A model for sorted circles as self-organized patterns

Sorted circles emerge as self-organized patterns from a laterally uniform a ctive layer that becomes laterally sorted as frost heave deforms the interf ace between a stone layer and an underlying soil layer. In a three-dimensio nal, cellular model of the active layer, cyclic freezing and thawing drives transport of stone and soil particles by (1) addition of ice particles rep resenting soil expansion by frost heave, (2) removal of ice particles repre senting 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 surfa ce morphology by soil creep and stone avalanche, and (6) vertical sorting o f stones and soil by illuviation. These transport processes give rise to so rted circles, which are characterized by a mean spacing of 3.6 m, a 2.4 m w ide soil domain surrounded by a 1.0 m wide, 0.3 m high annulus of stones, a nd 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 stone-soil interface grow upward as soil plugs by drawing in soil from the surrounding subsurface soil layer; soil plugs d evelop into sorted circles as they contact the ground surface, simultaneous ly elevating an encircling annulus of stones. Sorted circles are dynamicall y maintained by circulation within the stone and soil domains. Initiation o f soil plugs is driven by a positive feedback in which frost heave near the stone-soil interface pushes soil toward more compressible soil regions, wh ere the soil layer is thicker. The lateral component of these frost-heave-i nduced 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 s oil plugs and sorted circles is determined by an interplay between this pos itive 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 perturba tions on the stone-soil interface is accurately predicted using a linear st ability 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.