In this study we use the three-dimensional, free-surface, hydrostatic, primitive-equation global ocean general circulation model NEMO version 4.2 . This model includes a sea ice module as well as an ice shelf cavity module . The model is discretized on an Arakawa C-type grid where different variables are solved at different places on a 3D-grid cell (see Fig. a). For instance, temperature and salinity are computed at the centre of the cell (T-point or coordinate indices i,j) and the x-ward and y-ward velocity components, u and v, are computed at the cell boundaries (U-point of coordinate indices i+1/2,j for u, and V-point of coordinates i,j+1/2 for v).

We propose to homogenise the treatment of shear by solving the friction velocity in the first wet cell in both the dynamic and the thermodynamic equations, and to reduce the dependence of the drag coefficient in z by using the law of the wall (Eq. ). From Eqs. () and (), Cd takes the form 3Cd=κln⁡zz02 with z, half of the first wet cell thickness and z0, the hydrographic roughness length. In the logarithmic approach (LOG) case, z0 is constant everywhere while Cd is z-dependent. In the “topographic roughness” approach (ROUGH) case, z0 is given at T-point. The friction velocity, u* and the hydrographic roughness length (in the ROUGH case) are calculated at T-point, while velocity components (u,v) are computed at U- and V-point respectively. Since the difference between neighbouring cell thicknesses can be non negligible, particularly in the coarser cases (see Fig. b), we need to linearly interpolate Cd in between two cells and calculate the resulting u* at U-point (i+12,j,kti,j) and V-point (i,j+12,kti,j) respectively and take the mean at T-point. More details are given in Appendix . The sketch of Fig. a shows the difference between the treatment on the first wet cell and on the Losch layer for z* coordinate system with partial steps.

The choice of z0 is of course important as shown in Fig. a. Likewise, it is important to note the difference of Cd given the cell height (2–18 m for L121 and 2–60 m for L75). For z0=3×10-4 m, Cd=2.5×10-3 at z=1 m, Cd=2×10-3 at z=2 m and Cd=1.5×10-3 at z=15 m from the ice interface. This means that the change between 1 and 2 m is as large as the change between 2 and 15 m. This also means that, at high resolution, this formulation has important effects. These values of Cd are expected to lead to significantly different melt rates see Fig. 6 of, and this range of z values is common due to the use of partial cells and to the fact that fields are usually calculated at the cell centers (i.e., z is half the cell thickness). To make sure the difference between the two parameterisations is not only dependent on the choice of the first wet cell, we propose an experiment with a constant drag on the first wet cell.

We want to test the sensitivity of the parameterisation (LOG) as well as its vertical dependency in z compared to the reference (REF) and the sensitivity of a spatially variable drag dependent on the “topographic roughness” (ROUGH). The experiments are ordered in three different blocks:

the Losch layer approach – REF – the reference runs where the friction velocity u* used in the thermodynamic equations is computed in a 30 m thick Losch layer with a constant Cd,

To compare results, we choose to look at the melt rate (in m yr-1) and the total melt rate (in Gt yr-1) averaged over the year 2013. We do not conduct a thorough evaluation of the simulations as this was already done in .

We first compare REF and LOG experiments to show the effects of the new parameterisation. Second, we evaluate the vertical dependency of both parameterisations by comparing simulations with 121 vertical layers (REF and LOG) and 75 vertical layers (REF.L75 and LOG.L75), which is the standard number of layers in global Earth System Models using NEMO as an ocean model. Third, we compare them with simulations with constant Cd computed in the first wet cell for 75 and 121 vertical layers to evaluate the impact of the law of the wall. Fourth, we estimate the impacts of having a spatially variable z0 compared to LOG experiment. We use the “topographic roughness”, R, calculated from the three different sub-shelf topography resolution datasets described in Sect.  and shown in Fig. : 500 m in ROUGH500m , 100 m in ROUGH100m , and 32 m in ROUGH32m . Boxplots of ln⁡(z0) for each ice shelf are shown in Fig. a with the horizontal line being the value for ln⁡(z0) in the LOG. Figure b shows Cd as a function of ln⁡(z0) for different half thickness z. All experiments are summarized in Table . Finally, we estimate the impacts of an increase of ice shelf roughness, as a potential future damage evolution, by increasing the roughness length, z0. We choose to have a mean roughness, R, equal to the maximum roughness of the ROUGH experiment in LOGx5.

Figure a shows the basal melt rates produced in the REF experiment. Higher melt rates are generally present in regions below 500 m depth, which corresponds to where the modified Circumpolar Deep Water (mCDW) is entering the cavities in the ASE and where the freezing point temperature is lower (compare Figs.  and a). In shallower regions, melt rates are lower except at places where the buoyant melt-enriched water is following the slope of the ice shelf, generally concentrated in meltwater channels.

We chose a z0 so that the total melt rates of the ASE region in LOG and REF are similar (see Fig.  and Table ) with an overall regionally averaged relative difference of 2.5 %. Ice shelves in the LOG experiment are generally experiencing more melt than in REF but the relative differences are variable from one ice shelf to the other. Total melt rates of Pine Island ice shelf from LOG and REF are very close (-0.8% difference) but they differ greatly in the case of Thwaites ice shelf (55 % difference).

The absolute difference between basal melt rates (see Fig. b), reveals spatially interesting features, which are not random as is the thickness of the first wet cell (Fig. ). Even though total melt rates are similar, the difference can be non negligibly positive or negative (∓ 15–20 myr-1 difference at places of high melt). This probably means that spatial pattern differences tend to be compensated, with the exception of Thwaites ice shelf where most of the extra freshwater is produced close to the grounding line and particularly in the western sector.

In order to compare the spatial differences between LOG and REF relative to the mean, we standardize the data (i.e., we calculate the deviation from the ice shelf mean and divide it by the SD so that data is in SD units). It is particularly interesting for Thwaites ice shelf since the mean difference of basal melt rates is so high. With standardized melt rates, it is possible to also see that there is a spatial pattern in melt rates differences for Thwaites ice shelf (Fig. c) but the areas with lower melt rates do not compensate for the much higher rates near some parts of the grounding line. Remarkably, melt rates in Thwaites and Dotson basal channels are lower in LOG than in REF.

In the three equation model (Eq. ), melt rates depend on the friction velocity (through the drag coefficient, Cd, the mean horizontal current speed, UM, the thermal driving, T*=TB-TM and the salinity, SM). In Fig. d and e, we can see the differences of the standardized value of the two most important contributors, the friction velocity and the thermal driving (salinity difference is negligible). The spatial pattern of melt rate differences and their magnitude is very similar to the spatial pattern of friction velocity. This was expected as the difference between these two experiments is in the treatment of the friction velocity. Nevertheless, thermal driving differences are not negligible.

Knowing that friction velocity is the main difference contributor, it is interesting to compare both treatments of friction velocity (from the logarithmic and the Losch layer approach) within one single experiment. Figure a shows the friction velocity differences between LOG and REF and Fig. b shows the difference of friction velocity treatments within the LOG experiment. Spatial patterns are once again similar. To go further, we can look at the difference of mean horizontal current speed, UM, as shown in Fig. c. At a first order, the differences of friction velocity between LOG and REF (Fig. a) are similar to the differences between current speed in the first wet cell, u1wc, and averaged in the Losch layer, utbl although it does not apply to the Eastern sector of Thwaites ice shelf close to the grounding line. Melt rates are therefore directly related to the UM treatment change in the three equation parameterisation.

It is interesting to note that the current speed in the first wet cell can be higher than the average in the Losch layer. This is due to the fact that, in the current version of NEMO, using the Losch layer approach, the mean horizontal current speed, used in the calculation of the friction velocity, is the magnitude of the component averages and not the average of the magnitudes: 4UM2=uu,tbl2+uv,tbl2 meaning that a change of speed direction in the column could induce a lower UM in the Losch layer than in the first wet cell. Also, if the vertical resolution is coarse the friction velocity might be over- or underestimated depending on the actual velocity profile. Finally, if the first wet cell is thicker than the default Losch layer, this effect can be even stronger and the use of a constant Cd, representative of a constant Losch layer depth, is misleading.

To assess the vertical resolution dependence of the drag coefficient treatement in ice shelf melt, we compare “global Earth System Model type” vertical resolution (75 levels) to an “ice shelf model type” vertical resolution (121 levels) as described in Sect. . Particularly, we look at the melt differences between the two vertical resolutions for experiments with the Losch layer approach (REF.L75 vs. REF) and with the logarithmic approach (LOG.L75 vs. LOG).

In Table , we can see very strikingly that the Losch layer approach is more dependent on the vertical resolution, with 22 % melt rate difference for the ASE region, than the logarithmic approach, with 6 % melt rate difference. Each ice shelf has a different dependency with Pine Island having the most differences between the two resolutions (43 % for the REF and 29 % for the LOG). Thwaites melt estimated in REF.L75 is much greater than in REF (43 %) and somewhat lower in LOG.L75 than in LOG (-14%). Dotson, Crosson and Getz ice shelves show a diverse but relatively high dependency to vertical resolution in REF (15 %, 36 % and 18 % respectively) but this dependency is around 5 % in LOG. The spatial differences are presented in Fig.  showing most differences close to the grounding line, particularly at Pine Island ice shelf and on the eastern part of Thwaites ice shelf.

With 121 levels, the Losch layer contains at least two cells (Fig. a) with the sum of their thickness close to 30 m (Fig. b). On the contrary, with 75 levels, the Losch layer may only contain one or two cells, and while the first wet cell is rescaled and might be smaller than the Losch layer thickness, the second cell thickness is already greater than 30 m below 300 m depth and increases exponentially to the grounding line. Moreover, if the first wet cell thickness is equal to or thicker than the Losch layer thickness, we are back to the same problem as described by . Essentially, the cell thickness is variable but the drag coefficient is constant. If the first wet cell thickness is thinner than the Losch layer, with a coarse resolution, we use the weighted velocity of the next cell below (as explained in Appendix ), which could have a thickness much thicker than the Losch layer. In this context, depending on the velocity profile, the Losch layer speed could be under- or overestimated.

To better evaluate if the law of the wall used in the LOG parameterisation is important or if it is as good as using a constant drag coefficient in the first wet cell, we compare NOLOSCH, LOG and REF experiments for 75 and 121 layers. With 121 layers, melt rates for NOLOSCH.L121 are closer to LOG.L121 than to REF.L121 for Thwaites, Crosson, Pine Island and Dotson ice shelves and slightly above for Getz ice shelf. On the contrary, with 75 levels, the total melt difference for all ice shelves is 0.1 % between NOLOSCH.L75 and REF.L75 % and 13 % between NOLOSCH.L75 and LOG.L75 with clear discrepency for individual ice shelves. So in summary, NOLOSCH is slightly closer to LOG than to REF with 121 layers and much closer to REF than to LOG with 75 layers.

Moreover, and most importantly, the difference between NOLOSCH.L75 and NOLOCH.L121 (32.2 % for all ice shelves) is much higher than between LOG.L75 and LOG.L121 (6 %) and even REF.L75 and REF.L121 (22 %). This means that it is absolutely not equivalent to have a constant or variable drag coefficient in the first wet cell.

To understand better why melt rates from NOLOSCH are closer to LOG when there is 121 levels and to REF with 75 levels, we plot the histogram of drag coefficient (Fig. b) and the histogram of cell half thickness (Fig. c) for all ice shelves for 121 and 75 levels. For 121 levels, cell half thickness is always smaller than 10 m and the drag coefficient is between 1.4×10-3 and 2.6×10-3 while for 75 levels the cell half thickness can be up to 30 m and the drag coefficient down to 1.0×10-3. In other words, LOG.L121 drag coefficient is closer to 2.5×10-3 than LOG.L75 and therefore NOLOSCH.L121 melt rates are closer to LOG.L121 although the difference is not that strong for all ice shelves. On the contrary, LOG.L75 drag coefficient can be much lower than 2.5×10-3 while the first wet cell thickness is closer to the Losch layer thickness used in REF.L75. For that reason, NOLOSCH.L75 is closer to REF.L75. It would have been interesting to run a NOLOSCH experiment with a drag coefficient close to the average of the LOG experiment. Nonetheless it shows that a variable drag coefficient, depending on the cell thickness, gives finer results and can be used the same way in any configuration, regardless the vertical resolution. In a nutshell, because of the high range of cell thickness (from 1 to 20 m for 121 levels and up to 60 m for 75 levels), having a constant or variable drag coefficient on the first wet cell is not equivalent.

Here, we use the “topographic roughness”, R, as a proxy to a variable hydrographic roughness length, z0 with different initial topography resolutions (ROUGH500m, ROUGH100m and ROUGH32m) as presented in Table . It is important to note that, even if the mean hydrographic roughness lengths for all considered ice shelves for the ROUGH experiments are equal to the constant hydrographic roughness length used in the LOG experiment, it does not imply that the mean hydrographic roughness length for an individual ice shelf is the same between the ROUGH experiments as can be seen in Fig. a. Here, we are therefore mostly interested in melt rates differences relative to each ice shelf's mean melt rates so that the standardized variable is preferred.

In Table , we see the relative difference of total melt rates between the ROUGH experiments and LOG. The total melt rates of the region are lower by 3 %–4 % for the ROUGH experiments compared to the LOG experiment. Dotson, Crosson and Getz ice shelves have similar total melt rates across roughness resolution (1 %–3 %, 6 %–7 % and 8 %–9 % difference respectively) while Pine Island ice shelf (1 %–9 % difference) experiences more melt in the coarsest resolution (500 m) and Thwaites ice shelf (7 %–17 % difference) on the two finer resolutions (100 and 32 m).

Figure a shows the difference of standard basal melt rates between the three ROUGH experiments and the LOG. From the comparison between the hydrographic roughness length, z0=αR in Fig.  (plotted as ln⁡(z0)) and the melt rates differences in Fig. a, at a first glance, we get higher melt rates in places of higher z0. ROUGH32m shows sharper contrast in places of finer topographic patterns (see Fig. ), particularly in channels or at damaged zones. In Fig. , we can see for instance that at 500 m resolution, the shear margin of Pine Island ice shelf is not as well represented as in 100 or 32 m resolution. The hydrographic roughness length of the western part of Thwaites and the eastern part of Crosson are more pronounced for higher resolutions.

As the ocean melts the ice below ice shelves and the ice is flowing, the geometry and the subgrid scale topography of the ice will change in time, due to ice shelf thinning and increased damage, and might have a very different pattern in the future. If we assume that drag is a function of “topographic roughness”, as we do in this study, it is worth looking at the sensitivity of melting to an increase of “topographic roughness” on melt. Here we imagine that the geometry is unchanged but the mean hydrographic roughness length of the region has increased to the maximum value of today from ROUGH500m (18 % total change of ln⁡(z0)). In this case, the total melt rates of the region would increase by 16 % (27 % for Pine Island ice shelf, 34 % for Thwaites ice shelf, 20 % for Dotson ice shelf, 35 % for Crosson ice shelf and 8 % for Getz ice shelf). The increase in melt rates averaged for the region is almost proportional to the amount of change in the logarithmic roughness length (16 % and 18 % respectively). However, when looking at each ice shelf independently, the same increase of z0 is producing different effects on melt rates. In Fig. b, the drag coefficient is plotted against ln⁡(z0) for different cell half thickness z. This is only giving a partial picture of what the melt rates would look like because of its dependency on velocity, temperature, salinity. Nonetheless, one may see it as a coarse way to speculate on the roughness sensitivity.

Six limits can be identified in this study and should be considered when using this drag parameterisation or to go further in the direction of implementing a variable hydrographic roughness length in the melt equation.

The first one is inherent to the parameterisation of the melt (the three-equation parameterisation) as it has been established for well-mixed turbulence regimes and is known to poorly perform under certain conditions by overestimating melt rates when stratification is strong , or when fluid velocity is weak .

In the framework of shear melting, the second caveat comes from the use of a theoretical law (the law of the wall) that is only valid in the logarithmic layer and in certain conditions (surfaces with no slope, neutrally-stratified flow) . This is designed for constant-stress boundary layers adjacent to an interface that is hydraulically smooth. It can cope with a variable roughness length, as used here, but only as long as that topographic roughness is not larger than the viscous boundary layer depth, such that it would start to disturb the boundary layer with lee eddies, etc.

Vertical mixing in the model is parameterized using a Turbulent Kinetic Energy (TKE) closure scheme, which determines the vertical eddy viscosity and diffusivity coefficients, a lower-bound cut-off is applied to ensure numerical stability and physical consistency, through a prognostic equation for the specific TKE. Its time evolution is governed by a balance between shear-induced production, buoyant destruction via stratification, vertical diffusion, and a Kolmogorov-type dissipation . At high resolution, it is therefore unlikely to have a similar velocity profile from one grid resolution to another.The third caveat is in the fact that NEMO, and our configuration does not resolve the log-layer. The top cell thickness is usually more than 1 m, and even with a thin top partial cell, the thickness of the cells below is generally ∼20 m so that there is no way to correctly resolve the log layer even if the viscosity parameterisation was suitable. Therefore, the log formulation is applied to all model cells beneath ice shelves, it is imposed exactly because NEMO does not produce such vertical log profiles.

Fourth, and other studies only use averaged temperature and salinity in the Losch layer while we also use averaged velocities in the standard experiments. It is therefore possible that the added value of the new parameterisation is lower in models that do not average velocities in the Losch layer, although a part of the added value is not related to the Losch layer itself.

Fifth, the definition of the “topographic roughness” is arbitrary and other equations could be considered such as:

a roughness estimation based on the obstacle height, the silhouette area and the specific area .

Finally, the “topographic roughness” in this study is static in time and based on an observed topographic map. In reality, the geometry and the “topographic roughness” changes in time, shaped by the melt, the flow and the calving rate. A coupled approach is therefore necessary. An interesting line of research for a coupled ocean-ice sheet model would be to use a damage variable based on shear stress although correlation between damage and roughness is still to be proven .