A Predictive Theory for Heat Transport Into Ice Shelf Cavities

Antarctic ice shelves are losing mass at drastically different rates, primarily due to differing rates of oceanic heat supply to their bases. However, a generalized theory for the inflow of relatively warm water into ice shelf cavities is lacking. This study proposes such a theory based on a geostrophically constrained inflow, combined with a threshold bathymetric elevation, the Highest Unconnected isoBath (HUB), that obstructs warm water access to ice shelf grounding lines. This theory captures ∼ 90% of the variance in melt rates across a suite of idealized process‐oriented ocean/ice shelf simulations with quasi‐randomized geometries. Applied to observations of ice shelf geometries and offshore hydrography, the theory captures ∼80% of the variance in measured ice shelf melt rates. These findings provide a generalized theoretical framework for melt resulting from buoyancy‐driven warm water access to geometrically complex Antarctic ice shelf cavities.


Introduction
The mass loss of Antarctic ice shelves has been accelerating for the past four decades (Paolo et al., 2015;Shepherd et al., 2018).This mass loss has been attributed to the basal melt on the underside of floating ice shelves, which is driven by oceanic heat fluxes (Pritchard et al., 2012;Shepherd et al., 2004).The most vigorous basal melt in Antarctica comes from the intrusion of a subsurface warm water mass, Circumpolar Deep Water (CDW), into ice shelf cavities (Jacobs et al., 1996;Jenkins et al., 2010;Nakayama et al., 2019;Rignot et al., 2019).The depth and temperature of CDW vary around Antarctica (Schmidtko et al., 2014).Ice shelves with shallower (i.e., a thicker intrusion of) CDW and deep troughs tend to have higher melt rates (Nitsche et al., 2017) (see also Figure S1 in Supporting Information S1).
There are various controls on the supply of CDW from the open ocean to the continental shelf.Wind stresses over the continental slope lead to cross-slope Ekman transport that has been linked to variability of CDW heat fluxes across and along the shelf in observations (Assmann et al., 2013;Greene et al., 2017) and models (Dotto et al., 2020;Spence et al., 2014;Tamsitt et al., 2021;Thoma et al., 2008).Wind forcing over the continental shelf can also lead to vigorous deep mixing which erodes the thickness of CDW on the shelf (Caillet et al., 2023;Moorman et al., 2023).Surface buoyancy losses, for example, due to sea ice formation in coastal polynyas, are also able to erode the thickness of CDW across the shelf by deepening the mixed layer (Caillet et al., 2023;Webber et al., 2017).In some regions these polynyas produce High Salinity Shelf Water (Nicholls et al., 2009) that fills the ice shelf cavities, blocking the intrusion of CDW (Gwyther et al., 2014;Hazel & Stewart, 2020;Hellmer et al., 2017).In other regions, precipitation onto the ocean in front of the ice shelves can enhance stratification and lead to more lateral transport of CDW to ice shelf faces (Flexas et al., 2022).
Among the various influences on CDW intrusions, previous studies have consistently emphasized the role of bathymetry (Heimbach & Losch, 2012;Klinck & Dinniman, 2010;Nakayama et al., 2019).In particular, deep troughs have been shown to allow CDW to flow mostly unimpeded from offshore into ice shelf cavities in models (Haigh et al., 2023;Schodlok et al., 2012;St-Laurent et al., 2013) and in observations (Assmann et al., 2013;Rintoul et al., 2016).Modeling studies have similarly shown that raising CDW above the height of the main bathymetric obstacles is a necessary condition for pushing cold shelves like the Filchner-Ronne from a low-melt state to a high-melt state (Daae et al., 2020;Hazel & Stewart, 2020).
There have been attempts to link the net melt rate of ice shelves to the bulk properties of the CDW layer and ice shelf cavity geometry (Holland et al., 2008;Lazeroms et al., 2018;Little et al., 2009;Pelle et al., 2019;Reese et al., 2018) but they have all almost exclusively focused on parameterizing the ice ocean boundary layer or plume processes.Burgard et al. (2022) evaluated existing basal melt parameterizations in a regional model that included ice shelves and found that the parameterizations' error was often on the order of the signal.Lazeroms et al. (2018) found that a plume-based melt parameterization could approximately replicate the observed spatial patterns of ice shelf melt, but only with the aid of a tuning parameter that was specific to each ice shelf.
In this study we present a new dynamical framework that determines area-averaged ice shelf melt rates based on a geostrophic constraint on the transport of warm water into the ice shelf cavity (Section 2), rather than based on processes occurring at the ice-ocean boundary.This allows us to predict the average ice shelf melt rate from the hydrographic conditions outside of an ice shelf cavity.We combine this theory with a novel quantification of the bathymetric obstruction of CDW access, referred to as the Highest Unconnected isoBath (HUB, Section 3).We then test our theory against a suite of idealized model simulations (Section 4) and against observed ice shelf melt rates (Section 5).

Theory of Geostrophically Constrained CDW Heat Flux Into Ice Shelf Cavities
In this section we formulate a theoretical framework for estimating ice shelf cavity melt based on hydrography external to the cavity and its geometry.Previous studies have qualitatively shown that when CDW floods an ice shelf cavity, it fills the cavity horizontally but is deflected downwards to the ice shelf's grounding line by the boundary layer plume that forms at the ice-ocean interface (Nakayama et al., 2019).The change in interface height of CDW inside the ice shelf cavity drives a geostrophic flow parallel to the grounding line until it reaches a wall of the cavity, at which point it is directed toward the grounding line of the ice shelf in a boundary current.This flow regime can be seen in idealized models (e.g., De Rydt et al., 2014;Zhao et al., 2019), as well as in regional models (e.g., Dutrieux et al., 2014;Nakayama et al., 2019).Zhao et al. (2019) showed quantitatively in an idealized model that the transport in this flow regime parallel to the ice shelf grounding line, and subsequently in a boundary current toward the grounding line, could be constrained by the geostrophic velocity driven by the change in depth of the CDW layer inside the cavity.This is analogous to previous scaling theories for buoyancydriven circulation in enclosed basins in the open ocean (Gnanadesikan, 1999;Nikurashin & Vallis, 2012;Youngs et al., 2020).We will adapt the constraint introduced by Zhao et al. (2019) to estimate the net heat transport associated with the flow of CDW into an ice shelf cavity.
To formulate our theory, we idealize the ice shelf cavity circulation as a two-layer flow, comprised of a fresh cold melt layer overlying a warm salty layer (Figures 1a and 1b).We have labeled the lower layer in our schematic as CDW, although, depending on the specific ice shelf, this could represent other water masses (Thompson et al., 2018).Assuming vertically uniform flow in each layer, the cross-cavity geostrophic transport of CDW may then be formulated as where y is an along-cavity coordinate, h CDW is the thickness of the CDW layer, and u CDW is the cross-cavity CDW velocity.Here we have scaled the cross-cavity flow by the geostrophic shear, that is, where s CDW is the slope of the isopycnal interface between CDW and the overlying waters in the direction from the grounding line to the ice-shelf front, f is the Coriolis parameter, and g′ in = g(σ CDW σ surf )/ρ 0 is the reduced gravity determined by the potential density of the CDW layer and surface layer (σ CDW and σ surf , respectively).To further simplify (1), we assume that the interface between the two density layers approximately follows the shape of the ice draft due to melting and mixing processes at the ice-ocean boundary, or equivalently that the gradient of upper layer thickness is much smaller than the gradient of the ice interface, that is, s CDW ≈ s ice , (see Figure 1a and Section 4).Note that because we assume the ice shelf is floating in isostatic equilibrium, gradients in ice shelf thickness exert no horizontal pressure gradient force on the fluid.Taking L to be a representative distance from the grounding line to the ice front, we scale (1) as Here H CDW is a representative CDW layer thickness, which we assume to be limited by bathymetry between the grounding line and the continental shelf break (see Figure 1 and Section 3).
To estimate the amount of melt which occurs due to this inflow of CDW, we assume (a) that the net transport of CDW into the cavity is balanced by return flow of freezing-temperature meltwater, and (b) that the net advective heat transport into the cavity is balanced by heat lost to the ice shelf via basal melting.The latter assumption holds provided that the cavity is in steady state, that is, over time scales much longer than the cavity flushing time scale (Holland, 2017).Neither assumption takes into account the role of subglacial discharge, which has been shown to be regionally important to basal melt rates (Goldberg et al., 2023;Gwyther et al., 2023).The resulting heat balance can be expressed as where W is the cross-cavity width, ṁ is the melt rate per unit area, C p is the specific heat capacity of seawater, ρ 0 is a reference ocean density, ρ i is the reference density of ice, I f is the latent heat of melting, θ CDW is the temperature of the CDW, and θ surf is the surface freezing temperature.Substituting (1) into (3) and rearranging leads to the following scaling for the area-averaged melt rate, Here we introduce a non-dimensional scaling parameter α, the interpretation of which is discussed further in Section 6.
A shortcoming of this scaling is that in cavities with realistic geometries, the length L and width W are ambiguous.However, in our simulations (in which the ice shelf cavity does have well-defined dimensions; see Section 4) we find that the stratification in the interior of the cavity varies approximately linearly with width, that is, g′ in /W ∼ g′ out / W 0 , where W 0 ≈ 100 km is a constant reference width and g′ out is the reduced gravity outside the cavity.This relationship yields a predicted area-averaged melt rate that is independent of both the cavity width and length, consistent with the findings of Little et al. (2009), In the last equality of 5 we have contracted all constant parameters into a single constant of proportionality C. Note that Equation 5relates the area-averaged melt rate to quantities derived either from the stratification external to the cavity (θ CDW θ surf , g′ out ), the geometry of the cavity (s ice ) or a combination of the two (H CDW ), and thus serves as our theory for ice shelf melt rates.

Quantifying Bathymetric Obstructions to CDW Inflows: The Highest Unconnected isoBath (HUB)
To apply our theory from the previous section in three dimensions we must calculate the thickness of the CDW layer (H CDW ), and the temperature of the CDW (θ CDW ) at the entrance of the cavity in complex three-dimensional geometries.Because previous studies have shown that the deepest entry points to ice shelf cavities play an important role mediating heat transport (e.g., St-Laurent et al., 2013;Walker et al., 2007), it is crucial that our estimates of CDW thickness and temperature account for these deepest entry points.
To generalize this concept across all Antarctic ice shelves, we formulate a new metric called the Highest Unconnected isoBath (HUB), which may be defined for any reference location on the continental shelf.The HUB may be understood as follows: Consider an ocean that is completely drained of its water, and then slowly fills from its deepest point in such a way that the water is always approximately stationary and in gravitational equilibrium.For any given reference location on the continental shelf, the HUB is defined as the elevation that the water must rise to in order for the reference location to be immersed.More precisely, we can define the HUB for any reference location x = (x 0 , y 0 , z 0 ) on the sea floor of the Antarctic continental shelf.The HUB is equal to the deepest elevation z HUB ≥ z 0 such that (x 0 , y 0 , z 0 ) can be connected by a three-dimensional path to the open ocean without traversing any depths shallower than z HUB and without traveling through bathymetry.Further discussion of the HUB, including a topological definition, is provided in the Supporting Information.
Figure 1a provides a two-dimensional visualization of the HUB.In this example, all points along the continental shelf highlighted in green share the same HUB, corresponding to the elevation z HUB .CDW must rise to an elevation of at least z HUB in order to reach any of the points highlighted in green.For a real world example, consider the Filchner-Ronne ice shelf; Figure 1c shows the HUB for a reference location x situated at the Filchner-Ronne ice shelf grounding line.This reference location has a HUB of around 605 m (green line).CDW would need rise to an elevation of at least 600 m (red line) in order to reach the reference location from offshore, but would not flood the reference location at a depth of 605 m (green line).

Predicting Melt in Idealized Ice Shelf Cavity Simulations
To test our theory of warm water inflows (Section 2), we conduct idealized ocean-ice shelf simulations that span a wide range of cavity geometries and offshore hydrographies (see Figure 2).Our simulations utilize the MIT general circulation model (Marshall, Adcroft, et al., 1997;Marshall, Hill, et al., 1997) to evolve the state and circulation of the ocean resulting from the ocean's thermodynamic and mechanical interactions with a static ice shelf (Losch, 2008) (see Supporting Information for more details).To focus on the buoyancy-driven inflow of CDW, we omit other drivers of ocean circulation such as sea ice, tides, and atmospheric forcing.We prescribe an analytical profile of potential temperature and salinity at the northern and eastern boundaries of the model domain (see Figures 2a and 2b) and the Supporting Information), motivated by climatological observations of warm ice shelf cavities (Boyer et al., 2018).
We illustrate the geometry and forcing of our reference case in Figure 2a.This ice shelf has dimensions resembling ice shelves in the Amundsen Sea embayment (Morlighem, 2020), being approximately 150 km long and 100 km wide, with an ice front depth of 250 m and a grounding line depth of 1,000 m.The ice shelf slope is linear, and equal to s ice ≈ 0.005.The HUB of the reference case is approximately 650 m.
We conduct a series of experiments with different ice shelf/bathymetric geometries by varying the continental shelf slope, the ice shelf slope, the cavity width and the extent of the ice shelf front.A full list of the model geometries used in this study is given in the Supporting Information (Tables S1 and S5-S8 in Supporting Information S1).For all but the reference case we add pseudo-random noise to the sea floor to create more realistic bathymetries with deeper trough-like access pathways.The random noise has a peak wavelength of 62.5 km which is roughly the width of troughs in the Amundsen (Dinniman et al., 2011;Walker et al., 2007).The noise is scaled by the water column height (before the noise is applied) in order to prevent the bathymetric variations from closing off portions of the grounding line.For each ice shelf geometry, we conduct three simulations in which we set the depth of the subsurface temperature to 300 m deeper than, at the same depth as, and 125 m shallower than the HUB.In all experiments we use a horizontal grid spacing of 2 km horizontal to adequately resolve mesoscale eddies (St-Laurent et al., 2013;Stewart & Thompson, 2016), although the instantaneous flow fields suggest that the flow is not in a strongly eddying regime.We use a vertical grid consisting of 91 geopotential levels, with resolution varying smoothly from 2 m at the surface to 200 m at the sea floor.The vertical spacing is approximately 20 m at the depth of the ice shelf grounding line.All simulations reach a quasi-steady state by 2.5 years of integration, and are then run for 7.5 additional years for analysis.
We calculate our estimate of area average basal melt rate (Equation 5) in each simulation using the model's offshore hydrography and cavity geometry.We calculate H CDW by subtracting the HUB from the elevation of the pycnocline depth.The ice slope s ice is determined by the model geometry.We define the CDW temperature θ CDW as the temperature on our prescribed offshore hydrographic profile at the depth of the HUB.Finally, we determine the coefficient C (and thus α) via linear regression using the diagnosed area-averaged melt rates across our entire suite of simulations.This linear regression yields an α of 0.187.Because this factor is constant across all runs it does not change the correlation with the diagnosed melt rate but rather scales the parameterization output to the correct magnitude.
To evaluate our theory, we compare the predicted ( ṁpred ) and diagnosed ( ṁmodel ) area-averaged ice shelf melt rates in Figure 2c.We find that the predicted melt rates explain 91% of the variance in the diagnosed melt rates across all simulations.Experiments with the same geometry (which have the same marker shape/color in Figure 2c) show increasing predicted and diagnosed melt rates in simulations with higher offshore CDW.The ability of our parameterization to predict the diagnosed melt rate suggests that the geometric aspects of the cavity that are of first order importance are the large scale ice shelf slope and the deepest depth of CDW access (the HUB).These results indicate that our theory is successfully capturing the leading order dynamics of warm water inflows in this idealized model.

Predicting Observed Ice Shelf Melt Rates
The parameterization from Section 2 is able to accurately predict melt in a geometrically simple model designed to isolate the dynamics of warm water inflows (Section 4).We now test our prediction of basal melt using observations around Antarctica.We draw on observations of near-Antarctic hydrography, as synthesized in the World Ocean Atlas, 2018; Boyer et al., 2018;Locarnini et al., 2018;Zweng et al., 2019) annual climatology, and on satellite-derived estimates of ice shelf melt from Adusumilli et al. (2020).
The theory encapsulated by Equation 5 assumes a simplified geometry that contrasts with the complex geometries of natural ice shelf cavities; for example, the depth of real ice shelf grounding lines vary spatially, as does the slope of the ice.In order to generalize the theory to real ice shelf cavity geometries, we compute bulk estimates of the different parameters in our theory (Equation 5).Specifically, for a given ice shelf we identify all points from the Bedmachine (Morlighem, 2020) 500 m resolution grid which contain grounded ice and are adjacent to floating ice as grounding line points, and then estimate the hydrographic parameters H CDW , g′ out and θ CDW θ surf for each grounding line point.We then group those grounding line points by ice shelf and average each parameter separately to formulate our prediction of the area-averaged melt rate, where 〈⋅〉 denotes an average over all grounding line points within the ice shelf and ⋅ denotes an average over the whole ice shelf area.We treat the ice shelf slope s ice differently because this parameter is related to the geometry of the whole cavity, rather than external hydrographic properties.The Supporting Information specifies how we choose an appropriate offshore hydrographic cast at the 1,500 m isobath for each grounding line point using the HUB, and how we calculate the temperature of the CDW layer (θ CDW ), the thickness of the CDW layer (H CDW ), the exterior reduced gravity g′ CDW ) , and the bulk ice shelf slope s ice .6).On the x-axis the grounding line-averaged hydrographic terms, 〈H CDW 〉〈g′ out 〉〈θ CDW θ surf 〉〈| f 1 |〉, and on the y-axis the cavity-averaged ice shelf slope s ice .Antarctic ice shelves' locations in this parameter space are indicated by white circles.
In Figure 3c we compare the melt predicted by our theory Equation 6 against the satellite-derived estimates of basal melt and accompanying uncertainty from Adusumilli et al. (2020).We determine the constant prefactor C via linear regression, which yields α = 0.094 (see Equation 5).We find that our theoretical prediction explains ∼81% of the variance in the observed melt rates.This can be contrasted with Figures S4 and S5 in Supporting Information S1 which show the correlation between melt and just the thermal forcing term and just the slope term of our parameterization.This suggests that, for ice shelves in which the melt rates are driven by CDW inflows, variations in these melt rates are accurately accounted for by our geostrophic constraint on the inflow of CDW into the cavity.As expected, the theory does poorly at predicting the melt rate in "cold" cavities in which CDW inflows do not dominate the melt rate.Note that in "cold" ice shelf cavities, the error bars on observations are often nearly the same magnitude as the signal.
In Figure 3d we use our theory to estimate the relative importance of ice draft slope versus external hydrography in the predicted ice shelf melt rates.Specifically, we map the melt rates in a parameter space defined by two parts of Equation 6: the cavity-averaged ice shelf slope, s ice , and the rest of the equation, 〈H CDW 〉〈g′ out 〉〈θ CDW θ surf 〉〈| f 1 |〉.This decomposition shows that ice shelves with similarly high rates of melt may have an abundance of warm CDW that has access to the cavity, for example, Dotson ice shelf, or from a relatively steep ice draft, for example, Drygalski ice shelf.Furthermore, neglecting changes in ice shelf slope, the theory predicts that ice shelves with gentle slopes (e.g., the eastern Ross) would exhibit little change in melt rate even if CDW was to rise significantly, in contrast to steeply sloping ice shelves like the Totten.

Discussion and Conclusion
This study presents a novel constraint on the net heat transport into ice shelf cavities, and thus, indirectly, on the area-averaged melt rates of the ice shelves.The guiding principle of our theory (Section 2) is that if CDW is shallower than the dominant bathymetric obstacle blocking the cavity, its flow into the cavity is geostrophically constrained by the along-cavity density gradient established by the interface between CDW and meltwater within the cavity.Applying scaling arguments, we obtain a relationship (Equation 5) between the area-averaged melt, the slope of the ice shelf draft, and the thickness, temperature and density anomaly of CDW.Motivated by previous findings that the deepest troughs in the continental shelf play a key role in funneling CDW toward ice shelves (e.g., St-Laurent et al., 2013;Walker et al., 2007) we further introduce a new metric called the Highest Unconnected isoBath that identifies the minimum depth which offshore water must reach before access to ice shelf cavities is possible (Section 3).We use the HUB to determine the waters that can access a given ice shelf cavity, which in turn constrains the along-cavity density gradient and thus the net heat transport in our theory.We evaluate our theoretical prediction across a suite of idealized model simulations (Section 4), and find that it explains 90% of the variance of the diagnosed melt rates.Finally, we apply the theory to predict observational estimates of ice shelf melt rates (Adusumilli et al., 2020), and find that the theory explains 80% of the variance in melt rate across all Antarctic ice shelves (Section 5).Taken together, these findings indicate that our geostrophic constraint captures the leading-order dynamics of the net heat transport into warm Antarctic ice shelf cavities.
Our formulation contrasts from existing parameterizations of ice shelf melt by focusing on the transport of heat into the cavity using solely the offshore hydrographic properties and the morphology of the ice shelf rather than the dynamics of melt once warm water reaches the ice shelf face.This means that our theory predicts only one area averaged basal melt rate for an ice shelf cavity, and does not produce spatially varying maps of ice shelf melt.
In deriving and applying our theoretical estimate of the heat flux into ice shelf cavities (Equation 5) we have made a number of simplifying assumptions, discussed in Section 2. One is that we neglect the effects of wind and surface buoyancy forcing, whereas previous observational and modeling studies indicate that these effects may play a key role in controlling ice shelf melt rates (Guo et al., 2022;Hattermann, 2018;Silvano et al., 2022;Thoma et al., 2008;Webber et al., 2017).We also assume that the cavity circulation is in equilibrium with the external oceanic conditions, that is, that the net heat transport into the cavity is completely used for ice shelf melt.We might expect this assumption to fail on time scales shorter than the flushing time scale of the cavity (Holland, 2017), on which transient heat storage in the cavity and ice shelf boundary layer/plume dynamics more directly dictate the melt rate (Lazeroms et al., 2018).Our theory also predicts that the melt rate is entirely determined by the ice shelf geometry and the external hydrography, in contrast with previous studies showing that circulation within ice shelves can exhibit bi-stable states (Caillet et al., 2023;Hellmer et al., 2017;Moorman et al., 2023).Future work is required to reconcile our theory with previous theories for bi-stability of ice shelf cavity circulation and melt rates (Hazel & Stewart, 2020).Our model configuration (Section 4) is reflective only of warm ice shelves by virtue of the prescribed offshore hydrography and lack of dense water formation.Future work is needed to understand if cold shelves are similarly geostrophically constrained.
An outstanding question from this study is the extent to which other processes influencing the ice shelf-ocean boundary layer (or parameterizations thereof) are compatible with our geostrophic theory.For example, tides have been shown to increase melt rates across Antarctica (Richter et al., 2022), simulated basal melt has been shown to be dependent on vertical resolution (Schodlok et al., 2016), and melt has been shown to be sensitive to the parameterization of turbulent transfer into the ice-ocean boundary layer (Jourdain et al., 2017).Such processes could conceivably change elements of the physics encapsulated by the scaling prefactor α, that is, the partitioning of the geostrophic shear between the CDW and melt water layers, the cavity width-dependent relationship between external and internal reduced gravity, and/or the change in CDW thickness between the shelf break and the ice shelf front.In this case we might expect that including a dependence of α on the tides, vertical resolution, and turbulent transfer parameterization to yield more accurate predictions of melt rate.However, it is not yet clear whether incorporating such dependencies into α is necessary: an alternative hypothesis is that changes the processes occurring in the modeled/observed ice-ocean boundary layer lead to feedbacks on the stratification outside the cavity, such that the melt rate remains consistent with our geostrophic constraint.We propose further experiments in a regional ocean/sea ice/ice shelf model configuration to explore factors that determine α more thoroughly.
To our knowledge, this is the first time satellite-derived melt has been successfully estimated using offshore hydrographic observations without a tuning for every ice shelf.The framework succeeds despite observational error in the bathymetric, hydrographic, and basal melt measurements.We argue this could lead to improved parameterizations with better predictive capabilities.The theory we introduce also provides insight into the relative importance of geometry and hydrographic forcing in ice shelves around Antarctica.This material is based in part upon work supported by the National Science Foundation under Grant OCE-1751386 and OPP-2220968, and by the National Aeronautics and Space Administration ROSES Physical Oceanography program under Grant 80NSSC23K0357.This work used the Extreme Science and Engineering Discovery Environment (XSEDE, Towns et al. (2014)), which is supported by National Science Foundation Grant ACI-1548562.Without implying their endorsement, the authors thank Clara Burgard and Ken Zhao for various discussions that improved this study.We would also like to thank the reviewers (David Gwyther and an anonymous reviewer) for their comments in the peer review process.

Figure 1 .
Figure 1.(a) A schematic representation of the highest unconnected isobath (HUB; see Section 3) in two dimensions.All points colored green underneath the ice shelf share the same HUB depth of z HUB (b) An illustration of the proposed watermass structure which is assumed by the theory presented in Section 2. (c) A map of the bathymetry of the Filchner-Ronne ice shelf (FRIS).Regions with grounded ice are filled in gray.The green contour (z = 605 m) surrounds the reference point x but is closed at the shelf break.This means that for water from the open ocean to reach x, it must rise shallower than z = 605 m.The red contour (z = 600 m) is open at the shelf break and contains location x, meaning that this is the shallowest depth that CDW must reach in order to access x.This means the HUB depth for x is z = 605 m (note that the resolution of our HUB depth calculation is 5 m).

Figure 2 .
Figure 2. (a) Reference run (ref) model geometry with bathymetry (brown), ice shelf (blue), and boundary temperature forcing colored along the eastern edge of the model domain.(b) Time average cross section of temperature from model run in the same geometry.(c) Linear regression of predicted melts from Equation5against diagnosed area-and time-averaged melt rates across our suite of simulations.Experiments with the same marker and color have the same model geometry, but differing temperature maximum depths: 300 m deeper than, at the same depth as, and 125 m shallower than the HUB.The legend provides the simulation names which can be referenced in Supporting Information S1 (TableS1).(d) Depth of 0.75°C isotherm is plotted in the background with white arrows denoting the time depth average horizontal velocity below that isotherm.The HUB of the grounding line of this model geometry is shown in red dotted line, and the icefront is shown in the solid orange line.

Figure 3 .
Figure3.Application of our theory to predict circum-Antarctic ice shelf melt rates.(a) An illustration of the off-shore hydrographic cast selection methodology for a single point on the Amery ice shelf grounding line.The bathymetry of the Amery Ice shelf is colored in blue and green, floating shelf ice in translucent white and grounded ice in gray.The red line depicts the HUB depth for the starred grounding line point (GL).The WOA hydrographic cast that is used to estimate heat transport toward point "GL" is labeled "WOA", and is selected as described in Section 5. (b) The hydrography at the point labeled "WOA" in panel (a), with the HUB for point "GL" marked by a red line, and the calculated pycnocline marked by a blue line.(c) The linear regression of predicted melt rate from Equation 5 against observed melt rates fromAdusumilli et al. (2020).Error bars are estimates of observational error fromAdusumilli et al. (2020).(d) Predicted melt rate (colors and white contours) as a function of different parameters in our theory (Equation6).On the x-axis the grounding line-averaged hydrographic terms, 〈H CDW 〉〈g′ out 〉〈θ CDW θ surf 〉〈| f 1 |〉, and on the y-axis the cavity-averaged ice shelf slope s ice .Antarctic ice shelves' locations in this parameter space are indicated by white circles.