The path-independent $M$ Integral around R\"{o}thlisberger channels

Abstract:

R\"{o}thlisberger channels are essential components of subglacial hydrologic systems. Deviations from the Nye creep closure of the ice around a R\"{o}thlisberger channel have been long recognized and enhancement factors or a more complex rheology for ice have been suggested as ameliorations to account for channels closing faster than predicted. Here we use the $M$ integral, a path-independent integral of the equations of continuum mechanics, with a Glen power-law rheology to unify descriptions of creep closure under a variety of stress states surrounding the R\"{o}thlisberger channel. The advantage of this approach is that the $M$ integral around the R\"{o}thlisberger channel is equivalent to the integral around the far field. In this way, the creep closure on the channel wall can be determined as a function of the far-field loading, e.g. antiplane shear as well as overburden pressure. We start by analyzing the case of axisymmetric creep closure and we see that the Nye solution is implied by the path-independence of $M$ integral. We then examine the effects of antiplane shear in several geometries and derive scalings for the creep closure rate based on the $M$ integral. The results are compared to observations for tunnel closure measurements in a variety of stress states and it is shown that the additional stress components can account for the deviations from the Nye solution. Furthermore, creep closure can be succinctly written in terms of the path-independent $M$ integral and the variation with applied shear can be found via scalings, which is useful for subglacial hydrology models.