An enthalpy method for moving boundary problems on the earth's surface

Purpose - To present a novel moving boundary problem related to the shoreline movement in a sedimentary basin and demonstrate that numerical techniques from heat transfer, in particular enthalpy methods, can be adapted to solve this problem. Design/methodology/approach - The problem of interest involves tracking the movement (on a geological time scale) of the shoreline of a sedimentary ocean basin in response to sediment input, sediment transport (via diffusion), variable ocean base topography, and changing sea level. An analysis of this problem shows that it is a generalized Stefan melting problem; the distinctive feature, a latent heat term that can be a function of both space and time. In this light, the approach used in this work is to explore how previous analytical solutions and numerical tools developed for the classical Stefan melting problem (in particular fixed grid enthalpy methods) can be adapted to resolve the shoreline moving boundary problem. Findings - For a particular one-dimensional case, it is shown that the shoreline problem admits a similarity solution, similar to the well-known Neumann solution of the Stefan problem. Through the definition of a compound variable (the sum of the fluvial sediment and ocean depths) a single domain-governing equation, mimicking the enthalpy formulation of a one-phase melting problem, is derived. This formulation is immediately suitable for numerical solution via an explicit time integration fixed grid enthalpy solution. This solution is verified by comparing with the analytical solution and a limiting geometric solution. Predictions for the shoreline movement in a constant depth ocean are compared with shoreline predictions from an ocean undergoing tectonic subsidence. Research limitations/implications - The immediate limitation in the work presented here is that "off-shore" sediment transport is handled in by a "first order" approach. More sophisticated models that take a better accounting of "off shore" transport (e.g. erosion by wave motion) need to be developed. Practical implications - There is a range of rich problems involving the evolution of the earth's surface. Many of the key transport processes are closely related to heat and mass transport. This paper illustrates that this similarity can be exploited to develop predictive models for earth surface processes. Such models are essential in understanding the formation of the earth's surface and could have a significant impact on natural resource (oil reserves) and land (river restoration) management. Originality/value - For the most part the solution methods developed in this work are extensions of the standard numerical techniques used in heat transfer. The novelty of the work presented rests in the nature of the problems solved, not the method used. The particular novel feature is the time and space dependence of the latent heat function; a feature that leads to interesting analytical and numerical results.