Mathematics Colloquium

The mathematics of moving boundaries is typically ascribed to Josef Stefan’s 1889 paper "On some problems of the theory of heat conduction”.  Stefan, Boltzmann’s advisor and discoverer of the radiation law associated with the pair, did not mention the origins of the problem arose from data taken during the Austro-Hungarian Polar Expedition of 1872-74.  Thus, although moving boundary, or “Stefan-problems”, abound in applied mathematics, their origins are in the growth and decay of floating polar ice, which is at the center of modern climate and global change research.  However, such ice does not just grow and decay, but it buckles, breaks and rafts, processes central to the net mass balance and whose understanding requires appeal to the Föppl–von Kármán equations of thin plate theory.  Whilst the mosaic of the ice pack is influenced by both elastic failures and growth and decay, how to incorporate these in the large scale leads to an intransigent multi-scale problem, the solution to which involves stochastic concepts from the realm of the microscopic.  I discuss the key historical waypoints and the solution to a four decade old problem described by Thorndike (1975) that incorporates all scales in a relatively simple manner that allows one to compute the climatological evolution of the mass of the ice pack.  Further simplifications lead to analytical solutions using approaches from quantum mechanics