Earthquakes nucleate as accelerating slip over a region of finite size.  Understanding what controls the length and time scales of this process is important for assessing earthquake early-warning systems and for interpreting data obtained by monitoring earthquake-generating faults within tens to hundreds of meters of the source (such as at the 2.5-km-deep San Andreas Fault Observatory at Depth, or SAFOD; see Figure 1). 

Theoretical models of earthquake nucleation require coupling the equations of elasticity with a constitutive law for the evolving strength of the fault surface.  For more than 2 decades the most complete constitutive laws have been various incarnations of “rate- and state dependent friction”, meaning that the frictional strength depends upon the micromechanical state of the fault surface as well as the current slip speed.  Despite this lengthy history, it is most likely that no numerical simulation has ever employed the “correct” constitutive law.  This is because no proposed law fits even all the available experimental data, let alone the conditions of temperature, pressure, and fluid chemistry that might be appropriate in situ.  To make matters worse, the strongly nonlinear nature of friction has made it very difficult to obtain an intuitive understanding of the differences between the underlying equations that people use.  This combination of poorly-constrained constitutive laws and opaque equations is a significant impediment to extrapolating from numerical simulations to fault slip in the Earth.

Over the last several years, the work I have done with post-doc Jean-Paul Ampuero (now an assistant professor at Caltech) has gone a log way toward developing this intuitive understanding.  Using standard methods of fracture mechanics, we now have analytic expressions for the length and time scales of nucleation under the most commonly-used law for the evolution of state (the “aging” law; see Figure 2).  Under this law, nucleation zones can grow to be so large that they might often be detectable from the Earth’s surface.  However, our analytical solutions also let one see immediately that the properties of the aging law that generate these large nucleation zones are directly contradicted by lab experiments. This has led to a collaboration with Chris Marone (Penn State), where we recently showed that lab data relevant to earthquake nucleation are much more consistent with the “slip” evolution law (Figure 3). 

Figure 2. Snapshots of slip speed, from 10-11 to 1 m/s, from a numerical simulation of nucleation under the “aging” evolution law.  Linf is our analytical prediction of the nucleation length for these parameters. The normalizing length scale Lb could be anywhere from centimeters to many meters.

Figure 3. Measured change in friction (black curve) as a function of slip, for step velocity increases and decreases of first 1 and then 2 orders of magnitude, on simulated quartz gouge (glass beads). Data were obtained in Chris Marone’s lab at Pen State. The symmetric response to velocity increases and decreases, and the similar decay distance for the different magnitudes of velocity step, are consistent with the slip law (red curve) but not the aging law.

Our work shows that if lab results can be safely extrapolated to the Earth (a very big if!), the “slip” law is the proper law for studies of earthquake nucleation. Nucleation under the “aging” and “slip” laws is entirely different, taking the form of an expanding crack in the first case and a unidirectional slip pulse in the second (Figure 4). The extent of this difference is surprising in that both laws have been advertised as being adequate at some level. However, this difference is understandable in terms of the much larger fracture energy at the edge of the expanding nucleation zone implied by the “aging” law. Nucleation under the “slip” law is much less likely to be observable by surface instruments.