A mathematical model of differential frost heave

The O’Neill-Miller model of frost heave, which takes account of a partially frozen fringe between the frozen and unfrozen soil, is used to study the mechanism of differential frost heave, which is a possible cause of earth hummocks and stone circles. In order to facilitate this study, the model must firstly be generalised to three dimensions, which requires a modification, due to Gilpin, of Miller’s concept of regelation; secondly, four key simplifications, variously introduced in previous work by Holden, Fowler and Krantz, must be made to render the computation of the model tractable. With these simplifications, and with the assumption that frozen soil deforms viscously, the model can be reduced to a coupled set of partial differential equations for the frozen soil temperature and velocity. A quasi-stationary stability analysis of the uniform heaving state is conducted on a simplified version of this reduced model to examine whether spatial instabilities can occur in physically realistic conditions. I give an explicit parametric criterion for the occurrence of differential frost heave.