## 1. Introduction

Ocean-driven melting of ice shelves around Antarctica has the potential to play an important role in accelerating sea level rise (Jacobs et al. 2002; Rignot and Jacobs 2002; Rye et al. 2014; Harig and Simons 2015). Ice shelves are the floating extensions of ice sheets that act to buttress land-bounded ice and prevent it sliding into the ocean. The thinning of ice shelves can reduce the resistance to the flow of ice upstream (Schoof 2007; Gudmundsson 2013) or melt basal channel cavities that weaken the entire shelf (Rignot and Steffen 2008; Alley et al. 2016), resulting in calving events and land ice moving into the ocean, thereby raising the sea level. The regions near Antarctic ice shelves are also important for the modification of water masses, such as in the formation of the densest water mass in the ocean (Antarctic Bottom Water), which feeds the downwelling limb of the global meridional overturning circulation (Nicholls et al. 2009; Purkey and Johnson 2012). Changes in the interaction between ice sheets and the ocean could affect the dense water formation rate and influence the global transport of heat and hence the climate (Snow et al. 2018). Key to predicting future climate scenarios is understanding the processes governing the ice shelf melt rates and response to changes in ocean circulation.

Observations of ice shelf melt and the underlying ocean circulation show contrasting behavior at different locations around Antarctica. Data taken by drilling through the Larsen C ice shelf on the Antarctic peninsula show well-mixed profiles of temperature and salinity up to 20–30 m beneath the basal surface with an underlying weakly stable stratification, a high current speed, and a strong tidal signal (Nicholls et al. 2012). The temperature difference between a few meters depth and the ice–ocean interface, known as the thermal driving, is small (Δ*T* = 0.08°C) and the basal melt rate is modest at 1.9 m yr^{−1}. This picture of energetic flow with a weak stratification has also been observed beneath the Ronne ice shelf (Jenkins et al. 2010), Fimbul ice shelf (Hattermann et al. 2012), and Ross ice shelf (Arzeno et al. 2014). In contrast, the water column beneath the George VI ice shelf is highly stratified with a low current speed and a weak tidal signature (Kimura et al. 2015). Here, the thermal driving is large (Δ*T* = 2.3°C) but the melt rate, measured using upward-looking sonar, remains modest at 1.4 m yr^{−1} (Kimura et al. 2015). Borehole measurements near the grounding line of the Ross ice shelf also show strong stratification in quiescent flow and low melt rates (Begeman et al. 2018). Other strongly stratified layers have been observed beneath the Pine Island Glacier ice shelf, where data from an autonomous underwater vehicle (AUV) show a sharp temperature gradient maintained close to the ice shelf and a slow horizontal current speed (Kimura et al. 2016). In a different area under the Pine Island Glacier ice shelf, borehole measurements also show a stratified boundary layer, but here the flow is dominated by melt-generated buoyancy acting on the sloping base of the ice shelf (Stanton et al. 2013). The extreme Antarctic environment means that observations are sparse and lack the resolution to fully characterize the processes controlling the melt rate when the oceanic boundary layer is turbulent compared to when it is more strongly stratified.

The structure of the ocean boundary layer beneath the ice is often characterized by an interfacial sublayer (on the order of millimeters to centimeters) where molecular viscosity or roughness dominates the flow, followed by a surface layer (a few meters) where the logarithmic “law of the wall” scaling applies, and finally an outer planetary boundary layer (tens of meters) where Earth’s rotation limits the mixing length (Holland and Jenkins 1999; McPhee 2008). If the flow is strongly stratified, the law-of-the-wall scaling will not hold in the surface layer and stratification will limit the maximum mixing length in the outer layer. In cases of very strong stratification and weak shear, the dynamics may be dominated by free convection (Martin and Kauffman 1977; Keitzl et al. 2016) or double-diffusive layers, the latter of which is theorized to apply to regions of the ocean boundary layer below the George VI (Kimura et al. 2015) and Ross (Begeman et al. 2018) ice shelves. The picture becomes more complicated when there is a buoyancy-driven plume adjacent to the ice, which can occur when the ice is significantly sloped such as near the grounding line, and entrainment into the plume determines the heat transferred to the ice and hence the melt rate (Jenkins 2016; McConnochie and Kerr 2018). Here, we focus on the ISOBL without a significant slope to be consistent with ice shelf observations further from the grounding line (e.g., Nicholls et al. 2012; Kimura et al. 2015).

In most ocean models, computational limitations mean that the ISOBL cannot be fully resolved and must be parameterized to achieve a realistic melt rate. There are a wide range of parameterizations, but none completely capture the dynamics of the ocean boundary layer and its response to the melt rate. One common parameterization, known as the three-equation model, is based on the relatively simple concept of parameterizing the turbulent fluxes of heat and salt into transfer coefficients (Holland and Jenkins 1999; McPhee 2008). Comparing the parameterization against observations, the three-equation model works reasonably well in some locations such as the Ronne ice shelf (Jenkins et al. 2010). However, the three-equation model does not work in other locations such as the George VI ice shelf where it overestimates the true melt rate by more than an order of magnitude (Kimura et al. 2015). This is likely because the influence of stratification on turbulence is not included in this parameterization. The three-equation model is also known to poorly estimate the melt rate in regions where the ice is significantly sloped and there is a buoyancy-driven plume (McConnochie and Kerr 2017).

Monin–Obukhov similarity theory was formulated to describe the influence of stratification effects on a turbulent boundary layer (Monin and Obukhov 1954). The Obukhov length is a measure of the distance away from the ice where stratification starts to dominate the flow (Obukhov 1946). Here, building on previous work (McPhee 2008; Deusebio et al. 2015; Scotti and White 2016; Zhou et al. 2017), we find that the Obukhov length provides a useful way to characterize the influence of density stratification on turbulence in the law-of-the-wall region of the ISOBL. Large values of the Obukhov length imply that stratification does not affect the near-ice flow, while small values imply that more of the ISOBL is susceptible to stratification effects. If the Obukhov length is comparable to the viscous sublayer thickness, then there is no region of the flow free of either viscous or stratification effects, both of which damp out turbulence (Pope 2000; Flores and Riley 2011). In this case the law-of-the-wall scaling is not expected to hold and the flow is susceptible to laminarization. The ratio of the Obukhov length to the thickness of the viscous sublayer has been used to describe the transition between turbulent, intermittent, and laminar flow in a stable atmospheric boundary layer (Flores and Riley 2011) and stratified plane Couette flow (Deusebio et al. 2015; Zhou et al. 2017).

The present study is motivated by ocean-driven melting beneath ice shelves. We use large-eddy simulations (LES) with a state-of-the-art turbulent parameterization (Rozema et al. 2015; Abkar et al. 2016) to examine steady, unidirectional flow with an unstratified free stream as a model of a small region near the ice. As outlined in section 2 the model is designed to resolve the viscous sublayer and surface layer, only parameterizing the smallest scales of turbulence. Our focus is on turbulence very near the ice. Our computational domain can be viewed as a small region embedded within the deeper planetary boundary, so for simplicity we do not include Earth’s rotation. The majority of simulations use a flat ice base, perpendicular to the direction of gravity. Scaling theory for the viscous sublayer and surface layer is outlined in section 3, along with the three-equation parameterization. The results in section 4 explore different far-field currents that generate shear turbulence, and a range of imposed far-field temperatures. The focus is on understanding the influence of stratification associated with the input of meltwater on turbulence and the subsequent feedbacks on the melt rate. A summary of the results is in section 5. In section 6 we discuss the applicability of our results to the ocean. While the motivation for this study was the ice shelf–ocean boundary layer, the simulations are idealized enough that they also have implications for other applications, including the boundary layer beneath sea ice.

## 2. Model design

Here, we model the ocean boundary layer under an ice shelf in a rectangular domain of length *h* (Fig. 1). The flow is bounded from above by the base of the ice shelf which is assumed to be flat. The upper and lower boundaries are impenetrable, while the two horizontal directions are periodic. A no-slip condition is imposed on the upper boundary (the ice base) and a free-slip condition on the lower boundary. For most of the simulations, we assume that the ice shelf is horizontal with gravity perpendicular to the ice–ocean interface and no rotation term. Simulations with small basal slope angles are discussed in appendix A and give very similar results to the simulations with a flat ice base.

*t*is time,

*p*is pressure,

*T*is temperature,

*S*is salinity,

*ρ*from the reference value

*g*= 9.81 m s

^{−2}is the gravitational acceleration,

*x*and

*z*directions, and

*α*= 3.87 ×10

^{−5}°C

^{−1}and

^{−1}are the coefficients of thermal expansion and saline contraction, respectively (Jenkins 2011). We use realistic values of the molecular viscosity

^{2}s

^{−1}and the molecular diffusivity of heat

^{2}s

^{−1}(Prandtl number

^{2}s

^{−1}(Schmidt number

*x*direction. In Eq. (1) this constant driving force appears as

*b*refers to the ice–ocean boundary. By imposing a pressure gradient, we are effectively setting the wall shear stress and hence the friction velocity

^{−1}and

^{−1}, which result in far-field velocities of

^{−1}. Here, the far-field means the maximum depth in the domain of

*τ*to chosen far-field temperature

^{−1}such that

The governing equations [Eqs. (1)–(5)] are discretized using Fourier modes in the two horizontal directions and second-order finite differences in the vertical direction (see Taylor 2008). Note that Eqs. (1)–(5) are the grid-filtered equations where *T*, and *S* are the resolved fields. A recently developed LES parameterization known as the anisotropic minimum dissipation model (Rozema et al. 2015; Vreugdenhil and Taylor 2018) is used to evaluate the subfilter stress tensor

*m*, temperature

*w*refers to parameters corresponding to water and subscript

*i*to parameters corresponding to ice. The specific heat capacity of water is

^{−1}°C

^{−1}, the latent heat of fusion is

^{−1}, and

^{−1}are coefficients in a linearized expression for the freezing point of seawater (Jenkins 2011). The locally hydrostatic background pressure due to the depth of the ice base below sea level

*P*= 350 dbar is chosen to be broadly consistent with the Larsen C ice shelf (Nicholls et al. 2012).

The domain size for all runs was set to ^{−1} case was 128 × 128 × 145 and for the ^{−1} case was 256 × 256 × 289. These grids were chosen to be consistent with the criteria outlined in Vreugdenhil and Taylor (2018) for resolved LES. One exception was that a 1/8 vertical-to-horizontal grid cell aspect ratio at the edge of the viscous layer was found to work just as well as a 1/4 aspect ratio, thus the former was chosen to allow more grid stretching in the vertical direction. The vertical grid was stretched to place more grid cells adjacent to the ice to resolve the near-ice conductive and diffusive sublayers, which are thin because of the realistic values of *k* is the grid cell number, ^{−1} cases and ^{−1} cases.

A range of far-field temperatures *T* = 0.005°–0.43°C (Table 1). The far-field salinity was set to *g* = 0). These runs were designed to examine the transport of heat and salt when the scalars do not influence the flow. The simulations were run with chosen values of *T* and a particular melt rate. However, as outlined in section 4, in the passive scalar case the melt rate is dependent only on Δ*T* (for a particular *T* and the melt rate have not been included for the passive scalar cases in Table 1 because the runs apply more generally.

Run summary varying friction velocity *g* = 0.

Each melting scenario is initialized from an equilibrated fully turbulent flow, with uniform temperature and salinity profiles set to the chosen far-field values

## 3. Scaling theory

### a. Viscous, conductive, and diffusive sublayer scaling

### b. Law-of-the-wall and Monin–Obukhov scaling

Further away from the ice, at the edge of the viscous layer, small-scale turbulent structures form and drive larger-scale turbulent eddies in the “surface layer.” The solid boundary of the ice influences the size of the turbulent eddies in the surface layer. When the effects of stratification are weak, the shear *z*. Dimensional analysis then gives

*L*is the Obukhov length,

*ξ*,

### c. Three-equation parameterization

## 4. Results

### a. Mean flow properties and melt rate

Vertical profiles of horizontally averaged velocity, temperature, and salinity show the influence of the imposed far-field temperature on the flow structure (Fig. 2). Immediately below the ice lies the interfacial sublayer where the viscous scaling is consistent with the measured velocities (Fig. 2a). The lower edge of the viscous boundary layer is an important region for the formation of small-scale turbulent phenomena which go on to produce turbulence throughout the flow. Farther away from the ice, the case with weaker thermal driving (dark blue line) has a velocity profile similar to the logarithmic law-of-the-wall scaling (dashed) but with a modest increase in the far-field velocity. The increase in far-field velocity is very large in the case with stronger thermal forcing (cyan line). Increases in far-field temperature lead to a stronger temperature stratification (Fig. 2b) and hence larger thermal driving. This increases the melt rate, freshening the water and producing a stronger salinity stratification (Fig. 2c). The density stratification is dominated by the salinity component in all the runs presented here. Hence, the stabilizing salinity stratification damps out some of the small-scale turbulence at the edge of the viscous boundary layer, and as a result the drag decreases. However, as the friction velocity is prescribed (via imposing the pressure gradient) the equilibrated state wall shear stress must remain the same no matter the imposed far-field temperature, and so the reduction in drag results in an acceleration of the far-field velocity.

Vertical profiles of velocity, temperature, and salinity are plotted in terms of wall units in Figs. 2d–f where the results all closely match their respective sublayer scalings, indicating that the resolution is sufficient to fully resolve these sublayers. It is important to adequately resolve the sublayers to ensure that the resulting melt rate is correct. The Monin–Obukhov scaling in Eqs. (17)–(19) does reasonably well predicting the velocity profiles, even when the flow is strongly influenced by the stratification (Fig. 2d). The scaling is consistent with the temperature profile for weak stratification but departs significantly from the strongly stratified profile (Fig. 2e). For the salinity profiles, the scaling is reasonable for the passive scalar results (not shown here) but departs from the LES results for even the most weakly stratified case (Fig. 2f). Note that the Monin–Obukhov scaling for the salinity profile in Eq. (19) is dominated by the huge Schmidt number in the

At the ice base there can be large instantaneous spatial variability in the melt rate (Fig. 3) with peaks of up to 5 times the mean. These peaks are correlated with small-scale turbulent structures that form at the edge of the viscous boundary layer. Turbulent structures such as near-wall streaks are effective at transporting heat across the viscous boundary layer and hence a signature of these structures appears in the melt rate snapshots.

The mean melt rate is shown in Fig. 4 for all the runs in Table 1. The melt rates have been horizontally averaged across the ice base and averaged in time for 10 h, except for runs 1–3 and 10–11 which were averaged for >50 h. The passive scalar cases (*g* = 0) are included as lines in Fig. 4 since these results apply for any imposed Δ*T* (for a particular *T* are compensated for by a linear increase in

For stronger thermal driving, the melt rate departs from the value for passive scalars as the stable stratification inhibits turbulence and its ability to mix heat toward the boundary and melt the ice. At very strong thermal driving, the melt rates appear to become largely independent of Δ*T*. The point at which thermal driving and the stable salinity stratification become strong enough to damp turbulence is dependent on the friction velocity—higher friction velocities have more energetic turbulence and so stronger stratification is required to reduce the heat transfer and melt rate.

### b. Evolution of boundary layer turbulence

The response of the flow at early times in the simulations (Fig. 5) provides insight into the boundary layer turbulence. Recall that the initial condition consists of fully turbulent flow with uniform temperature and salinity,

One intermittently turbulent case (run 3) is shown in more detail in Fig. 6 to better understand the nature of the turbulent bursts. The TKE and friction velocity are both small during intervals of laminar flow before rapidly increasing when the flow goes turbulent. The bulk flow accelerates when the flow is laminar and the turbulence and friction velocity are small and exert less drag on the far-field current. The trace of TKE through time with friction velocity and driving temperature (Fig. 6g) begins when the flow is laminar. At the time immediately before a turbulent burst (*t* = 338.3 h) the stratification near the ice is very weak (Fig. 6f), allowing turbulent structures to form at the edge of the viscous sublayer. When a turbulent burst begins, the friction velocity and TKE rapidly increase to their maximum values (*t* = 339.3 h). The turbulence mixes more heat across the sublayer, increasing the temperature at the ice base *t* = 340 h). The melt rate increases in response to the increase in heat. In the salinity field (which dominates the density) the increased melt rate results in a decrease in the salinity at the boundary, resulting in a decrease in density near the ice as shown in Fig. 6f. As the turbulence continues, the density at the boundary reduces further until eventually the stable stratification is strong enough to damp turbulence. The trailing edge of the loop at smaller Δ*T* (*t* = 354 h) shows the continued smaller levels of turbulence which eventually die out as the system becomes laminar again. As the turbulence intensity decreases, less heat is transferred to the ice and so Δ*T* begins to slowly increase (*t* = 366 h). The density at the boundary slowly increases toward the preturbulent maximum, weakening the stratification under the ice again to eventually set off another turbulent burst.

Similar turbulent events occur in runs 1, 2, 10, and 11, although when thermal driving is very strong the turbulent portion of the trajectory in Δ*T*, ^{−1} that have large thermal driving (runs 10 and 11) it is more computationally expensive to run these for long intervals, hence there are fewer events to examine and the time interval of reoccurrence is unclear. The intermittently turbulent runs show that the TKE is not just a function of friction velocity and that the time history matters.

In an effort to quantify whether the system is fully or intermittently turbulent, we calculate the time-averaged TKE along with the standard deviation away from this mean (Fig. 7). For the smaller thermal driving (runs 4–8 and 12–15), the flow is fully turbulent and the TKE has small standard deviation. For larger thermal driving, the standard deviation increases significantly and there is a decrease in the total TKE as the flow becomes intermittently turbulent.

### c. Three-equation parameterization and Obukhov length

Here we examine whether the turbulent fluxes can be approximated by transfer coefficients as assumed in the three-equation model. For each simulation the drag coefficient in Eq. (26) and the transfer coefficients for heat

The passive scalar cases are shown as horizontal lines in Figs. 8a–c. There is very little dependence of

*L*to the viscous length scale,

The variance in TKE is plotted as a function of the time-averaged

Crucially,

For fully turbulent flow with large *T*. Using the maximum limiting values of *T* and *T* and

The *T* ≈ 5°C) where the heat flux starts to noticeably contribute to the buoyancy flux in ^{−1} for ^{−1} and 0.9 m yr^{−1} for ^{−1}, the latter of which is close to geophysically relevant values (Nicholls et al. 2009; Kimura et al. 2015). Comparing the predicted

Using smaller values of either

The three-equation model with the upper limits of

## 5. Summary

Large-eddy simulations were used to model the upper region of the ocean boundary layer beneath a melting ice shelf. Increases in thermal driving enhance the melt rate until the flow becomes strongly stratified in salinity. Turbulence is then suppressed by the stable stratification and no longer efficiently mixes heat across the interfacial sublayer, causing the melt rate to plateau with further increases in thermal driving. At this point the flow becomes intermittently turbulent in time, with long periods of laminar flow followed by abrupt turbulent bursts.

The transition between turbulent and intermittent regimes is well-described by the ratio of the Obukhov and viscous layer thicknesses

The

## 6. Discussion

Applying the ^{−1} the flow remains turbulent even at large thermal driving.

The Obukhov to viscous length ratio *λ* that has been used in ice–ocean studies (McPhee 2008). The mixing length is hypothesized to increase with depth until it saturates at a maximum value

The turbulent transfer coefficients for heat and salt diagnosed from our fully turbulent simulations with weak stratification are in good agreement with those empirically inferred from beneath the Ronne ice shelf (Jenkins et al. 2010). The drag coefficient is a factor of three smaller in the simulations compared to the Ronne ice shelf observations. This could be due to additional processes such as ice roughness which can increase the friction velocity or because, as Jenkins et al. (2010) notes, the drag coefficient is less well constrained than the transfer coefficients for this set of observations. Observations of turbulent flow under sea ice also give transfer coefficients consistent with the simulations (Sirevaag 2009). Note that the friction velocity (or drag coefficient) needs to be prescribed in the three-equation model, but it is difficult to observe and can vary significantly in space and time. Uncertainty around the friction velocity is perhaps the most difficult step in applying our results to observations or ice-melt parameterizations in ocean models.

The turbulent transfer and drag coefficients in the LES are consistent with those predicted by Monin–Obukhov similarity scaling, but the scaling significantly overestimates the salt transfer in stratified conditions. An improved model may require a modification to the Monin–Obukhov function

The intermittently turbulent simulations are thought to be dynamically different from the highly stratified ISOBL observed in the ocean. This is because the prescribed pressure gradient in the simulations accelerates the far-field current for cases with strong thermal driving. In contrast, the strongly stratified flow under the George VI ice shelf is observed to have low current speeds with evidence for double-diffusive steps (Kimura et al. 2015). Work in the atmospheric boundary layer community may give insight into other dynamical processes that could become important when the flow is strongly stably stratified (see review by Mahrt 2014).

Our focus has been on simulating regions of ice shelves that do not have a significant slope. In the weakly sloped case of a few degrees away from the horizontal, plume theory predicts that there will be negligible effects of an upslope current (Kerr and McConnochie 2015; McConnochie and Kerr 2018). Here, small slopes were found to have very little effect on the flow turbulence (see appendix A), making our results applicable to small slope angles. Steeper slopes occur near the grounding line which is an important region for ice-sheet dynamics. In such cases an upslope plume may be the primary source of turbulence and is likely to influence ice–ocean interactions (McConnochie and Kerr 2017; Mondal et al. 2019).

The present study was motivated by the ice shelf/ocean boundary layer. However, many results from the simulations can apply more generally to other ice–ocean interactions including land-fast and drifting sea ice. The formation of ice from seawater can result in a small ice salinity, commonly observed to be 3–7 psu for land-fast ice (Gerland et al. 1999; Vancoppenolle et al. 2007). Increasing the ice salinity in the simulations from the fresh ice shelf to saltier fast ice values is expected to modestly increase the melting temperature, but otherwise the results and conclusions will be very similar. It would be reasonably straightforward to include a constant

Future work will focus on simulations with larger thermal driving and friction velocities to get closer to real-world scenarios. There are also many other processes that are likely to affect the melt rate such as roughness of the ice, tides and basal slope. The simulations here were designed to model a subset of the larger planetary boundary layer—future work could include Earth’s rotation and to have both a surface layer and an outer layer. While it is significantly more difficult to simulate, the changing topography of the melting ice and the formation of channel cavities will be important in directing the melt outflow. We have not considered effects such as allowing the thermal expansion coefficient to vary with temperature, however this is unlikely to have much influence unless temperature differences become large. Other complicated flow phenomena such as double-diffusive layers will also be relevant for ice melting.

The NERC Standard Grant NE/N009746/1 is gratefully acknowledged for supporting the research presented here. This work used the ARCHER U.K. National Supercomputing Service (http://www.archer.ac.uk). Supporting data such as the DIABLO source code and input files to reproduce the simulations are available at https://doi.org/10.17863/CAM.39569. The authors are grateful to P. Holland for his helpful comments on an earlier version of the manuscript and to A. Jenkins, K. Nicholls, P. Davis, L. Couston, R. Patmore, and L. Middleton for their thoughts and discussion on this work. Thanks also to the editor and two anonymous reviewers for their comments that helped to improve the manuscript.

# APPENDIX A

## Sloped Runs

*x*direction,

*y*direction,

*F*can be viewed as the upslope component of an imposed hydrostatic pressure gradient. Therefore, it is just the feedback between

*changes*in mean density and the upslope buoyancy force that are neglected. This is not expected to have a strong effect in cases with small slopes, especially where the flow is dominated by shear turbulence such as the cases examined here.

Summary of additional runs with slope of ice changed from horizontal. Parameters are as in Table 1 with magnitude and direction of slope change also indicated.

Three fully turbulent runs from Table 1 were selected as base cases, with the direction of gravity angled to produce an ice slope of either 1° or 5° from the horizontal in the streamwise *x* or cross-stream *y* direction. The tilt of gravity does not have much, if any, influence on the turbulence in this system, as is shown by the results in Table A1. Future work will be to simulate the full boundary layer including the upslope acceleration for more strongly sloped cases. We note that there can be important feedbacks between melting and slope that act on larger scales (Jenkins 2016) that has not been ruled out here.

# APPENDIX B

## Anisotropic Minimum Dissipation Model for Large-Eddy Simulations

*δ*to ensure that the resulting eddy dissipation properly counteracts the spurious kinetic energy transferred by convective nonlinearity) gives subgrid scale viscosity,

*C*is a modified Poincaré constant,

*δ*we follow the suggestion of Verstappen (2016) to use

*k*is the grid cell (Verstappen 2016). In the two horizontal directions the grid is discretized using Fourier modes and a 2/3 dealiasing rule is applied moving from Fourier back to physical space. The filter widths are then

*i*and

*j*are the grid cells and

# APPENDIX C

## Implementation of Melting Boundary Conditions

In the vertical direction *z* the numerical solver has a grid for the vertical velocities (named *G* for base grid) and a staggered grid (named GF for fractional grid) for the horizontal velocities and scalars (Taylor 2008). The staggered grid is halfway between neighboring points of the base grid such that, for grid point *k*, the staggered grid is *N* grid points in the vertical direction, along with ghost cells at the base and top (0 and *m* are also defined at

## REFERENCES

Abkar, M., and P. Moin, 2017: Large-eddy simulation of thermally stratified atmospheric boundary-layer flow using a minimum dissipation model.

,*Bound.-Layer Meteor.***165**, 405–419, https://doi.org/10.1007/s10546-017-0288-4.Abkar, M., H. J. Bae, and P. Moin, 2016: Minimum-dissipation scalar transport model for large-eddy simulation of turbulent flows.

*Phys. Rev. Fluids*,**1**, 041701, https://doi.org/10.1103/PhysRevFluids.1.041701.Alley, K. E., T. A. Scambos, M. R. Siegfried, and H. A. Fricker, 2016: Impacts of warm water on Antarctic ice shelf stability through basal channel formation.

,*Nat. Geosci.***9**, 290, https://doi.org/10.1038/ngeo2675.Arzeno, I. B., R. C. Beardsley, R. Limeburner, B. Owens, L. Padman, S. R. Springer, C. L. Stewart, and M. J. Williams, 2014: Ocean variability contributing to basal melt rate near the ice front of Ross Ice Shelf, Antarctica.

,*J. Geophys. Res. Oceans***119**, 4214–4233, https://doi.org/10.1002/2014JC009792.Begeman, C. B., and Coauthors, 2018: Ocean stratification and low melt rates at the Ross Ice Shelf grounding zone.

,*J. Geophys. Res. Oceans***123**, 7438–7452, https://doi.org/10.1029/2018JC013987.Bradshaw, P., and G. P. Huang, 1995: The law of the wall in turbulent flow.

,*Proc. Roy. Soc. London***451A**, 165–188, https://doi.org/10.1098/rspa.1995.0122.Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux-profile relationships in the atmospheric surface layer.

,*J. Atmos. Sci.***28**, 181–189, https://doi.org/10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2.Dean, R., 1978: Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow.

,*J. Fluids Eng.***100**, 215–223, https://doi.org/10.1115/1.3448633.Determann, J., and R. Gerdes, 1994: Melting and freezing beneath ice shelves: Implications from a three-dimensional ocean-circulation model.

,*Ann. Glaciol.***20**, 413–419, https://doi.org/10.1017/S0260305500016785.Deusebio, E., C. P. Caulfield, and J. R. Taylor, 2015: The intermittency boundary in stratified plane Couette flow.

,*J. Fluid Mech.***781**, 298–329, https://doi.org/10.1017/jfm.2015.497.Eicken, H., H. Oerter, H. Miller, W. Graf, and J. Kipfstuhl, 1994: Textural characteristics and impurity content of meteoric and marine ice in the Ronne Ice Shelf, Antarctica.

,*J. Glaciol.***40**, 386–398, https://doi.org/10.1017/S0022143000007474.Flores, O., and J. J. Riley, 2011: Analysis of turbulence collapse in the stably stratified surface layer using direct numerical simulation.

,*Bound.-Layer Meteor.***139**, 241–259, https://doi.org/10.1007/s10546-011-9588-2.Foken, T., 2006: 50 years of the Monin–Obukhov similarity theory.

,*Bound.-Layer Meteor.***119**, 431–447, https://doi.org/10.1007/s10546-006-9048-6.Gayen, B., R. W. Griffiths, and R. C. Kerr, 2016: Simulation of convection at a vertical ice face dissolving into saline water.

,*J. Fluid Mech.***798**, 284–298, https://doi.org/10.1017/jfm.2016.315.Gerland, S., J.-G. Winther, J. Børre Ørbæk, and B. V. Ivanov, 1999: Physical properties, spectral reflectance and thickness development of first year fast ice in Kongsfjorden, Svalbard.

,*Polar Res.***18**, 275–282, https://doi.org/10.1111/j.1751-8369.1999.tb00304.x.Grosfeld, K., R. Gerdes, and J. Determann, 1997: Thermohaline circulation and interaction between ice shelf cavities and the adjacent open ocean.

,*J. Geophys. Res.***102**, 15 595–15 610, https://doi.org/10.1029/97JC00891.Gudmundsson, G., 2013: Ice-shelf buttressing and the stability of marine ice sheets.

,*Cryosphere***7**, 647–655, https://doi.org/10.5194/tc-7-647-2013.Harig, C., and F. J. Simons, 2015: Accelerated West Antarctic ice mass loss continues to outpace East Antarctic gains.

,*Earth Planet. Sci. Lett.***415**, 134–141, https://doi.org/10.1016/j.epsl.2015.01.029.Hattermann, T., O. A. Nøst, J. M. Lilly, and L. H. Smedsrud, 2012: Two years of oceanic observations below the Fimbul Ice Shelf, Antarctica.

,*Geophys. Res. Lett.***39**, L12605, https://doi.org/10.1029/2012GL051012.Holland, D. M., and A. Jenkins, 1999: Modeling thermodynamic ice–ocean interactions at the base of an ice shelf.

,*J. Phys. Oceanogr.***29**, 1787–1800, https://doi.org/10.1175/1520-0485(1999)029<1787:MTIOIA>2.0.CO;2.Jacobs, S., C. Giulivi, and P. Mele, 2002: Freshening of the Ross Sea during the late 20th century.

,*Science***297**, 386–389, https://doi.org/10.1126/science.1069574.Jenkins, A., 2011: Convection-driven melting near the grounding lines of ice shelves and tidewater glaciers.

,*J. Phys. Oceanogr.***41**, 2279–2294, https://doi.org/10.1175/JPO-D-11-03.1.Jenkins, A., 2016: A simple model of the ice shelf–ocean boundary layer and current.

,*J. Phys. Oceanogr.***46**, 1785–1803, https://doi.org/10.1175/JPO-D-15-0194.1.Jenkins, A., and A. Bombosch, 1995: Modeling the effects of frazil ice crystals on the dynamics and thermodynamics of ice shelf water plumes.

,*J. Geophys. Res.***100**, 6967–6981, https://doi.org/10.1029/94JC03227.Jenkins, A., K. W. Nicholls, and H. F. J. Corr, 2010: Observation and parameterization of ablation at the base of Ronne Ice Shelf, Antarctica.

,*J. Phys. Oceanogr.***40**, 2298–2312, https://doi.org/10.1175/2010JPO4317.1.Kader, B., and A. Yaglom, 1972: Heat and mass transfer laws for fully turbulent wall flows.

,*Int. J. Heat Mass Transfer***15**, 2329–2351, https://doi.org/10.1016/0017-9310(72)90131-7.Kaimal, J., J. Wyngaard, D. Haugen, O. Coté, Y. Izumi, S. Caughey, and C. Readings, 1976: Turbulence structure in the convective boundary layer.

,*J. Atmos. Sci.***33**, 2152–2169, https://doi.org/10.1175/1520-0469(1976)033<2152:TSITCB>2.0.CO;2.Keitzl, T., J. P. Mellado, and D. Notz, 2016: Reconciling estimates of the ratio of heat and salt fluxes at the ice-ocean interface.

*J. Geophys. Res. Oceans*,**121**, 8419–8433, https://doi.org/10.1002/2016JC012018.Kerr, R. C., and C. D. McConnochie, 2015: Dissolution of a vertical solid surface by turbulent compositional convection.

,*J. Fluid Mech.***765**, 211–228, https://doi.org/10.1017/jfm.2014.722.Kimura, S., K. W. Nicholls, and E. Venables, 2015: Estimation of ice shelf melt rate in the presence of a thermohaline staircase.

,*J. Phys. Oceanogr.***45**, 133–148, https://doi.org/10.1175/JPO-D-14-0106.1.Kimura, S., A. Jenkins, P. Dutrieux, A. Forryan, A. C. Naveira Garabato, and Y. Firing, 2016: Ocean mixing beneath Pine Island glacier ice shelf, West Antarctica.

,*J. Geophys. Res. Oceans***121**, 8496–8510, https://doi.org/10.1002/2016JC012149.Mahrt, L., 2014: Stably stratified atmospheric boundary layers.

,*Annu. Rev. Fluid Mech.***46**, 23–45, https://doi.org/10.1146/annurev-fluid-010313-141354.Martin, S., and P. Kauffman, 1977: An experimental and theoretical study of the turbulent and laminar convection generated under a horizontal ice sheet floating on warm salty water.

,*J. Phys. Oceanogr.***7**, 272–283, https://doi.org/10.1175/1520-0485(1977)007<0272:AEATSO>2.0.CO;2.McConnochie, C., and R. Kerr, 2017: Testing a common ice-ocean parameterization with laboratory experiments.

,*J. Geophys. Res. Oceans***122**, 5905–5915, https://doi.org/10.1002/2017JC012918.McConnochie, C., and R. Kerr, 2018: Dissolution of a sloping solid surface by turbulent compositional convection.

,*J. Fluid Mech.***846**, 563–577, https://doi.org/10.1017/jfm.2018.282.McPhee, M. G., 2008:

*Air-Ice-Ocean Interaction: Turbulent Ocean Boundary Layer Exchange Processes*. Springer, 216 pp.McPhee, M. G., G. A. Maykut, and J. H. Morison, 1987: Dynamics and thermodynamics of the ice/upper ocean system in the marginal ice zone of the Greenland Sea.

,*J. Geophys. Res.***92**, 7017–7031, https://doi.org/10.1029/JC092iC07p07017.McPhee, M. G., J. Morison, and F. Nilsen, 2008: Revisiting heat and salt exchange at the ice-ocean interface: Ocean flux and modeling considerations.

,*J. Geophys. Res.***113**, C06014, https://doi.org/10.1029/2007JC004383.Mondal, M., B. Gayen, R. W. Griffiths, and R. C. Kerr, 2019: Ablation of sloping ice faces into polar seawater.

,*J. Fluid Mech.***863**, 545–571, https://doi.org/10.1017/jfm.2018.970.Monin, A. S., and A. M. Obukhov, 1954: Osnovnye zakonomernosti turbulentnogo peremesivanija v prizemnom sloe atmosfery (Basic laws of turbulent mixing in the surface layer of the atmosphere).

,*Tr. Geofiz. Inst., Akad. Nauk SSSR***24**, 163–187.Nicholls, K. W., S. Østerhus, K. Makinson, T. Gammelsrød, and E. Fahrbach, 2009: Ice-ocean processes over the continental shelf of the southern Weddell Sea, Antarctica: A review.

,*Rev. Geophys.***47**, RG3003, https://doi.org/10.1029/2007RG000250.Nicholls, K. W., K. Makinson, and E. Venables, 2012: Ocean circulation beneath Larsen C Ice Shelf, Antarctica from in situ observations.

*Geophys. Res. Lett.*,**39**, L19608, https://doi.org/10.1029/2012GL053187.Obukhov, A. M., 1946: Turbulentnost’v temperaturnoj-neodnorodnoj atmosfere (turbulence in an atmosphere with a non-uniform temperature).

,*Tr. Akad. Nauk. SSSR Inst. Teorel. Geofiz.***1**, 95–115.Oerter, H., J. Kipfstuhl, J. Determann, H. Miller, D. Wagenbach, A. Minikin, and W. Graft, 1992: Evidence for basal marine ice in the Filchner–Ronne Ice Shelf.

,*Nature***358**, 399, https://doi.org/10.1038/358399a0.Orszag, S. A., 1971: Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representation.

,*Stud. Appl. Math.***50**, 293, https://doi.org/10.1002/sapm1971504293.Pope, S. B., 2000:

*Turbulent Flows*. Cambridge University Press, 806 pp.Purkey, S. G., and G. C. Johnson, 2012: Global contraction of Antarctic Bottom Water between the 1980s and 2000s.

,*J. Climate***25**, 5830–5844, https://doi.org/10.1175/JCLI-D-11-00612.1.Rignot, E., and S. S. Jacobs, 2002: Rapid bottom melting widespread near Antarctic ice sheet grounding lines.

,*Science***296**, 2020–2023, https://doi.org/10.1126/science.1070942.Rignot, E., and K. Steffen, 2008: Channelized bottom melting and stability of floating ice shelves.

,*Geophys. Res. Lett.***35**, L02503, https://doi.org/10.1029/2007GL031765.Rozema, W., H. J. Bae, P. Moin, and R. Verstappen, 2015: Minimum-dissipation models for large-eddy simulation.

*Phys. Fluids*,**27**, 085107, https://doi.org/10.1063/1.4928700.Rye, C. D., A. C. N. Garabato, P. R. Holland, M. P. Meredith, A. G. Nurser, C. W. Hughes, A. C. Coward, and D. J. Webb, 2014: Rapid sea-level rise along the Antarctic margins in response to increased glacial discharge.

,*Nat. Geosci.***7**, 732–735, https://doi.org/10.1038/ngeo2230.Schlichting, H., and K. Gersten, 2003:

*Boundary-Layer Theory*. Springer, 800 pp.Schoof, C., 2007: Ice sheet grounding line dynamics: Steady states, stability, and hysteresis.

,*J. Geophys. Res.***112**, F03S28, https://doi.org/10.1029/2006JF000664.Scotti, A., and B. White, 2016: The mixing efficiency of stratified turbulent boundary layers.

,*J. Phys. Oceanogr.***46**, 3181–3191, https://doi.org/10.1175/JPO-D-16-0095.1.Sirevaag, A., 2009: Turbulent exchange coefficients for the ice/ocean interface in case of rapid melting.

,*Geophys. Res. Lett.***36**, L04606, https://doi.org/10.1029/2008GL036587.Snow, K., S. Rintoul, B. Sloyan, and A. M. Hogg, 2018: Change in dense shelf water and Adélie Land bottom water precipitated by iceberg calving.

,*Geophys. Res. Lett.***45**, 2380–2387, https://doi.org/10.1002/2017GL076195.Stanton, T. P., and Coauthors, 2013: Channelized ice melting in the ocean boundary layer beneath Pine Island Glacier.

,*Science***341**, 1236–1239, https://doi.org/10.1126/science.1239373.Taylor, J. R., 2008: Numerical simulations of the stratified oceanic bottom boundary layer. Ph.D. thesis, University of California, San Diego, 212 pp., https://escholarship.org/uc/item/5s30n2ts.

Vancoppenolle, M., C. M. Bitz, and T. Fichefet, 2007: Summer landfast sea ice desalination at Point Barrow, Alaska: Modeling and observations.

,*J. Geophys. Res.***112**, C04022, https://doi.org/10.1029/2006JC003493.Verstappen, R., 2016: How much eddy dissipation is needed to counterbalance the nonlinear production of small, unresolved scales in a large-eddy simulation of turbulence?

,*Comput. Fluids***176**, 276–284, https://doi.org/10.1016/j.compfluid.2016.12.016.Vreugdenhil, C. A., and J. R. Taylor, 2018: Large-eddy simulations of stratified plane Couette flow using the anisotropic minimum-dissipation model.

*Phys. Fluids*,**30**, 085104, https://doi.org/10.1063/1.5037039.Williams, M., R. Warner, and W. Budd, 1998: The effects of ocean warming on melting and ocean circulation under the Amery Ice Shelf, East Antarctica.

,*Ann. Glaciol.***27**, 75–80, https://doi.org/10.3189/1998AoG27-1-75-80.Wyngaard, J. C., 2010:

*Turbulence in the Atmosphere*. Cambridge University Press, 406 pp.Yaglom, A., and B. Kader, 1974: Heat and mass transfer between a rough wall and turbulent fluid flow at high Reynolds and Peclet numbers.

,*J. Fluid Mech.***62**, 601–623, https://doi.org/10.1017/S0022112074000838.Zhou, Q., J. R. Taylor, and C. P. Caulfield, 2017: Self-similar mixing in stratified plane Couette flow for varying Prandtl number.

,*J. Fluid Mech.***820**, 86–120, https://doi.org/10.1017/jfm.2017.200.