The Amery Ice Shelf Ocean Research (AMISOR) project was launched to investigate ice–ocean interaction processes, the interaction with the interior grounded ice sheet and the properties of oceanic water masses beneath the ice shelf (Allison, 2003). As a part of the AMISOR project, from 2001 onwards, six boreholes, named AM01–AM06 (Fig. 1), were hot-water-drilled on the AIS (Craven et al., 2014). Sites AM01, AM04 and AM05 are located on approximately the same ice flowline where basal marine ice is present, and we name it the Jetty Peninsula flowline (hereafter JP flowline for simplicity) in this study. It originates from what we term the West Tributary Glacier (WG) of the AIS (Fig. 1). Sites AM02, AM03 and AM06 are in areas without basal marine ice where we determine another two specific flowlines (Fig. 1). The AM03 flowline originates from Mellor Glacier (MG), passing through the AM03 borehole. The AM06 flowline, from Kronshtadtskiy Glacier (KG), passes through the AM06 borehole and passes close by the AM02 borehole.

After hot-water drilling, the boreholes were kept open for several days to make observations in the ocean cavity, and deploy oceanographic mooring instruments (Craven et al., 2004) and thermistor strings (or optical fibres) for long-term measurements. Two thermistor strings were deployed within and through each of the earlier boreholes, AM01–AM04. One was used to measure the internal ice shelf temperature, and the other was for tracking the ice–ocean interface. All the internal thermistor data points are used in this study, while only a few characteristic thermistor data points at the ice–ocean interface are selected, since the interface thermistors are closely distributed along the cable, and the differences between readings for those thermistors are insignificant. The Sensornet Oryx instruments (distributed temperature sensors) and optical fibres were deployed at AM05 and AM06 sites, which provided continuous profiles of the temperature distribution along the fibre cable with a spatial sampling interval of 1.015 m. Temperatures within the top 10 m of the firn layer were recorded by automatic weather stations (AWSs) of the Australian Antarctic Programme at AM01 and AM02 boreholes and at the Amery G3 site (70.891∘ S, 69.871∘ E), which is approximately 42 km south of AM03, and these showed clear seasonal signals. Similar signals were also observed within the top 10 m of the firn layer on the McMurdo Ice Shelf (Kobs, 2014). To eliminate these near-surface seasonal signals and derive “steady-state” vertical temperature profiles at the borehole sites for comparison with the simulations, some temperature data near the top surface at AM01–AM04 have been carefully selected and temporally averaged. Temperatures at 10 m depth at AM01 and AM02 are temporally averaged from the collocated AWS records in the corresponding time interval (Table 1), while the near-surface temperatures at AM03 and AM04 are estimated with reference to all the available AWS data in the corresponding time interval and a multi-year average surface temperature field over 1979–1998 (Comiso, 2000). At AM05 and AM06, the DTS data within 20 m of surface are not considered, due to strong seasonal signals. Details about the temperature data, including the depths of the measuring instruments, are presented in Table 1. We note that the temperature profiles of AM01 and AM04 based on thermistor string data have been published in Craven et al. (2009) and Treverrow et al. (2010).

After the temperature-measuring instruments are deployed, the water in the boreholes refreezes in a relatively short time, while the borehole temperatures take a much longer time to fall back to equilibrium. This cooling process can be detected within each borehole, assuring thermal disturbance produced by the drilling has basically dissipated and the borehole thermal regime is in approximate equilibrium. The internal thermistors at AM01 recorded a rapid drop of borehole temperatures during 15 d after instruments were deployed, and after 40 d they were still slowly decreasing at a rate of ∼-0.04 ∘C d-1, illustrating the long-term adjustment to the original temperature regime before hot-water drilling. The temperature observations at AM01–AM04 sites lasted for several years until 2009. However, during this period, battery exhaustion and surface logger failures resulted in multiple measurement interruptions, and individual thermistor failures also led to the loss of data in space over the long-term measurement. Since January 2010, the distributed temperature sensors recorded borehole temperature along the optical fibres at AM05 and AM06 until 2013, but data gaps in these time series also exist due to operational difficulties. The non-equilibrium data disturbed by drilling work and the data with errors due to equipment failures are eliminated in this study. The temporal coverages of near-equilibrium temperature data used are listed in Table 1. A frozen-in device, either a thermistor string or a distributed temperature sensor, is a “Lagrangian” measuring instrument, advected horizontally and vertically by the flow of the ice shelf. The locations of the six boreholes during the initial drilling process are used for subsequent analysis in this study, presented in Table 1.

For our 3-D modelling study, we carry out a series of simulations for the entire Lambert-Amery Glacial System (LAGS), which involves modifying the dynamic boundary condition at the base of the ice shelf to impose four different BMB fields. The main aim of our 3-D simulations is to explore the sensitivity of the thermal structure of the AIS to different BMB fields. We then compare the simulated 3-D temperature distributions of the AIS with the borehole observations to evaluate our simulation results. The simulations presented here build on a larger study concerning optimisations in a regional Antarctic ice sheet model (methodology described by Gladstone and Wang, 2022), and we take simulations from that work for our starting point, as detailed in the next section. All the simulations are implemented using the Elmer/Ice model (Gagliardini et al., 2013), a finite-element, full-Stokes ice sheet/shelf model, which also has the capacity to calculate the englacial temperature distribution.

From that starting point (Gladstone and Wang, 2022), we first optimise the ice flow dynamics across the LAGS for each choice of ice shelf BMB forcing. We do this by optimising spatial distributions of basal resistance and ice viscosity using adjoint inverse methods (Gillet-Chaulet et al., 2012), with the observed horizontal surface velocities (Rignot et al., 2017) as our optimisation target. After these diagnostic simulations of the dynamics, we perform simulations for the 3-D steady-state temperature distribution using the newly optimised ice dynamics (the strain rates and 3-D velocity fields). The complete simulation workflow is shown in Fig. 2, where each simulation uses the optimised parameters from the previous stage.

To explore the formation of the vertical thermal structure in the areas with and without basal marine ice, we also conduct 1-D column simulations based on time-stepping to follow columns of ice along the JP flowline and AM03 flowline (Fig. 1), respectively. A pioneering application of this approach was made by Macayeal and Thomas (1979) to interpret the formation of the measured temperature profile at J9 borehole on the Ross Ice Shelf. In the current study, a model of a vertical ice column with 100 equally spaced layers is constructed using Elmer/Ice. A series of key sites along the two flowlines, as shown in Fig. 1, are determined as key time stamps. The time intervals between each time stamp are derived according to the spatial locations of the key sites and the surface velocity field of the AIS (Rignot et al., 2017). Figure 5, as a schematic diagram, demonstrates the evolution of an ice column along each flowline, with related column parameters and boundary conditions as shown. Each 1-D experiment consists of a series of simulations using temperature solvers in Elmer/Ice and involves two stages: initial spin-up and forward transient simulations.

The initial spin-up is a steady-state temperature simulation used to initialise the vertical thermal regime of a column of grounded ice at the start of each flowline (WG and MG; see Figs. 1 and 5). Starting from the initial steady state, we then perform forward transient simulations representing evolution of the ice column as it is advected through the ice sheet and ice shelf (Fig. 5). The transient simulations also implicitly involve a steady-state assumption, since our ice columns are assumed to move strictly along the ice streamline of the reference horizontal velocity field (Rignot et al., 2017), and the boundary conditions we use do not incorporate temporal records but represent the appropriate location along the streamline.

We impose boundary conditions, including the mass balances and temperatures at the upper and lower surfaces of the column model, advancing by 1-year time steps between the key time stamps. The surface temperature is determined at each key time stamp (Fig. 5), based on all available AWS observations and multi-year surface mean temperature dataset (the same as used in our 3-D simulations; Comiso, 2000), and is linearly interpolated between time stamps. Similarly, the basal temperature at each key time stamp is taken from borehole observations (where available) or as the calculated in situ pressure melting temperature (Fig. 5), also linearly interpolated between time stamps. According to the location of the ice column at each time step, the SMB is interpolated from the gridded 1979–2016 mean data of RACMO2.3p2 (a regional atmospheric model; Van Wessem et al., 2018). Similarly, the BMB within the floating sector is extracted from Adusumilli et al. (2020) (i.e. BMB_CAL), while zero BMB is imposed for the grounded ice. Ice thus flows vertically across the upper and lower surfaces of the column according to these imposed mass balance (melting/freezing rates) in the column simulations. The ice thickness of the column model at each key site (time stamp) is also fixed, based on borehole measurements and BedMachine Antarctica (Morlighem, 2019; Morlighem et al., 2020), while the vertical strain rates for each interval between key sites (always strain thinning; marked in Fig. 5) are selected to adjust the column thickness variations, in conjunction with imposed mass balance, to fit the prescribed ice thicknesses at the key sites. The vertical velocity in the ice column, taken as varying linearly with depth, is thus determined by the prescribed SMB, BMB and vertical strain rate.

In general, the initial setup of the spin-up and the boundary conditions for the transient simulations are inferred from a variety of available data. In addition to the steady-state assumption, there are a series of other assumptions:

The ice density of the column is taken as constant everywhere and will not change with the vertical strain process. The heat capacity and conductivity are functions of in situ ice temperature. The ice accreted at either surface in the simulations is assumed to have the same material properties. Detailed physical parameters are shown in Table 2.

The measured temperature profiles within the boreholes are relatively stable over the selected observation periods. The thermistors in AM01 and AM03 boreholes show a slight decrease in temperature within 0.05 ∘C during the temporal coverages (Table 1), which we attribute to continuing adjustment towards the original ice temperatures after hot-water drilling. The DTS time series in AM05 and AM06 suggest random fluctuations (noise) of ±0.05 ∘C in individual observations at any given depth, which are reduced to ±0.01 ∘C by averaging consecutive measurements. This may be regarded as indicating the precision that the DTS can provide. However, the averaging process also reveals small systematic differences between adjacent depths, which points to limits on accuracy, although usually accuracy is limited by availability of precision calibration data. Borehole thermal regimes are derived from averaging all those measurements (Table 1) selected to represent the thermal equilibrium state. To achieve continuous temperature profiles, spline interpolation is used to smoothly connect the discrete thermistor points to compare with DTS profiles. Figure 6 shows the borehole temperature profiles at AM01–AM06 grouped as with or without basal marine ice, where the profiles of AM02, AM03, AM05 and AM06 boreholes are published here for the first time. Our time-averaged temperature profiles of AM01 and AM04 boreholes are consistent with the previously published temperature profiles (Craven et al., 2009; Treverrow et al., 2010).

AM01, AM04 and AM05, approximately located on the same flowline in the marine ice band (the JP flowline; Fig. 1), show similar profile patterns (Fig. 6a). The temperature profiles at these sites show nearly isothermal basal layers up to 120 m thick, which are closely related to the in situ pressure-dependent freezing point of seawater. The correlation is well reflected in the DTS profile at AM05 (subplot of Fig. 6a), where the profile for 530–624 m depth is well approximated by a constant salinity in situ pressure freezing line. The observed pressure-dependent temperature suggests that the marine ice below 530 m depth maintains a hydraulic connection with the ocean below. There is a slight step at 530 m depth. Fresher water from the drilling process was observed above this level prior to borehole freeze-up. The profile above 530 m depth no longer matches the pressure freezing line, implying the termination of the hydraulic connection, and at 500 m there is an abrupt change in the temperature gradient. Therefore, we estimate that at AM05, the interface between permeable and impermeable marine ice (corresponding to the hydraulic connection depths observed at AM01 and AM04 by Craven et al., 2009) is around 530 m depth (marked in Fig. 6a). The temperature at the interface between meteoric and impermeable marine ice drops from ∼-6.2 ∘C at AM04 to ∼-6.8 ∘C at AM01 over the period of 102.6 years. If it is assumed that the interface temperature drops at the same rate from AM05 to AM04, the temperature of the meteoric–marine ice interface at AM05 can be estimated to be -5.8 ∘C. Combined with the observed temperature profile of AM05, the interface depth can then be estimated as 447 m (marked in Fig. 6a), corresponding to a marine ice thickness of 177 m at AM05. This estimation is in good agreement with the marine ice thickness expected on the basis of vertical strain thinning and basal accretion from AM05 to AM04. The internal temperature gradually increases along the JP flowline, while the internal temperature gradients remain stable and relatively high (Table 3) above the isothermal zone at these three sites. At AM05, the upper 200 m of the meteoric ice is nearly isothermal, and the corresponding layer is thinned to ∼ 100 m at AM04, then approximately dissipated at AM01.

AM02, AM03 and AM06, in the area without marine ice, show large temperature gradients within 100 m of the lower surface layer (Fig. 6b). AM02 and AM03 reveal significantly colder ice in the interior of the ice shelf column than at the upper surface, hereafter referred to as “cold cores” in the ice shelf. The temperature gradients at the base of the meteoric ice, as well as the corresponding heat flux, derived from the borehole temperature profiles are shown in Table 3. At sites AM02, AM03 and AM06 without a marine ice layer, the heat flux represents that across the ice shelf–ocean interface.

The basal resistance and viscosity inversions performed in each of our ice shelf BMB experiments produced very similar distribution patterns to those of experiment E3 (Gladstone and Wang, 2022) shown in Fig. 3a and b. The level of agreement of the modelled surface velocities with observations is consistent with the result of E3 (Fig. 3c). The quality of the fits to surface velocities achieved across the four BMB experiments shows very little variation. For example, in the BMB_CAL experiment, the magnitude of surface velocity mismatch between simulations and observations (Rignot et al., 2017) for 90 % of the surface nodes on the ice shelf is less than 20 m a-1, and the average magnitude of the velocity mismatch is 8.8 m a-1. This suggests that the dynamic basis of the temperature simulation results is reliable.

A set of 3-D steady-state temperature distributions are computed using the results of the dynamical simulations with the four different BMB datasets described in Sect. 2.2. The simulated temperature profiles at the six borehole sites are extracted from these 3-D temperature fields (Fig. 8). Normalised depth, D‾=D/H (where D is the depth below the upper surface and H is the ice thickness), is used to facilitate comparisons between the simulated and measured vertical temperature profiles. This is more convenient due to thickness differences between borehole measurements and the BedMachine data (Morlighem, 2019; Morlighem et al., 2020) used in the model (Fig. 7). Within the marine ice band, the total ice thickness at sites AM01, AM04 and AM05 in the model is ∼ 60 m less than the borehole measurements, while the differences at sites AM02, AM03 and AM06 are significantly smaller, no more than 20 m.

The extracted simulated temperature profiles for the different BMB choices show different shapes at the six sites (Fig. 7), which reflect significant differences between simulated temperature fields. Differences between the various simulations and the borehole measurements at the upper surface are found at each site, especially for AM05, because the upper surface temperature in the simulations is fixed by the Antarctic surface temperature dataset (Comiso, 2000). As shown in Fig. 7, the simulation with BMB_ISMIP6 provides reasonable fitting at AM02 and AM06 (within the basal-melt area) but poor fitting at the sites with basal marine ice (AM01, AM04 and AM05) compared with other experiments, which is expected since BMB_ISMIP6 purely represents basal melting in those marine ice regions (Fig. 3a). The simulation with BMB_ROMS fits the borehole temperature profiles slightly better at the sites with marine ice, since basal accretion is considered in BMB_ROMS, while at AM02 and AM06 the effect of higher melt rates (see Fig. 3b) leads to poorer agreement than for BMB_ISMIP6. The BMB_ISMP6 and BMB_ROMS simulations give very similar but poor matches at AM03. The simulation with BMB_CAL provides better fitting results at most of the borehole sites but still does not reconstruct the near-isothermal marine ice layer at the bottom. In contrast, the simulation with BMB_CAL2 shows a much closer agreement in the lower part of the marine ice layer, which is visually close to the pressure freezing temperature line in the permeable marine ice layer. However, the manually increased basal accretion in BMB_CAL2 also leads to a severe overestimation of temperatures for the colder ice above the near-isothermal layer. For the sites outside the marine ice band (AM01, AM03 and AM06), the simulated temperature profiles from BMB_CAL and BMB_CAL2 show little difference. In general, the temperature field simulation with BMB_CAL best fits most of the borehole temperature profiles visually, which suggests that BMB_CAL is more representative of the real mass balance at the ice–ocean interface. Therefore, we mainly conduct detailed analysis and discussion on the simulations with BMB_CAL in the remainder of the paper.

To visualise the simulated steady-state temperature field, the depth-averaged temperature distribution and a series of temperature sections along and across the three flowlines are extracted from the simulation of BMB_CAL, as shown in Figs. 8 and 9. We note that the three flowlines here are derived from the simulated velocity field of the inversion simulations, which is almost the same as those obtained from the MEaSUREs Antarctica ice velocity data (Rignot et al., 2017). The distribution pattern of the depth-averaged temperature is strongly aligned with the ice shelf flow (Fig. 8). The depth-averaged ice temperature gradually increases downstream along the flow, which is the clearest along the AM03 flowline.

All the simulated temperature sections (Fig. 9) show internal cold ice advected from upstream inlet glaciers, originally formed due to downward advection of ice from cold, high-elevation regions far inland. The minimum internal temperature on the AM03 flowline is ∼-28 ∘C, lower than that of the JP flowline (∼-22 ∘C) and AM06 flowline (∼-24 ∘C). The cold ice is gradually warmed along the flowlines by heat conduction from warmer ice both above and below, as it propagates toward the ice front. Along the three flowlines, the tens of kilometres near the grounding line reflect the transition of the thermal structure from grounded ice sheet to floating ice shelf, all of which experience the steepening of the basal temperature gradient, associated with active basal melting. Along the JP flowline (Fig. 9a), the internal cold core, warmer than that of the AM03 flowline, gradually warms up and dissipates around JP (Fig. 9a, d). Marine ice starts to accrete around 150 km downstream of the grounding line (see the BMB curve in Fig. 9a), where the basal temperature gradient starts to decrease. The vertical temperature regime in the marine ice zone downstream of AM05 maintains a consistent profile (Fig. 9a). The simulated (and observed) vertical temperature profiles at AM05, AM04, and AM01 are all similar (Fig. 7a, b, c), essentially just scaling as the ice shelf thins despite the continuing accretion of marine ice. Along the AM03 flowline (Fig. 9b), the ice thickness decreases rapidly within 100 km downstream of the grounding line, accompanied by a significant steepening of the temperature gradient of the lower part of the ice shelf, where it experiences considerable basal melting (see the BMB curve in Fig. 9b). The large basal temperature gradient gradually eases until 200 km downstream of the grounding line where a BMB close to zero is reached. The cold core, approximately 30 km wide at AM03 (Fig. 9d), is mainly composed of the cold continental ice from the Mellor and Lambert glaciers, flowing through the southern grounding line of the AIS. Along the AM06 flowline (Fig. 9c), the temperature section illustrates the formation of cold core ice within the ice shelf. Internal temperature of the ice shelf upstream of AM06 is close to the surface temperature (Fig. 9c, d). The surface temperature increases significantly downstream of AM06, at which time the internal temperature is lower in comparison, resulting in the formation of a cold core. It can be found at AM02 (Fig. 9c, e) and propagates downstream all the way to the ice front. The basal temperature gradient along the AM06 flowline is relatively small upstream of AM06, associated with a gentler basal melting than the other two flowlines.

The transverse temperature sections exhibit a great variation in the thermal structure across the ice flow (Fig. 9d, e). Based on the AM01–AM02 transverse temperature section (Fig. 9e), in addition to the cold core on the AM03 flowline and at AM02, there is also internal cold ice (at 50–70 km along the transverse section) that has entered from Scylla and Charybdis glaciers to the west. The basal warm ice layer at AM01 is ∼ 30 km wide (75–105 km along the same section), which correlates well with the pattern of the marine ice band (Fig. 8). Similarly, the basal warm layer to the west, ∼ 30 km wide (15–45 km along the section), also corresponds to the distribution of the western marine ice band (Fig. 8).

To quantify the influence of englacial temperatures on ice viscosity (Eq. 2) we introduce a temperature-dependent ice stiffness factor: B(T′)=A(T′)-1/n. Motivated by Humbert (2010), we calculate the distribution of the depth-average of B(T′) across the AIS, B(T′)‾, using the steady-state temperature field of the BMB_CAL simulation (details presented in Appendix B) to illustrate the spatial variation of this aspect of the ice shelf dynamics. The pattern (Fig. B1) is very similar to that of the depth-averaged temperature (Fig. 8). The depth-averaged ice stiffness factor achieves its maximum east of the AM03 flowline, approximately 70 km downstream of the grounding line, once the warmer lower layers of ice flowing in from the continent have been melted away (Fig. 9b). As the ice keeps progressively warming along the flowlines downstream (Figs. 8; 9a, b, c), the B(T′)‾ decreases, corresponding to softening of the ice, while there is also significant lateral variation in stiffness across the ice flow. In most areas of the AIS (86 % in area), B(T′)‾ is between 1×108 and 1.8×108 Pa s1/3, with an average value of 1.48×108 Pa s1/3.

The use of the column model allows the specification of both the vertical velocity profile at each time step (or location) and, implicitly, the horizontal velocity along the flowline, in contrast to the 3-D modelling, where vertical velocities emerge from the 3-D dynamical simulations. A series of column simulations provides solutions for the time-stepping of the vertical 1-D advection–diffusion equation with specific boundary conditions (as discussed in Sect. 2.3). Vertical temperature profiles for each borehole on the two flowlines are extracted and compared with those from the 3-D simulation of BMB_CAL and observations (Fig. 10). The column simulations achieve slightly better fitting results than the 3-D simulation at AM01, AM03 and AM05. There is also no difference in thickness between the column model and the observations at the borehole sites, which is not the case for the 3-D model. Since the boundary temperature conditions for the 1-D model are taken directly from the field observations at the borehole sites, the temperature profiles from the 1-D simulations strictly match the observations on the upper and lower surfaces (Fig. 10). From AM05 to AM04, the prescribed surface temperature warms by 1.8 ∘C (see Fig. 5) and is reflected in the modelled cold core within 200 m of the upper surface in AM04 temperature profiles (Fig. 10b).

Distinct thermal structures are evident for the areas with or without a basal marine ice layer from the observed borehole temperature profiles and the simulations. Vertical advection, determined by surface and basal mass balance (melting and freezing) and vertical strain rates, strongly affects the vertical thermal regime at each location. Thermal conduction is also significant in the vertical direction, smoothing the temperature profile. Horizontal advection transports the local thermal regime from one location to another, thereby establishing the spatial pattern of the temperature distribution. For the marine ice layer, its distinct material properties and the hydraulic interaction between the porous layer and the ocean below dominate the local thermal regime.

Focussing on vertical advection, basal melting creates downward ice advection deep in the ice shelf. The internal ice, with lower temperature, is therefore advected closer to the base where the ice is warmer due to the ocean contact, resulting in a significant increase in temperature gradient near the base of the ice shelf. This effect can be seen in both borehole temperature profiles and the simulations. Large basal temperature gradients are observed at the AM02, AM03 and AM06 sites in the basal ablation zone (Fig. 6b; Table 3). In the simulated temperature field for BMB_CAL, the maximum basal temperature gradient of -0.8 ∘C m-1, occurs in the southern part of the AIS, where basal melting is ∼ 7 m a-1 and downstream of a region where melt rate exceeds 15 m a-1. Large gradients have also been observed at the base of other Antarctic ice shelves, such as at the S1 site on the Fimbulisen (Orheim et al., 1990a, b; modelled in Humbert, 2010) and the McMurdo Ice Shelf (Kobs et al., 2014). For McMurdo Ice Shelf, the average observed temperature gradient at the base of the borehole is -0.38 ∘C m-1, associated with an estimated basal melt rate of 1.05 m a-1. In contrast, basal refreezing creates an upward advection of warm accreted marine ice and decreases the basal temperature gradients, a feature illustrated by the simulated temperature profiles at AM05, AM04 and AM01 within the marine ice band (Figs. 7a, b, c; 9a). In the temperature section along the JP flowline (Fig. 9a), the dissipation of the cold core at JP is associated with the upward advection of basal warm ice. Accretion of marine ice dominates the thermal regime of the lower part of the ice shelf downstream of JP. Similarly, the SMB (accumulation/ablation) also causes vertical advection of temperature and affects the pattern of the vertical thermal regime. However, the SMB is always smaller in magnitude than the BMB and has less spatial variation across the AIS. According to the RACMO model data (Van Wessem et al., 2018), the surface accumulation rates of the AIS averaged from 1979 to 2017 range from approximately -0.03 to +0.6 m a-1 ice equivalent. In contrast, the range of the BMB is approximately -30 to +3 m a-1 (Adusumilli et al., 2020). Considering the magnitude of the two mass balances, the effect of SMB on thermal structure should be significantly less than that of the BMB. We note that the nearly isothermal profile of the upper meteoric ice observed at AM05 (Fig. 6a) cannot be explained by general surface accumulation and that horizontal ice advection is the more significant ingredient as discussed below. Vertical strain also contributes to the vertical temperature profile. Vertical strain results from horizontal divergence in the depth-averaged flow regime and in ice shelves is mainly due to extensional flow. As ice flows towards the calving front, it generally accelerates and thins. Strain thinning acts to compress the ice column and steepen the vertical temperature gradient in the ice shelf (Craven et al., 2009). Comparing the borehole temperature profiles at AM01 and AM04 (Fig. 6a, and Table 3) above the marine ice layer, the internal temperature gradient at AM01 is significantly greater than at AM04, which basically reveals the effect of strain thinning.

Horizontal ice advection transports cold ice originally deposited at high elevations on the grounded ice sheet to the downstream ice shelf and gives rise to the formation of the core of cold ice, well reflected in the temperature sections from some of the 3-D simulations (e.g. BMB_CAL in Fig. 9) and observed at AM03. Along the AM03 and AM06 flowlines (Fig. 9b, c), the cold continental ice from the upstream tributaries persists to the ice front and dominates the internal temperature regime along the ice shelf. Compared with the cold core observed at AM02 (Fig. 6b), that of AM03 is proportionally much lower in the vertical column, which is determined by the origin of the coldest ice and the influences of surface accumulation and basal melting, as well as the evolution of ice surface temperature. Our results indicate that the evolution of thermal structure along the flowlines is accompanied by the warming of the meteoric ice, contributed by internal thermal conduction.

The porous structure of the lower part of the marine ice and its hydraulic connection with the ocean below give rise to the near-isothermal basal layer (Fig. 6a). Craven et al. (2009) regarded the hydraulic connection depth encountered in drilling as an approximation to the effective pore close-off depth. Beneath the hydraulic connection depth, the permeable marine ice has interconnected channels and cells, filled with seawater (Craven et al., 2009). The relatively free movement of the seawater within the pores keeps the ice–seawater mixture at the in situ pressure-dependent seawater freezing temperature (McDougall et al., 2014), as shown in the subplot of Fig. 6a. Above the hydraulic connection depth, there is still apparently residual brine trapped in the pores of the impermeable marine ice, observed through borehole video imagery and in ice core samples (Craven et al., 2005, 2009). These brine inclusions decrease in volume by freezing at the walls, becoming saltier and lowering the freezing point of the residual brine, allowing the two-phase material to further cool. Available salinity measurements on ice core samples recovered from the AM01 borehole reveal that the total salinity of the upper impermeable layer is very low (Craven et al., 2009). Above the permeable layer, the consolidated marine ice gradually cools, which is confirmed by the temperature drop of the meteoric–marine ice interface from AM04 to AM01 (Craven et al., 2009).

Both the 1-D and 3-D simulations produce imperfect fits to the observed vertical temperature profiles at the six borehole sites (Figs. 7, 10). The discrepancies are most notable in the regions that have a marine ice layer. Our modelling approach contains assumptions, limitations and sensitivities that are pertinent to consider when interpreting the model outputs, some of which may contribute to this model–data discrepancy. We discuss these limitations and some possible avenues to address them before proceeding to the implications of our present modelling studies.