Greenland Ice Sheet : dissipation , temperate paleo-firn and cryo-hydrologic warming

1Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie (VAW), ETH Zurich, 8093 Zurich, Switzerland 2Institute for Geophysics, The University of Texas at Austin, Austin, Texas, 78758, USA 3Dept. of Geological Sciences, The University of Texas at Austin, Austin, Texas, 78713, USA 4Dept. of Earth Sciences, Dartmouth College, Hanover, New Hampshire, 03755, USA 5Fluid Dynamics and Solid Mechanics Group, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA 6NASA Goddard Space Flight Center, Code 615, Greenbelt, Maryland, 20770, USA *now at: University of Zurich, Zurich, Switzerland


Introduction
Vertical ice temperature profiles of the Greenland Ice Sheet (GrIS) carry information on past upstream surface temperature and accumulation rates.This information is modified by vertical and horizontal stretching of the ice during flow, smoothed by diffusion, and altered by subglacial and englacial heat sources.
Previous work from Jakobshavn Isbrae demonstrates that comparison of modeled and measured ice temperature profiles provides a means to infer ice-dynamical characteristics such as vertical stretching of basal ice as it enters Jakobshavn Isbrae (Iken et al., 1993;Funk Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | et al., 1994).In this case, the validity of the assumed initial and boundary conditions was assessed by an almost perfect match of measured and modeled ice temperatures at drill site DUCK at the margins of Jakobshavn Isbrae (Fig. 1) (Lüthi et al., 2002).It is noteworthy that the temperature profile at 500 km from Funk et al. (1994) agrees well with : is ::::::: similar :: to the profile at Swiss Camp (Thomsen et al., 1991) whereas all other temperature profiles from the Paakitsoq area (Thomsen et al., 1991) are considerably warmer.These warmer temperatures must be caused by heat sources not accounted for in the model.Such heat sources are likely flowing water in moulins and englacial pathways, or freezing water in crevasses, processes recently referred to as "cryo-hydrologic warming" (Phillips et al., 2010(Phillips et al., , 2013)).Future increases in melt area extent might therefore lead to more cryo-hydrologic warming, a mechanism considered to cause rapid future warming of the ice sheet (Phillips et al., 2013).
In this study we present full-thickness temperature profiles from four new boreholesat drill sites FOXX and GULL, located on a flow line downstream of Swiss Camp with temperature profile TD5.Using a numerical heat flow model we calculate expected ice temperature profiles for our study sites.Comparison of modeled profiles with measurements shows excess heat ::::::: warmer ::::::::::::: temperatures of the ice in the ablation area.Possible sources of this excess ::::: extra heat are investigated and discussed.
2 Data and methods
For measurements close to the surface, sets of two thermistor strings were deployed in two boreholes at site FOXX, and in one borehole at site GULL.Each string consisted of a 18-core cable with nine NTC thermistors (Fenwal 135-103FAG-J01) that were shielded from pressure and moisture, and for which individual calibration curves were determined in a calibration bath at six to eight reference temperatures in the range of −17.5 to 10 • C with a digital multi-meter.Reference thermistors, calibrated to an absolute accuracy of 20 mK by the Swiss Federal Office of Metrology, were used to determine the bath temperature.The maximum difference between reference temperatures and the calibration curve was 40 mK.In the field, resistances were measured with a full bridge circuit logged by a Campbell CR-1000 data logger, and occasional readings with a digital multi-meter over the course of two years.With the setup described above, the absolute accuracy of all measured temperatures is estimated to be better than 70 mK.
Figure 2 shows ice temperature measured at sites FOXX, GULL and TD5 (Swiss Camp; Thomsen and Thorning, 1992).Ice temperature increases from the highest site towards the margin.A notable exception is profile FOXX2 which was recorded in 86 m distance from FOXX1, and which is even colder than GULL.

Heat sources
Heat sources within the ice are due to dissipation (strain heating), or related to flowing or freezing water.Here we calculate order of magnitude estimates of these effects.For this purpose we assume ice with a density of ρ i = 900 kg m −3 : , : heat capacity C i = 2093 J kg −1 and latent heat of freezing L = 333.5 kJ kg −1 .

Freezing water
The heat energy from freezing water needed to raise the temperature of 1 m 3 of ice by To produce this amount of heat by freezing, 5.6 kg of water are required.Such water is readily available at the glacier surface during the ablation season, and also in the lower reaches of the accumulation area where permanent storage of water within the firn was recently discovered, although 250 km south of our sites (e.g.Harper et al., 2012;Forster et al., 2013).Water seeping through cracks can freeze and provide an extensive heat source down to the bottom of such cracks.

Strain heating
The volumetric heat production rate P due to dissipation (strain heating) under shear deformation is calculated under the assumption of the shallow ice approximation ::: and :::::: n = 3 as where the shear stress σ xz = ρ i gh tan α is calculated from the density ρ i , gravity g, surface slope α and depth h below the surface.The value of P therefore varies with current surface slope, and depends on depth and current temperature.The temperature dependent flow rate factor A(T ) is taken from Cuffey and Paterson (2010) which agrees well with measurements of ice deformation at sites FOXX and GULL (Ryser et al., 2014b).At a shear stress of 0.1 MPa and with A(−5 • C) = 29.3MPa −3 a −1 the heating power is P = 0.006 MJ m −3 a −1 , equivalent to a heating rate of 0.31 K per century.

Heat flow model
The heat flow model is adapted to model the lower accumulation and upper ablation areas in our region of study.The model tracks a flowline with origin at the ice sheet center (0 km; Fig. 7b in Funk et al., 1994).Figure 3 shows our area of interest, located on this flowline between coordinate 450 and 530 km.In this coordinate system, the 1990 drill site TD5 (Thomsen et al., 1991) at Swiss Camp is located at coordinate 498 km at the equilibrium line.The two 2011 drill sites FOXX and GULL are located at coordinates 520 km and 530 km.
The initial temperature profile F450, located 450 km along the flow line, is taken from (Funk et al., 1994, Fig. 8b).This profile was obtained by solving the advection-diffusion equation with a discrete phase-change boundary along the flowline from the ice sheet center, with prescribed surface and basal velocity.In our model, the block is assumed to move at prescribed horizontal velocities, also taken from Fig. 4 in Funk et al. (1994).These velocities are similar to those from recent satellite imagery feature tracking (Joughin et al., 2008).

Model results
In the following we compare model runs of the heat flow model with borehole data.The model runs are driven with different internal heat sources, and a modified surface boundary condition.These assumptions are independent from each other, and are combined to investigate the origin of measured temperatures profiles.Designations of model runs implementing these assumptions are given in Table 1.The relative importance of these contributions is evaluated by comparison of modeled to measured vertical temperature profiles from Swiss Camp (TD5), GULL and FOXX (both profiles).
Figures 5-7 show comparisons of measured temperature profiles to model results from the reference run mD, and for runs with additional heat sources.The reference model run mD (dissipation only) reveals that measured ice temperatures are considerably higher than

Discussion
Any shape of temperature profile within the ice can be obtained by carefully positioning heat sources during certain time spans within the ice body.The aim of this study is not to perfectly match the observed ice temperature distribution but to investigate how several simple source mechanisms might contribute to the observed temperature profiles.The lower panels of Figs.5-7 indicate the amount of extra heat per volume needed to match the measured temperatures.This heat is likely provided by several heat sources: dissipation (strain heating), temperate paleo-firn, and cryo-hydrologic warming.For the ice near the base, strain heating is a crucial process to provide the required 10 MJ m −3 of heat (Fig. 5).Calculations with Eq. ( 2) show that for the considered ∼ 900 years of ice flow the warming is of the order 3 K which is sufficient to explain the measured temperature of the lowest 200 m.
The relatively warm ice between 100 and 300 m depth at site Swiss Camp (profile TD5; Fig. 5) can be explained by temperate firn conditions in the past.Setting the surface temperature of the accumulated firn between horizontal coordinate 460 and 475 km to melting temperature (Fig. 3) yields the temperature profile shown in Fig. 6.This time span coincides with reconstructed warm temperatures in 1570-1730 (Dahl-Jensen et al., 1998).We assume that during this warm period surface melt events were frequent in the accumulation zone, and therefore the firn was considerably warmed, or even at melting temperature throughout the year.Such conditions with perennial water within the firn have recently been found in the accumulation area in southern Greenland (Humphrey et al., 2012;Harper et al., 2012).Currently, firn temperatures are again warming in the accumulation zone upstream of the study area (Polashenski et al., 2014), and will leave their imprint in the thermal state of the ice sheet.Warm paleo-firn only affects the upper ablation area, and explains only measured temperatures of borehole TD5, but is not sufficient to reproduce warm temperatures at depth at sites GULL and FOXX.
For the central ice body an important heat source is needed to provide the 25 MJ m −3 heat, corresponding to the observed 12-14 K temperature difference at GULL and FOXX (Fig. 5).Strain heating due to horizontal stress gradient ::::::::: gradients : is one possible heat source.Caterpillar-like horizontal extension and compression on diurnal and longer time scales has been observed in the study area (Ryser et al., 2014a).This heat source is, however, not strong enough: A quick calculation with Eq. ( 2), A(−10 • C) and a continuous, very high horizontal longitudinal stress gradient of 0.1 MPa yields a heat production of only 0.0011 MJ m −3 a −1 which is one order of magnitude lower than the shear strain-induced heating of basal ice discussed above.
To explain the high temperature at GULL and FOXX, the only conceivable heat source at depth is advection of heat through flowing, or ponding and freezing water between Swiss Camp and GULL.A major crevasse zone halfway between the sites, and moulins draining water from the surface have been mapped (Thomsen et al., 1988;Phillips et al., 2011).The extent and amount of such heat sources was investigated with several model runs with simple source geometries and durations.Assuming a single crevasse advected to the drill site, its depth has to be of the order of 400 m, and it needs to be provide 0 • C temperature for 100 years.Figure 7 shows results from a model run assuming a series of crevasses of 400 m depth and 100 m spacing which provide 0 • C temperature for 50 years.
Very deep crevasses (300-400 m) with tens of years of activity are only possible if they are water-filled (Van der Veen, 1998), but do not penetrate to the bottom.While there is hydrofracturing as a mechanism to create a :::: deep : crevasse, there must be a mechanism to stop their depth penetration.Limited water supply for hydrofracturing is one possible cause, which is very likely within the small upstream surface area within a crevasse field.Another possible limiting factor for crevasse penetration to the bed is tougher ice at the bottom.Fracture toughness ::: The ::::::: critical :::::::: crack-tip :::::::: loading :::: rate at −20 • C is orders of magnitude lower than at temperatures approaching the melting point (Schulson and Duval, 2009, Table 9.1).We therefore suggest that water-driven crevasses stop their downwards growth once they reach warmer ice.If such very deep, water-filled crevasses indeed exist is unknown, although some observational evidence of strong englacial reflectors was detected in the study area (Catania et al., 2008;Catania and Neumann, 2010).These strong reflectors are likely due to water stored within the glacier ice.

Discussion
Paper | Discussion Paper | Discussion Paper | Discussion Paper | with temperature T , density ρ i , specific heat capacity C i , heat conductivity k and volumetric heat production rate P .The model domain consists of a block discretized with rectangular Quad4 elements with Galerkin weighting (linear approximation of temperature).Dirichlet boundary conditions (prescribed temperature) were applied at the upper and lower boundary.Implicit time stepping was implemented with a standard Crank-Nicolson scheme.Each model run consisted of the deformation of the block of ice (the model domain) during horizontal motion over the distance of 100 km (from 450 to 550 km; Fig.3).The horizontal velocity was taken fromFunk et al. (1994), resulting in a travel time from Swiss Camp to Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | FOXX of ∼ 900 years.During motion along the flow line the model domain (the block) is horizontally and vertically stretched or compressed to comply with the local ice thickness.
The initial dimensions of the model domain were 1000 m length and 2000 m thickness.The mesh was refined around the vertical coordinate 1670 m corresponding to the vertical position of the initial surface.Elements above the current surface were flagged to belong Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | to the inactive subdomain.The heat flow Eq. ( Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | modeled throughout the ice column (Fig. 5).The maximum excess :::: extra : heat energy of 27 MJ m −3 corresponds to a temperature difference of up to 14 K.Following Eq. (1), correcting the temperature difference between the modeled (mD) and observed temperature profiles in the upper part of the model domain would require additional heat from refreezing water equivalent to 9 % by mass.Adding enhanced dissipation through enhanced shearing ice deformation in the Wisconsin ice (model run mE; Fig. 5) explains the observed excess :::: extra : heat between 500 and 700 m depth.The vertical extent of Wisconsin ice was taken from Ryser et al. (2014b) but the enhancement factor had to be set to 5 instead of the measured (and commonly assumed) value of 3. Since the whole model domain moves at the same velocity, the basal ice moves faster than in reality, and therefore has less time to warm.A model run with half surface velocity (not displayed) shows that enough heat is produced at an enhancement factor of 3 to explain measured temperature in the lowest 200 m.The effect of accumulation of temperate firn (run mF) is shown in Fig. 6.In this model run the surface temperature between 460-475 km was set to 0 • C, which corresponds to the 160 years time span 1570-1730 C.E.This seemingly arbitrary horizontal extent of temperate firn conditions coincides with reconstructed warm temperatures at the deep drill sites GRIP and Dye3 (Dahl-Jensen et al., 1998) (see Sect. 4).

Discussion
Paper | Discussion Paper | Discussion Paper | Discussion Paper | −2 • C at GULL, but colder near-surface temperatures at downstream drill site FOXX.It is likely this difference in surface temperature, and therefore the distribution of dust (dirty ice vs. cryoconite Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | holes Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Figure 1 .
Figure1.The study site indicated with red area within Greenland outline is illustrated with a MODIS satellite image (NASA/GSFC, 2010).The drill sites FOXX and GULL are located on a flowline downstream of Swiss Camp, and North of Jakovshavn Isbrae.Site DUCK is from 1995(Lüthi et al., 2002).

Figure 2 .Figure 3 .
Figure 2. Measured ice temperatures.(a) shows near-surface temperatures at sites GULL and FOXX.Holes FOXX1 and FOXX2 are 86 m apart.(b) measured ice temperatures in four boreholes at the drill sites TD5 (Swiss Camp; data from Thomsen et al., 1991), GULL and FOXX.The dashed line indicates the pressure melting temperature.

Figure 4 .Figure 5 .
Figure 4.The gradual deformation of the computing mesh along the flow line to accommodate for the varying ice thickness.The ice surface is marked with blue lines between meshes, and inactive elements are indicated with gray bars (red elements on top).The precise position of the surface with respect to the mesh is tracked with adaptive mesh refinement.Modeled ice temperatures are indicated with colors.The greenish elements close to the surface are accumulated during flow (thickest at Swiss Camp at the equilibrium line) and slowly removed from the top due to ablation (sites GULL and FOXX).Note that computational mesh shown is for illustration only, actual model runs were performed on considerably finer meshes.