The slow slip of viscous faults

We examine a simple mechanism for the spatio-temporal evolution of transient, slow slip. We consider the problem of slip on a fault that lies within an elastic continuum and whose strength is proportional to sliding rate. This rate dependence may correspond to a viscously deforming shear zone or the linearization of a non-linear, rate-dependent fault strength. We examine the response of such a fault to external forcing, such as local increases in shear stress or pore fluid pressure. We show that the slip and slip rate are governed by a type of diffusion equation, the solution of which is found using a Green's function approach. We derive the long-time, self-similar asymptotic expansion for slip or slip rate, which depend on both time $t$ and a similarity coordinate $\eta=x/t$, where $x$ denotes fault position. The similarity coordinate shows a departure from classical diffusion and is owed to the non-local nature of elastic interaction among points on an interface between elastic half-spaces. We demonstrate the solution and asymptotic analysis of several example problems. Following sudden impositions of loading, we show that slip rate ultimately decays as $1/t$ while spreading proportionally to $t$, implying both a logarithmic accumulation of displacement as well as a constant moment rate. We discuss the implication for models of post-seismic slip as well as spontaneously emerging slow slip events.