A model for sorted circles as self-organized patterns

Sorted circles emerge as self-organized patterns from a laterally uniform active layer that becomes laterally sorted as frost heave deforms the interface between a stone layer and an underlying soil layer. In a three-dimensional, cellular model of the active layer, cyclic freezing and thawing drives transport of stone and soil particles by (1) addition of ice particles representing soil expansion by frost heave, (2) removal of ice particles representing soil consolidation during thawing, (3) addition of void particles (a discrete abstraction of soil compressibility) representing soil expansion by water absorption, (4) removal of void particles representing compaction and desiccation of underlying soil by frost heave, (5) relaxation of surface morphology by soil creep and stone avalanche, and (6) vertical sorting of stones and soil by illuviation. These transport processes give rise to sorted circles, which are characterized by a mean spacing of 3.6 m, a 2.4 m wide soil domain surrounded by a 1.0 m wide, 0.3 m high annulus of stones, and a 750 year period of circulation in the soil domain, all consistent with measured characteristics of sorted circles in western Spitsbergen. In the model, instabilities on the stonesoil interface grow upward as soil plugs by drawing in soil from the surrounding subsurface soil layer; soil plugs develop into sorted circles as they contact the ground surface, simultaneously elevating an encircling annulus of stones. Sorted circles are dynamically maintained by circulation within the stone and soil domains. Initiation of soil plugs is driven by a positive feedback in which frost heave near the stone-soil interface pushes soil toward more compressible soil regions, where the soil layer is thicker. The lateral component of these frost-heave-induced displacements is not reversed during thaw because soil consolidation (as ice-rich soil melts and drains) and soil expansion (as desiccated and compacted soil hydrates) displace soil vertically. Further development of soil plugs and sorted circles is determined by an interplay between this positive feedback and amplitude dependent negative feedbacks that result from decoupling of the freezing front from the stone-soil interface. Parameters outside the range in which sorted circles form can result, for example, in stone islands and labyrinthine patterns. The initial wavelength of perturbations on the stone-soil interface is accurately predicted using a linear stability analysis, but increase in this wavelength through time reflects the nonlinearities that control the spacing of soil plugs and sorted circles, namely, interactions and mergers between neighboring forms.