Earthquake nucleation on rate and state faults aging and slip laws

We compare 2-D, quasi-static earthquake nucleation on rate-and-state faults under both "aging'' and "slip'' versions of the state evolution law. For both versions mature nucleation zones exhibit 2 primary regimes of growth: Well above and slightly above steady state, corresponding respectively to larger and smaller fault weakening rates. Well above steady state, aging-law nucleation takes the form of accelerating slip on a patch of fixed length. This length is proportional to b(-1) and independent of a, where a and b are the constitutive parameters relating changes in slip speed and state to frictional strength. Under the slip law the nucleation zone is smaller and continually shrinks as slip accelerates. The nucleation zone is guaranteed to remain well above steady state only for values of a/b that are low by laboratory standards. Near steady state, for both laws the nucleation zone expands. The propagating front remains well above steady state, giving rise to a simple expression for its effective fracture energy G(c). This fracture energy controls the propagation style. For the aging law G(c) increases approximately as the square of the logarithm of the velocity jump. This causes the nucleation zone to undergo quasi-static crack-like expansion, to a size asymptotically proportional to b/(b-a)(2). For the slip law G(c) increases only as the logarithm of the velocity jump, and crack-like expansion is not an option. Instead, the nucleation zone grows as an accelerating unidirectional slip pulse. Under both laws the nucleation front propagates at a velocity larger than the slip speed by roughly mu'/b sigma divided by the logarithm of the velocity jump, where mu' is the effective elastic shear modulus. For this prediction to be consistent with observed propagation speeds of slow slip events in subduction zones appears to require effective normal stresses as low as 1 MPa.