Intermittent Granular Dynamics at a Seismogenic Plate Boundary

Earthquakes at seismogenic plate boundaries are a response to the differential motions of tectonic blocks embedded within a geometrically complex network of branching and coalescing faults. Elastic strain is accumulated at a slow strain rate on the order of ${10}^{-15}{\mathrm{s}}^{-1}$, and released intermittently at intervals $>100\mathrm{yr}$, in the form of rapid (seconds to minutes) coseismic ruptures. The development of macroscopic models of quasistatic planar tectonic dynamics at these plate boundaries has remained challenging due to uncertainty with regard to the spatial and kinematic complexity of fault system behaviors. The characteristic length scale of kinematically distinct tectonic structures is particularly poorly constrained. Here, we analyze fluctuations in Global Positioning System observations of interseismic motion from the southern California plate boundary, identifying heavy-tailed scaling behavior. Namely, we show that, consistent with findings for slowly sheared granular media, the distribution of velocity fluctuations deviates from a Gaussian, exhibiting broad tails, and the correlation function decays as a stretched exponential. This suggests that the plate boundary can be understood as a densely packed granular medium, predicting a characteristic tectonic length scale of $91\pm 20\mathrm{km}$, here representing the characteristic size of tectonic blocks in the southern California fault network, and relating the characteristic duration and recurrence interval of earthquakes, with the observed sheared strain rate, and the nanosecond value for the crack tip evolution time scale. Within a granular description, fault and blocks systems may rapidly rearrange the distribution of forces within them, driving a mixture of transient and intermittent fault slip behaviors over tectonic time scales.

Figure 1

(a) Vector field of 1106 interseismic GPS velocities from across the Pacific–North American plate boundary in California [15], where magnitude is given by color and size. The $x$ axis is approximately parallel to the trace of the San Andreas Fault shown here, while the $y$ axis runs across $L=565\mathrm{km}$ from the Pacific to the North American plate. (b) Average velocity profile as a function of $y$ position in (a). Standard deviation in grey. (c) The full fault system: a geometrically complex network of branching and coalescing faults which accommodates the deformation between the Pacific and North American plates [14], spread across $>100\mathrm{km}$.

Figure 2

Plot of the distributions of the velocity fluctuations $P(\mathrm{\Delta }{v}_x)$ (the red circles) and $P(\mathrm{\Delta }{v}_y)$ (the blue squares). We collapse the distributions by plotting them multiplied by the standard deviation ${\sigma }_x$ or ${\sigma }_y$ accordingly, as a function of the absolute value of velocity fluctuations rescaled by the standard deviation [17]. We use a semilogarithmic scale. A Gaussian distribution (the dashed line) is plotted for comparison, highlighting the heavy tails of the distributions, which clearly deviate from a Gaussian at high velocities.

Figure 3

Equal-time spatial correlation function of velocities fluctuations calculated as in Eq. (2), $C(r)=⟨\delta \mathbf{v}(r)·\delta \mathbf{v}(0)⟩/⟨\delta {\mathbf{v}}^2⟩$ (the dotted line), and fit to the stretched exponential function in Eq. (4), $e^{-(r/\xi {)}^{\beta }}$ (the solid line). The fit yields $\beta =0.75$ and $\xi =92\mathrm{km}$, with $R^2=0.92$ of the least squares fit. (Inset) Plot of the correlation function on a semilog scale, with an exponential (the dashed blue line) shown for comparison.