The temperature at the Antarctic Ice Sheet bed and the
temperature gradient in subglacial rocks have been directly measured only a
few times, although extensive thermodynamic modeling has been used to
estimate the geothermal heat flux (GHF) under the ice sheet. During the last
5 decades, deep ice-core drilling projects at six sites – Byrd, WAIS
Divide, Dome C, Kohnen, Dome F, and Vostok – have succeeded in reaching or nearly reaching the bed at inland locations in Antarctica. When temperature
profiles in these boreholes and steady-state heat flow modeling are combined
with estimates of vertical velocity, the heat flow at the ice-sheet base is
translated to a geothermal heat flux of 57.9

The Antarctic geothermal heat flux (GHF), an important boundary condition for ice-sheet behavior, can influence sea-level changes (Golledge et al., 2015) considering its significant influence on the viscosity of basal ice and meltwater content at the ice–base interface. What are the basal ice temperature and mechanical properties? How does GHF control basal melt and affect the internal deformation of the ice sheet? How old is ice at different locations? These questions can be answered only by applying reliable GHF measurements or estimates.

The average global surface GHF is

Some studies use remote methods to estimate the GHF underneath the Antarctic Ice Sheet. For example, satellite magnetic data showed that the GHF
underneath the ice sheet varies from 40 to 185 mW m

Direct temperature measurement obviously produces the most reliable GHF
estimates and can be used to verify results of preliminary thermal modeling
and geological–geophysical studies. While over 10 000 heat flow measurements
have been made globally, 90 % are from Europe, North America, and southern
Africa. South America, Asia, and Australia have far fewer measurements,
while Antarctica has virtually none (Davies, 2013). Drilling through thick
ice is extremely complicated, time-consuming, and expensive; therefore,
direct temperature measurements in Antarctic subglacial till and bedrock
environments have only been conducted twice so far, both under the WAIS: at
the subglacial Lake Whillans (285

GHF derived in the present study (P.S.) from basal temperature
gradients in deep ice boreholes (green bars) compared with modeling. Red
circles show locations of deep-ice-drilling sites (Byrd, WAIS Divide,
Vostok, Dome C, Kohnen, and Dome F) discussed in the present study. Black
squares show locations of boreholes drilled in Antarctic margins in which
borehole temperature measurements were carried out and GHF values were
estimated (

More reliable GHF estimates under the Antarctic Ice Sheet can be made from available temperature profiles in ice boreholes. During the last 5 decades, deep ice-core drilling projects at six sites – Byrd (Ueda and Garfield, 1970), WAIS Divide (Slawny et al., 2014), Dome C (Augustin et al., 2007), Kohnen (Wilhelms et al., 2014), Dome F (Motoyama, 2007), and Vostok (Lukin and Vasiliev, 2014; Vasiliev et al., 2011) – have succeeded in reaching or nearly reaching the ice-sheet bed at inland locations in Antarctica. Reported drill site conditions – snow accumulation rate, mean annual surface air temperature, ice-sheet surface velocity, ice thickness, and drilling depth – are summarized in Table 1.

Information for Antarctic deep-ice-drilling sites.

The Byrd and Kohnen holes encountered water at the base, which welled up into the holes. The borehole at Vostok penetrated the subglacial Lake Vostok at 3769.3 m, and here, water rose from the lake to a height of more than 340 m. Drilling of the other holes was stopped within 10–50 m of the bed. All these holes were temperature-logged and provide a good opportunity to fill the gap in our knowledge of the GHF under the Antarctic ice.

Temperatures in the Byrd, WAIS Divide, Vostok, Dome C, Kohnen, and Dome F boreholes were measured using different devices and different methods. All boreholes were mechanically drilled and filled with kerosene-based drilling fluid. Temperature profiles were then obtained by logging with custom-made borehole loggers (Dome C, Kohnen, Dome F, WAIS Divide, and Vostok) or a thermistor embedded in the drill (Byrd).

Measured temperature profiles in four of the boreholes (Vostok, Dome C,
Kohnen, and Dome F) increase almost linearly with depth, as expected at
locations with minimal annual snow accumulation and hence small vertical
velocities (Fig. 2). In contrast, vertical advection is much greater at the
Byrd and WAIS Divide sites in West Antarctica; at these locations the upper
part of the ice sheet is nearly isothermal, but at depth the temperature
gradient is nearly the same as that at the other sites. Temperature
gradients at the bed are 2.02–3.12

Smoothed measured temperature profiles in Antarctic deep ice
boreholes. Pressure melting point temperature

Thermophysical properties at the base of the Antarctic Ice Sheet at sites of deep ice drilling estimated in this study.

Temperature profiles in deep ice-core drilling boreholes are approximated
closely by polynomials with correlation factors of

Polynomial approximations of borehole temperature

A one-dimensional time-dependent energy-balance equation (Dahl-Jensen et
al., 2003; Johnsen et al., 1995) is usually used to model the temperature
distribution through the ice as a function of the climate conditions on the
surface and the GHF from the bedrock:

The temperature measured at the six drill sites can be considered at thermal
steady state in their near-base portion. Three drill sites (Dome C, Dome F,
Vostok) are in close vicinity to ice divides where horizontal advection and
horizontal heat conduction are assumed to be minimal, and the environment
approximates a steady state (Cuffey and Paterson, 2010). At first approximation, we
also assume WAIS Divide is in a steady state. Byrd and Kohnen are in the
slow-moving areas of the interior of the Antarctic Ice Sheet with a relatively
smooth bed, where horizontal conduction is much lower than vertical
conduction (Hindmarsh, 1999, 2018), and horizontal advection and horizontal
heat conduction can be neglected (Robin, 1955; Van Liefferinge et al.,
2018). This assumption reduces the non-steady-state heat-transfer equation
to

Substitution of Eq. (5) into Eq. (4) and integrating on the assumption that

The least squares method was used to fit measured borehole temperatures with
this equation. In fitting, the initial values of the unknown parameters

In the GA, the crossover fraction is set to be 0.9, while the migration fraction is 0.2 (Reeves and Rowe, 2002). To obtain an accurate solution and save calculation time, we set the population size to be 8000 and the number of generations to be 20. Usually, after 15 generations of iteration, the optimal solution can be found. All the calculations were performed using MATLAB software. The GA provides results for the first generation of the optimal solution in a wide range based on a random combination of the fitting parameters. Thus, for each deep borehole, the fitting experiments were trialed 5 times to avoid random error of the GA caused by the initial random parameter combination. Then, the average value from the five fitting experiments was used as the GHF from bedrock into the ice sheet at the selected site.

Equation (4) can also be re-expressed as follows:

In our method, the temperature in the lower portion of the ice sheets is
assumed to be in steady state, and the GA is used to fit the
measured temperatures in deep ice-core drilling boreholes by varying the
four key parameters influencing the temperature distribution: the surface
temperature, surface accumulation rate, basal melt, and basal temperature
gradient. All these parameters are suggested by the algorithm in order to obtain
the best-fitting curve. We assume that the main uncertainties in our fitting
model come from temperature measurements, variability of the form factor

Interpretation of temperature measurements in mechanically drilled deep boreholes filled with drilling fluids is complicated by several factors (Clow, 2008). First, the temperature is measured in the borehole fluid, not in the surrounding ice; therefore, an important consideration is the need for thermal equilibration of the ice wall and the borehole fluids following drilling and prior to measurement. Second, the heat produced during drilling needs to be dissipated from the borehole, or the thermal drilling disturbance needs to be accounted for (Clow, 2015). Third, increasing temperature with depth can cause convective mixing in the borehole. Fourth, the depth of temperature measurements has an inherent uncertainty due to cable slippage in the counting assembly and cable elongation. Thus, all successful temperature measurements in deep boreholes obey a logging protocol in terms of logger tripping speed, measurement direction, borehole settling time, and so on to minimize the effects of these complicating factors. Temperature measurement errors from sensor accuracy and calibration are found to be within the tolerance for large-scale GHF estimates for our six boreholes to interpret ice-sheet basal dynamics.

The temperature in the Byrd borehole was measured with an accuracy of 0.1

Selection of the appropriate form factor

To set up the vertical velocity profile at Dome C, Fischer et al. (2013)
performed three runs with

We assume that the ice-sheet thickness at the studied sites has kept constant at the present-day height; however, it has varied in the past. The 3D thermomechanical model and the simple 1D model showed that the maximum variation in ice-sheet thickness at Dome C and Dome F was less than 250 m in the past (Parrenin et al., 2007a). In general, the typical difference in the ice thickness in the glacial and interglacial periods at Dome C was 150 m (Passalacqua et al., 2017). At the Kohnen site, the local elevation variation is on the order of 100 m (Huybrechts et al., 2007). The ice-thickness variation at Vostok, located in the central part of the East Antarctica Plateau, exhibits a similar range as at Dome F and Dome C (Ritz et al., 2001).

The best evidence for ice-sheet elevation change in the interior of the West
Antarctic Ice Sheet comes from the Ohio Range, to the south of the WAIS
Divide site at a height of 1600 m a.s.l., and from Mt. Waesche, to the north
of the WAIS Divide site at a height of 2000 m a.s.l. (Ackert et al., 1999,
2007). Moraines at Mt. Waesche were

Comparison with the modern ice-thickness value indicates that the variation
in ice thickness is small, and its influence on ice temperature distribution
can be neglected, in particular, on the lower portion of the ice borehole. For
example, assuming a 150 m thickness increase from the last glacial maximum (LGM) to 15 ka leads to
the change in the reconstructed LGM temperature by less than 0.2

For each deep borehole, the fitting experiments were repeated 5 times for
the best value of the form factor

GHF estimates were made using the following ice parameters:

density of 918 kg m

specific heat capacity of

thermal conductivity of

specific latent heat of

Temperatures measured in Antarctic deep ice boreholes compared with best-fit temperature profiles for the deepest 1500 m.

We performed five runs for estimating GHF with

GHF (mW m

GHF values with the highest correlation factor and smallest RMSE are highlighted by bold.

Comparison of the measured age scales (Ahn and Brook, 2008; Bazin
et al., 2013; Bereiter et al., 2012; Blunier and Brook, 2001; Kawamura et
al., 2007; Neftel et al., 1988; Parrenin et al., 2007b; Sigl et al., 2016;
Staffelbach et al., 1991; Veres et al., 2013) and modeled age scales with

The temperature profiles show that the heat flow through the ice at six deep
drilling sites in Antarctica must be

The surface temperature time curve for the upper
bound of the present-day accumulation rate at Vostok corresponds to a GHF of
53 mW m

At this stage we are not yet able to predict GHF at the bed of 600 m thick
subglacial Lake Vostok because the temperature profile in the lake is still
indefinite. However, the DNA detection of thermophile bacteria in the
near-base accretion ice suggests the existence of near-bottom warm waters
with temperatures as high as 50

The inverse approach to retrieving GHF from radar-inferred distribution of wet and dry beds at the EPICA drilling site
(Passalacqua et al., 2017) gave 54.5

The model with a standard GHF of 54.6 mW m

A previously estimated GHF of 59 mW m

Unfortunately, age scales for the Byrd borehole for
all modeled

A preliminary estimate of the GHF at this
site suggested a high value in the range of 200–230 mW m

The preliminary GHF estimate (Clow et al., 2012) was based on the first temperature log in 2011 in the borehole before it reached its final depth. The reasons why the preliminary GHF estimate may be so high are that (i) temperatures in the borehole were still thermally disturbed in 2011, and (ii) the bottom of the 2011 temperature log was still far from the base of the ice sheet. The borehole was relogged in 2014, and temperature data were obtained much closer to the bed. In addition, Clow et al. (2012) also did not account for horizontal flow effects, and the GHF estimation could have been lower than the one they produced. Further investigations on ice dynamics through WAIS Divide borehole tilt measurements can allow us to determine in-depth stress and velocity distributions and estimate horizontal flow effects on temperature.

Although a steady-state model is used in the lower portion of the boreholes to describe the temperature distribution, it is worth noting that the measured modern temperature is the cumulative effect of historical climate forcing. Therefore, the best-fitting parameters obtained by the GA are not the real parameters occurring during the ice sheet's history. They can be considered as “equivalent” parameters, which are used for calculating the modern temperature profile by eliminating the historical climate changes. Processing back, the “equivalent” vertical velocity, modern accumulation rate, and temperature can be calculated from the GA results. Estimated vertical velocity profiles are shown in Fig. 5. Table 5 lists values of “equivalent” snow accumulation rate and temperature at the ice-sheet surface, which were derived from GA calculations. In all cases, “equivalent” accumulation rates are higher than the modern rates, while the “equivalent” surface temperatures are very close to the modern ones. This can be explained by the fact that the high “equivalent” accumulation rates are used by the GA to eliminate the colder climate effects on the ice temperature profile during the glacial period.

Estimated vertical velocities at drilling sites in West Antarctica

Equivalent thermophysical parameters used by the GA in comparison with published data.

Both transient thermal models (e.g., Dahl-Jensen et al., 2003; Engelhardt, 2004; Martos et al., 2017; Passalacqua et al., 2017; Van Liefferinge et al., 2018) and steady-state models (e.g., Martin and Gudmundsson, 2012; Mony et al., 2020; Parrenin et al., 2017; Price et al., 2002; Zagorodnov et al., 2012) were used intensively in the past and are still used for GHF estimates in Antarctica. Obviously, an exact steady state never occurs in reality, and thus transient models would be expected to give more precise results than steady-state models. However, the answer is not as simple as it is supposed to be.

It is important to recognize that, first, in both cases the models will produce GHF “estimates”, not “measurements”, and second, the thermal gradient can be affected by processes other than GHF, creating local anomalies that may coincide with the point estimate. In order to use a transient model, the accumulation rate and surface temperature in the past should be known. For some of the discussed drill sites these data are available from ice-core studies, while for other sites they are not.

To evaluate the possibility of using a transient model, the GHF at WAIS Divide was estimated by using the accumulation rate and surface temperature in the past provided by Buizert et al. (2015). In these calculations, we assume that the history of the ice sheet at WAIS Divide is about 68 kyr long. The governing equation for the transient model was solved using the finite-difference method. The equation was discretized by both the central-difference and upwind-difference methods and then solved using MATLAB. To find the best solution, the GA was still used. The central-difference method and upwind-difference method demonstrated the same temperature profile. Therefore, here we present the calculation results obtained via the upwind-difference method.

Unfortunately, the calculation results with the transient model showed the
best-fit GHF value of

Paleo-temperature profiles at WAIS Divide based on transient model
(

Both models lack additional heat sources (i.e., shear heating, heat
advection, and basal frictional heating) that might be generated at the
bottom of the ice sheet. Thus, the results of both modeling approaches
strongly depend on the selected initial parameters, in particular, from the
selected value for the form factor

Most numerical models of the EAIS basal conditions assume the GHF to be
42–65 mW m

Variability of crustal thickness, hydrothermal circulation (Seroussi et al.,
2017), magmatic intrusion (Van Wyk De Vries et al., 2017), and thermal-conductivity variability are the main contributors to the elevated and
highly variable values of GHF in West Antarctica (Begeman et al., 2017). One
of the first pieces of evidence for an “unreasonably high” GHF
(

Prediction of the future behavior of the Antarctic Ice Sheet undeniably requires accurate ice-sheet models. However, GHF models based on seismic tomography, radar data, magnetic field observations, the tectonic age, and geological structure of the bedrock yield mixed results at sites of deep ice-core drilling in Antarctica. We suggested to estimate GHF from ice-borehole temperature profiles using a one-dimensional steady-state energy-balance equation and the genetic algorithm (GA) for determining the optimal solution of temperature fitting. To our knowledge, we used the GA approach for the first time in ice thermodynamics. Comparison of modeled and measured depth–age scales shows that our model is able to assess the variation in GHF estimates from ice-borehole temperature profiles if in-depth horizontal ice velocities are low and can be ignored. The correlation analyses at the EAIS sites indicate that all of them can be adequately approximated by the steady-state model. However, horizontal velocities and their variation over ice-age cycles are much greater at WAIS than at the EAIS sites. Thus, the steady-state model cannot precisely describe temperature distribution here.

At three studied EAIS sites (Dome C, Dome F, and Kohnen), the GHF is higher than that predicted by other models. We assume that this elevated GHF can represent regional value and can be used as a reference point for regional modeling. More precise GHF estimates and explanations for an elevated GHF would be possible after temperature logging and subglacial rock studies from deep boreholes that are required to drill in Antarctica in the distant future. Finally, the proposed method of GHF estimates can be used at other sites in Antarctica and Greenland where the steady-state model is acceptable.

The data that support the findings in this study are available from the corresponding authors.

PT and YL developed the concept, wrote the manuscript, and drew all figures. YL also designed and performed all GHF estimations. JH processed all temperature profiles. GDC has made important recommendations and edited the final version of the paper. LA, GDC, EL, AM, HM, and CR provided observed borehole temperature data. All authors contributed to discussion and interpretation of the results.

The authors declare that they have no conflict of interest.

We are grateful to Herbert Ueda (retired from USA CRREL) for providing original Byrd borehole temperature log data. We thank Dorthe Dahl-Jensen (Physics of Ice, Climate and Earth group, Niels Bohr Institute, University of Copenhagen, Denmark) for fruitful discussion and useful comments. We also would like to thank the editor Alex Robinson and both anonymous reviewers for fruitful comments and advice.

This research has been supported by the National Natural Science Foundation of China (grant nos. 41327804 and 41806220) and the Fundamental Research Funds for the Central Universities (grant no. 2017TD-24).

This paper was edited by Alexander Robinson and reviewed by two anonymous referees.