Scaling behavior of coarsening Faraday heaps

When a layer of sand is vertically shaken, the surface spontaneously breaks up in a landscape of small conical "Faraday heaps," which merge into larger ones on an ever increasing time scale. We propose a model for the heap dynamics and show analytically that the mean lifetime of the transient state with N heaps scales as N(-2). When there is an abundance of sand, such that the vibrating plate always remains completely covered, this means that the average diameter of the heaps grows as t(1/2). Otherwise, when the sand is less plentiful and parts of the plate get depleted during the coarsening process, the average diameter of the heaps grows more slowly, namely as t(1/3). This result compares well with experimental observations.