The motion of wires pulled transversely through ice has long been explained in terms of pressure melting at the front of the wire and regelation behind it, the speed of the process being controlled by the rate of conduction of the heat of fusion through the wire and the ice. Treated quantitatively, this simple picture predicts wire speeds that are directly proportional to driving stress, defined as driving force per unit length divided by half the circumference of the wire. Experimental observations, however, show much more complicated behaviour. The observed speeds increase nonlinearly at all but the lowest driving stresses, and at a stress of about 1 bar (10
5
Pa) jump sharply, but continuously and reversibly, by an amount that ranges from six-fold for Nylon wires to 60-fold for copper wires. Above this transition the speeds of highly conductive wires, such as copper, are as low as one-eighth of those predicted, though those of poorly conductive wires, such as Nylon and Chromel, are about the same as predicted. Below the transition the speeds of all wires are much less than predicted. Surprisingly, all wire speeds are significantly reduced by the presence of air bubbles in the ice. The wires leave behind a trace that below the transition consists of widely scattered, generally tiny bubbles of water, but above it grades from numerous bubbles of water and of vapour in the case of highly conductive wires to a central tabular layer of water in the case of poorly conductive ones. Measurements of the fractional volume of water in the trace show that above the transition heat flows to the moving wire from the surrounding ice. The nonlinearity and low speed below the transition are due to the presence of accumulated solutes in the water layer around the wire, which concentrate toward the rear, lowering the freezing temperature there and hence the rate of heat flow toward the front. The transition occurs when the temperature at the rear reaches the triple point, which fixes the pressure there, so that with increasing driving stress the mean pressure around the wire increases and hence the mean temperature decreases, causing heat flow to the wire and formation of the trace, which carries away the dissolved solutes. The trace of highly conductive wires is bubbly, rather than tabular, because of the Frank instability of the freezing surface, which permits fingers of water and vapour to grow until pinched off by surface tension. For poorly conductive wires the nonlinearity above the transition is mainly due to the additional melting at the front of the wire and the change in pressure distribution around the wire associated with the formation of the trace. For highly conductive wires the nonlinearity and unexpected slowness above the transition are mainly due to the supercooling required for a finite rate of freezing, which, like the presence of dissolved solutes, lowers the freezing temperature at the rear of the wire. When modified to take approximate account of these effects, the simple quantitative treatment predicts wire speeds that, considering the uncertainties about the parameters describing the solute content and the required supercooling, are in good agreement with the experimental observations.
Symplectic flow for the square root of the negative Laplacian