Near-surface ice temperatures were measured through time in seven shallow boreholes at three different field sites (Fig. 2). Although hot-water drilling methods temporarily warm ice near the instrumentation, the ice around these shallow boreholes cools to its original temperature within days to weeks. Measured temperatures are spatially variable between sites. The mean value from the lowermost sensor (analogous to T0) is -3.2 ∘C at 27km-11, -8.6 ∘C at 46km-11, and from -9.7 to -8.1 ∘C at 33km-14. In all cases, measured T0 values are warmer than the mean annual air temperature. Temperature gradients are calculated by fitting a line to the mean temperature of the four lowermost sensors for each string. These gradients are also variable, typically being between -0.15 and 0.0 ∘C m-1 but +0.16 ∘C m-1 at 27km-11 (positive being increasing temperature with depth below the surface). As expected, the direction of the temperature gradients measured here correlate with those measured in the uppermost ∼100 m for full-thickness temperature profiles (Harrington et al., 2015; Hills et al., 2017).

Even the five temperature profiles measured at 33km-14 exhibit some amount of spatial variability. Three temperature strings, T-15a, b, and c, are all similar, having strong negative temperature gradients (approximately -0.14 ∘C m-1) and cold T0 temperatures (-9.6 ∘C). Close to the surface, these three temperature strings are cold compared to the others. However, those strings stopped collecting measurements in May 2017 and did not yield a full year of data. The missing summer period explains the strong positive temperature gradient near the surface for those three strings. T-16 is the shortest string, extending to only 9.5 m depth. This short string exhibits the smallest range in temperatures throughout a season with the coldest surface temperatures not even reaching -15 ∘C. In terms of mean temperature, T-16 is similar to T-14, having a small negative temperature gradient and warm temperatures in comparison to those of T-15a, b, and c. Based on our observations, spatial variability in near-surface ice temperature at 33km-14 is controlled on the scale of hundreds of meters. Proximal observations from the nearby T-15a, b, and c strings are similar to one another, but greater variability is observed when including the more distant strings, T-14 and T-16.

Closer inspection of the measured temperature record through time reveals the transient nature of near-surface ice temperature (Fig. 3). As expected, these data show a strong seasonal oscillation near the surface. During the melt season, the ice surface quickly drops as ice is warmed to the melting point. Just below the surface, the winter cold wave persists for several weeks into the summer season. In string T-14 we observe delayed freeze-in behavior in one sensor (Fig. 3b) and transient heating events during the melt seasons (Fig. 3c, d, e). Similar heating events were observed in string T-16 (Fig. 4) but not in any other. The events range in magnitude, but in one instance ice is warmed from -10 to -2 ∘C in 2 h (Fig. 3c). We can only speculate on the origins of these events and address this below in Sect. 5.2.

Meteorological data from 33km-14 were observed over 3 years (Supplement Fig. S3). Air temperatures are normally at or above the melting temperature during the summer but fall to below -30 ∘C in winter months. The measured ablation rate is 2–3 m yr-1 and maximum snow accumulation is only up to 0.5 m. Net radiation is less than zero in the winter (net outgoing because thermal emission in the infrared wavelengths dominates over atmospheric inputs) but over 100 W m-2 (daily mean) on some days in the summer.

The mean air temperature over the entire measurement period at 33km-14 (-10.5 ∘C) is cold in comparison to measured ice temperatures at that site (Fig. 2; T-14, T-15a, T-15b, T-15c, and T-16). This warm anomaly between the ice and air temperature is also observed at 27km-11 and 46km-11, where ice is warmer than the measured air temperature and significantly warmer than the reference from a regional climate model (Meierbachtol et al., 2015). Interestingly, we measure almost no winter snowpack at 27km-11 and 46km-11 due to low precipitation and strong winds during the time period over which those data were collected (2011–2013). Our observations are thus in contradiction to the inferences made by Hooke et al. (1983) in Arctic Canada, where the offset between air and ice temperature appeared to be primarily a result of snow insulation.

Overall, the 3 years for which meteorological data were collected are significantly different. The 2014–2015 winter was particularly cold, bringing the mean air temperature of that year more than a degree lower than the other two seasons. Snow accumulation was approximately doubled that winter in comparison to the other two. Also, the summer melt season is longer in 2016 than in 2015. In comparison with past trends from the nearby PROMICE station, KAN_L, the second year is more typical for this area (van As et al., 2012). To model a representative season, data from that second year (July 2015 to July 2016) were chosen as annual input for the model case study.

The foundation for quantifying impacts of near-surface heat transfer processes is a one-dimensional thermodynamic model. We argue that the processes tested here are close enough to being homogeneous that they can be adequately assessed in one dimension. The one exception is the measured heating events, which are transient and spatially discrete; these are discussed in Sect. 5.2 and are not included in the model analysis. Our model uses measured meteorological variables as the surface boundary condition and simulates ice temperature to 21 m, a depth chosen for consistency with measured data. The ice temperature at the depth of zero annual amplitude, T0, is output from the bottom of the domain for each model experiment and used as a metric to compare net temperature changes between simulations. The model, its boundary conditions, and the experiments are all designed to test heat transfer processes within the ice itself. To maintain focus on ice processes, we ignore any atmospheric effects above the ice surface such as turbulent heat fluxes. The model does not, nor is it meant to, simulate the surface mass balance.

We implement an Eulerian framework, treating the z dimension as depth from a moving surface boundary so that emerging ice moves through the domain and is removed when it melts at z=0. We use a finite element model with a first-order linear element and 0.5 m mesh spacing refined to 2 cm near the surface. For a seamless representation of energy across the water/ice phase boundary, we implement an advection–diffusion enthalpy formulation (i.e. Aschwanden et al., 2012; Brinkerhoff and Johnson, 2013), (∂t+u∂z)H=∂z(α∂zH)+ϕρi. Here, ∂ is a partial derivative, t is time, u is the vertical ice velocity with respect to the lowering ice surface, z is depth, H is specific enthalpy, α is thermal diffusivity, ϕ is any added energy source, and ρi is the density of ice. The diffusivity term is enthalpy dependent, αH=kiρiCpνρicold,H<Hmtemperate,H>Hm, where ki is the thermal conductivity of ice which we assume is constant over the small temperature range in this study (∼25 ∘C), Cp is the specific heat capacity which is again assumed constant, ν is the moisture diffusivity in temperate ice, and Hm is the reference enthalpy at the melting point (all constants are shown in Table 2). Aschwanden et al. (2012) include a thermally diffusive component in temperate ice (i.e. ki∂z2TmP). However, since we consider only near-surface ice, where pressures (P) are low, this term reduces to zero. Using this formulation, energy moves by a sensible heat flux in cold ice and a latent heat flux in temperate ice. We assume that the latent heat flux, prescribed by temperate ice diffusivity (ν/ρi) in Eq. 2), is an order of magnitude smaller than the cold ice diffusivity (ki/ρiCp). We argue that this is representative of the near-surface ice when cold ice is impermeable to meltwater.

The desired model output is ice temperature. It has been argued that temperature is related to enthalpy through a continuous function, where the transition between cold and temperate ice is smooth over some “cold-temperate transition surface” (Lüthi et al., 2002). On the other hand, we argue that cold ice is impermeable to water except in open fractures (which we do not include in these simulations), so we use a stepwise transition, TH=H-HmCp+TmTmcoldtemperate. Additional enthalpy above the reference increases the water content in ice, ωH=0H-HmLfcoldtemperate, where Lf is the latent heat of fusion. If enough energy is added to ice that its temperature would exceed the melting point, excess energy goes to melting. In our case study, we limit the water content based on field observations of water accumulation in the layer of rotten ice and cryoconite holes. This rotten cryoconite layer extends to approximately 20 cm depth and as an upper limit accumulates a maximum 50 % liquid water. Therefore, we limit the water content in the rotten cryoconite layer, 0.0≤ω≤0.5, with any excess melt immediately leaving the model domain as surface runoff.

The two boundary conditions are (1) fixed to the air temperature at the surface, Tsurface,t=Tair, and (2) free at the bottom of the domain, ∂T∂zbottom=0.0. Both boundary conditions are with no liquid water content, ω=0. The surface boundary condition is updated at each time step to match the measured air temperature. The bottom boundary condition is fixed in time. This bottom boundary condition is also changed for some model experiments to test the influence of a temperature gradient at the bottom of the domain (Sect. 4.2.4).

Four separate model experiments are run, each with a new process incorporated into the physics and each guided by observational data. All simulations use the enthalpy formulation above rather than temperature in order to track the internal energy of the ice–water mixtures that are prevalent in the ablation zone. The results from each experiment are referenced to an initial control run, which is simple thermal diffusion of the measured air temperature in the absence of any additional heat transfer processes. Meteorological data are input where needed for an associated process in the model. These data are clipped to 1 full year and input at the surface boundary in an annual cycle. The model is run with a 1-day time step until ice temperature at the bottom of the domain converges to a steady temperature. A description of each of the model experiments follows below. These experiments build on one another, so each new experiment incorporates the physics of all previously discussed processes.

The control model run of simple thermal diffusion predicts that ice temperature converges to approximately the mean annual air temperature of the study year (-9.9 ∘C) by about 15 m below the ice surface. This result is in agreement with the analytical solution (Carslaw and Jaeger, 1959) but is slightly different from the mean air temperature (-9.6 ∘C) because the air can exceed the melting temperature in the summer, while the ice cannot. Other atmospheric effects such as turbulent heat fluxes and the thermal inversion could also cause a difference between measured air temperature and ice surface temperature, but these are not considered here. For each model treatment, 1–4, the incorporation of an additional physical process changes the ice temperature. Differences between model runs are compared using T0 at 21 m. Again, the model experiments are progressive, so each new experiment includes the processes from all previous experiments. Key results from each experiment are as follows (Fig. 5):

Diffusion alone results in T0=-9.9 ∘C, whereas observed temperatures range from -9.7 to -8.1 ∘C at 33km-14.

The strongest result from our model case study was a drop in T0 by -2.5 ∘C associated with the imposed subsurface temperature gradient. While it was important to test this scenario for one case, the temperature gradient we used was representative but somewhat arbitrary. In reality, the observed temperature gradients are widely variable from one site to another and even within one site (Fig. 2). Interestingly, full ice thickness temperature profiles show similar temperature gradients, both positive and negative (Harrington et al., 2015; Hills et al., 2017). Hence, the limiting cases were added to show simulation results over the range of measured gradients from our temperature strings. The resulting T0 values span a range of 19 ∘C.

The majority of the subsurface temperature gradients that we measure are negative, and theoretically the gradient should be negative. Consider that fast horizontal velocities (∼100 m yr-1) advect cold ice from the divide to the ablation zone, and the air temperature lapse rate couples with the relatively steep surface gradients so that the surface warms rapidly toward the terminus. These conditions lead to a vertical temperature gradient below the ice surface that is negative (Hooke, 2005; pp. 131–135), as in our model example. The one exception is in the case of deep latent heating in a crevasse field (Harrington et al., 2015; sites S3 and S4), where the deep ice temperature is warmer than the mean air temperature rather than colder.

Overall, our results demonstrate that the effect of the subsurface temperature gradient is coupled to that of surface lowering. With respect to the surface, the temperature gradient below is advected upward as ice melts. There is competition between surface lowering and diffusion of atmospheric energy into the ice: as near-surface ice gets warmer, it can be removed quickly and a new boundary is set. Therefore, our conceptualization of temperature in the near-surface ice of the ablation zone should not be a seasonally oscillating upper boundary with purely diffusive heat transfer (Carslaw and Jaeger, 1959), but one with advection and diffusion (Logan and Zlotnik, 1995; Paterson, 1972). This conceptualization is unique to the ablation zone because of the rapid rate of surface lowering, whereas a diffusive model for near-surface heat transfer is much more appropriate in the accumulation zone.

The disconnect between air and ice temperature implies that the near-surface active layer in the ablation zone is shallow (i.e. less than 15 m) and could be skewed toward the subsurface temperature gradient. Therefore, the surface boundary condition has weak influence on diffusion for ice well below the surface. This is in contrast to the accumulation zone where new snow is advected downward, so the surface temperature quickly influences that at depth. Under these conditions, it is no surprise that we see spatial variability in near-surface ice temperature even within one field site. That variability is simply an expression of the deeper ice temperature variations, which are hypothesized to exist from variations in vertical advection (Hills et al., 2017), and do not have time to completely diffuse away before they are exposed at the surface.

We observe heating events in two temperature strings, the largest case being 8 ∘C in 2 h between 3 and 8 m below the ice surface (Fig. 3c). These events are transient, they are spatially discrete, and they are generated at depth. All of these factors are most easily explained by the refreezing of liquid water in cold ice. Similar refreezing events have been observed in firn (Humphrey et al., 2012), where they are not only important for ice temperature but could also imply a large storage reservoir for surface meltwater (Harper et al., 2012). However, unlike firn, solid ice is impermeable to water unless fractures are present (Fountain et al., 2005). Two persistently warm features are also observed between 5 and 10 m depth into the winters of 2015 and 2017 (Fig. 3a). We interpret these as a nearby latent heat source, either with running or ponded water that does not freeze for an extended time.

In Greenland's ablation zone, prior work has demonstrated the importance of large-scale latent heating in open crevasses (Phillips et al., 2013; Poinar et al., 2016). Additionally, water-filled cavities have been observed in cold, near-surface ice on mountain glaciers (Jarvis and Clarke, 1974; Paterson and Savage, 1970). In our case, however, an explanation for refreezing water is not obvious. While the field site has occasional millimeter-aperture “hairline” cracks, there are no visible open crevasses at the surface for routing water to depth. As far as we know, this work is the first to report evidence of short-term transient latent-heating events in cold ice that is not obviously linked to open surface fractures.

While the hairline fractures could perhaps move some water, to permit much water to move meters through cold ice they would need to be large enough that water moves quickly and does not instantaneously refreeze. For example, a 1 mm wide crack in ice that is -10 ∘C freezes shut in about 45 s (Alley et al., 2005; Eq. 8). That amount of time could be long enough for small volumes of water to move 5–10 m below the surface but would require a hydropotential gradient to drive water flow. Thus, top-down hairline crevassing does not seem a plausible explanation for the events we observe.

Importantly, several independent field observations in this area, including hole drainage of water during hot-water drilling, ground-penetrating radar reflections, and borehole video observations, all point to the existence of subsurface air-filled and open fractures with apertures of up to a few centimeters (see Supplement). That they are open at depth, but are narrow or nonexistent at the surface, could be linked to the colder ice at depth and its stiffer rheology. Nath and Vaughan (2003) observed similar subsurface fractures in firn, although in their case density controls the stiffness rather than temperature. On rare occasions, we argue that the aperture of the fractures open wider to the surface, where there is copious water stored in the cryoconite layer (Cooper et al., 2018) that can drain and refreeze at depth. While the events seem to happen in the springtime and it would be tempting to assert that fracture opening coincides with speedup, our measurements of surface velocity at these sites show that this is not always the case. This may be due to that fact that the spring speedup coincides with early melt rather than peak melt and copious water in the cryoconite layer.

Latent heating in the form of these subsurface refreezing events is an obvious candidate for a source for the “extra” heat we observe in our temperature strings relative to simulations. Our data show that refreezing in subsurface fractures has the potential to warm ice substantially over short periods of time, and apparently this can occur in places where open crevasses are not readily observed at the surface. Furthermore, the difference between measured and modeled temperatures (up to 3 ∘C) is the equivalent of only ∼1.7 % water by volume. Our simplified one-dimensional model would not be well suited to assess the influence of these latent heating events. Instead, we provide a simple calculation for energy input from the events by differencing the temperature profiles in time and integrating for total energy density (Fig. 7a–c), ϕmeasured=ρiCpΔz∫ΔTdz, where Δz is the total depth of the profile, and ΔT is the differenced temperature profile. Only sensors that are below the ice surface for the entire time period are considered. To calculate the total water content refrozen in the associated event, we remove the conductive energy fluxes from the total energy density calculated above. We do this by calculating the temperature gradients at the top and bottom of the measured temperature profile at each time step as in Cox et al. (2015): ϕconductive=-kiΔz∫∂T∂ztop-∂T∂zbottomdt. The resulting energy sources are then converted to a volume fraction of water by ωmeasured=ϕmeasured-ϕconductiveρwLf, where ρw is the density of water. Results show that each year some fractions of a percentage of water are refrozen (Fig. 7). Through several seasons that amount of refreezing could easily add up to the ∼3 ∘C anomaly that we observe.

Unfortunately, without a more thorough investigation, we do not have enough evidence to show that these refreezing events are more than a local anomaly. Of our seven near-surface temperature strings, only T-14 and T-16 demonstrated refreezing events, so we are not confident that they are temporally or spatially ubiquitous.

The only other logical mechanism for the warm offset between measurements and model results would be warming from below through a positive subsurface temperature gradient. While it is tempting to associate deep warm ice with residual heat from the exceptionally hot summers of 2010 and 2012 (Tedesco et al., 2013), this scenario is unlikely because the ablation rates are so high that any ice warmed during those years has likely already melted. Deeper latent heating from an upstream crevasse field is a more plausible alternative; however, full-depth temperature profiles from 33km-14 do not show deeper ice to be anomalously warmed except in one localized case (Hills et al., 2017).