The Slow Slip of Viscous Faults

We examine a simple mechanism for the spatiotemporal 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 nonlinear, 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 nonlocal 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 and a constant moment rate. We discuss the implication for models of postseismic slip as well as spontaneously emerging slow slip events.