We built a high-resolution numerical model of the 79NG fjord (Fig. ) using the coastal ocean model GETM . GETM is a hydrodynamic model that computes currents and transports with the three-dimensional equations of motion under the Boussinesq approximation. In particular, GETM resolves the flow in the glacier cavity below the floating ice tongue. Since the 79NG tongue has a gentle slope of only about 2 % on average, and a much greater horizontal than vertical extent, it is appropriate to use the classical hydrostatic mode of GETM, instead of the computationally more demanding non-hydrostatic extension . The glacier tongue is implemented by accounting for the melt fluxes at the ice–ocean interface, the friction between ice and ocean, and the additional pressure due to the weight of the tongue. Note that we do not employ an ice sheet model and use a constant-in-time ice thickness, since the timescales over which the ice evolves are longer than those of the oceanic flow . Nevertheless, the vertical position of the floating ice and the free surface may change in response to propagating long waves and varying seawater density. The implementation details of glacier ice in GETM are explained by .

The employed melt formulation is a numerically consistent implementation of the classical three-equation model for volume and temperature fluxes across the ice–ocean interface . Numerically consistent means that the melt fluxes converge to their analytical values and the velocity directly at the ice converges to zero for increasing resolution. This is achieved by using a resolution-dependent formulation of the drag coefficient at the ice base, derived from the law of the wall: 1cd=κln⁡12hkmax+z0,icez0,ice2, where hkmax is the thickness of the uppermost model layer, z0,ice=0.01m is the ice roughness length, and κ=0.4 is the von Kármán constant. For increasing vertical resolution with hkmax→0, the drag coefficient cd increases infinitely, such that the velocity in the uppermost model layer converges to zero. See for further details; regarding the ice roughness length sensitivity, see also .

For an accurate computation of the melt rate, the vertical resolution should be finer than 2 m in the subglacial plume flowing along the ice . Such a high resolution is achieved in GETM by using adaptive vertical coordinates . These topography-following coordinates automatically adjust the distribution of the vertical layers in response to the current state of the system . The vertical resolution increases dynamically in places of interest: the ice–ocean interface, the seafloor, and – importantly – stratified areas . This is implemented by letting the vertical distribution of the model layers evolve over time, where the layer spacing is a function of the vertical density gradient and the vertical distance from the boundaries, see and for the mathematical details. This results in a “zooming toward stratification”, which allows resolving the meltwater currents at the ice–ocean boundary with about 1 m vertical resolution over the whole ice tongue, and also the inflowing plume of warm water is well-resolved. Our setup uses 100 adaptive vertical coordinate layers.

In the horizontal, our model uses a regular latitude–longitude grid with a resolution of about 500 m (precisely: (1/240)° = 0.00417° in latitude, (6/240)° = 0.025° in longitude). The model grid consists of 312×273 cells, i.e., about 85 000 grid cells in total, 56 % of which are water points (Fig. b). The timestep of our model is 2 s for the barotropic (vertically integrated) mode, in accordance with the CFL stability condition, and 30 s for the baroclinic (vertically resolved) mode, i.e., the split factor between the two modes is M=15. The setup uses the Smagorinsky parameterization of the horizontal momentum diffusion and state-of-the-art vertical turbulence closure with GOTM .

The bottom topography of our setup (Fig. b) comes from RTopo-2.0.4 . In the creation of this dataset, particular attention was paid to the bathymetry of the 79NG fjord. Importantly, data of a recent bathymetric survey covering the fjord mouth were incorporated in RTopo-2.0.4. Furthermore, we verified that the RTopo-2.0.4 bathymetry fits the seismic depth soundings of the ice-covered 79NG cavity . RTopo has a horizontal resolution of 1/120°, corresponding to a meridional resolution of about 930 m and a zonal resolution of 155 to 175 m at 79NG.

Regarding ice thickness, RTopo is rather smooth and lacks details, so instead we use the more detailed BedMachine Version 5 for the topography of the floating ice tongue (Fig. c). BedMachine has a nominal horizontal resolution of 150 m, includes more recent measurements of the ice thickness and shows more features in the ice than RTopo. In particular, melt channels in the glacier tongue are visible in BedMachine, which are an important feature impacting currents and melting at the underside of the floating ice tongue . These channels are typically between 500 m and a few kilometers wide , so the larger basal channels are resolved in our model. In the vicinity of the grounding line, BedMachine contains smoothed data as a result of blending datasets from different sources: a mass conservation approach was used for fast-flowing grounded ice, gravity inversion was used for the floating ice tongue, and both data sources were connected smoothly by interpolation . This results in a less detailed ice topography and no channels visible in this area (see the western end of the ice tongue in Fig. c). Since the well-established BedMachine dataset provides a consistent topography of the whole floating ice tongue, we did not blend the data with recent high-resolution ice thickness measurements around the grounding line .

Our model domain extends from the grounding line in the southwest through the whole 79NG fjord, including the northern branch Dijmphna Sound, and ends at a three-sided open boundary on the continental shelf (Fig. b). The open boundary is located in the south at 79.2° N, in the north at 80.3° N, and in the east at 15° W. At these boundaries, we prescribe realistic long-term averaged temperature and salinity conditions (Fig. d, e) that come from a global run of the ocean model FESOM2.1 with increased resolution in the 79NG fjord . The FESOM data were averaged over the ten-year period 2011–2020, to have steady boundary conditions that are similar to the present-day situation. We focus in this paper on the description of the typical circulation in the 79NG fjord, for which a steady forcing seems appropriate. How the results could change under time-varying forcing is discussed in Sect. .

We do not prescribe tides or velocities at the open boundary, because the tidal velocity in the several hundred meters deep cavity is small compared to the velocity of the plumes, so tides have only a minor impact on the 79NG melt rate . This is different from some ice shelves in Antarctica, where melting can be strongly impacted by tides . Also note that tides affect the floating and grounded 79NG ice by modifying the glacier's sliding speed and increasing the strain, so tidal forcing is relevant to model the flow and deformation of the glacier ice itself . In our model, the oceanic flow below the ice is simulated, but not the ice evolution happening on longer timescales . Since we focus on processes in the glacier cavity, which is isolated from the atmosphere by the ice tongue, we do not include atmospheric forcing or sea ice in the model (see also Sect. ).

Oceanographic measurements taken with moorings at the 79NG fjord entrance in 2016/2017 suggest that the subglacial runoff discharged at the grounding line is about Qrunoff=0.07mSv (milli-Sverdrup, 1mSv=103m3s-1) in annual mean . This freshwater input is included in our setup, uniformly distributed along the 25 km-long grounding line (Fig. b). Even though subglacial discharge is more likely to cross the grounding line through discrete channels , rather than widely distributed , we did not use channelized discharge in our model, since the BedMachine ice topography does not show channels in the grounding zone (see Sect. ). However, we discuss the possible implications of this simplification in Sect. . Both the subglacial discharge at the grounding line and the ocean-driven basal melting of the ice tongue are implemented in GETM as freshwater fluxes that increase the volume of the corresponding water column . This implementation is more realistic than the often employed alternative of using virtual salt fluxes .

Initiating the setup with conditions from the same FESOM simulation that is used as boundary data, we let the simulation spin up for five years. At the end of the spin-up phase, the model has reached a quasi-steady state, where melt rate, surface elevation, and barotropic kinetic energy do not fluctuate much anymore. The spatial mean melt rate has converged to a range of 11.2 to 11.5myr-1. In this study, we show results averaged over one simulation year after the five-year spin-up. As the model forcing comes from a realistic global ocean simulation, we consider this one-year average after the spin-up to be a representative quasi-steady state of the 79NG fjord at present-day. Averaged results were computed on the model timestep, i.e., numerically exact. Standard deviations were computed on hourly model snapshots, i.e., using 8760 data points.

Our simulation shows an inflow of dense water as a bottom gravity current, as expected from previous studies (see Sect. ). To quantify the properties of this inflowing plume, we follow the method by : We consider the inflow as a 2.5-layer system with a well-mixed plume layer at the bottom, mixing with a middle layer above it; the upper part of the water column forms the 0.5-layer not directly impacted by the inflow, as it is separated by the middle layer. The two interfaces between the layers are chosen as two isopycnals, the deeper one at 27.5kgm-3 separating the well-mixed bottom current from the stratified interior, the lighter one at 27.2kgm-3 being relatively horizontal in a rather quiescent part of the water column. Note that this choice of isopycnals is specific for the inflow transect shown in Figs.  and . We verified that our findings still hold if the density interfaces are varied by up to ±0.1kgm-3, and show the plume properties also for a second isopycnal in Fig. , to give an idea of their sensitivity.

With this definition, the plume in the transect consists of all water with densities of 27.5kgm-3 and above, and the plume thickness Dplume(x,y) is the height above ground of this isopycnal. The density of each layer can be computed by integrating the density ρ(x,y,z) over the layer and dividing by the layer thickness, e.g., for the plume density: 2ρplume(x,y)=1Dplume∫-H-H+Dplumeρ(x,y,z)dz, where H=H(x,y)>0 is the depth of the seafloor, and similarly for the density of the middle layer, ρ0(x,y) . Given the average density of each layer, the plume buoyancy is 3b(x,y)=-gρplume-ρ0ρ0 with the gravitational acceleration g=9.81ms-2. The buoyancy is negative, since the inflowing plume is denser than the ambient water.

We can then compute the Froude number of the plume as 4Fr=uplume-bDplume. This non-dimensional number relates the plume velocity along the transect uplume, computed analogously to Eq. (), to the phase speed of long waves traveling at the interface between plume and middle layer . Froude numbers larger than one, Fr>1, mean that the plume flows faster than gravity waves propagate in the opposite direction. This is called supercritical flow and shows that the inflow is limited by hydraulic control. Hydraulic control at the fjord mouth can limit the heat transport into the glacier cavity, with implications for melting and glacier stability , see Sects.  and .

To analyze the overturning circulation in the 79NG fjord, we compute the overturning stream function, integrated meridionally from the southern to the northern fjord wall. We consider the stream function in depth coordinates as well as in tracer coordinates for the tracers salinity and temperature. Both tracers generally increase with depth at 79NG.

The depth–longitude stream function is defined by 5Qz(x,z)=∫ysouthynorth∫-Hzu(x,y,z′)dz′dy, where u is the zonal velocity and x, y, z denote the eastward, northward, upward coordinates, respectively. Analogously, the tracer–longitude stream function of a tracer C is defined by 6QC(x,C)=∬c>Cu(x,y,z)dA, as the integral over the area in the y–z plane where c>C, i.e., the area where the tracer concentration c(x,y,z) is greater than the tracer coordinate C. Some authors e.g., employ a definition similar to Eq. () but with c≤C, in which case the sign of the stream function changes. Here, we integrate over areas above a given tracer value, so that the stream functions in tracer coordinates have the same signs as in depth coordinates.

To analyze the zonal exchange flow between cavity and ocean in temperature–salinity (T–S) space, we first define a stream function that considers both tracers simultaneously: 7QST(x,S,T)=∬t>T,s>Su(x,y,z)dA. In the following, we omit the explicit notation of the eastward position x for simplicity, keeping in mind that the equations hold for any given meridional transect. Then, the volume transport per salinity and temperature class is analytically defined as 8q(S,T)=-∂2QST(S,T)∂S∂T, i.e., the second derivative of the two-dimensional tracer stream function with respect to each tracer . Drawn in a T–S diagram, q(S,T) shows how much each water mass contributes to the exchange flow, see Sect. .

Note that q(S,T) is the boundary transect term in the water mass transformation framework of and . Also note that the 2D tracer stream function of Eq. () differs from the thermohaline stream function defined by and , despite the fact that both are stream functions that depend on salinity and temperature. In contrast to their global analysis resulting in one stream function for the entire World Ocean, we use one stream function for each meridional transect of the fjord, considering the transport perpendicular to the transect.

Numerically, the volume transport per salinity and temperature class, q(S,T), is computed from hourly, three-dimensional model output. At each longitude, the zonal volume transport of each grid cell is sorted into temperature–salinity bins, using the Python package pyTEF, which implements the methods described by and .

To quantify the exchange flow between the 79NG fjord and the open ocean with simple bulk values, we apply the Total Exchange Flow (TEF) analysis framework in salinity and temperature coordinates. The TEF framework is often used to analyze estuaries, and glacier fjords can be considered a special type of estuary . TEF combines the tracer-space analysis of with the bulk-value approach of .

The bulk values of in- and outflow are found by evaluating the extrema of the one-dimensional tracer stream function QC defined in Eq. (), see . The minimum and maximum points of the tracer stream function are called dividing tracer values, because they mark the transition(s) between in- and outflowing water masses. If the exchange flow in tracer coordinates consists of exactly two layers, then there is a single dividing tracer value between in- and outflow. At 79NG, the inflow is primarily westward, so u<0 for inflowing water (opposite sign than in typical estuarine analyses). In this case, the dividing tracer value Cdiv is given by the minimum point of the stream function: QC(Cdiv)=min⁡(QC).

The definition of the stream function QC in Eq. () implies that

QC(Cmax)=0, i.e., the stream function vanishes for the maximum tracer value (which is usually at the seafloor);

The barotropic (vertically integrated) flow in the 79NG fjord is, to first order, in geostrophic balance, which means that the primary flow direction is along contours of f/D (Fig. ). Since the Coriolis frequency f does not vary much over the fjord extending less than 1° in latitude, contours of f/D are essentially parallel to contours of 1/D, where D is the water column thickness, i.e., the difference between the depth of the seafloor and the draft of the ice tongue. The strongest barotropic inflow into the cavity passes the main calving front near the sill at the fjord entrance (Fig. ). Behind the calving front, the barotropic flow follows D-contours toward the north of the cavity. The 100–300 m-contours cross the northern calving front and a small part of the barotropic flow leaves the fjord through Dijmphna Sound, while deeper contours and most of the barotropic flow turn south. The flow recirculates in the cavity, forming a large cyclonic (anti-clockwise) vortex below the ice with a north–south extent of about 25 km, as wide as the fjord mouth, and elongated in western direction toward the grounding line. Along the southern fjord wall, the barotropic flow aligns once more with the contours of water column thickness. It splits near the calving front with one part recirculating in the cavity, whereas the other part leaves the fjord along the southern fjord wall. Deviations of the barotropic flow from geostrophic balance visible in Fig.  are mainly due to the baroclinic meltwater plumes, which are described in detail in Sect.  below.

As most of the barotropic flow recirculates in the cavity, the volume transport (the difference of in- and outflow) across the main calving front is with 5.99 mSv much smaller than the maximum of the barotropic stream function, which is 78 mSv in the center of the vortex, corresponding to the volume transport of the vortex. The outflow through Dijmphna Sound, the northern branch of the fjord, is 6.70 mSv. Thus, the combined transport through both fjord branches is a net volume transport out of the fjord of 0.71 mSv. This value is consistent with the measured outflow of (0.63±0.21)mSv estimated from mooring data in 2016/2017 . The net outflow out of the fjord is, of course, the combination of the subglacial discharge at the grounding line (Qrunoff=0.070mSv, see Sect. ) and the integrated basal melting in the glacier cavity (Qmelt=0.638mSv), the distribution of which is described in the following Sect. .

The average basal melt rate of the floating 79NG tongue computed by our 3D fjord model is (11.4±0.1)myr-1. Integrated over the floating ice tongue, this corresponds to a melt flux of Qmelt=(0.638±0.003)mSv=(20.1±0.1)km3yr-1. These values lie well within the error bounds of oceanographic in situ measurements over a one-year period in 2016/2017 that gave (10.4±3.1)myr-1 or (17.8±5.2)km3yr-1 . Our melt rate falls within the higher part of the measurement uncertainty range, presumably due to a slight positive temperature bias in the boundary data used to force our model . Tracer-based measurements yielded a lower melt rate estimate of (8.6±1.4)myr-1 , but these measurements were conducted during a time of particularly low AIW temperatures, which may explain part of the discrepancy .

The basal melt rate varies over the ice tongue in along-fjord (approximately west–east) and in across-fjord (south–north) direction. Regarding the along-fjord variability in our simulation (Fig. a), most of the melting occurs at or in the first few kilometers after the grounding line, with a peak melt rate of 104myr-1. The melt rate decreases toward the calving front and is essentially zero beyond the 100 m depth contour of the ice draft. There are no negative melt rates in our temporally averaged, quasi-steady state model result; negative melting (i.e., refreezing) only occurs at individual timesteps and only of small amplitude in our simulation. The distribution of basal melting in our model fits well with satellite-based observations by , , and , see Fig. b–d. Apart from the southwestern part of the ice tongue, the melting appears to be more intense in our model than in the satellite data, but the melt patterns are similar. The hinge zone, which is the part of the ice tongue near the grounding line where the ice is not freely floating, is generally excluded from satellite measurements. However, the hinge zone is also the part where the most extreme melt rates occur , so – as pointed out by – the area-averaged melt rate derived from satellite data rather underestimates the total melting. The area-averaged melt rates are 8.2myr-1 , 4.6myr-1 , and 8.0myr-1 , and thus smaller than the above-mentioned in situ measurements and our model results. Reasons for the difference are the absence of measurements in the hinge zone, the presence of negative melt rates in the satellite products, and the different temporal ranges covered (Fig. ).

Also across the fjord, the melting is not uniformly distributed, but high melt rates are focused along specific lanes. This can be seen clearly in high-resolution satellite data (Fig. b–d). Our model reproduces these features in the melt distribution and shows that strong melting occurs at the slopes (side walls) of basal channels in the ice tongue (Fig. a). This is where subglacial plumes drive the melting, which is explored further in the following Sect. .

Basal melting cools and freshens the top layer of the water column that is in contact with the ice. The water rises up along the ice tongue due to its higher buoyancy, entrains ambient water, and forms the subglacial meltwater plume . The subglacial plume is fueled by meltwater (and by subglacial discharge at the grounding line), but also influences the melting. On the one hand, the plume can cause more melting by mixing up ambient heat toward the ice and exerting friction on the ice–ocean interface . On the other hand, the relatively cold plume water isolates the ice from the warmer ambient water in the cavity, potentially reducing melting. Due to this pivotal role in basal melting, we analyze the subglacial plume in our simulation in detail.

For a first overview of the plume, we consider the average velocity in the 10 m just below the ice (Fig. a), since this is a typical thickness of the subglacial meltwater plume . Thanks to the adaptive vertical coordinates employed in our model, the upper 10 m of the water column are resolved by at least 6 and on average 12 model layers, so our model achieves the vertical resolution necessary for an accurate melt flux computation . The flow speed (Fig. a) shows similar spatial patterns as the melting (Fig. a), particularly in the part of the cavity where the ice is thick. This confirms that melting and plume strongly influence each other. Three parts of the glacier tongue have particularly fast under-ice flow (labeled p1–p3 in Fig. a), which we consider as distinct subglacial plumes and describe further in the following subsections.

The heat that melts the floating ice tongue of 79NG is provided mainly by relatively warm and salty Atlantic Intermediate Water (AIW), which is the densest water flowing into the fjord. We analyze this inflow in our model by looking at a map of the inflow (Sect. ) and at a vertical transect along the inflow (Sect. ).

The inflowing AIW plume and the outflowing meltwater plumes constitute together an estuarine exchange flow, i.e., a zonal overturning circulation in the 79NG fjord. Since the stratification in estuaries is primarily set by salinity, the exchange flow is commonly analyzed in salinity coordinates, by transforming the vertical coordinate from depth to salinity . The persistent temperature stratification at 79NG allows doing the same analysis also in temperature coordinates. For the mathematical details of the exchange flow analysis presented in this section, see Sect.  and above.

The zonal velocity, meridionally integrated across the fjord at each depth (-z), shows an inflow through the fjord mouth coming from the east below 200 m depth (Fig. a). The inflow passes over the 325 m-deep sill near 19.5° W and flows down into the cavity to about 500 m depth. Downstream of the sill, the inflow is more spread out vertically. It propagates west toward the grounding line and moves down to about 625 m, the maximum depth of the grounding line.

Near the grounding line, the outflow going eastward above the inflow shows transport maxima at several depths (Fig. a). These maxima correspond to the different subglacial meltwater plumes (Sect. ). They start at about 500 m depth and intensify as the plumes flow eastward and upward along the ice tongue. The outflow leaves the cavity between 70 and 200 m below sea level, with maximum volume transport at about 130 m depth, consistent with observations in front of 79NG . The glacier tongue near the calving front reaches only 50 to 80 m deep, so the outflow is at greater depth than the ice base.

In salinity coordinates (Fig. b) as well as in temperature coordinates (Fig. c), the circulation appears mostly as a 2-layer system with an inflow at higher salinities/temperatures and an outflow at lower salinities/temperatures. The streamlines are more closely spaced in the inflow than in the outflow, showing that the inflow occurs over shorter salinity and temperature ranges than the outflow (Fig. b, c), despite the inflow covering more depths in the central cavity than the outflows (Fig. a). The inflow is thus mostly AIW with temperatures above 1 °C and salinities above 34.2gkg-1, while the outflow consists of different mixtures of AIW, meltwater and subglacial discharge. The bulk temperature and the bulk salinity of the inflow (computed with Eq. ) decrease slightly as the inflow passes over the sill (dashed cyan graph in Fig. b, c), showing again that not all the warm and salty AIW can enter the cavity. Regarding the outflow, its bulk values decrease eastward, from about 34.0gkg-1 and +0.8°C at the grounding line to 33.6gkg-1 and -0.5°C at the calving front. This is due to two effects. Firstly, the outflowing plume accumulates more and more meltwater as it flows to the east, making it colder and fresher. Secondly, the eastward flowing plume rises along the ice tongue, passing through the stably stratified water in the cavity. Therefore, the ambient water that is entrained into the plume becomes lighter toward the east .

The outflow at the fjord mouth of Qout=35.7mSv is larger than the inflow of |Qin|=35.0mSv, where the difference of 0.7mSv is explained to 90 % by meltwater, Qmelt=0.638mSv (Sect. ), and to 10 % by subglacial discharge, Qrunoff=0.070mSv (Sect. ). The total freshwater flux leaving the cavity, Qmelt+Qrunoff, contributes 2 % to the cavity overturning |Qin|, and the mixing completeness, -Qin/Qout , is 98 %, meaning that most of the water volume in the outflow originates from inflowing AIW. The other 2 % of the outflowing water volume are freshwater from ocean-driven melting and subglacial runoff, which are mixed by entrainment of AIW to almost ocean salinity before leaving the 79NG fjord. These results agree with mooring data from 2016/2017 within the measurement uncertainties, which gave an overturning strength of (46±11)mSv and a contribution of 1.4 % from the total freshwater flux . The overturning strength is more than 5-times larger than the barotropic flow across the calving front (Sect. ), showing that the cavity circulation is strongly baroclinic and three-dimensional.

To analyze the plume properties, we map the zonal transport in temperature–salinity (T–S) space (see Sect. ), and find that the strongest exchange flow in the 79NG cavity falls on a straight T–S line (Fig. , left column). Generally, a straight line in T–S space is referred to as a mixing line, as it is the effect of turbulent mixing between two water masses . In a glacier cavity, the lighter water mass is basal meltwater and the slope of the meltwater mixing line (or Gade line) can be computed analytically from the properties of the ice and the ambient water . At 79NG, the ambient water mass is AIW and the resulting Gade slope is about 2.8 °C(g/kg)-1, consistent with our model results (Fig. e). Deviations from the straight line are due to the influence of other water masses, in particular subglacial discharge and Polar Water offshore the calving front .

Near the grounding line, mixing with subglacial discharge creates outflows of lower salinities than the mixing line Fig. a;. These are parts of the southern and northern subglacial plumes (p1 and p3, see Sect. ). Since the northern plume starts at the grounding line (Fig. a), the subglacial discharge directly influences its properties. This can explain why it has the lowest salinities, but note the discussion on subglacial discharge in Sect. . Further away from the grounding line, the influence of subglacial discharge on the T–S properties of the exchange flow is reduced, but still visible in deviations from the Gade line toward lower salinities (Fig. c). This looks similar to CTD profiles taken directly in front of the 79NG ice tongue . Apart from the northern and southern plumes, all other water volumes (and transports) in the cavity lie on the meltwater mixing line, in particular the (primarily outflowing) water masses below the central part of the ice tongue, and the (primarily inflowing) water below 300 m depth (Fig. a, b). Note that the divisions marked in panels (a) and (b) of Fig.  are only meant as an orientation. Both panels show data temporally averaged in their respective coordinate systems, so there is no one-to-one correspondence between a point in T–S space and a point in physical space.

Comparing a transect landward of the sill (Fig. c, d) with a transect on its seaward side (Fig. e, f), we see again the effect of the sill in controlling the warm water inflow. The AIW inflow (shown in blue) contains water with temperatures above 2 °C and salinities above 34.5 gkg-1 before (Fig. e), but not behind the sill (Fig. c), so this warm and salty water seems to be held back by the sill. Both inflow and outflow are mostly bottom-attached and have the land on their right-hand side, indicating geostrophically balanced flow (Fig. d, f). Outside the cavity, the cold water mass with salinities well below 33 gkg-1 almost at the freezing point (Fig. e) corresponds to Polar Water near the sea level . This water mass generally does not enter the cavity, as it is shallower than the ice base.

Our simulation shows that the sill at the fjord entrance has two effects on the inflow of warm and salty AIW into the 79NG cavity. On the one hand, the sill is a physical barrier that limits the inflowing AIW volume and hinders dense water from entering the cavity (Figs. a–c, , c, e). On the other hand, the sill hydraulically controls the inflow, so that the plume becomes supercritical on the downstream slope (Fig. i). In consequence, vertical mixing across the plume interface increases strongly (Fig. d). The resulting mixing with ambient water cools the inflow, such that less heat is brought into the deep part of the cavity, where the inflow comes near the ice (Fig. a, b). Thereby, the hydraulic control reduces ocean-driven basal melting , which seems to be a reason why 79NG is one of the few glaciers around Greenland still having a floating tongue.

These results are in line with previous observational and modeling studies at 79NG, which identified the role of the sill for constraining inflow volume and heat supply. Furthermore, used a conceptual 2-layer model to explain how hydraulic control increases mixing in the inflow and thereby reduces the thermal forcing at the base of the glacier, compared to cases in which a sill is absent or too shallow to control the inflow. Building on these results, studied the effect of the sill for several different sill depths and subglacial discharges in a two-dimensional MITgcm fjord model. The here-presented three-dimensional GETM setup now enables us to resolve these processes in a realistic setting. In particular, the state-of-the-art turbulence closure with GOTM used in our model shows the increase of turbulent mixing due to the hydraulic control, which results in the reduction of heat supply to the glacier base.

The subglacial plume flowing along the ice base is the driver of basal melting and transports meltwater out of the fjord toward the open ocean . Our fjord model shows that there is not one, but there are several plumes flowing along different parts of the ice tongue. We classified them as three different plumes in the south, north and center of the ice tongue. The southern plume is the most intense (Fig. ) and corresponds to the highest melt rates (Fig. a). The northern plume contains the freshest water mass (Fig. a), presumably due to subglacial discharge (see also Sect. ). The central plume is different from the other two, as it is confined to the upper few meters just below the ice and does not continue along the whole ice tongue, but instead merges with the southern plume (Fig. a).

The path of all three plumes is determined by the ice topography and the Coriolis effect. They flow primarily in channels at the ice-shelf base and are most intense along the side walls of the channels (Fig. c). This is the effect of the Coriolis force, deflecting the plumes to the right (in the flow direction). Accordingly, the highest melt rates are not found in the center of the basal channels, but along their right-hand side in flow direction (Fig. a), resulting in an asymmetric channel geometry see Fig. c–e;. This is analogous to the situation below Antarctic ice shelves, where the plume is deflected to the left and causes higher melt rates on the left side of ice-shelf channels .

Using a coupled model to simulate ice shelf evolution, showed that basal channels form due to the melting caused by the subglacial meltwater plume. In our simulation, the central plume flows initially in the along-fjord direction, but then turns and reverses (Fig. a). The size of these turns is approximately equal to the Rossby deformation radius, i.e., determined by the Coriolis effect. This seems to indicate that the subglacial plume does not only create channels, but might also be responsible for the cone-shaped features in the 79NG ice topography. However, this cannot be said with certainty yet, since our model does not compute the ice shelf evolution happening on longer time scales , but only simulates the oceanic flow and ice melting in response to the given topography.

showed with a two-dimensional (x–y) plume model of 79NG that the simulation of the meltwater plume depends strongly on the representation of the ice base, with important implications on the melt rate estimation. If the employed ice topography is smooth, the flow spreads out and the melted area broadens; in contrast, when the ice is channelized, the melt pattern is more focused and spatially heterogeneous . Our computed melt distribution (Fig. a) lies between the smooth and channelized experiments of . Thus, it is possible that more than three meltwater plumes would appear if our simulation used a finer horizontal grid spacing, resolving more basal channels. On the other hand, models using the smooth ice topography of RTopo-2.0.4 show only one plume below the 79NG tongue , since this dataset lacks most basal channels.

While the subglacial meltwater plume is initially at the ice base, we see in our simulation that it can detach and flow in a distance from the ice. In a two-dimensional (x–z) simulation of an idealized 79NG fjord, showed that the plume detaches from the ice because it reaches neutral buoyancy. It may even overshoot its neutral level, before it propagates away from the ice. With the more complex topography in our realistic simulation, the southern and northern plumes may (partly) detach from and reattach to the ice a number of times (Fig. ). When propagating out of the cavity below the calving fronts, both plumes are detached from the ice tongue and transport the meltwater out at depth (Fig. a), as seen in observational data .

The overturning circulation presented in Sect.  is similar to the 2D-vertical model of the 79NG fjord by , but with two noteworthy differences. First, the maximum strength of the overturning stream function, Q, is with 36 mSv clearly weaker in 3D than in 2D, where its maximum is over 80 mSv in absolute value. The reason is that in a 2D setup without cross-fjord resolution, the sill at the fjord entrance is as wide as the fjord, allowing for a much greater volume to pass over it, while in the here-presented realistic 3D model, the sill is only a few kilometers wide and allows much less water volume to flow into the cavity.

Second, the inflow and particularly the outflow are much more spread out vertically in 3D than in 2D, due to the more complex topography in 3D: The subglacial meltwater plumes covering parts of the ice tongue are at multiple depth levels in each meridional slice of the 3D fjord (Figs.  and ), but this cannot be represented in a 2D-vertical model. Furthermore, there is no Coriolis force deflecting the flow in the two-dimensional model by . The Coriolis effect appears in idealized three-dimensional models , enhancing melt in the south and reducing melt in the north of the cavity. However, idealized topographies are not sufficient to represent the details of melt distribution, see the comparison between idealized and realistic 3D models by . Therefore, while idealized simulations get the principal dynamics in the 79NG cavity right, there are important features of the circulation that only appear in a three-dimensional model with realistic topography resolved on a sufficiently fine grid.

Our model uses adaptive vertical coordinates, while earlier modeling studies of glacier fjords often employed z-coordinates . For plumes flowing over sloping topography, z-coordinate models generally have larger effective vertical diffusivities due to their staircase manner of resolving sills, which makes gravity currents dissipate too fast, even with high vertical resolutions. This problem does not occur in topography-following coordinates, as they almost eliminate flow across layers for topography-following currents. Numerical mixing is further reduced with stratification-aware vertical coordinates, like those used in GETM . The effect is that the inflowing plume stays narrow and focused in our simulation (Fig. ) and does not diffuse much through spurious mixing as it flows down the slope, allowing the water to propagate further into the cavity. The same applies to the meltwater plumes, which are well-resolved by the adaptive vertical coordinates used in our model (Fig. ). This close-to-reality depiction of the in- and outflowing currents in our simulation is important for an accurate representation of ocean-driven basal melting , which is necessary to analyze glacier stability. Nevertheless, it is worth noting that also models with z-coordinates generally compute realistic melt rates , presumably because the effects of unresolved processes can cancel each other to some extent. A possible explanation can be that without a well-resolved meltwater plume, the water at the ice base is too warm, but the friction at the ice–ocean interface is too low, which may result in a melt rate similar to that induced by a colder plume exerting more friction.

While the simulation of the plumes works well in adaptive vertical coordinates, the calving front presents a challenge. This front is almost vertical in reality, which blocks barotropic flow from entering the cavity . However, the ice front stretches out over a few grid cells in models with topography-following coordinates like ours, facilitating barotropic flow into the cavity. This does not seem to cause big issues in our simulation, because the barotropic transport across the calving front is low compared to the strength of the baroclinic inflow (Sect.  and ). But this might change in the presence of wind-driven circulation, when atmospheric forcing is included in the model (see Sect. ). On the other hand, barotropic blocking may also be reduced in reality, which can make the sloping front in the model acceptable. Observations in Antarctica showed that melting at the ice front creates a wedge of fresher water, reducing the blocking effect of the vertical wall and allowing currents to enter the cavity more easily . It is plausible that this wedge appears at 79NG, too, which would further reduce possible issues at the calving front.

The model of the 79NG fjord presented here uses realistic topographies for the ice base and the seafloor (Sect. ), combined with stationary forcings (Sect. ): The prescribed conditions at the open ocean boundaries (Fig. d–e) and the initial conditions come from a realistic global model averaged over a ten-year period (2011–2020); the subglacial discharge is the one-year average from mooring data (2016/2017). This forcing was used to analyze the steady state circulation in the glacier cavity at present day. Our results, which are generally consistent with observations and previous studies, should thus be seen as a description of the one- or multi-year average situation in the fjord. The temporal variability of the circulation is, however, not represented in our model and would require time-resolved forcings, as explained in the following two subsections.