Ice rises are areas of locally grounded, slow-moving ice adjacent
to floating ice shelves. Temperature profiles measured through ice rises
contain information regarding changes to their dynamic evolution and
external forcings, such as past surface temperatures, past accumulation
rates and geothermal heat flux. While previous work has used borehole
temperature–depth measurements to infer one or two such parameters, there
has been no systematic investigation of parameter sensitivity to the
interplay of multiple external forcings and dynamic changes. A
one-dimensional vertical heat flow forward model developed here examines how
changing forcings affect temperature profiles. Further, using both synthetic
data and previous measurements from the Crary Ice Rise in Antarctica, we use
our model in a Markov chain Monte Carlo inversion to demonstrate that this
method has potential as a useful dating technique that can be implemented at
ice rises across Antarctica. However, we also highlight the non-uniqueness
of previous ice rise formation dating based on temperature profiles, showing
that using nominal values for forcing parameters, without taking into
account their realistic uncertainties, can lead to underestimation of dating
uncertainty. In particular, geothermal heat flux represents the dominant
source of uncertainty in ice rise age estimation. For instance, in Crary Ice
Rise higher heat flux values (i.e. about 90 mW m
Present-day englacial temperatures are the product of the millennial-scale histories of ice flow and thermal boundary conditions experienced by an ice sheet (Robin, 1955). Temperature measurements from boreholes drilled through ice sheets have been widely used to extract important palaeoclimatic archives, such as surface temperature and accumulation history, as well as information about the conditions at the base of an ice sheet (i.e. glacial thermal regime and geothermal heat flux), both in Antarctica and Greenland (see Dahl-Jensen and Johnsen, 1986; Dahl-Jensen et al., 1998; Engelhardt, 2004a; Orsi et al., 2012; Cuffey et al., 2016). In Antarctica, the ice sheet contains ice rises – regions of slow-flowing, locally elevated grounded ice embedded within or adjacent to fast-flowing, floating ice shelves; one way they form is through an ice-shelf grounding on marine bed (e.g. Martin and Sanderson, 1980; Matsuoka et al., 2015; Wearing and Kingslake, 2020). This shift in boundary condition at the base of an ice sheet results in a transient evolution of the temperature–depth profile within an ice rise (MacAyeal and Thomas, 1980; Bindschadler et al., 1990). Therefore, due to their proximity to the marine ice-sheet periphery and negligible horizontal flow, ice rises can retain an imprint of past grounding line migration on millennial timescales, a record that is otherwise largely inaccessible beneath the ice sheet or its fringing ice shelves (e.g. Conway et al., 1999; Matsuoka et al., 2015; Kingslake et al., 2018; Neuhaus et al., 2021).
Past work used temperature–depth measurements within ice rises in Antarctica to estimate the timing of ice-shelf grounding. For instance, Bindschadler et al. (1990) developed an advection–diffusion thermal model of Crary Ice Rise, West Antarctica (Fig. 1). The model calculated the initial steady-state ice-shelf temperature profile from the specified set of parameters, including ice thickness, surface temperatures and accumulation rates. The steady-state ice-shelf profile was perturbed using thermal properties of the bed (geothermal heat flux, diffusivity and conductivity of the bedrock) and a specified vertical ice velocity function to calculate transient thermal evolution after ice-shelf grounding. The modelled temperature profiles were then compared to in situ borehole measurements at Crary Ice Rise, and minimising the mismatch between the measured and synthetic profiles yielded the best age estimate of the ice rise in its thickest part to be 1100 years. Recent work by Neuhaus et al. (2021) built upon the model of Bindschadler et al. (1990) to evaluate the timing of grounding at three sites in the Ross Sea sector of Antarctica, where previous measurements showed anomalously high basal temperature gradients (Engelhardt, 2004a). These results largely corroborate hypotheses of late Holocene re-advance in the region (Kingslake et al., 2018) and associated grounding at these sites between 1100 and 500 years ago (Neuhaus et al., 2021).
Present-day distribution of key parameters affecting temperature–depth profiles in Antarctica.
Thus, the methods used in these studies have potential if future boreholes are drilled at Antarctic ice rises in locations suspected of undergoing significant dynamic changes. Yet, the uncertainties inherent in these approaches must be carefully assessed to target drilling, maximise the utility of borehole drilling and increase the accuracy of ice dynamics inferences and palaeoclimatic inferences. Previous work has included sensitivity tests where some predefined variables, such as accumulation, ice thickening and melt rate, were assigned several different values to examine how they affected the final temperature profile and their relation to inferred timing of grounding (Bindschadler et al., 1990; Neuhaus et al., 2021). Yet, there is a lack of systematic investigations of temperature profile sensitivity to the cumulative effects of multiple external forcings and dynamic changes, particularly given that some parameters (e.g. geothermal heat flux) have considerable uncertainties (e.g. Fudge et al., 2019). In addition, time-variable parameters, such as ice thickness, accumulation and surface temperature may significantly increase the dimensionality of the problem, solutions to which need optimised inversion methods, as opposed to exhaustive global search algorithms where highly dimensional inversion tasks become computationally unfeasible (Mosegaard and Tarantola, 1995).
Here we use forward modelling to investigate how the interplay between forcings and parameters (i.e. surface temperature, accumulation rate, heat flux and thickness history) affects the englacial temperatures. Using previous measurements from the Crary Ice Rise, West Antarctica (Fig. 1), we also implement a Markov chain Monte Carlo inversion to explore the contributions of multiple uncertainties to the inferred timing of ice rise formation.
In this section we outline the numerical forward model and inversion method. The forward model builds upon and extends the models used by Alley and Koci (1990), Orsi et al. (2012), and Neuhaus et al. (2021). The inversion method is similar to that previously used to reconstruct past surface temperatures in Greenland and Antarctica (e.g. Dahl-Jensen et al., 1998, 1999). Extending previous work, we include the effects of different vertical velocity functions, we use an optimised set of some parameters (e.g. pressure melting/freezing point of ice/seawater), and we introduce temporal variability to ice thickness, surface temperature and accumulation rates, as well as multiple phases of grounding and ungrounding and corresponding changes to the boundary conditions.
We use the following form of the vertical diffusion–advection equation to
simulate the time and depth evolution of temperature
For sensitivity experiments, we implement three analytical approximations
for vertical velocity
Finally, a simplified approximation where vertical velocity varies linearly
with depth from zero at the base to a maximum value at the surface,
following Bindschadler et al. (1990), was implemented for sensitivity
experiments and estimation of associated uncertainties:
We define a one-dimensional spatial domain that extends vertically from the
bedrock base to the ice surface. The vertical coordinate is
In the ice-shelf scenario (Fig. 2a) boundary conditions at the ice surface
are set by time-variable surface temperatures
Geometry of the ice shelf and ice rise heat flow model. The simple one-dimensional
vertical model is implemented along a stationary crest of hypothetical ice
rise (
When modelling temperatures within a grounded ice sheet (i.e. an ice rise; see
Fig. 2b), the boundary condition at the base of the ice column is set such
that the vertical gradient in
Parameters characterising the properties of ice and the bedrock are assumed to be constant (Table 1). Values of geothermal heat flux at specific locations are sampled from the Antarctic-wide heat flux data compilation derived from spectral analysis of airborne magnetic data (Martos et al., 2017). Surface temperature evolution, accumulation rate histories and ice-thickness histories are sampled from distributions provided by an ensemble of simulations using the Parallel Ice Sheet Model (PISM) (Fig. 1g–i; Kingslake et al., 2018; Albrecht et al., 2020a, b). PISM is a three-dimensional, thermomechanical ice-sheet model that solves a hybrid shallow approximation of Stokes flow. PISM produces continental-scale (16 km grid cell size), long-term (multi-millennia length) simulations of ice-sheet thickness. Associated with these fields are surface temperatures reconstructed from the West Antarctic Ice Sheet (WAIS) divide ice core (Cuffey et al., 2016), scaled by modelled ice-surface elevation and accumulation patterns simulated by the regional climate model RACMOv2.1 (Ligtenberg et al., 2013) and scaled by 2 % per degree of temperature change from present (Kingslake et al., 2018). Other specifications and parameters used for the chosen PISM model output, including mantle viscosity and flexural rigidity, are described in detail in Kingslake et al. (2018). Despite inherent uncertainties associated with large-scale numerical models, which have relatively coarse resolution compared to the length scales of ice rises, PISM outputs provide first-order estimates of prior information about the time-variable parameters (i.e. temperature, accumulation and ice thickness) at any specified location on the Antarctic Ice Sheet (AIS) (Fig. 1e–f).
Numerical values of the parameters used in the simulations.
Prior to imposing time-variable forcings, we allow the simulation to reach a
steady state using surface temperatures, thicknesses and accumulation rates
equal to their values at the beginning of the chosen simulation period (Fig. 1e–g). The transient temperature profile is then simulated using the
histories of surface temperatures, ice thickness and accumulation rates
throughout the specified period. During the transient simulation, once a
grounding/ungrounding event is introduced, the boundary conditions at the
base of the ice column switch accordingly (see Sect. 2.1.2). The
calculated temperature profile at the last time step of the chosen period
represents the final product of a single simulation. For forward sensitivity
experiments, we perform an ensemble of simulations in which uncertain
parameters are perturbed. The resultant temperature profiles are then
compared to examine how altering our prior parameters affected the final
temperature profiles. In order to evaluate the fit between two temperature
profiles, we used a root-mean-squared mismatch:
To increase computation efficiency in our forward simulations where the Monte Carlo approach is used (i.e. where parameters are repeatedly randomly sampled from a range of their prior distributions), we use Dask, a flexible library for parallel computing in Python, which is implemented using cloud-based computing clusters provided by the Pangeo project (Odaka et al., 2019). Pangeo is a developing ecosystem of scalable, open-source tools for cloud-based parallel computation and interactive large-scale computation and data analysis. Automatic parallelisation on tens to hundreds of workers significantly increases computation performance, as compared to a standard approach using a desktop computer, and is particularly applicable to tasks that are easily parallelised, such as forward Monte Carlo simulations.
Where borehole temperature–depth measurements are available, they can be used to infer the history of dynamic change and evolution of boundary conditions (e.g. Dahl-Jensen et al., 1999) via inversion (e.g. Mosegaard and Tarantola, 1995). In this section, we outline what observational data and inversion methods we use to infer past ice rise evolution.
Among numerous ice rises mapped across the AIS, only a few have been sampled
by borehole temperature measurements (Matsuoka et al., 2015). These sites
include Siple Dome and Crary Ice Rise of the Ross Ice Shelf, as well as Mill
Island of the Shackleton Ice Shelf in East Antarctica (e.g. Koci and
Bindschadler, 1989; Engelhardt, 2004b; MacGregor et al., 2007;
Roberts et al., 2013). We digitised Bindschadler et al.'s (1990) temperature
observations from Crary Ice Rise and used them as input data for our
Bayesian inversion method (i.e. probabilistic data analysis that involves
using the prior information and computing the posterior probability
distribution for the parameters of the model; see Sect. 2.2.2). Due to the
quality of available data, digitisation implies inherent uncertainties (up
to 0.1
The Markov chain Monte Carlo (MCMC) method tests randomly selected
combinations of prior variables using a random walk through a
high-dimensional parameter space. The variables are assumed to be “unknown”
parameters and are prescribed prior distributions (or simply a range of
realistic prior values). The forward model uses these parameters from their
prior distributions as inputs, and its output is compared to the measured
(or synthetic) temperature profiles. In each step of random walk with
predefined length, a perturbed model of the current model is proposed. The
MCMC then uses a likelihood function to estimate the agreement between the
modelled and measured profiles (Mosegaard and Tarantola, 1995; Dahl-Jensen
et al., 1998):
The perturbed model is either “rejected”, in which case a new random perturbation is applied from the same starting location in parameter space, or “accepted”, in which case this location becomes the new starting location for the next random perturbation. Even when a new perturbation yields better fit to the observations, the model may be rejected, or if the new perturbation produces larger misfit, the model may be accepted. Whether a model is accepted is based on an acceptance probability (see Dahl-Jensen et al., 1999), which introduces a degree of stochasticity, ensuring that the random walk avoids entrapment in local likelihood maxima (Mosegaard and Tarantola, 1995). Eventually, the paths converge towards the regions corresponding to model parameters that yield the lowest misfit between modelled and observation temperature profiles.
In practice, MCMC can be used to infer any inputs to the forward model. For
instance, in the Dye 3 borehole drilled through the 3 km thick Greenland Ice
Sheet, the unknown temperature history has been divided into 125 intervals,
which, together with heat flux (also assumed to be unknown), yielded a
126-dimensional parameter space (Dahl-Jensen et al., 1999). Here, we focus
on inferring past dynamic changes (i.e. timing of grounding) but also
assume heat flux, temperature, accumulation and thickening rates to be
unknown in order to explore the possible combinations of realistic parameter
values that could provide a close fit between the field observations and
modelled temperature–depth profiles. To perform the MCMC we used
Prior to using actual borehole measurements, we tested the MCMC inversion
method on a synthetic temperature profile calculated using a simplified
forward model that uses the Lliboutry vertical velocity function and assumes
a hypothetical 1000 m thick, 1 kyr old ice rise (i.e. formed when an ice
shelf grounded 1000 years ago), forced by a heat flux of 50 mW m
Inversion of the synthetic temperature profile data. Synthetic
temperature–depth profile (dotted red line in panel
To implement our MCMC inversion with englacial temperature data from Crary
Ice Rise, we prescribe a wide range of prior values for model parameters.
This allows us to probabilistically evaluate the possible combinations of
realistic parameter values that could closely match the measured
temperature–depth profiles (Fig. 4). For example, instead of assigning a
heat flux of 77 mW m
Inversion of the temperature–depth measurements near Crary Ice Rise
Site D. Grey cloud of points illustrates the random walk in the parameter
space, showing the tested combinations of parameter values that yielded a
less than 0.3
Within the assigned prior parameter limits, the results show a wide range of
posteriors that fit the observed temperature–depth measurements to within
measurement errors reported by Bindschadler et al. (1990) (i.e. rms
The distributions of the coloured points in the scatter plots in Fig. 4
provide insight into the dependence of each parameter and its uncertainty on
the other parameters. For example, as with synthetic data inversion (Sect. 3.2.1), the inferred age of ice rise formation is strongly dependent on heat
flux (Fig. 4c). Higher heat flux (i.e. about 90 mW m
To explore temperature profile sensitivity to model parameters and to identify which parameters are particularly important for ice rise dating and borehole thermometry, a series of forward simulations (Figs. 5a–c, 6) and Monte Carlo experiments (Fig. 5d–f) were conducted for each parameter under consideration. The resultant temperature profiles (Fig. 5a–c) were then directly compared for different ice thicknesses (Fig. 5d–f).
Englacial vertical temperature profile sensitivity to model
parameters.
Englacial vertical temperature profile sensitivity to vertical
velocity function and accumulation/surface temperature temporal variability.
Temperature effects of changing heat flux and surface temperatures by a
fixed value are similar for ice of a given thickness (e.g. for
We also investigate how different velocity functions and temporal
variability of thickness, accumulation and surface temperatures may affect
the distribution of temperature within grounded ice of various thicknesses
(Fig. 6). For these simulations, variable temperature and accumulation for
the last 40 kyr were obtained from PISM simulation outputs (Fig. 1g–i). Advection effects of vertical velocity approximations on englacial
temperature profiles are illustrated in Fig. 6a and d. Temperatures are
affected most in the lower part of the ice column (Fig. 6a). Deviations
between profiles produced with Dansgaard and Johnsen (1969) and Lliboutry (1979) velocity approximations also increase with ice thickness. For
example, in 500 m thick ice the rms difference between two profiles is
0.18
Inversion experiments described in Sect. 3.1 assumed constant surface
temperatures,
Overall, forward sensitivity experiments outlined in this section provide insight into what model parameters have the strongest effect on the ice column, how these effects manifest in the shape of temperature profile and how this is affected by ice thickness. For thick ice, more likely to be found in the ice-sheet interior, parameters such as accumulation rates and the vertical velocity profile affect the temperature profile more strongly compared to thinner ice, more characteristic of ice-sheet periphery, including ice rises (Fig. 6). These results highlight the importance of careful vertical velocity parameterisation for borehole thermometry experiments in thick ice-sheet settings. In contrast, knowledge of ice-sheet thickening history is more important for dating shallow ice rises (Fig. 6c).
Bayesian inversion of englacial temperature–depth profiles indicates that
inferences of ice rise age (i.e. timing of grounding) may significantly
vary depending on the values of other forcing parameters. Among these,
geothermal heat flux may have a significant effect on the inferred timing of
grounding, with lower heat flux yielding earlier age estimates, along with
much smaller corresponding uncertainties (that is, narrower range of
solutions acceptable within a prescribed degree of misfit; see Figs. 3, 4). In
the case of the Crary Ice Rise inversion experiment, we infer a range of
possible ice rise age that encompasses the value of 1100 years previously
reported by Bindschadler et al. (1990) but increases from 500
The effect of heat flux may be significant when prior knowledge about its
values is poor or unconstrained. For instance, in the synthetic data
inversion experiment with prescribed grounding timing of 1000 years, heat
flux of 30 mW m
Uncertainties associated with ice rise age inversion from
temperature–depth profiles.
The uncertainties in ice rise age estimates are also determined by the
degree of accepted misfit between predicted and measured temperature
profiles, which in turn relies on the accuracy of englacial borehole
thermometry. Depending on the instrumentation used, previous
temperature–depth data have been collected with typical uncertainties of
around 0.1
Higher accuracy of englacial thermometry implies that the grounding-induced
evolution of the temperature profile can be detected with two measurements
separated by a few decades, which could potentially be utilised in
previously drilled boreholes (Fig. 7b). In the case of Crary Ice Rise, a
series of forward models shows that a 30-year increase in the timing of
grounding (e.g. difference between temperature profiles within 1030 and
1000-year-old ice rise) may yield a difference in englacial temperatures
that could be detected in the lower part of the ice column, as well as the
upper section of the underlying sediment/bedrock (Fig. 7b). Therefore, if
a sufficiently deep borehole is drilled, measurements through the underlying
bedrock may provide useful additional constraints on both the heat flux and
the timing of ice rise formation. However, following grounding, rapid attenuation of temperature differences with time implies that this method could only be applicable for ice rises that experienced recent grounding (i.e. less than
around 1 ka), assuming measurement uncertainties of 0.02
The Bayesian inversion method presented here has potential use in providing information about other model parameters, including time-variable thickness, temperature and accumulation rates, as previously done in multiple locations in Greenland and Antarctica (e.g. Dahl-Jensen et al., 1998, 1999; Waddington et al., 2005; Orsi et al., 2012; Cuffey et al., 2016). Since the addition of temporal variability to external forcings significantly increases the dimensionality of the inverse problem, with associated exponential growth of the computation cost, here we largely focus on thermal effects of dynamic ice rise evolution. However, our forward simulations show the variable impact of these forcings on englacial temperature distribution and their variability with ice thickness and depth within the ice column (Figs. 5, 6). For example, the effects of relatively small perturbations to accumulation rates are generally greater for thicker ice, whereas the opposite is true for timing of grounding and thickening rates, which play a more important role when ice is thin. Anomalies associated with changes to most parameters under consideration typically increase with depth (insets of Figs. 5a–c and 6a–d), with the exception of surface temperatures, which exert a strong influence on the temperature–depth distribution in the upper part of the ice column (e.g. Dahl-Jensen et al., 1998). This implies that the upper part of the ice column is likely to contain a more recent and less diffused record of past surface temperatures (e.g. Orsi et al., 2012).
Matsuoka et al.'s (2015) Antarctic ice rise inventory shows that ice rises rarely exceed 500 m in thickness (Fig. 1d), suggesting that these features may store a stronger signal of past dynamic changes and thickening history, whereas thick, permanently grounded ice domes may retain more information about accumulation and temperature histories and are thus more appropriate locations for palaeoclimatic inferences from englacial temperature measurements (e.g. Dahl-Johnsen et al., 1999; Engelhardt, 2004b; Orsi et al., 2012; Cuffey et al., 2016). Forward model experiments with different vertical velocity approximations show that the impact of vertical ice flow parameterisation becomes more significant for thicker ice (Fig. 6a, d). Yet, previous studies from Greenland and Antarctica have also shown that analytical ice flow approximations from Dansgaard and Johnsen (1969) and Lliboutry (1979) cannot fully capture the nonlinearity of vertical velocity profiles in ice divide/ice rise settings, and phase-sensitive radar measurements can provide useful additional constraints on vertical ice-sheet velocities (e.g. Gillet-Chaulet et al., 2011; Kingslake et al., 2014; Buizert et al., 2021). Therefore, integration of these techniques would help improve inferences of external forcings from borehole temperatures, in particular surface temperature and accumulation histories from deep boreholes.
Building on Bindschadler et al.'s (1990) foundational work and Neuhaus et al.'s (2021) more recent study, we have demonstrated how this method has potential as a useful dating technique that can be implemented at ice rises across Antarctica where direct geological sampling methods are inaccessible (e.g. Bentley et al., 2010; Spector et al., 2018). Integrating this technique with other methods, such as (1) indirect estimates of timing of grounding from radar observations and modelling (e.g. Schroeder et al., 2014; Kingslake et al., 2016; Wearing and Kingslake, 2019); (2) parameterisation of vertical velocities (Kingslake et al., 2014); and (3) adoption of more tightly restricted, informative prior constraints from geochemical ice-core data (e.g. for past temperature proxies; Cuffey et al., 2016) will allow for more accurate inferences of dynamic ice-rise/ice-sheet evolution and grounding line migration (e.g. Orsi et al., 2012). Moreover, we have demonstrated an approach for better quantifying uncertainties in these inferences. Borehole measurements through the upper tens of metres of underlying sediment/bedrock could place additional important constraints on both the geothermal heat flux and ice-rise evolution. This technique could even provide insights into dynamic ice-sheet evolution if future boreholes are drilled through floating ice and sediment in the vicinity of the grounding line, in places where recent ungrounding has left a pronounced vertical temperature anomaly within both ice and sediment/bedrock columns.
Antarctic ice rises and their suitability for dating of the potential Holocene ice-sheet regrounding. Circles correspond to the locations of ice rises from an Antarctic-wide inventory presented in Matsuoka et al. (2015). Sizes of the circles correspond to the mean present-day accumulation rates at their locations. Basemap shows distribution of basal geothermal heat flux presented in Martos et al. (2017). Areas where ice rise inverse dating techniques could be best applied are characterised by low heat flux, high ice thickness and low accumulation rates.
Our results prompt the question of what characteristics make a location favourable for borehole drilling and measuring temperature–depth data within an ice rise. Due to the diffusive nature of the englacial thermal signal, and as synthetic data experiments have shown, the temporal resolution decreases with time since grounding (Figs. 3, 5b, e, 7a). Therefore, ages of ice rises that are over 4 kyr old may be difficult to determine accurately, subject to other parameters like heat flux and thickening rates. Areas that are located close to the present-day grounding line, (and thus more likely to have been formed relatively recently) with well-constrained, low values of heat flux and low thickening rates, could represent optimal locations for implementation of this method and could yield accurate (i.e. on the order of 10 %) ice rise age estimations. Kingslake et al. (2018), Venturelli et al. (2020) and Neuhaus et al. (2021) have shown evidence of potential regrounding across large areas of West Antarctica. Preliminary investigations, juxtaposing these maps of potential grounding line migrations (Kingslake et al., 2018; Albrecht et al., 2020b) with Matsuoka et al.'s (2015) ice rise inventory (Fig. 1a–d) and Martos et al.'s (2017) heat flux model (Fig. 8) shows several ice rises where ice is relatively thick and accumulation rates are relatively low, for example Korff Ice Rise in the Ronne Ice Shelf area (Kingslake et al., 2016) and small ice rises to the east of the Weddell Sea. These ice rises could prove to be optimal locations for application of this technique. Future work could systematically quantify the suitability of these locations following this approach.
In this paper, we combine Bayesian inversion and forward modelling to make
an evaluation of uncertainties inherent in inferences of ice rise dynamic
evolution from temperature–depth profiles. Tested with both synthetic
datasets and borehole temperature measurements from Crary Ice Rise, Ross Sea
Embayment, our method explores the interplay between surface temperature, rates
of accumulation and thickening, geothermal heat flux, and parameterised
vertical velocities. We show that depending on the accuracy of borehole
thermometry, the same temperature profile (within the accuracy of
measurements) may result from a range of forcing parameters, of which
geothermal heat flux through underlying bedrock plays a particularly
important role. The key implication is that careful model parameterisation
and evaluation of uncertainties are essential to infer dynamic ice rise
evolution from borehole thermometry. We highlight that uncertainties in
inferred ice-formation time may increase significantly with ice rise age.
Accuracy of inversion relies on the low measurement uncertainties (i.e.
The code and data related to this article is available online at:
The supplement related to this article is available online at:
AM co-designed this research, performed the analysis and wrote the manuscript. JK co-designed this research and contributed to the writing and editing of the manuscript.
The contact author has declared that neither of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Aleksandr Montelli is grateful to the Schmidt Science Fellows for funding this research. We thank Thorsten Albrecht for granting access to the PISM model outputs and Nicholas Holschuh for helpful discussions of temperature measurements from Crary Ice Rise.
This research has been supported by the Schmidt Science Fellows to Aleksandr Montelli.
This paper was edited by Nanna Bjørnholt Karlsson and reviewed by Tyler J. Fudge and one anonymous referee.