Constraining Antarctic coastal ocean properties is a formidable challenge given that the Southern Ocean remains a huge data desert . In particular the continental-shelf regions are sparsely sampled, with large biases toward the sea-ice-free summer season. Ice shelf cavities are even more sparsely sampled and are not included in continental- or global-scale datasets. Therefore, observation-based products often have biases near the coast, particularly when they have been interpolated or extrapolated to fill data gaps. Ocean reanalyses and model products also have trouble properly representing coastal water masses, mostly because of scales that are not properly resolved .

Meanwhile, significant advances in hydrographic observations around the Antarctic continent have been made through the use of sensor-equipped marine mammals that yielded thousands of temperature and salinity profiles in coastal waters, including significant spatial coverage during wintertime conditions . Whereas data from Argo floats, ship cruises, and satellites are incorporated into most traditional ocean climatology products, observations from marine mammals were not yet included in these products at the time this project began. Fortunately, the Marine Mammals Exploring Oceans from Pole to Pole (MEOP) community had recently released a publicly available, standardized, and quality-controlled global dataset .

Thus, to obtain an improved estimate of present-day, three-dimensional fields of temperature and salinity of the coastal ocean around Antarctica, we begin by combining data from two traditional ocean climatologies: a prerelease from September 2018 of the NOAA World Ocean Atlas 2018 dataset WOA18p; and the Met Office EN4 subsurface ocean profiles (EN4; ) with the complementary MEOP data. We use the “statistical mean” (average of all available values at each standard depth level in each 1∘ square) rather than the “objectively analyzed mean” (interpolation from irregularly spaced locations to a fixed grid) values for WOA18p and EN4 because we wanted to perform interpolation and extrapolation ourselves after combining the datasets. The WOA18p data have already been binned by the creators of the dataset on the native WOA18p grid (0.25∘ bins in latitude and longitude). We interpolate these data (first conservatively in the vertical and then bilinearly in the horizontal) to the ISMIP6 standard grid. Since the EN4 and MEOP data are provided at their original locations without binning, we are able to bin-average these datasets directly on the standard grid. A simple average of the three datasets is then performed. Since the WOA18p and EN4 products rely to some extent on the same data source, some data sources may be double-counted and have extra influence on the bin average. This is not likely to have a significant adverse affect on the results as double-counting would only be a problem in areas with an abundance of data, whereas the larger problem we face is the large data gaps when we use any one of these datasets on its own. The result is a combined climatology on the standard grid that still contains significant data gaps (though less than we would have with any one of the three data sources on its own). The combined product is then interpolated and extrapolated to generate continuous fields that also extend inside the ice shelf cavities, as described below.

We note that the temporal coverage of the datasets differ from one another, possibly skewing the temporal coverage of the climatology toward the second half of the 23 years spanned by the three datasets. For the WOA18p and EN4 datasets, we only use data from 1995 to 2017, while the MEOP record spans from 2004 to 2018. We further note that there is a mismatch between this time period and the 2003–2009 time frame, over which the satellite-derived basal melt observations used to calibrate the melt parameterization were obtained . Estimates of interannual variability of ocean properties exist only for a handful of coastal regions around Antarctica. Based on , we believe that the uncertainties due to temporal variability between these two time periods are smaller than those due to the spatial interpolation and extrapolation. Additionally, due to the higher frequency of summer observations, the combined climatology likely has significant seasonal biases. However, we expect these effects to have limited influence on the decadal-scale variability of subsurface ocean properties, which are expected to have the most influence on melt rates. The uncertainty due to large spatial gaps in observations and the resulting need for interpolation and extrapolation is likely to swamp the error resulting from any temporal bias.

Data density maps and comparisons of the merged product with objectively analyzed WOA18p fields show significantly increased data coverage and reduction in bias in certain regions (Fig. ). In particular, large parts of the narrow continental-shelf region surrounding East Antarctica show a subsurface warm bias on the order of 1 ∘C in the WOA18p data, while a similar cold bias is evident in the Bellingshausen Sea sector. Obviously, those biases will affect parameterized melt rates that largely depend on the coastal ocean temperatures. In July 2019, well after this project was underway, a final version of the World Ocean Atlas 2018 was released that also incorporated the MEOP observations. While the new WOA18 statistical means are close to those in our merged dataset, the objective analysis provided in WOA18 does not seem able to account for the strong horizontal gradients over the very narrow continental shelf of East Antarctica, which suggests that our merged dataset may be more adequate for providing continental-shelf properties for these regions (Fig. S1 in the Supplement).

Besides providing a reference of contemporary Antarctic coastal ocean temperature and salinity, the climatology obtained in the above method is used when computing the projected thermal forcing based on the CMIP future model anomalies. For this purpose, CMIP potential temperatures are converted to in situ temperature using the Gibbs SeaWater Oceanographic Toolbox of TEOS-10 . Then, temperature and salinity fields are interpolated onto the standard grid. Anomalies are calculated as the difference between CMIP annual means and the CMIP 1995–2014 average and added to the observed climatology. This general methodology can be used to obtain ocean temperature and salinity anomalies for any CMIP model under any emission scenario, while the general strategy to select the CMIP5 models used in the ISMIP6 experiments is described in .

The CMIP ocean model components typically include a coarse representation of the open ocean on the Antarctic continental shelf but do not explicitly resolve the circulation inside ice shelf cavities. Basal melt parameterizations such as those we propose for ISMIP6 require knowledge of the “ambient” ocean properties (i.e., ocean temperature and salinity unaffected by interactions with the melt plume), preferably as functions of depth within the ice shelf cavity. In addition, bathymetric features are known to control ocean properties in ice shelf cavities ; deep troughs will make it easier for warmer, deeper water masses to reach into the cavity, while sills will tend to block them. Our goal is to allow temperature and salinity fields from the observed climatology and CMIP projections to flood into the ice shelf cavities and regions below sea level that are presently covered by glacial ice while accounting for topographic barriers.

To accomplish this, ocean model and observational data are first conservatively interpolated in the vertical to a 20 m regular grid, then bilinearly remapped in the horizontal onto the ISMIP6 standard grid. Next, the extrapolation algorithm described below is applied first in the open ocean (outside of present-day ice shelf cavities) and then in ice-covered regions of each of the 16 independent sectors (see Sect.  and Fig. ). We assume that ice shelf cavities in separate sectors will remain disconnected from one another over the timescales of ISMIP6 runs so ocean properties are not interpolated across sectors. Note that because we perform extrapolation first in the open ocean and then in each basin, we do not use the portions of each basin in Fig.  that have been extended into the open ocean. The basins have only been extended for use by ISMIP6 ice sheet modelers, who may need the basins as part of the parameterizations described in Sect. .

In each basin, and separately in each horizontal layer, we convolve the resulting fields with a 2-D Gaussian kernel with a 1σ radius of 8 km to smooth the data at the grid scale and fill in missing values in open ocean. We allow this smoothing to extend the reach of the valid data by up to 12 km. This “flooding” only applies to regions where the bedrock topography, taken from Bedmap2 , is below the depth of the layer, meaning that bathymetric sills can block access of water masses deeper than the sill. This extrapolation via Gaussian smoothing is performed repeatedly, each time extending the reach of “valid” data by an additional 12 km, until no further cells with missing data can be reached. “Valid” data are only smoothed once in this process and are held fixed in subsequent iterations of the extrapolation process. We discovered that extrapolation by more than ∼12 km in one iteration results in unphysical mixing of qualitatively different water masses over narrow topographic features, including across the Antarctic Peninsula.

Deep ice shelf cavities blocked by sills will not be reached by purely horizontal extrapolation, so these deeper regions are filled in by copying the in situ temperature and salinity from overlying layers. Since ice sheet models will not necessarily use the Bedmap2 topography on the ISMIP6 standard grid as we have, we also copy ocean properties vertically below the bathymetry. This ensures that valid (and reasonable) values of ocean properties are available at all depths. To reduce the size of the final dataset, we conservatively interpolate from the 20 m vertical grid to a 60 m vertical grid.

We note that, in retrospect, we should have performed vertical extrapolation (that is, copying) using conservative temperature rather than in situ temperature because conservative temperature is the more appropriate quantity to remain constant with vertical advection. However, we estimate that this difference introduces an error in thermal forcing of no more than 0.1 K, which is certainly much smaller than other sources of uncertainty (observational errors, extrapolation, projection uncertainty, approximation error in the melt parameterization, etc.).

The extrapolation algorithm provides continuous, three-dimensional ocean fields that can be interpolated to any possible depth of the ice shelf base for use in the basal melt parameterization of the ISMIP6 models. This method keeps the same vertical structure inside ice shelf cavities as in the “ambient ocean” of observations or CMIP5 anomalies, which omits several physical processes. For example, the extrapolated “ambient” temperature inside of some of the large ice shelf cavities (e.g., Ross and Ronne–Filchner) are typically warmer than observed temperatures, which are often below the surface freezing point as a result of the high pressure and entrainment of meltwater. These processes are not represented in CMIP ocean models, which have no cavities, and there is no simple way of accounting for these in our extrapolation. Furthermore, the heat loss and freshwater input from ice shelf melting itself, the topographic steering by the ice draft topography, and the interaction of the buoyancy-driven flow in the cavity with the circulation outside of the cavity may result in feedback mechanisms that may increase or decrease the on-shore heat transport as a response to ice shelf melting e.g.,. Again, there is no simple way of including these processes in this effort. Finally, poor knowledge of bathymetry beneath many ice shelves may affect the accuracy of the extrapolated ocean properties e.g.,. The resulting thermal forcing along the current ice shelf drafts is shown in Sect. .

Given that melt rates have been estimated for more than 60 individual ice shelves , we could in theory calibrate the parameterizations with different parameters in each cavity. However, ice sheet models have evolving cavities, and their present-day ice shelves and ice flows do not necessarily correspond to the observed ones, depending on their initialization method . Furthermore, two initially distinct ice shelves may merge at some future time (e.g., Pine Island and Thwaites), leading to melt discontinuities if parameters are set at the scale of individual drainage basins. Therefore, we choose to calibrate parameters at the scale of larger sectors.

We start from the 18 sectors used in the latest ice sheet mass balance inter-comparison exercise (IMBIE) assessment and based on drainage basin boundaries defined from satellite ice sheet surface elevation and velocities . To obtain continuous melt rates underneath all the ice shelves, we merge the two sectors feeding the Ross ice shelf and the two sectors feeding the Ronne–Filchner ice shelf (Fig. ). To allow simulated ice shelves to be larger than currently observed, the 16 remaining sectors are then extended into the open ocean over the full ISMIP6 standard grid by finding the basin of the closest ice-covered point to a given point in the open ocean.

Melt rates in the common ISMIP6 experiments are derived using a slightly modified version of the nonlocal quadratic parameterization proposed by . The parameterization is explicitly defined over regional sectors rather than for a single ice shelf, and it includes a temperature correction: 1m(x,y)=γ0×ρswcpwρiLf2×TF(x,y,zdraft)+δTsector×〈TF〉draft∈sector+δTsector, where TF(x,y,zdraft) is the thermal forcing at the ice–ocean interface, and 〈TF〉draft∈sector is the thermal forcing averaged over all the ice shelves of an entire sector. The uniform coefficient γ0, with units of velocity, is somewhat similar to the exchange velocity commonly used to calculate ice–ocean heat fluxes e.g.,. The temperature correction δTsector for each sector is needed to reproduce observation-based melt rates (at the scale of a sector) from observation-based thermal forcing. The other constants are given in Table .

The temperature correction is introduced to account for biases in observational products, ocean property changes from the continental shelf to the ice shelf base (not accounted for in the aforementioned extrapolation), tidal effects, and other missing physics. A similar correction was used by . Because of the quadratic dependency of parameterized melt to thermal forcing, the melting sensitivity to ocean warming (i.e., the derivative) linearly depends on the present-day thermal forcing, which is a strong incentive to apply a temperature correction. Without temperature correction, a sector-dependent γ coefficient would be required to simulate the observation-derived melt rates in each sector. This approach would lead to γ ranging from 500 to 60 000 myr-1 in the different sectors, which seems difficult to justify with physical arguments. Differences across ice shelves in how efficiently available heat is converted to melting are expected as the melt-induced circulation may respond differently to a given thermal forcing depending on its specific geometry e.g., and on regional contrasts in the amplitude of tides . However, tides and cavity geometry unlikely account for efficiencies across ice shelves that differ by 2 orders of magnitude. Furthermore, as γ explains most of the melt sensitivity to increasing thermal forcing (see Eq. ), an approach with sector-dependent γ would produce sensitivities to future ocean warming that would be strongly influenced by the regional biases in the observational products used to estimate γ. For these reasons, we think that a constant γ0 for all of Antarctica is preferable for ISMIP6.

The nonlocal parameterization involves spatial integration, which may not be straightforward to implement for all the modeling groups. For those models which cannot implement the nonlocal parameterization, we propose a local version: 2m(x,y)=γ0×ρswcpwρiLf2×maxTF(x,y,zdraft)+δTsector,02, in which the max function is preferred to the absolute value of the last term on the right in order to avoid extreme melt rates when adjusting parameters in areas with both melting and refreezing.

We propose two calibration methods that provide both present-day, sector-integrated melt rates equal to observational estimates and different melt patterns and sensitivities to ocean warming. These two calibration methods are applied to both the local and nonlocal parameterizations. For both methods, the calibration is done in two stages. First, it is assumed that δTsector=0 in every sector, and we estimate γ0 based on observational constraints specific to each method. Then we calibrate δTsector to obtain present-day, sector-integrated melt rates equal to observational estimates. For all these estimates, we use temperatures and salinity from the climatological dataset described in Sect.  and the ice shelf geometry from Bedmap2 .

In the “MeanAnt” method, γ0 is calibrated in such a way that the parameterization reproduces the total Antarctic melt rate with no thermal forcing correction, i.e., 1325±175 Gtyr-1 or 1193±163 Gtyr-1 , where ± indicates standard deviation. To estimate a distribution of possible γ0 values, we take 105 random samples in both the total Antarctic melt rate and the error in thermal forcing using normal distributions based on the aforementioned melt values (equally sampling Rignot and Depoorter's datasets; ; ) and assuming an uncertainty of 0.17 K for the ocean thermal forcing. Using 105 samples gives percentiles of the γ0 distribution that converge with three significant digits. The 0.17 K uncertainty was calculated as the average temperature standard deviation at 500 m depth, between 80 and 60∘ S, only considering locations with more than three valid points in the original dataset (Sect. ) and neglecting the uncertainty in salinity in the calculation of freezing temperature. The random error applied to the ocean thermal forcing is sampled once per sector, i.e., we assume coherent errors at the scale of a sector, and the sector random error is added to the grid point thermal forcing. The MeanAnt method is summarized in Fig. .

The idea behind the second method, hereafter “PIGL”, is that total Antarctic melt rate may be less relevant than melt rates near deep grounding lines for ice sheet dynamics . We assume that the highest melt rates of Antarctica, found near Pine Island's deep grounding line, as well as the relatively high number of ocean observations in the Amundsen Sea provide a constraint on how Antarctic melt rates could respond to strong future ocean warming. In the PIGL method, we therefore use the spatial pattern of melt rates provided by (2013; here version 2.1 of their product is used), and we estimate γ0 by randomly sampling one of the 10 grid points with the highest melt rates (with equal probability) and associated thermal forcing (normally distributed uniform error over the entire Amundsen basin) underneath the Pine Island ice shelf. This is repeated 105 times to obtain the median, 5th, and 95th percentiles of γ0. The PIGL method is summarized in Fig. . The values obtained through the PIGL method are an order of magnitude greater than through the MeanAnt method, and the two distributions do not overlap (Table ). The PIGL median and 5th percentile γ0 values are 3 times higher for the nonlocal than for the local parameterization, which can be explained by the presence of refreezing in the first case, requiring a large γ0 to compensate small sector-averaged thermal forcing. For PIGL local, the 95th percentile γ0 takes values as large as the nonlocal case because δT corrections become strongly negative and make the max function in Eq. () produce zero melt at numerous grid points.

In the following, the parameterizations and calibration methods are sometimes referred to as “nonlocal MeanAnt” and “nonlocal PIGL” and similarly for the local version.

Then, we determine δT in each sector by iterations. For each specific γ0 value (e.g., median), we estimate δT 105 times by randomly sampling the sector-integrated melt rates (Gt yr-1) in normal distributions. Random errors are sampled independently for each ice shelf within a sector when using the melt rates from , with a normal distribution defined by the means and standard deviations given in their Table 1. When using melt rates from , random errors are sampled independently for each sector described in their supplementary Table 1. The median δT values corresponding to the median of the γ0 distributions are shown in Fig. . We note that MeanAnt δT values are positive and negative by construction, while PIGL δT values are negative, as expected if this correction represents changes in water mass properties along the ice draft (keeping in mind that it also likely accounts for missing physics). Median δT values corresponding to the 5th and 95th percentiles of the γ0 distributions are provided in the ISMIP6 repository (see data availability section).

After this two-stage calibration, we have a distribution of γ0 for the whole ice sheet and distributions of δT for each sector. The ISMIP6 protocol (Nowicki et al., in preparation) explicitly requires exploration of the sensitivity of ice sheet projections to γ0 using various γ0 values from Table  but taking the median δT values obtained for a given value of γ0. These experiments will thus highlight the uncertainty in the sensitivity of melting to ocean warming but for experiments that all start from the median observed melt rates. To further explore parameter uncertainty (beyond ISMIP6 experiments), it could be interesting to randomly sample δT independently in each sector and for each γ0 value to obtain a range of possible melt rates for their initial states, which would require running a much larger number of experiments to really sample the uncertainty in the different sectors.

To summarize, we suggest using either the nonlocal or the local quadratic parameterization in ISMIP6. For any of them, we recommend using two sets of parameters to account for the large uncertainty in parameterized melt rates: (i) the MeanAnt parameters, giving a sensitivity to ocean warming based on the present-day relationship between temperatures and the mean Antarctic melt rate, and (ii) the PIGL parameters, giving a sensitivity to ocean warming based on present-day high thermal forcing and melt rates near Pine Island's grounding line.

To illustrate the differences between the two calibration methods, we first consider the example of the nonlocal parameterization applied to the Ronne–Filchner and Amundsen sectors that are cold and warm, respectively (Fig. a). Without applying the thermal forcing correction (i.e., considering δT=0), the MeanAnt method produces melt rates in good agreement with observations between 400 and 1000 m in the Ronne–Filchner sector (dashed light-blue line in Fig. b) but underestimates melt rates at all depths in the warm cavities of the Amundsen sector by 1 order of magnitude (dashed red line in Fig. b) and in the deepest parts of Ronne–Filchner. Adding δT=1.07 K brings the Amundsen Sea sector-averaged value close to the observational estimate (solid light-red line in Fig. b), while no substantial correction is needed for Ronne–Filchner. Without δT correction, the PIGL method fits the highest melt value of Pine Island but overestimates melt rates in all cavities, imposing δT<0 almost everywhere (Fig. ). The MeanAnt method underestimates melt rates near the deepest parts of grounding lines for both the Amundsen and Ronne–Filchner sectors even after the thermal forcing correction. On the other hand, the PIGL method produces higher melt rates in the deepest parts of ice shelf cavities, which is in better agreement with observational estimates but yields to significantly underestimated melt rates at shallower depths (Fig. c).

We now assess parameterized melt rate patterns for the entire Antarctic ice sheet in comparison to the observational melt patterns from , as shown in Fig. a. The general picture is that nonlocal MeanAnt produces relatively uniform melt rates within individual sectors, with maximum present-day melt rates below 25 myr-1 (Fig. b). Nonlocal PIGL produces sharper gradients, with significantly larger maximum values near grounding lines (up to 54 myr-1; Fig. c). As in the observational product, nonlocal PIGL produces refreezing areas in some sectors but generally not at exactly the same location (e.g., Ronne, Dröning Maud Land). In the Bellingshausen Sea, areas of significant refreezing (3–4 myr-1) are produced by nonlocal PIGL, while observations suggest no significant refreezing in that sector. The local version produces sharper gradients than the nonlocal with no refreezing (by construction in Eq. ), and maximum melt rates reach 43 myr-1 for the MeanAnt method (Fig. d) and 93 myr-1 for the PIGL method (not shown) in the Amundsen Sea sector.

While melt patterns are often used to assess basal melting parameterizations, assessing their ability to capture the interannual variability of melt rates is also valuable. The only region where both measured T,S profiles and observational estimates of cavity melt rates are available for multiple years (thus allowing an assessment of interannual variability) is the Amundsen Sea, in particular the Pine Island and Dotson ice shelves. We use these T,S profiles to parameterize melt rates based on the aforementioned methods and compare them to observational estimates (based on meltwater fluxes estimated from T,S sections under the geostrophic assumption). This comparison is not a perfect evaluation of the ISMIP6 protocol as there is no observation to estimate the average thermal forcing over the entire Amundsen sector, and we therefore need to apply the parameterization over a single cavity rather than a larger sector. This affects the sector-averaged thermal forcing in the nonlocal case, with 1.40 ∘C and 1.09 ∘C for Pine Island and Dotson, respectively, vs. 1.07 ∘C for the Amundsen sector as a whole. Therefore, δT needs to be recalibrated to get the present-day melt rate and the correct sensitivity to ocean warming (see Sect. ). Another reason to recalibrate δT in this comparison is that the observational conductivity, temperature, and depth (CTD) profiles give much higher cavity-mean thermal forcing (2.27 and 1.62 ∘C for Pine Island and Thwaites, respectively) than in the new climatological dataset (1.40 and 1.09 ∘C). Using the ISMIP6 thermal corrections would lead to an overestimation of both present-day melting and sensitivity to temperature variations. The comparison is nonetheless useful to assess the behavior of such simple quadratic formulations and the γ0 values used in the ISMIP6 protocol. For Pine Island, there is little agreement between the parameterized and observational variabilities, e.g., the parameterizations do not reproduce the observational peak in 2007 (Fig. a). In contrast, the parametrizations capture the increasing melt rate from 2000 to 2009 in Dotson, followed by a decrease and relatively stable melt rate over 2012–2016 (Fig. b). For both cavities, the MeanAnt method significantly underestimates the amplitude of interannual variability, while the PIGL method is close to the observational amplitude. These results are somewhat consistent with , who tuned γ0 without thermal correction to reproduce the mean melt rate of an idealized warm cavity and obtained γ0 values in between the MeanAnt and PIGL values.

Here we illustrate the ocean forcing protocol by deriving future melt rates under the Antarctic ice shelves from six CMIP5 models, considering the r1i1p1 ensemble member of CCSM4, CSIRO-mk3-6-0, HadGEM2-ES, IPSL-CM5A-MR, MIROC-ESM-CHEM, and NorESM1-M. These projections are to be considered as zero-order approximations because the depth and extent of ice shelves in the ice sheet simulations are expected to change in response to the evolving ice dynamics as well as basal and surface mass balance, which is not taken into account here. The changing geometry will be accounted for in the final ISMIP6 ice sheet projections, with all forcings combined to ice sheet dynamics (Nowicki et al., in preparation).

As illustrated in Fig.  for a single CMIP5 model NorESM1-M;, the mean Antarctic ice shelf melt rate can be estimated throughout the historical period (1850–2005) and for Representative Concentration Pathway (RCP) scenarios (2006–2100), accounting for parameter uncertainty. The early part of the historical period is close to the preindustrial state and can be used for the long spin-up period that is sometimes needed to initialize ice sheet models. For the NorESM1-M model, the mean melt rate remains close to the observational estimate (0.85 m w.e. yr-1) under the rcp26 scenario. In contrast, the mean melt rate is strongly enhanced at the end of the 21st century under rcp85, reaching ∼6 m w.e. yr-1 with nonlocal MeanAnt and ∼40 m w.e. yr-1 with nonlocal PIGL (median values).

Returning to Fig. , we can see how the calibration method affects projected melt rates in the Ronne–Filchner and Amundsen sectors, which both undergo ∼0.75 ∘C warming at depth: as for present day, the PIGL method produces much stronger future melt rates at depth than the MeanAnt method (43 vs. 5 myr-1 for Ronne–Filchner at 1800 m depth and 150 vs. 39 myr-1 for Pine Island at 900 m depth).

We now consider projections from six CMIP5 models under the rcp85 scenario in four different regions where ice shelves buttress a large volume of ice grounded below sea level, here only considering the nonlocal parameterization (Fig. ). For Pine Island and Thwaites, all models but IPSL-CM5A-MR indicate an increase in mean melt rate by a factor of ∼1.5 (MeanAnt) to ∼4.5 (PIGL) in 2100. The relative increase for the three other regions is much larger: based on PIGL parameters, a majority of CMIP5 models give projected melt rates exceeding the present-day Pine Island and Thwaites mean values (∼17 myr-1) before 2100, even exceeding 100 myr-1 for some models. The MeanAnt parameters give weaker increases for these three regions, with melt rates underneath Ronne–Filchner remaining below 3 myr-1 in all models but HadGEM2-ES and half of the models reaching ∼10 myr-1 underneath Cook and Ninnis and ∼20 myr-1 underneath Totten and Moscow University by 2100.

Comparing these results to projections from ocean models that represent ice shelf cavities can be used to assess projections from the proposed parameterizations. Such model projections were done for CMIP3 and CMIP5 models by and , respectively, using the Finite Element Sea Ice–Ocean Model (FESOM). The FESOM simulations produce realistic present-day melt rates beneath Ronne–Filchner, and their projected melt rates are close to the low end of the CMIP5 distribution for nonlocal MeanAnt. By contrast, half of the CMIP5 models used through nonlocal PIGL give melt rates above 20 myr-1, which is far above FESOM projections for 2100 (Fig. a), and still far above the 6 myr-1 in 2200 under much warmer conditions . This could suggest that our melting parameterization is too sensitive to warming (overestimated γ0), although this is not what is suggested by our previous analysis of Dotson and Pine Island interannual variability (Fig. ). Alternatively, it could mean that projected ocean warming is largely overestimated by some CMIP5 models in the Weddell Sea.

In contrast to Ronne–Filchner, present-day melt rates produced by FESOM were significantly underestimated in other cavities, as reported by and . For example, the underestimation reached a factor of 10 beneath Pine Island and Thwaites as well as beneath Totten and Moscow University (see dotted lines in Fig. ), mostly due to overly strong convection and to the subsequent presence of cold water in these regions, as discussed by . These strong present-day biases make it difficult to use these ocean simulations as a reference to assess projections from the proposed parameterizations.

FESOM is the only model so far that has been used to build CMIP-based projections of ice shelf melting. have run more idealized projections, adding +0.5 ∘C to the initial state and lateral boundary conditions of a regional ocean simulation centered on the eastern Amundsen Sea. We use their simulation to evaluate the sensitivity of melt rates to warming beneath Thwaites glacier (Fig. ). The comparison is made by adding +0.5 ∘C to the present-day thermal forcing in the entire Amundsen sector and using the standard ISMIP6 values for γ0 and δT. It is again not easy to conclude which parameterization performs better. First of all, the present-day parameterized values are underestimated compared to and to the observations. MeanAnt seems to be in better agreement with in terms of relative increase for both average and high-end values, but the warm state of nonlocal PIGL is quite close to the warm state in . The change in melt rates is probably more relevant for ISMIP6 than the initial value as the ISMIP6 results are presented with respect to control simulations , which would suggest that the MeanAnt sensitivity is correct in the Amundsen Sea.