Cryo-Hydrologic (CH) warming is proposed as a potential mechanism for rapid thermal response of glaciers and ice sheets to climate warming. We present a simple parameterization to incorporate CH warming in thermal models of ice sheets using a dual-continuum concept, which treats ice and the cryo-hydrologic system (CHS) as overlapping continua with heat exchange between them. The presence of liquid water in the CHS due to surface melt leads to warming of the ice. The magnitude and time-scale of CH warming is controlled by the average spacing between elements of the CHS, which is often of the order of just 10's of meters. The corresponding time-scale of thermal response is of the order of years-decades, in contrast to conventional estimates of thermal response time-scales based on vertical conduction through ice (∼102–3 m thick), which are of the order of centuries to millennia. We show that CH warming is already occurring along the west coast of Greenland. Increased temperatures resulting from CH warming will reduce ice viscosity and thus contribute to faster ice flow.
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 During the melt season, significant volumes of water flow through the cryo-hydrologic systems (CHS) in ice sheets and glaciers. Although the influence of these hydrologic flows on enhancing basal sliding is recognized [e.g., Fountain and Walder, 1998], their influence on the thermodynamics of glaciers is not very well recognized. Classical studies of temperature profiles [e.g., Budd, 1969; Paterson, 1994] have clarified that heat released by refreezing melt water in the firn horizon leads to warmer temperatures at shallow depths in the accumulation zone than in the ablation zone. It is typically assumed that a similar warming influence does not occur in the ablation zone because there all snow and some ice are lost during summers.
 In this paper we propose a mechanism for warming induced by englacial water in the ablation zones of glaciers and ice sheets. Englacial water flows through a complex network of crevasses, fractures, moulins (vertical conduits of 10–100 m depth through which melt water drains) and conduits that constitute the CHS [Fountain and Walder, 1998; Fountain et al., 2005]. The occurrence of water in the CHS during the melt season implies that its temperature is at or above the pressure melting point. Thus heat can be conducted from the edges of the CHS into glacier ice, causing a warming effect whose magnitude will depend on the spacing, surface area and geometry of CHS features. At the end of the melt season, water retained in the CHS can serve as a “latent heat buffer” that moderates wintertime cooling. We present a simple parameterization for incorporating the above effects into thermal models for glaciers and ice sheets. Our approach involves a dual-column model in which the first column is a modified column model [Budd, 1969; Hooke, 1977] for glacier ice to include cryo-hydrologic (CH) heat exchange, and the second column represents the CHS and ice proximal to it. The conceptual framework and heat transport equations for the model are presented in Section 2. In Section 3, we present illustrative applications of the model to Sermeq Avannarleq on the Greenland ice sheet. Our results suggest that the time scale of thermal response of ice sheets can be greatly accelerated by CH warming.
2. Conceptual Framework and Heat Transport Equations
 The conceptual framework for the dual column model is illustrated schematically in Figure 1. In principle, a combination of Spring and Hutter's  and fracture-mechanics equations [e.g., van der Veen, 1998; Alley et al., 2005] may be used to describe the dynamics of the CHS over a melt season. Correspondingly, a high-resolution three-dimensional heat transfer model may be used to track heat fluxes between the ice and CHS. However, the poorly constrained geometry of the CHS and computational limitations would render such an approach intractable. Thus we propose a simplified parameterization for incorporating the thermal influence of the CHS into ice sheet models.
 We view the ice and CHS columns as overlapping continua, as in the dual-porosity models for flow in fractured rock [Barenblatt et al., 1960]. At larger scales, equations for spatially averaged temperatures may be written for each continuum, incorporating an aggregated exchange flux with the other. These exchange fluxes are controlled largely by the thermal properties of the continua and the density/spacing of CH features within the ice. Where there are closely spaced fractures carrying water [Fountain et al., 2005], typical dimensions of an ice block bounded by CH pathways are small, resulting in a stronger warming influence. Conversely, if the CH pathways are far apart, their thermal influence will be smaller and slower. Field observations on Sermeq Avannarleq (outlet glacier northeast of coastal town Ilulissat, west coast of Greenland) have shown that in addition to moulins (about 6–8/km2) and crevasses (25–50 m spacing in crevasse fields and 100–150 m elsewhere), the ablation zone contains a dense network of fractures (∼10 m spacing) even outside crevasse fields (K. Steffen et al., unpublished observations, 2008). There is also evidence that moulins do not progress vertically downward to the bed of the glacier or ice sheet, but drain into englacial water bodies and feed a three-dimensional network of conduits and fractures [Fountain and Walder, 1998; Fountain et al., 2005; Holmlund, 1987] that can efficiently warm the surrounding ice.
 Based on the above motivation, we propose heat transport equations for vertical temperature variations using a dual-column model. The temperature variables that appear in these equations should be viewed as spatial averages over a “representative” area whose horizontal dimensions are large enough to incorporate a representative number of CH pathways, by analogy with the concept of “representative elementary volume” for porous continua [Bear, 1972]. Alternatively the representative area may be viewed as the horizontal area of a grid block (∼10 km2) in an ice sheet model. The heat transport equation for the ice column is written by adding a CH heat exchange term to the conventional column model [Budd, 1969; Hooke, 2005]:
In (1), t is time, x and z are the horizontal and vertical directions; θi(z,t) and θCH(z,t) are the temperatures in the ice and CH column respectively; ρi, Ci and ki are respectively the density, specific heat and thermal conductivity of the ice; u(z) and w(z) are the horizontal and vertical velocities in the ice. The first term on the right side represents the cooling effect of horizontal advection and is approximated as −u(z)* λ* α [Hooke, 2005; Budd, 1969], where λ is the lapse rate and α is the surface slope. Since this representation is approximate, we explore its sensitivity by varying it over a significant range in the computations. The second term on the right, Q ≈ 2A(ρi(Z − z)α)n+1 is the heat source from deformational heating based on the shallow ice approximation [Hooke, 2005], where Z is the total thickness of the ice, A (1e-15 to 1e-16 [Pa−3/a]) is the flow law parameter and n = 3 is the exponent in Glen's flow law. The last term on the right side of (1) represents heat exchange with the CH column, where R is an average spacing between CH pathways. In general, R is expected to vary with depth at a given location and across the horizontal extent of a glacier, reflecting the local geometry of the CHS. Furthermore, R may also exhibit seasonal variations. However, in this first evaluation of CH warming, we approximate R as invariant with depth and time. The quantity (ρiCiR2)/(4ki) may be viewed as a characteristic time scale for heat exchange with the CHS.
 The heat transport equation for the CH column is represented using the following rationale: During the melt season the temperature is simply set to the pressure melting point (θPMP). In principle, an energy equation including the influence of viscous/turbulent heat generation [Spring and Hutter, 1982] could be used to model the CHS. However, for quantifying the warming influence of the CHS on the ice, all that matters is that the interface between them is brought to θPMP as long as there is liquid water in the CHS.
At the end of the melt season, the CH column is assumed to contain some fraction of water entrapped in englacial water bodies [Fountain and Walder, 1998; Catania and Neumann, 2010] and in the temperate ice matrix near CH pathways. During winter, the CH column looses heat to the ice column. However, until all the liquid water retained within the CH column freezes, θCH = θPMP. To account for phase change, we use an enthalpy-based energy equation in the CH column [Lunardini, 1981]:
In (3), (z,t) = (1 − ϕw(z,t))ρiCiθCH(z,t) + ϕw(z,t)(ρwCwθCH(z,t) + L) denotes the bulk enthalpy per unit volume of the mixture of ice and water in the CHS, with ϕw = volume fraction of liquid water and L = the latent heat of fusion. The effective thermal conductivity of the CH column is = (1 − ϕw)ki + ϕwkw. The water content ϕw is assigned an initial value ϕw0 at the end of the melt season. Since θCH = θPMP as long as ϕw > 0, (3) leads to an equation for the reduction in ϕw by refreezing in early winter. Once ϕw = 0, the CH column can cool below θPMP as (3) reduces to a standard heat conduction equation.
 The coupled heat transport equations in the ice and CH column were solved using a semi-implicit finite-difference approach to calculate the temperature and water content. Temperature dependence of the thermal properties was incorporated in the model. The model also accounts for snow cover, which insulates the ice surface from cold winter temperatures. The temperature at the surface of the ice or snow was assumed to be equal to the air temperature. The basal boundary condition was set to a temperature gradient based on the estimated geothermal heat flux or a fixed temperature based on a measured temperature.
 We present two applications of the dual column model to the Sermeq Avannarleq glacier on the Greenland Ice Sheet. In the first example, we consider a location initially just above the historic equilibrium line altitude (ELA) and model the transient evolution of ice temperatures after the equilibrium line moves inland to a higher elevation, which will initiate CH activity in the region. In Sermeq Avannarleq, the equilibrium line is estimated to have risen from ∼1050m in the late 1980s [Thomsen, 1988] to 1250m [Fausto et al., 2009]. The surface temperatures were specified based on measurements from 5 Greenland Climate Network (GC-Net) automatic weather stations (AWS) in the Sermeq Avannarleq region [Steffen and Box, 2001], ranging in elevation from 250m to 2000m. Measured temperatures for each station were analyzed to obtain the mean, maximum and minimum annual temperatures. These values were interpolated to obtain corresponding temperatures at the location of interest (1100 m elevation, just above the 1991 ELA). A modified sinusoidal function was used to represent the annual temperature cycle (similar fitted functions accurately matched the measured temperature cycle at GC-Net stations). The duration of the melt season was estimated using these temperature values in a positive-degree-day approach, and was typically around 8 weeks. Due to the lack of site-specific measurements, we used a temperature gradient of 0.0227 K/m at the bed as suggested by Hooke  based on geothermal heat flux estimates for Greenland. To represent conditions when the region is still in the accumulation zone, a cold ice temperature profile was calculated (without CH heat exchange) for an accumulation rate of 0.3 m/a. The horizontal surface velocity was taken as 80 m/a [Rignot and Kanagaratnam, 2006] with no basal sliding and the shallow-ice approximation was used to derive the horizontal velocity profile. A lapse rate of 0.7 K/100m [Fausto et al., 2009] and a surface slope of 0.5% were used to compute the horizontal advection term in (1). To represent conditions after the equilibrium line has risen, an ablation rate of 0.4 m/a was used, based on recent measurements [Box et al., 2006]. At the base of the ice sheet the vertical velocity was set to zero and a linear function was used for the vertical velocity profile. A profile based on mass conservation [Hooke, 2005] did not result in significantly different simulated temperatures. The volume fraction of liquid water remaining in the CHS at the end of the melt season (ϕw0) was assumed as 0.5%. This is in agreement with measurements of Duval and LeGac  in temperate ice.
Figure 2 shows the evolution of the ice and CHS temperatures after the initiation of CH activity, using a value of 20 m (consistent with observed fracture spacing) for R. The annual temperature cycle is evident at shallow depths in both the ice and CH columns in Figure 2. At greater depths, the CH column reaches a steady state temperature of θPMP within a few years, because the liquid water in the CHS at the end of the melt season does not entirely refreeze during the winter. Englacial water bodies could be detected using ground-penetrating radar on the Sermeq Avannarleq in 2008 [Catania and Neumann, 2010], suggesting that englacial water persists throughout the year. Because the deeper portions of the CH column stay temperate, the ice column temperatures at depths >10m become warmer and do not exhibit seasonality after a few years. The warming trend in the ice temperatures is strongly influenced by R. Figure 3 shows the sensitivity of the ice column temperatures at 100 m depth to R and the influence of the initial ice temperatures before initiation of CH warming. For small R, the warming influence is larger and the ice temperatures reach a new steady state faster. The initial ice temperature has little influence on the new steady-state temperature and the response time scale. As noted above, the characteristic time scale for CH warming is quantified by (ρiCiR2)/(4ki). With R = 20 m, (ρiCiR2)/(4ki) is of the order of 3 years, and the ice temperature attains a new steady state in about 5 years. Although deep temperature measurements are not available at this location, a proposed field program (M. Luethi, personal communication, 2009) will help to further evaluate the role of CH warming. The ∼10 K higher temperatures at 100 m depth when R = 20 m correspond to an increase in the flow law parameter (A) by about a factor of 5. CH warming can thus contribute indirectly to faster ice flow through the temperature-dependence of A [see also Fountain et al., 2004].
 The second application is motivated by the recognition that CH activity is probably already occurring in some regions. If the ablation zone is already temperate by upward advection of ice [Hooke, 2005], CH warming will not have a significant role. Thus, the question is whether there are present-day regions in ablation zones where the ice is cold (not temperate), but warmer than predicted by conventional ice temperature models, potentially indicating the role of CH warming. We present an example of such a region on Sermeq Avannarleq. A borehole (TD3) was drilled at 50°00′W and 69°29′N in 1988 as part of an ice temperature survey [Thomsen and Thorning, 1992]. The ice surface elevation was at 615 m and the ice thickness was 350m. The mean annual air temperature estimated by interpolation from the GC-Net stations was 266K. The surface velocity at the location was taken as 40 m/a with 6 m/a of basal sliding, and the assumed ablation rate was 1.5 m [Rignot and Kanagaratnam, 2006]. A surface slope α = 0.5% and lapse rate λ = 0.7 K/100m were used in the computations. The measured temperature profile and several alternative simulated profiles are shown in Figure 4. At the base of the borehole there should be no melt, as the basal temperature of 272.3 K is below the pressure melting point of 272.74 K.
 Using the conventional column model without CH heat exchange and specified temperature gradients between −0.0227 K/m and 0.0227 K/m at the bed, unrealistic temperature profiles resulted (not shown in Figure 4 for this reason). Using a fixed temperature equal to the measured bed temperature, we obtained realistic temperature profiles (Figure 4). However, these temperatures do not match the measured temperatures very well. A sensitivity analysis of the temperatures simulated by the conventional model was carried out to determine upper and lower bound temperature profiles (“max” and “min” in Figure 4). The “max” curve was obtained by reducing the horizontal advective cooling term to of the base-case value and the “min” curve by using a surface velocity of 80 m/a with 6 m/a basal sliding. The measured temperatures are outside the range of temperature values covered by the “min” and “max” profiles. Only with the inclusion of CH heat exchange could realistic temperature profiles be obtained when a temperature gradient of 0.0227K/m was used at the bed. According to Thomsen's  map, TD3 was at the edge of a crevasse field in the 1980s. WorldView imagery of the area for 2009 showed that the area is heavily fractured with an average distance of ∼25m between crevasses; thus R = 25m was used in the dual column model. Temperature profiles simulated using the dual column model with either a temperature gradient specified at the bed, or a specified bed temperature equal to the measured temperature, match the observed profile reasonably well. Sensitivity studies using the dual-column model showed little variation from the simulated profiles in Figure 4 due to the regulating influence of the CHS. We therefore conclude that the temperature profile in the TD3 borehole cannot be explained without invoking the role of CH warming, and that this is a location where CH warming is currently active.
4. Concluding Remarks
 We propose CH warming as a potential mechanism for relatively rapid response of an ice sheet to a warming climate and demonstrate that CH warming may already be occurring in some regions on Sermeq Avannarleq. The essence of this mechanism is as follows: As the equilibrium line rises in response to climate warming, a CHS develops in regions of the ice sheet that are transitioning from the accumulation to ablation zone. Subsequently, small quantities of liquid water retained in the CHS after the melt season do not completely refreeze during the winter. Thus, there is a sustained warming influence on ice temperatures, with a characteristic response time scale of (ρiCiR2)/(4ki), controlled by the spacing R between CH pathways. With R = 20–50 m, the response time scale is of the order of 5–20 years. In the absence of CH warming, the time scale of thermal response of an ice sheet to a warming climate in the vicinity of the equilibrium line is controlled by vertical conduction, i.e., (ρiCiZ2)/(4ki) (where Z is the ice thickness of the order of 102–3 m), which would be of the order of centuries to millennia. Current ice sheet models do not incorporate CH warming in evaluating the thermal response to a rising equilibrium line. An important consequence of the relatively rapid thermal response resulting from CH warming is that the concomitant increase in the temperature-dependent flow-law parameter (A) can accelerate the response of ice flow velocities to climate warming. In future work, we will investigate the influence of CH warming on temperature and ice flow by incorporating the dual-continuum concept into higher-dimensional ice sheet models. Future field observations will help to better understand the influence of CH warming on ice sheets in transition.
 This research was supported by NASA Cryopshere Science Program grants NNX08AT85G and NNX07AF15G. We thank three anonymous reviewers for their constructive comments and a faculty fellowship from the University of Colorado.