The formation of some forms of patterned ground, notably earth hummocks and stone circles, is associated with seasonal freezing and a spatial instability in the resulting frost heave. We analyse the Miller model of frost heave for such spatial instability, by incorporating three-dimensional heat and mass transfer, and allowing the frozen soil to deform as a viscous medium. We find that the heaving process is stable, unless account is taken of a surface snow cover, when the insulating thermal properties of the snow predict that instability will occur if a dimensionless parameter N>0.05. The parameter N is given by N=n_fv_s/(p_fgd²), where n_f is the frozen soil viscosity, v_s is the surface heave rate, p_f is the frozen soil density, g is gravity, and d is the depth of the freezing front. This implies that the propensity for differential frost heave depends on the soil heaving characteristics, as well as the rate of frost penetration.

A.C. Fowler
Mathematical Institute
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E-mail: fowler@maths.ox.ac.uk