Burridge-Knopoff model: Exploration of dynamic phases

Slider-block models are often used to simulate earthquake dynamics. However, the models’ origins are more conceptual than analytical. This study uses Navier’s equations of an elastic bulk to derive a one-dimensional slider-block model, the Burridge-Knopoff model. This model exhibits a critical phase transition by varying the friction parameter. Accurate analytical estimates are made of event size limits for the small scale, large scale, and intermediate dynamic phases. The absence of large scale quasiperiodic delocalized events is noted for the parameter set investigated here. The time intervals between large scale events are approximately exponentially distributed for the system in its critical state, in agreement with the theory of nonequilibrium critical systems and earthquake dynamics.

Figure 1

Schematic of the earthquake system. $\nu$ is the velocity of the top surface of the bulk in the direction of the $x$ axis.

Figure 2

The dynamic frictional function of Eq. (14) with $\sigma =0.1$.

Figure 3

Schematic of the limits on the moment $M_l$ as a function of $v_f$ with $l=100$, and $\sigma ={10}^{-4}$. Three regions (bounded by the arrows) may be observed as a large scale moment region $A$, a small scale moment region $C$, and an intermediate region $B$ of moments spanning the small and large regions. The equivalent moment magnitudes $M_w$ for a fault defined by Eq. (19) are also given.

Figure 4

The 100 block systems’s moment PDD with $v_f=1$ is fitted by a stretched exponential function. The exponent, $\alpha$, being 0.5 with the scale in the distribution, $v_0=1.25\times {10}^{-4}$. The parameter $\gamma$ readjusts the normalization of the probability density function.

Figure 5

Shown here is the moment PDD for the 100 block system on a log-linear scale. The moment PDD follows an exponential distribution with scale $v_0=2.575\pm 0.061$.

Figure 6

This figure shows the moment PDD near the dynamic transition and where the large scale excess is present. For the dynamic transition that demonstrates no large scale excess $k_c=1$, $v_f=2∕3$, and $\sigma ={10}^{-4}$. When $k_c=144$, $\sigma ={10}^{-2}$, and $v_f\approx 1∕6$ the large scale excess is prominent.

Figure 7

(a) The delocalized events of the large scale excess are quasi-periodic in the sense of the shear force increments. Here the intervals between the $2509$ events out of ${10}^6$ with moment $⩾10$ are recorded. (b) The intervals between events with moment $⩾0.5$ deviating from the power law on the small scale of Fig. 6 are recorded. The fit to this distribution of intervals is exponential.

Figure 8

Schematic of the total velocity in time for the unstable creep events.