A note on hysteresis in glaciology

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Recent studies on the mechanism governing the Laurentide ice sheet oscillations of the Last Ice Age focus on the most critical effect of the basal hydraulic processes enhanced when the ice is sliding along soft deformable beds. To understand the import of this, we consider Fowler and Johnson's 0-D hydrological flow model describing the sudden and rapid movements forward (surges) of a till-based 1-D ice sheet sliding on a hat soft bed. The basic idea is that the interplay between the ice sheets dynamics, the basal drainage system, and the sliding law can generate a surging behaviour. Mathematically this means that a multiple valued relationship between the ice flux and the ice thickness arises and the mass conservation equation turns out to be of multivalued type for some special values of the dimensionless parameters involved in the model. Assuming that a multiple valued ice flux law of the Fowler and Johnson type herds, we prove the existence of a weak bounded discontinuous solution to the system which becomes periodic after a suitable time.
A.C. Fowler and C. Johnson, Hydraulic runaway: A mechanism for thermally regulated surges of ice sheets, J. Glaciol. 41, 554-561, (1995). A.C Fowler, Mathematical Models in the Applied Sciences, Cambridge University Press, (1997). H. Brezis, Operateur Maximaux Monotones et Semigroupes de Contractions Dans los Espaces de Hilbert, North-Holland, Amsterdam, (1973). A.C. Fowler and E. Schiavi, A theory of ice-sheet surges, Journal of Glaciology 44 (146), 104-118, (1998). J.I. Dfaz and E. Schiavi, On a degenerate parabolic/hyperbolic system giving rise to a free boundary, Nonlinear Analysis: Real World Applications Series B 38, 649-673, (1999).