The Amery Ice Shelf (AIS) is an embayed ice shelf in East Antarctica with an area of ∼ 62 000 km2 and some of the deepest Antarctic ice in contact with the ocean ∼ 2200 m;. Modelling studies suggest that HSSW is present beneath the AIS where it drives moderate melt rates along the eastern flank of the ice shelf cavity . The deep draft of the AIS allows HSSW to drive strong melting at the grounding line; a draft of 2200 m depresses the freezing temperature by almost 2 ∘C, resulting in melt rates that exceed 30 myr-1 . These elevated melt rates at the grounding line produce cold, fresh, buoyant meltwater called Ice Shelf Water (ISW). ISW ascends the underside of the ice shelf along the western flank of the cavity where it eventually becomes colder than the in situ freezing temperature, allowing frazil ice to form and accumulate on the underside of the ice shelf . ISW exits the cavity on the western flank of the AIS at depth, creating a cyclonic circulation within the cavity .

A sustained observational campaign has improved our understanding of circulation in Prydz Bay and beneath the AIS. The Amery Ice Shelf-Ocean Research (AMISOR) project has been monitoring the ocean beneath the AIS for nearly 2 decades from 2001 on . Oceanographic measurements were collected through six boreholes from profiling and the deployment of instrumented moorings for longer-term monitoring . These moorings confirmed the presence of both HSSW and ISW beneath the ice shelf. Moorings at the AIS calving front observed a modified version of CDW (mCDW) entering the cavity at intermediate depths during the austral winter , and coincident observations from under-ice mooring AM02 show ISW with a fresher source water mass during this time, suggesting that mCDW drives melting in some areas of the AIS cavity.

A recent remote sensing study estimated the area-averaged melt rate of the AIS to be 0.8 ± 0.7 myr-1 over the period 1994–2018 , consistent with earlier studies . Modelling and oceanographic studies report similar values of 0.74 myr-1 and 1.0 myr-1 . showed a seasonal cycle in area-averaged melt with a maximum of 0.8 myr-1 in winter and a minimum of 0.7 myr-1 in summer. The in situ data analysed in the present study were collected at site AM06 (Fig. ), which is on the eastern flank of the AIS cavity, and are the first such measurements of basal melting from the AIS.

The ice shelf–ocean boundary layer (ISOBL) regulates heat and salt exchanges between the ice and the far-field ocean and plays a crucial role in determining the rate at which the ice shelf melts. The ISOBL can be broken up into two regions: the diffusive sublayer adjacent to the ice, and the turbulent outer layer. In the narrow diffusive sublayer, the effects of viscosity are dominant and heat and salt are transported by molecular diffusion. The outer layer transport is dominated by turbulent fluxes. The source of this turbulence may be shear instability due to friction between the ocean and ice e.g. or convection due to buoyant meltwater . An extensive review of the roles of current shear and convection in ice–ocean interactions can be found in . Notably, they suggest more work on ice shelf basal roughness at all scales should be a future focus of ice–ocean research. The resolution of general circulation models e.g. and regional models e.g. is far too coarse to capture the ISOBL processes that regulate melting, and a subgrid-scale parameterisation is needed to estimate the melt rate.

Melting is parameterised in these ocean models through a system of equations balancing heat and salt fluxes to the ice–ocean interface with the latent heat and brine fluxes due to melting e.g.. Further, it is assumed that the interface temperature (Tb) is at the freezing temperature at interface salinity (Sb) and pressure (pb, Eq.  in Appendix A). This results in a system of three equations, which can be solved for Tb, Sb, and melt rate (m). The way in which oceanic heat and salt fluxes are modelled is the key point of difference between melting parameterisations, and we will examine these models next.

This study presents a set of in situ oceanographic and basal melting observations collected beneath the AIS in 2010. Using this unique dataset, we will seek to address these questions: (i) what is the ocean variability in the AIS cavity; (ii) how does this relate to measured melt rate in terms of mean and variation; (iii) how well do available parameterisations predict the basal melt rate in this situation, and (iv) how do these data compare to other published datasets of concurrent melt rate and ocean observations?

The borehole at AM06 (70∘14.7′ S, 71∘28.1′ E) was hot-water drilled during the 2009/2010 summer at a site that was predicted to be melting . The ice shelf at this location is 607 m thick, with 73 ± 2 m of freeboard, and the water column thickness is 295 m. The ice–ocean interface and seafloor are at 523 ± 2 and 837 ± 2 dbar, respectively.

Several conductivity–temperature–depth (CTD) casts were collected over the full depth of the cavity during a 2 d period using a Falmouth Scientific Instruments (FSI) 3′′ Micro CTD instrument (serial 1610). Pre-season laboratory calibrations of the FSI CTD temperature, pressure, and conductivity sensors were done at the Commonwealth Scientific and Industrial Research Organisation Division of Marine Research; however, no in situ calibrations were performed. Based upon the largest corrections from previous AMISOR sites (using the same instrument) the error is expected to be less than 0.005 ∘C, 0.3 psu, and 3 dbar for temperature, salinity, and pressure, respectively. A mooring, comprising three Seabird SBE37IM MicroCATs and one upward-looking RDI 300 kHz Workhorse acoustic Doppler current profiler (ADCP), was then deployed through the ice. Weights were affixed to the end of the mooring to apply tension to the cable and minimise motion of the instruments. All instruments sampled at 30 min intervals. In the vertical, the ADCP sampled 27 bins at 4 m resolution, where 23 of the bins were within the water column. The beam angles were 20∘ from the vertical. The location of each instrument with respect to the ice–ocean interface is outlined in Table . The duration of the ADCP record is 366 d, and thus we restrict the analysis of the MicroCAT data to the same period for this study. In situ temperature (T) and practical salinity (S) are converted to conservative temperature (Θ) and absolute salinity (SA) using the Gibbs Seawater MATLAB package .

The range from the ADCP to the ice shelf is used to map the ice shelf base and measure the local melt rate. To our knowledge, ours is the first study to use an ADCP in this manner beneath an ice shelf. To map the ice shelf base we use the bottom tracking functionality of the ADCP. A range measurement is obtained from each of the four ADCP beams, which are oriented at 20∘ to the vertical and at 90∘ to each other, and the range, heading, pitch, and roll data are used to map the interface position in x, y, and z. The instrument rotates about the mooring line due to tidal currents, allowing the ADCP beams to map out a circular swath of the underside of the ice. We find that the bottom tracking data are too noisy to recover a direct melt rate measurement. Instead, for the melt rate estimates, range is obtained by post-processing the echo amplitude (intensity) of the ADCP pings. Following the method outlined in , a modified Gaussian is used to approximate the surface reflection peak profile A(z): 1A(z)=a0exp⁡-z-h0δ2+a1z+a2, where a0, a1, a2, h0, and δ are obtained by a least-squares fit of Eq. () to the echo amplitude data in the vicinity of the surface peak. The fitted value of h0 is an estimate of the range from the ADCP to the ice shelf.

The fit is found for each of the four ADCP beams independently. Ideally, an average over the four beams would be used to decrease the statistical error . However, this was not possible due to the shape of the ice shelf base (in Sect.  we will show that the ice shelf draft changes by 20 m over a horizontal distance of only 40 m). All four ADCP beams are mapped to polar coordinates and binned in dϕ × dr = 4∘ × 3 m bins. While the ADCP samples at 30 min intervals, the sampling frequency for each bin is variable, as it depends on the rotation of the ADCP about the mooring line due to currents. In practice, some areas of the ice base are frequently sampled and others rarely. If a bin has > 5 range estimates in a month-long period, an average ice–ocean interface depth is returned (e.g Fig.  in Appendix B). Melt rates are calculated from a centred finite difference of the interface depth. Unfortunately, the surface reflection peak in Eq. () is not captured for ice ≳100 m from the ADCP. This constraint, combined with spatially and temporally inconsistent sampling, means that the depth of the interface can only be calculated using this method for headings -30≤ϕ≤46, and consequently melt rate estimates are limited to this area.

Our methods of determining the position of the ice–ocean interface (and melting) are sensitive to motion of the ADCP; we assume that the ADCP remains fixed in space (in a frame of reference moving with the ice shelf) in order to transform the range, heading, pitch, and roll measurements into an interface position in x, y and z. Fortunately, the pressure sensor of the middle MicroCAT (mCAT2), 40 m below the ADCP shows that the motion of the ADCP was relatively small. The maximum pressure anomaly is -0.9 dbar (associated with a vertical excursion of < 1 m), and excursions were typically much smaller than this. In order to avoid contamination of the interface position and melt rate measurements by this motion, we have excluded data when the pressure anomaly at mCAT2 is < -0.25 dbar, corresponding to < 2 % of data.

CTD casts collected before the deployment of the mooring at AM06 show a water column stratified in both temperature and salinity, with cooler, fresher water overlying warmer, saltier water (Fig. ). There is no systematic difference between the up and down casts of the CTD, and we include both here. The water column is stably stratified with depth-mean stratification N2∼ 3.5 × 10-6 s-2, where N=[-(g/ρ0)(∂ρ/∂z)]1/2 is the buoyancy frequency. There is no mixed layer beneath the ice, and the temperature gradient is especially strong in the upper 30 m of the water column. There is some evidence for a benthic mixed layer below ∼ 790 dbar, although this was not consistently sampled by the CTD.

Figure  shows the four-cast mean Θ-SA properties from the CTD. A defining feature of ocean conditions beneath ice shelves is the presence of meltwater from meteoric (fresh) ice, which causes ocean properties to evolve along nearly straight line in Θ-SA space e.g.. Under the assumption of equal eddy diffusivities for heat and salt, the gradient (dΘ/dSA) of this “meltwater mixing line” can be calculated . At local conditions Eq. (16) of gives dΘ/dSA = 2.38 Cg-1kg, and explains the Θ-SA properties well over the pressure range 560–620 dbar. Below 620 dbar, temperatures remain below the surface freezing temperature; however, Θ-SA properties do not follow a meltwater mixing line, suggesting mixing between two different meltwater-modified water masses. Below 750 dbar, the HSSW observed is essentially unmodified. Near the interface, the dΘ/dSA gradient steepens, deviating from the meltwater mixing line. A line of best fit over this 15 m thick layer has slope dΘ/dSA = 4.8 Cg-1kg. This implies that the turbulent diffusivities of heat and salt are not equal over this region. Stratification can alter the ratio of turbulent diffusivities ; however, stratified turbulence effects would be expected to decrease the dΘ/dSA gradient. The steep dΘ/dSA gradient could instead be indicative of diffusive convection, which is known to occur beneath ice shelves due to the presence of cold, fresh meltwater at the ice–ocean interface. The ratio Rρ=β/α(dSA/dΘ) describes the stability of a system to diffusive convection, where observations suggest that the water column is susceptible to diffusive convection for 1≤Rρ≤10, with stronger convection as Rρ approaches 1 . Using α = 3.8 × 10-5 (∘C-1) and β = 7.8 × 10-4 (gkg-1), we find Rρ = 4.3, indicating that the system is susceptible to diffusive convection. However, we note this value is high compared to the Rρ = 2 observed beneath George VI Ice Shelf, where diffusive convection was observed .

The time series of temperature and salinity measurements from the upper water column indicate that AM06 is a site of melting year-round. For the whole period sampled, the temperature recorded by the upper MicroCAT is greater than the in situ freezing temperature at the interface pressure (543 dbar) by ∼ 0.2 ∘C (Fig. ). Temperatures recorded at all depths are colder than the surface freezing temperature (-1.9 ∘C), indicating the presence of ISW, and show similar seasonality at all depths (Fig. ). The water column, which is warmest in summer and autumn, cools over winter and reaches a minimum temperature in spring. The cooling is coincident with freshening at all depths. In October, the cooling and freshening trend reverses and temperature and salinity increase rapidly towards their previous summer values.

MicroCAT temperatures and salinities fall on a melt-freeze line , demonstrating that the water masses arriving at AM06 have been modified by the addition of fresh water due to ice melt, with the highest fraction of meltwater at the MicroCAT nearest the ice base. The ISW present at AM06 follows a single melt–freeze line year round (Fig. ), suggesting a single ISW source water mass with SA∼ 34.68 gkg-1. Whilst it is not possible to use the melt–freeze relationship to unequivocally identify the source water masses of the ISW, the high salinity suggests that it is HSSW driving melt at AM06, where we follow in defining HSSW as having -1.85 ≤Θ≤ -1.95 and SA > 34.67 gkg-1. The Θ-SA minimum in spring is likely the result of a high degree of modification of HSSW by meltwater (Fig. ).

A multi-year composite of Θ-SA from mooring AM02 , situated midway (∼ 70 km) from AM06 to the calving front and spanning the years 2001 and 2003–2006, is also shown in Fig. . The data from AM02 MicroCATs at 561 and 770 dbar show similar seasonality and properties to AM06. At these depths, Θ-SA properties at the two mooring locations follow the same meltwater mixing line, suggesting the same source water mass, although the ocean is typically cooler and fresher at AM06 than at AM02 at an equivalent depth. This could imply a circulation from AM02 to AM06, with meltwater accumulating as the water mass moves deeper within the cavity. However, the warmer mCDW observed at AM02 at 339 dbar in July is not observed at AM06, likely due to the deeper ice shelf draft at AM06. In October, a water mass at the local freezing temperature with a range of salinities is observed at AM02 at the 339 dbar MicroCAT. This water mass is likely the result of a rising meltwater plume raised to the in situ freezing point by frazil ice formation e.g..

Currents at AM06 have speeds on the order of 10 cms-1 and consist of both tidal and mean components (Fig. ). To determine the dominant tidal constituents we perform a harmonic analysis of the depth-mean currents using the T_TIDE package , restricting the analysis to frequencies higher than semi-annual due to the limited duration of the ADCP record. The analysis shows that the tides have mixed semidiurnal and diurnal properties. The ellipse semi-major velocities of semidiurnal constituents S2 and M2 are 3.3 and 3.0 cms-1, respectively, which is larger than those of diurnal constituents K1 and O1, which are 1.9 and 1.8 cms-1, respectively (Table ). Ellipse semi-minor velocities for both diurnal and semidiurnal constituents are small, indicating that tides are close to rectilinear. Tides are oriented NNE–SSW, roughly normal to the AIS calving front. At AM06 the CATS2008 circum-Antarctic tide model an update to the model described in performs relatively well against the T_TIDE fits, although the CATS estimates are ≲ 25 % higher across the constituents examined (Table ). This could be due to errors in the bathymetry used in CATS, as tide modelling is sensitive to water column thickness and bathymetry is typically poorly constrained beneath ice shelves. In light of this, the CATS model performs remarkably well at AM06. Our observations of mixed semidiurnal–diurnal tidal currents are similar to previous results from barotropic tidal modelling, which showed magnitudes of ∼ 5 and ≲ 2 cms-1 for the semidiurnal and diurnal tides, respectively .

An estimate of the typical tidal current magnitude is given by 2Utyp=∑i=14ue,i2+ve,i21/2, where ue and ve are the magnitudes of the semi-major and semi-minor axes of the tidal ellipse, with subscript i representing the four main tidal constituents. Utyp is roughly the maximum current speed available from these four constituents. Using the constituents in Table  yields Utyp=9.8 cms-1 at AM06. Instantaneous current speeds at AM06 in excess of 15 cms-1 (Fig. ) are a consequence of the superposition of mean and tidal currents.

In order to consider the tidal and non-tidal currents separately, we remove the best-fit T_TIDE tidal velocities (uT, vT) from the observed depth-mean velocities, yielding the residual flow Ur=(u-uT)2+(v-vT)2. Figure  compares the total and residual flow speeds, where the residual flow is smoothed with a Gaussian filter with a half-width of 1 week. The residual flow is oriented into the cavity (220∘ N), has an annual mean speed of 3.2 cms-1, and varies seasonally and at higher frequencies. Notably, in the period August–December Ur is at its weakest and the water column is cold and fresh (Fig. ), suggesting increased residence time beneath the ice resulting in a higher meltwater fraction at this time. We also compare the monthly mean and root-mean-squared currents near the ice (U7–19) and deeper in the water column (U19–91), where near-ice currents are not shown prior to April due to poor data return near the ice in this period (Fig. c). We find that near-ice current speeds are typically weaker than deeper currents, excepting October–November period when currents are surface-intensified.

The conditions observed beneath the AIS are qualitatively similar to conditions observed in other large cold-type ice shelf cavities, the Ross and Filchner–Ronne ice shelves, where the water column away from the ice shelf front is dominated by cold ISW e.g.. The AIS is the only one of these three ice shelf cavities within which mCDW has been observed . However, the mCDW observed entering the AIS cavity has been heavily modified and is much colder (T≲ -1.6 ∘C) than the CDW in intruding beneath rapidly melting ice shelves such as Pine Island Ice Shelf T∼ 1.0 ∘C,. The mCDW was not observed at site AM06; however, the depth of the ice at AM06 (∼ 600 m) is likely to exclude mCDW, which only occupies depths ≳ 600 m at the calving front .

Our observation from site AM06 of HSSW-derived ISW with a mean flow of ∼ 3.0 cms-1 oriented into the cavity supports a three-dimensional model of AIS cavity circulation, consistent with observations from other AIS moorings and modelling results . These earlier studies suggested a pathway for HSSW from the calving front into the AIS cavity past AM02 and AM06, which are situated on the eastern flank. The HSSW cools and freshens due to basal melting, with a return flow on the western flank of the ice shelf driven by a buoyant ISW plume.

The ADCP-based estimate of the annual mean melt rate at AM06 is 0.51 ± 0.18 myr-1 (Fig. ). Monthly averaged melt rates range from a maximum of 0.8 myr-1 in May to a minimum of 0.2 myr-1 in August and September. The seasonality in melting is consistent with the seasonality of upper MicroCAT temperatures. Broadly speaking, when temperatures are warmer, higher melt rates are observed (Fig. ). However, the period of maximum melt in June does not coincide with the warmest ocean temperatures, which are observed in April for the upper MicroCAT. Mean and root-mean-squared currents are slightly reduced in the period August–November, coincident with the period of low melt rates (Fig. ).

The melt rate measured here in situ is consistent with the modelled melt rate in the vicinity of AM06 ∼ 1 myr-1;, and more broadly with AIS area-averaged melt rates from models 0.74 myr-1;, oceanographic proxies 1.0 myr-1;, and glaciological studies 0.5–0.8 myr-1;. However, the seasonal cycle in melting at AM06 is somewhat out of phase with the modelled cavity average, which has a maximum in July and a minimum in January .

Data from the ADCP bottom tracking function reveals a large topographic feature or “scarp” in the underside of the ice shelf at AM06 (Fig. ). The scarp has a vertical extent of ∼ 20 m and a maximum slope of 45∘. The along-scarp direction is roughly north–south, and the ice deepens moving from east to west. The mean current direction (220∘ N) is neither along-scarp, as may be expected if the basal topography was playing an important role in guiding the flow, nor cross-scarp, which could result in flow acceleration, separation, and/or blocking. This scarp is an interesting and unexpected feature of the site. There are very few surveys of shelf undersides at the 1–100 m scale due to difficulty of access. Consequently, the ice base is typically assumed to be smooth, at least at small scales. However, our observation of steep topography with vertical and horizontal scales of ∼ 20 and ∼ 40 m, respectively, adds to the small but growing number of studies demonstrating that the underside of ice shelves are not necessarily smooth and may exhibit “roughness” at the O(10 m) scale .

The limited spatial coverage of the ADCP makes is difficult to draw conclusions about the nature or origin of our “scarp”; however, we present several possibilities. For example, the scarp may be an isolated feature, one element of a “rough” patch of ice or part of a larger system, for example, a terrace on the flank of a basal channel or a suture zone with its origin upstream. A thorough investigation of the glaciological origins of this scarp is outside the scope of the present study; however, we note that the scarp is not closely aligned with the local ice flowline (Fig. ), as would be expected if the scarp was suture zone between two ice tributaries of different thickness, nor does it have a surface expression consistent with a larger channel system. The scarp could also be formed by ocean processes: for example, a convective process known as double-diffusive convection can drive differential melting of a vertical ice face . Evidence of double-diffusive convection has been seen in observations and models of the ocean beneath ice shelves ; however, it is likely that the currents at AM06 are too strong for this process to dominate circulation near the ice and produce such a significant feature through differential melting.

There are many other possible feedbacks between complex topography, flow, and melting. For example, acceleration of buoyant flow up slope, higher melting on steep slopes , and differential effects of stratification on melting for flat vs. sloping ice are all possible. Due to the background currents, phenomena such as flow blocking, acceleration, and separation can also be expected, depending on ocean stratification and the orientation of the flow with respect to the scarp. These effects will be maximised when the flow is across-scarp; however, flow at AM06 is primarily along-scarp (Fig. ). The effects of basal topography on flow and melting warrant further investigation using field measurements of velocity at high resolution (e.g. sub-metre) and complementary modelling studies.

Here we use the concurrent temperature, salinity, and velocity measurements from site AM06 to predict the local melt rate using three different melt rate parameterisations. We test two current shear-dependent parameterisations solving Eqs. ()–(), one with the constant transfer coefficients recommended in J10 and the other using Eqs. () and () from HJ99. We also test the convective parameterisation solving Eqs. (), () and () from MK18. Because the ADCP-derived melt rate estimates are relatively noisy, we consider averages over 2–3-month periods when comparing observed and parameterised melt rates (Table ). This reduces uncertainty in the observed melt rates, while retaining some of the observed seasonal variability. The mixed-layer temperature (TML) and salinity (SML) are taken from the upper MicroCAT. Because of the sloped ice–ocean interface, the positioning of this instrument with respect to the ice depends on the region of ice considered. Here we use the lower bound on the interface position (z = -541 m) as our reference depth. The upper MicroCAT is therefore situated at a depth of 4 m (Table ). Choosing the upper bound (z = -519 m) would put the upper MicroCAT at a depth of 26 m.

All parameterisations included in this study assume a water column structure where strong mixing in a boundary current or plume produces a well-mixed layer of water near the ice such that, so long as they are measured within the layer, temperature and salinity do not vary in depth. However, the CTD profiles collected at the start of the observational period do not convincingly demonstrate the presence of a mixed layer adjacent to the ice (Fig. ). As such, our results will be sensitive to the depth at which TML and SML are taken. Over the upper 10 m of the water column the temperature gradient is dΘ/dz = 0.0017 ∘Cm-1. We assess the sensitivity of the predicted melt rate to the depth at which the temperature is taken in Sect. .

In the melting parameterisations, Eqs. (), (), (), and () require the friction velocity (u∗) to estimate turbulent heat and salt transport to the ice. The Law of the Wall relates u∗ to the near-ice velocity structure via the logarithmic expression u(z)=(u∗/κ)ln⁡(z/z0), where κ=0.41 is von Kármán's constant and z0 is the roughness length. The velocity profiles recorded by the ADCP were analysed for a logarithmic profile; however, the vertical resolution proved to be insufficient to capture the log layer. As such, in Eq. () we model u∗ using Eq. () with Cd=0.0025, for which the flow speed (U) outside of the log layer is needed. Typically, the log layer occupies ∼ 10 % of the total boundary layer depth, and for the polar oceans is typically in the range 2–4 m deep . The Ekman layer depth scales as δ∼ u∗/|f|, where f is the Coriolis frequency. For a free stream velocity U = 5.0 cms-1, f = -1.4 × 10-4 s-1, and using Eq. () with Cd=0.0025 we find δ∼ 18 m and therefore a log layer depth of ∼ 1.8 m. The uppermost bin sampled by the ADCP is 3 m below the ice; however, this data point is likely to be contaminated as it falls within the upper 6 % of the instrument range . This point has therefore been discarded. As the upper water column velocity structure is relatively homogeneous in depth, we use the four-bin mean over 7–19 m (U7–19). This averaging increases data return, and does not bias the speed low or high compared to taking the velocity at 7 m only (Fig. ). There is a significant amount of missing velocity data in the upper water column from January through to early April. As such, the shear-dependent parameterisations are tested using data from April onwards. To test the constant transfer coefficient parameterisation from J10 we take the recommended values CdΓT = 0.0011 and CdΓS = 3.1 × 10-5.

To apply the MK18 parameterisation, the basal slope of the ice (θ) is needed. As the melt rate was measured over the section of ice bounded by -30∘ ≤ϕ≤ 46∘, θ is taken to be the average basal slope within this region, θ = 9∘ (Fig. ). It is worth noting that this is a large slope with respect to the overall ice shelf slope; the mean slope of this ice shelf will be on the order of tan⁡-1(H/L) = 0.2∘, where H (∼ 2 km) is the approximate thickness change and L (∼ 600 km) is the approximate length of the AIS.

At the ice–ocean interface, the heat flux from the ocean is balanced by latent heat loss due to melting and conductive heat loss to the ice shelf (Eq. ). In order to estimate the ice shelf heat flux, we model heat transport within the ice shelf as a balance between vertical advection and diffusion. We assume that the vertical velocity is equal to the basal melt rate and constant within the ice shelf and that the ice shelf is in a steady state, meaning all ice removed from the base is balanced by surface accumulation or ice convergence (for a thorough discussion around different ice shelf heat transport approximations, see HJ99). The advection–diffusion balance is given by 3∂2Ti∂z2+mκT,i∂Ti∂z=0.

We can solve Eq. () over an ice shelf of thickness H with surface temperature Ts and basal temperature Tb. At AM06 H = 607 m, Ts∼ -20 ∘C, and Tb∼ -2.1 ∘C. At the annual average melt rate of 0.51 myr-1, this model gives a heat flux into the ice of QiT = -0.5 Wm-2, while the ratio of heat lost to the ice (QiT) to the latent heat flux (Qlatent) as a function of melt rate m is constant and equal to 0.095 for m > 0.2 myr-1 (Fig. ), indicating that ∼ 10 % of the heat supplied by the ocean (QT) is lost to the ice. Based upon this result, QT∼ 1.095 Qlatent at AM06.

Before evaluating the parameterisations, we briefly describe the relationship between the observed melt rate (m) and ocean forcing (T∗, U) for the 2–3-month averaging periods (Table ). As expected, the lowest melt period (July–September) coincides with the weakest currents and cool temperatures, while high melting coincides with warmer temperatures and faster current speeds (e.g. April–June). However, while melting is nearly 3 times higher in the April–June period than the July–September period, T∗ and U are only 16 % and 35 % larger, respectively. Similarly, melting in the October-November period is nearly double that of the July–September period, despite extremely similar mean ocean conditions during the two periods.

The J10, HJ99, and MK18 parameterisations are applied to a short slice of observational data to show how they respond to ocean forcing (Fig. ). The shear-dependent parameterisations exhibit large variability in melting on short timescales due to the tidal flow at AM06, which varies between 1 and 14 cms-1 over a 2-week period (Fig. ). The convective parameterisation does not vary much on these timescales, as variability is driven solely by small amplitude temperature fluctuations. Both the J10 and HJ99 parameterisations predict much higher melt rates than are observed, while the MK18 convective parameterisation prediction is close to observations over this period.

The parameterisations are evaluated quantitatively by comparing the time-mean of the predicted melt rate and the observed melt rate (Table ). The parameterisation that best fits the observations is the convective parameterisation (MK18) based on the local slope angle, which is biased 20 % low over period February–November. However, despite predicting the annual average melt rate quite well, the seasonal variation in mMK18 is much smaller than in the observed melt rate.

The shear-dependent parameterisations (HJ99 and J10) are only evaluated over the period April–November due to poor velocity data return near the ice in the early months of 2010. J10 melt rates are 400 % larger than the observed melt rates, while HJ99 melt rates are roughly 200 % the observations. In all cases, the fit worsens by ∼ 6 % if the heat flux into the ice is neglected. Again, neither parameterisation captures the seasonal variation of the melting (Table ), indicating an issue with the functional form of the parameterisation. This is further highlighted for the J10 parameterisation by the best-fit Stanton number CdΓT, which is obtained by solving Eqs. ()–() given the observed melt rate and assuming ΓT/ΓS = 35. The best-fit Stanton number also varies considerably between the different averaging periods. For example, we find CdΓT = 0.00017 for the July–September period, while we find CdΓT = 0.00031 for April–June. The fact that one value cannot be used year-round suggests that the functional form of J10 is not appropriate for all AM06 conditions.

The best fit CdΓT for the period April–November is 0.00027, much lower than the recommended value from the Filchner–Ronne site of 0.0011 from . In this formulation, it is expected that ΓT is constant, whereas Cd is determined by the frictional properties of the ice base and is therefore site dependent. Based on their Filchner–Ronne data suggest ΓT = 0.011. In order to reconcile this value with our observed Stanton number we would need an extremely low drag coefficient (Cd = 0.0006) at AM06.

CTD profiles taken at the beginning of the observational period (Fig. ) measure a temperature gradient of dΘ/dz = 0.0017 ∘Cm-1 over the upper 10 m of the water column. Accordingly, if the melt rate calculated using J10 for Θ(d=4) is 2 myr-1, taking Θ(d=1) yields 1.95 myr-1 while Θ(d=10) results in 2.1 myr-1.

Despite the relatively good agreement between observed melting and the MK18 convective melting parameterisation, it is not altogether clear that this parameterisation is appropriate for AM06 due to the presence of tidal currents. The MK18 parameterisation does not depend on current speed (tidal or residual) and is not expected to hold in the presence of strong currents. In laboratory experiments, observe a transition from convectively controlled to shear-controlled melting at a velocity of 2–4 cms-1 for T∗∼ 0.5 ∘C. The typical time-mean flow speed at AM06 (U7–19) is ∼ 4.0 cms-1, while instantaneous speeds can be in excess of 15 cms-1, suggesting that AM06 should be well within the shear-controlled melting regime. However, the effects of a small slope angle and ambient stratification are not taken into account in this transition, which was determined for vertical ice.

In their study, found that the convective melting parameterisation for a vertical ice face from captured observed melt rates well at sites on the Ross and Filchner–Ronne ice shelves during times where the Reynolds number for the diffusive sublayer (Reδ=Uδν) was low, despite being applied to ice shelves with a shallow mean slope. Based on these sites, they suggest a critical Reδ for the convective-shear transition of ∼ 20. Following the same methodology see Sect. 6.3 of, we find Reδ∼ 13 for the time-mean conditions over the April–December period (Table ; row 5), lending support to the idea that convective melting may dominate at AM06.

An important consideration for parameterising melt as a function of ice shelf slope in regional ocean models is the ice shelf basal topography. Here, we apply the MK18 parameterisation to the local slope measured by the ADCP; however, features such as our basal “scarp” (Fig. ) or the basal terraces observed beneath Pine Island Glacier will be subgrid-scale in circumpolar or regional-scale ocean models that have kilometre-scale resolution beneath ice shelves e.g.. Consequently, melting would be underestimated. For example, if we use a basal slope of θ = 0.2∘ rather than 9∘ at the AM06 site, the MK18 parameterisation predicts a melt rate of ∼ 3 cmyr-1. As such, while the convective melt rate parameterisation is a relatively good fit for our observations, significant challenges remain for the implementation of a slope-dependent parameterisation in ocean models where small-scale variations in basal slope are not resolved.

Here we extend the comparison between observed and parameterised melt rates to include other published studies of ice shelf melt rate and in situ ocean observations from around Antarctica. Due to limitations in the data available this comparison is only made for the J10 parameterisation. The observed and parameterised melt rates, mean thermal forcing, and current speed at each location are presented in Table , where the data are sourced from the relevant publications. More detail on the observational data is provided in the footnote of Table .

The ratio of observed to parameterised melt rates mOBS/mJ10 is plotted as a function of the local thermal driving and flow speed in Fig. . The J10 parameterisation significantly overpredicts melt rates at many locations, particularly at warm and quiescent conditions. For example, beneath George VI Ice Shelf, where thermal driving is extremely high, predicted melt rates are ∼ 5000 % of the observed values (mOBS/mJ10 = 0.02). The exception to this is beneath the Larsen C Ice Shelf . This is unsurprising, since the Larsen C site is characterised by low thermal driving (T∗∼ 0.05 ∘C) and strong, tidally dominated flow, much like the Filchner–Ronne Ice Shelf site, to which the J10 transfer coefficients were tuned. Note that we cannot tune ΓTCd to make the J10 parameterisation fit all the observations in Fig. , indicating that a parameterisation of this functional form is not applicable over the full range of observed conditions (i.e. that the oceanic heat flux is not proportional to u∗(Tb-TML) as in Eq. ) and/or that ΓTCd cannot be assumed constant.

There are many reasons why J10, a shear-dependent parameterisation, may not accurately reproduce melt rates, especially under warm and quiescent conditions. For example, another mechanism such as convection or double-diffusive convection may be the dominant process driving melting. Stratification effects, a poorly constrained drag coefficient, or an inappropriate choice of input variables may also result in inaccurate melt rate predictions. In high-resolution models, double-diffusive convection has been shown to drive melting beneath ice shelves under warm, low shear conditions , forming a thermohaline staircase beneath the ice . Observations of a thermohaline staircase beneath George VI Ice Shelf , which is subject to extremely high thermal driving, suggest that double-diffusive convection may drive melting there. Similarly, double-diffusive convection may play a role in melting at the quiescent grounding line of the Ross Ice Shelf . Both these sites are characterised by strong thermal forcing relative to the current speed.

For shear-dominated melting, surface buoyancy flux due to meltwater can inhibit vertical fluxes, decreasing the efficiency of heat and salt transfer . Thus, stratification effects may be responsible for the misfit between the parameterised and observed melt rates at AM06 and other sites. For example, a decrease in the efficiency of heat transport could explain the poor performance of the J10 parameterisation for the Amery (this study) and Ross ice shelf sites which have similar current speeds to, but much higher thermal driving than, the Larsen C and Filchner–Ronne sites.

Another possible source of discrepancy between observed and parameterised melt rates is the drag coefficient. At AM06, a lack of information about the frictional properties of the ice base forces an arbitrary choice of Cd in order to apply the shear-dependent parameterisations to the oceanographic data. This issue is not just specific to our study – in general, Cd is extremely poorly constrained beneath ice shelves e.g.. Furthermore, Cd is often used as a tuning parameter when attempting to reconcile observed and parameterised melt rates e.g.. In ice–ocean models, drag coefficients in the range 0.0015–0.003 are typically used e.g.. Recent turbulence measurements beneath the Larsen C Ice Shelf were used to infer a drag coefficient of Cd = 0.0022 at a melting site with a cold, unstratified, tidally forced ISOBL . Beneath melting sea ice, values in the range 0.0025–0.01 have been measured , while values in excess of 0.01 have been observed beneath sea ice in the presence of rough platelet ice . The drag coefficient Cd = 0.0097 recommended by is within this range of estimates.

Poor agreement between the parameterised and observed melt rates may be a result of the depth at which the temperature, salinity and velocity are measured. At AM06 we observe that the boundary layer beneath the ice is stratified in both temperature and salinity, contrary to the paradigm of a well-mixed ISOBL on which the three-equation parameterisation is based. Other borehole measurements such as those in the McMurdo and George VI ice shelves also show stratification in temperature and salinity below the ice. Near the calving front of the Ross Ice Shelf, the absence of a well-mixed layer beneath the ice was found to reduce the fit between the three-equation parameterisation and the observed melt rates , and result in a sub-linear dependence of melt rate on temperature. The importance of measuring the current speed at an appropriate depth has also been demonstrated. Beneath the Larsen C Ice Shelf, found that at low flow speeds – when their fixed-depth velocity measurements were taken outside of the log layer – the drag relationship (Eq. ) did not estimate the friction velocity accurately, introducing large errors in the predicted melt rate. In the presence of steep basal topography, the problem of correctly identifying the depth at which Θ, SA and U should be sampled becomes even more challenging. For example, the basal scarp observed at AM06 may result in acceleration, stagnation, or separation of the flow. Consequently U7–19, which we use to predict melting, may not be representative of the flow speed next to the ice. Finally, we highlight that these issues are not limited to observational studies. The numerical models for which these parameterisations were developed are also sensitive to the choice of sampling depth, as well as the way in which the meltwater flux is distributed . Sampling and distributing meltwater at the upper grid cell introduces a dependence of the melt rate on the model grid resolution.