Melting temperature changes during slip across subglacial 1 cavities drive basal mass exchange 2

The importance of glacier sliding has motivated a rich literature describing the thermomechanical interactions between ice, liquid water, and bed materials. Early recognition of the gradient in melting temperature across small bed obstacles led to focussed studies of regelation. An appreciation for the limits on ice deformation rates downstream of larger obstacles highlighted a role for cavitation, which has subsequently gained prominence in descriptions of subglacial drainage. Here, we show that the changes in melting temperature that accompany changes in normal stress along a sliding ice interface near cavities and other macroscopic drainage elements cause appreciable supercooling and basal mass exchange. This provides the basis of a novel formation mechanism for widely observed laminated debris-rich Cambridge University Press Journal of Glaciology

expressed using the leading-order terms in a Taylor series describing departures from those 20 reference temperature and pressure conditions so that Here, the left side is the change in chemical potential in the liquid phase obtained by altering the temperature from T 0 to T eq and the liquid pressure from P 0 to P , while the 23 right side is the change in chemical potential in the solid ice obtained by altering the 24 temperature from T 0 to T eq and the ice pressure from P 0 to P i . The change in chemical 25 potential with temperature at constant pressure is the specific entropy s, and the change 26 S1 in chemical potential with pressure at constant temperature is the specific volume, which 27 is the inverse of density ρ. Making these substitutions and rearranging leads to where the subscripts l and i refer to properties of liquid water and ice, respectively. The 29 difference in specific entropy between liquid water and ice can be expressed as the ratio of 30 the latent heat of fusion L to the reference temperature T 0 , yielding (after further algebraic 31 rearrangements) Following substitution of equation (2) from the main text for the Clapeyron slope C 0 33 and the ice normal stress σ n for P i , equation (S1) generalizes to equation (1) in the main 34 text when the ice stress state at the interface cannot be approximated as hydrostatic.

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In non-hydrostatically stressed solids, the Gibbs free energy and its associated chemical 36 potential are not well defined (e.g. Kamb, 1961 Common glaciological examples of such non-hydrostatic stress states occur at the surface of 44 a collapsing borehole (e.g. Nye, 1953) or an R-channel containing liquid at a pressure that is 45 lower than the ice pressure (i.e. defined as one third the trace of the stress tensor) so that a 46 deviatoric radial stress drives creep closure at a rate that is compensated in steady state by turbulent melting (Röthlisberger, 1972). Importantly, in such a system the force balance ice-liquid pressure difference of (e.g. Rempel, 2008, Eq. 2) to asking that the ice somehow be supported by a surface traction that has no means of 100 exerting a net force against anything other than the adjacent ice on its boundaries (i.e. the 101 glaciological equivalent of "pulling itself up by its bootstraps").

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In the main text we define P T = σ n − P as the homogenized thermomolecular pressure 103 that is the net force per unit macroscopic basal area arising from ice-mineral interactions; 104 while P T is suitably defined and may be regarded as uniform over macroscopic areas, over 105 length scales of millimeters or smaller, the local thermomolecular pressure p T ( ) that is so that with the average liquid pressure defined as and the average thermomolecular pressure over the portion of the bed area that does not 116 contain macroscopic drainage elements defined as the leftmost and rightmost sides of equation (3) can be expressed as For the idealized case considered, with a homogeneous liquid pressure P =P and with 119 P T =P T , equation (S5) simplifies to equation (4) in the main text. This can be contrasted with the behavior that results from variations in P away from 127 drainage elements, where P T must also change to maintain the vertical force balance over 128 representative projected areas A that is described by equation (4) of the main text (or S5 129 above), so that Because the density difference between the phases is about 10% of the densities of either Fourier's law with a constant thermal conductivity, the evolution of T satisfies where κ is the thermal diffusivity. Introducing the similarity variable η = z/(2 √ κt), this 145 can be written as Integrating twice and applying the boundary conditions that T (0, t) = ∆T and T (∞, t) = 0 147 (with initial condition T (z > 0, 0) = 0) gives the perturbed temperature field as 148 T (z, t) = ∆T erfc z 2 √ κt , and the perturbation to the temperature gradient as Equation (8) in the main text expresses the perturbed temperature gradient at z = 0.

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The analysis is easily extended to consider the response of the temperature field to 151 any number of subsequent changes in the boundary temperature at z = 0. For example, 152 consider the case where, after applying a jump of ∆T at t = 0, the boundary temperature 153 subsequently changes discontinuously again by −∆T at t = t 1 . The perturbed temperature 154 field that we just determined satisfies the heat equation, so we can define T 1 (z, t ≥ t 1 ) = 155T (z, t ≥ t 1 ) − T (z, t ≥ t 1 ), and look for a similarity solution in the same manner, finding Hence the temperature field itself for t ≥ t 1 is For ice sliding at a fixed velocity u s , the time taken to reach location x is x/u s and the to form a liquid network that is in equilbrium with temperate ice (e.g. Nye and Frank, 1973). (while neglecting advective transport) leads to where L is the latent heat and C p is the specific heat capacity. Since englacial liquid 176 contents are expected to become appreciable only in temperate ice, we anticipate that For intuition, given a temperature change of ∆T , this modified formula produces a predic-190 tion for h 0 that is noticeably smaller than that from equation (9) of the main text if the 191 change ∆n l is comparable to or larger than C p ∆T /L, or about 0.06 % when ∆T = 0.1 • C.

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The reports from Blue Glacier suggest that such complications may be important in some 193 portions of at least some temperate glaciers, but the prevalance of such conditions is not 194 well constrained. 195 We note that in circumstances where the impurity content in the ice is sufficiently small 196 that the liquid fraction is controlled primarily by the Gibbs-Thomson effect rather than 197 by the temperature depression associated with colligative effects, the change in n l with T 198 might be expected to be nonlinear (e.g. Rempel, 2005), causing the sensitivity of diffusive 199 heat transport to latent heat effects to decrease as the temperature cools. Moreover, at 200 large values of n l , the permeability of the ice to liquid transport may further affect the 201 temperature field. As there is currently only very sparse quantitative data available to 202 constrain the absolute levels of n l in temperate basal ice, let alone changes in n l with T , 203 we leave further speculation over such potential effects to future work.