Self-similar slip pulses during rate-and-state earthquake nucleation

For a wide range of conditions, earthquake nucleation zones on rate-and statedependent faults that obey either of the popular state evolution laws expand as they accelerate. Under the "slip" evolution law, which experiments show to be the more relevant law for nucleation, this expansion takes the form of a unidirectional slip pulse. In numerical simulations these pulses often tend to approach, with varying degrees of robustness, one of a few styles of self-similar behavior. Here we obtain an approximate self-similar solution that accurately describes slip pulses growing into regions initially sliding at steady state. In this solution the length scale over which slip speeds are significant continually decreases, being inversely proportional to the logarithm of the maximum slip speed V_max, while the total slip remains constant. This slip is close to D_c(1-a/b)^-1, where D_c is the characteristic slip scale for state evolution and a and b are the parameters that determine the sensitivity of the frictional strength to changes in slip rate and state. The pulse has a "distance to instability" as well as a "time to instability," with the remaining propagation distance being proportional to (1-a/b)^-2 [ln(V_maxθ_bg/D_c)]^-1, where θ_bg is the background state into which the pulse propagates. This solution provides a reasonable estimate of the total slip for pulses growing into regions that depart modestly from steady state.