burridge-Knopoff (BK) model of earthquakes. Further to this, the Dieterich law is modified to allow slip rates of zero magnitude yielding a three parameter friction law that is included in the BK system. Dynamic phases of small scale and large scale events are found with a transition surface in the parameter space. Near this transition surface the event size distribution follows a power law with an exponent that varies as the transition is approached contrasting with the invariant exponent observed using the Carlson and Langer friction. This variability of the power-law exponent is consistent with the range of exponents measured in real earthquake systems and is more selective than the range observed in the Olami-Feder-Christensen model.

Figure 1

(Color online) The statistically steady-state event moment, $M$, of a single-block system as a function of $\gamma -1$ and $\kappa$, at fixed $s=1.6$. A perturbation of ${10}^{-8}$ to the threshold block stress is applied. Moment contours are indicated at the base of the figure and in Fig. 2.

Figure 2

(Color online) The contours of the moment surface in Fig. 1. The moment contours from 1 to ${10}^{-3}$ are nearby in parameter space, indicative of a sharp change in behavior.

Figure 3

(Color online) The transition surface with moment 0.01, determined by linear interpolation, separates the parameter space into a large scale event phase (below the surface) and a small scale event phase (above the surface).

Figure 4

(Color online) The state space trajectory, which is a stable limit cycle, for the single-block system with $s=1.6$ and $\kappa =1.75$, varying $\gamma ∊\left\{1.3,2,2.2,2.4,2.6\right\}$. As the transition surface is approached, the cycle is contained in a smaller and smaller region of state space.

Figure 5

Results from a 100 block system. Large scale and small scale event moment PDDs are observed with intermediate spanning behavior broadly following a power law. Single block events are not included in the statistical analysis.

Figure 6

Moment PDDs for the 10 and 100 block systems with $s=1$, $\kappa =3$, and $\gamma =2$.

Figure 7

${\psi }_{\Delta }$ as a function of $\gamma$ through the transition for the 10 and 100 block systems. A transition is clear but not near the estimated transition point of $\gamma =2$.

Figure 8

The moment PDDs for a 100 block system with $s=1$, $\kappa =3$, and increasing $\gamma$ from the point of power-law behavior toward small scale behavior. Power laws are fitted to the large scale of the PDDs showing a decreasing exponent magnitude with increasing $\gamma$.

Figure 9

The power-law exponents associated with the large scales of the moment PDDs.

Figure 10

The effect of different perturbations on the small scale feature. Shown are the PDDs for different perturbations using ${10}^7$ events. The PDDs have been vertically translated to show that the underlying power-law behavior is unmodified.