The role of subglacial geothermal heat in the mass balance and dynamics of glaciers and ice sheets has in recent years gained increased attention e.g.,. In Iceland, basal melting due to geothermal and volcanic activity makes a significant contribution to glacier mass balance , particularly where the glaciers cover the volcanic zones of Iceland (Fig. ). This applies to our study area, the Mýrdalsjökull ice cap at the southern coast of Iceland, covering the central volcano Katla , where the estimated basal melting by geothermal activity is ∼0.15 mw.e.yr-1, averaged over the entire ice cap .

Geothermal as well as volcanic heat sources under glaciers influence local ice flow patterns and create distinct surface depressions (ice cauldrons), which are characteristic for sustained basal melt e.g.,. Often, direct measurements of geothermal heat flux is impractical or even impossible due to the extensive ice thickness. Indirect estimations have been carried out using ice flow simulations e.g.,. Over 20 ice cauldrons have been identified in the surface of Mýrdalsjökull, all of them located at or within the rim of the ice-covered caldera of Katla . The basal melt along with surface meltwater sometimes accumulates beneath the ice cauldrons, which can result in hazardous jökulhlaups (glacier lake outburst floods). Since the mid-20th century, three jökulhlaups (in 1955, 1999 and 2011), all with peak flow likely exceeding 1000 m3s-1, have originated from underneath the ice cauldrons of Mýrdalsjökull, and they have destroyed roads, bridges and power lines . The risk of jökulhlaups has provoked regular monitoring of the ice cauldrons of Mýrdalsjökull as well as detailed mapping of the bedrock topography beneath the ice cauldrons . These efforts have provided unique datasets which allow us to extend the two-dimensional simulations described in to three dimensions in the attempt to resolve basal heat flux distributions. In this contribution, we present a novel, straightforward method to infer basal heat flux locations and distributions which utilizes three-dimensional ice flow simulations that account for basal melting. Input data to our method consist of ice surface topography, bedrock topography and specific glacier mass balance.

The results from our simulations are presented in Table  and Figs. –. In general, both approaches applied to compensate for elevation changes caused by surface mass balance (Eqs.  and ) yield similar results in terms of which parameters fit best. The fit with the validation data is highly dependent on the location of the heat source (Fig. ). Simulation 01 (Fig. a), corresponding to a shift in the heat source center by 100 m northwest of the simulation with best fit in terms of RMSE (sim. 04, shown in Fig. d), results in comparable RMSE to the simulation with no heat source (sim. 00, shown in Fig. a). Simulation 03 (Fig. c), with heat source 25 m northwest of the 04 heat source, visually appears even better than sim. 04 (Fig. d), due to smaller maximum deviation from validation data, but it does however give slightly higher RMSE for both surface mass balance approaches (Fig. e). Moving the heat source southeast of the 04 heat source does, however, result in both higher RMSE and larger maximum errors (Fig. e–f). Simulations with heat source center locations shifted off the cauldron mirror axis (not shown here) result in worse RMSE for heat source center more than 20 m from the best fit source (sim. 04). When using the best fit location but varying Q, the best fits were obtained for sim. 07 (Q=60 MW) and sim. 04 (Q=74 MW). The RMSE values are quite similar in both cases; applying constant surface mass balance (Eq. ) favors Q=60 MW, while applying variable surface mass (Eq. ) favors Q=74 MW. Likely, the actual power is somewhere between these values.

Our results are the least dependent on the different shapes of heat source tested for approximately fixed Q and location (Fig. ). For both approaches of estimating the effects of surface mass balance, there is barely a significant difference between using σ=30 m (Fig. a) and σ=60 m (Fig. b), both in terms of the RMSE (Fig. e) and visual comparison. The fit does however become significantly worse for σ=100 m, both for the RMSE and from visual comparison, revealing a stronger blue color in the sides of the cauldron, indicating modelled lowering that is too high (Fig. c). This becomes more pronounced for σ=200 m, which also results in the cauldron becoming too shallow as indicated by the darker red color in the cauldron center (Fig. d). When viewing the results of modelling validation, errors in the input glacier surface DEM in 2016, as well as the validation data consisting of the 2017 surface DEM and the estimated winter snow distribution, must be considered. The two DEMs cover around 190 km2 of common area (∼1/3 of Mýrdalsjökull ice cap). If the difference between them is inspected, areas of similar size as the outlined area of K6 typically reveal a difference with ∼0.2 m standard deviation for smooth glacier surface, contributed by random high-frequency pixel noise on top of gradual variation in the difference. Such errors have minor effects on the validation and are also likely to be similar for the RMSE of all simulations, hardly shifting the values by more than 1–2 cm. The 2017 DEM was co-registered to the 2016 DEM to ensure that the elevation difference pattern caused by the horizontal shift between the DEMs was minimal. Shifting one DEM relative to the other by even just one pixel horizontally (4 m × 4 m) clearly increases slope/aspect-dependent differences between the DEMs. If we still assume that up to one pixel shift between the DEMs is possible and redo all RMSE calculations (for all simulation in Table ), assuming variable surface mass balance (Eq. ) and taking into account all combination of -4, 0 and 4 m horizontal shift in easting direction and -4, 0 and 4 m in the northing direction, then the minimum RMSE is obtained for simulation 04 in all cases except one, in which it gave the second lowest RMSE.

Summarizing the results of the model validations and the sensitivity check for possible errors in the validation data, we claim that the location of the heat source center is fairly well established, likely with ∼25 m location uncertainty. The heat source beneath K6 had a mean power of Q=70±10 MW from autumn 2016 to autumn 2017, with the lower boundary corresponding to the best-fit value, when assuming spatially fixed surface mass balance (Eq. ). The shape of the heat source distribution is the most uncertain quantity we infer. Simulations with significantly higher RMSE for σ≥100 m compared to σ≤60 m do, however, indicate that most of the heat at the ice–bed interface is released over a relatively small area (A<∼100×100×π m2), spanning less than 2 % of the 1.8 km2 cauldron area outlined with a dashed red line in Figs. –.

The performance of our simulations, quantified by low zonal RMSE values, is highly sensitive to the location of the subglacial heat flux distribution. Shifting the heat source location by only 25 m (i.e., run 03 vs. 04) demonstrates clear changes in RMSE values, which highlights the need for high-resolution computational meshes to adequately model the effects of heat sources beneath ice cauldrons on ice dynamics and glacier surface changes. Even though we compute our results on a 20 m horizontal resolution grid, we are still able to pinpoint the location of the heat source with about the same level of accuracy. In contrast to observations from a GNSS station, operated at K6 in the summers of 2016 and 2017 (Fig. b), revealing seasonal water storage and drainage under the simulated cauldron, we assume continuous and instant water drainage underneath the glacier. This simplification is required for Eq. () to be applicable, as a persisting water body would substantially alter the heat transfer between the geothermal system and the ice. Our modelling approach utilizes the integrative nature of ongoing subglacial processes which all contribute to the observed annual glacier surface changes. Hence, by simulating nearly a complete year of glacier surface evolution, we are able to reproduce the accumulated mass changes within the system, even though we have simplified the surface evolution as such and the temporal nature of possible subglacial water storage changes.

Which heat source yields the best fit depends to some degree on how the effects of surface mass balance are estimated. This raises the question of which of the two approaches is more applicable. By comparison of the obtained bias corrections (Zbias1 and Zbias2) with existing surface mass balance data (unpublished data at Institute of Earth Sciences, University of Iceland) near K6, we can check how well these methods likely capture the average surface mass balance around K6. The value obtained for Zbias1, used in Eq. (), was for most simulations ∼6.5 m (Table ), slightly lower than 7.2 m net mass balance (ice equivalent) measured at mass balance site M1, ∼2.5 km southeast of cauldron K6 (see Fig. a) from September 2016 to September 2017. The value obtained for Zbias2, used in Eq. (), was for most simulations ∼0.4 m (Table ). During the summer of 2017, the ablation was measured at six sites at locations forming a cross shape over K6 (Fig. b). These measurements showed relatively low ablation of only 0.4–0.8 m with a mean value of 0.65 m (ice equivalent). If this was taken into account by adding the term with the summer ablation in 2017 to Eq. (), then the value would become ∼1.1 m. This is in good agreement with measured winter mass balance at M1, yielding 1.2 m higher mass balance in the winter of 2016­–2017 (7.8 mm ice equivalent) than in the winter of 2015–2016 (6.6 m ice equivalent) for which the snow was mapped in spring 2016 and used in Eq. ().

The above comparison with surface mass balance data therefore suggests that, even though both approaches show fairly good agreement with existing surface mass balance data, a slightly better agreement is obtained when assuming a spatially varying surface mass balance, with the same spatial pattern as in spring 2016 (Eq. ). However, when looking at the RMSE alone, it favors Eq. (), which gives a lower best-fit RMSE (1.351 m for sim. 10) than Eq. () (RMSE =1.535 m for sim. 04). This may indicate that the pattern of winter accumulation caused by snow drift of the governing easterly wind directions was much less prominent in the winter 2016–2017 than in 2015–2016 and again in 2017–2018 . Alternatively, the reason may be heat sources used in the simulations that are too simple, causing Eq. () to result in lower RMSE for the wrong reasons, whereas more complex heat sources such as a narrow line source or multiple Gaussian heat sources with different centers would give more optimal results with lower RMSE for Eq. (). In the absence of better data to constrain the winter mass balance in 2016–2017, we do, however, consider further simulation to constrain more complex heat distribution at the bed to be non-conclusive.

This study highlights how fast steep depressions in a glacier surface, such as the ice cauldron K6, can be filled by ice flow in the absence of geothermal heat (Fig. a and c). With the heat source turned off beneath K6, our modelling indicates a reduction in depth of ∼15 m in a single year, from ∼45 to ∼30 m. This further demonstrates the precaution needed when linking reduced cauldron depth to water accumulation . In areas of powerful subglacial geothermal activity such as on Mýrdalsjökull, reduction in geothermal heat beneath a cauldron can result in similar depth changes as caused by subglacial water accumulation beneath cauldrons. If a lake formed beneath a glacier in the absence of strong geothermal heat source drains, creating a sharp depression in the glacier surface above as has been observed on the Greenland ice sheet, e.g.,, the ice dynamics are bound to play a significant role in filling up the depression following the lake drainage.

In this contribution, we have demonstrated an efficient modelling approach to quantify subglacial heat source locations, distributions and heat fluxes. Even though we apply simplifications to subglacial melt processes as well as surface mass balance, we are able to locate subglacial heat sources accurately to the resolution of our computational grid. The methods applied focus on the overall mass changes within the system integrated over almost a whole year, which is sufficient to adequately quantify heat fluxes, despite not resolving seasonal variations. Our best fitting model (run 04) infers a Gaussian-shaped heat source distribution under K6, resulting in a total heat flux of Q=70±10 MW. In combination with the configuration used at K5 (Q=10 MW), we get a total heat flux output at K5 and K6 combined of Q=80±10 MW. This result agrees well with previous estimates of Q=70.3±38.5 MW . Additionally, we show that the area of the main heat source beneath K6 in 2016 to 2017 was <0.03 km2, which is less than 2 % of the areal extent of the resulting ice surface depression (∼1.8 km2).

The method presented here is not only suitable to indirectly measure geothermal heat fluxes below glaciers and ice-sheets, but it also has great potential for continuously monitoring subglacial geothermal systems and estimating their risk potential for infrastructure as well as humans.