Dynamic behaviour of a phase boundary under non-uniform surface loads

A method is outlined to determine the dynamic behaviour of a phase boundary in the Earth when non-uniform time-varying pressure and temperature conditions are assumed at the Earth's surface. An integral equation describing the phase boundary motion is derived and it is solved under a linearizing assumption. The solution is obtained in the form of a double integral transform. Short and long time-expansions of the solution can be obtained from series expansion and integration of the Laplace transform along a branch cut. The method is illustrated by considering a stepwise change in surface pressure conditions. For short times, the solution exhibits the same type of time dependence (i.e. the first-order term is in t1/2) as the one obtained in the one-dimensional case (i.e. uniform pressure perturbation at the Earth's surface). For long times, it is shown that the time dependence of the phase boundary motion is almost identical to the one derived for the one-dimensional case if the wavenumber kL of the surface excitation is such that κ k2L τ ≪ 1 (where τ is the relaxation time associated with the one-dimensional phase boundary motion and κ is the thermal diffusivity). If κ k2L τ > 1, then the relaxation time for the phase boundary motion in two dimensions is of the order of κ -1k-2L. When considering parameters that would be appropriate for a basalt to eclogite phase transition at Moho depth, the latter situation is met only when the load wavelength is smaller than 35 km.