Regimes and Transitions in the Basal Melting of Antarctic Ice Shelves

Madelaine G. Rosevear aInstitute of Marine and Antarctic Studies, University of Tasmania, Hobart, Tasmania, Australia
bOceans Graduate School, University of Western Australia, Perth, Western Australia, Australia

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Bishakhdatta Gayen cDepartment of Mechanical Engineering, University of Melbourne, Melbourne, Victoria, Australia
dCentre for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bengaluru, India
eThe Australian Centre for Excellence in Antarctic Science, University of Tasmania, Hobart, Tasmania, Australia

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Benjamin K. Galton-Fenzi eThe Australian Centre for Excellence in Antarctic Science, University of Tasmania, Hobart, Tasmania, Australia
fAustralian Antarctic Division, Kingston, Tasmania, Australia
gAustralian Antarctic Program Partnership, Institute for Marine and Antarctic Studies, University of Tasmania, Hobart, Tasmania, Australia

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Abstract

The Antarctic Ice Sheet is losing mass as a result of increased ocean-driven melting of its fringing ice shelves. Efforts to represent the effects of basal melting in sea level projections are undermined by poor understanding of the turbulent ice shelf–ocean boundary layer (ISOBL), a meters-thick layer of ocean that regulates heat and salt transfer between the ocean and ice. To address this shortcoming, we perform large-eddy simulations of the ISOBL formed by a steady, geostrophic flow beneath horizontal ice. We investigate melting and ISOBL structure and properties over a range of free-stream velocities and ocean temperatures. We find that the melting response to changes in thermal and current forcing is highly nonlinear due to the effects of meltwater on ISOBL turbulence. Three distinct ISOBL regimes emerge depending on the relative strength of current shear and buoyancy forcing: “well-mixed,” “stratified,” or “diffusive-convective.” We present expressions for mixing-layer depth for each regime and show that the transitions between regimes can be predicted with simple nondimensional parameters. We use these results to develop a novel regime diagram for the ISOBL which provides insight into the varied melting responses expected around Antarctica and highlights the need to include stratified and diffusive-convective dynamics in future basal melting parameterizations. We emphasize that melting in the diffusive-convective regime is time dependent and is therefore inherently difficult to parameterize.

Significance Statement

The purpose of this study is to investigate the processes that control ocean-driven melting of Antarctic ice shelves (100–1000-m-thick floating extensions of the Antarctic ice sheet). Currently, these processes are poorly understood due to the difficulty of accessing the ocean beneath ice shelves. Using an ocean model, we determine the melting response to different ocean conditions, including feedbacks whereby cold, fresh meltwater can enhance or suppress turbulent eddies beneath the ice, depending on the ocean state. Our results point the way to improvements in the representation of ocean-driven melting in ocean/climate models, which will allow more accurate predictions of future climate and sea level.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Madelaine G. Rosevear, madi.rosevear@gmail.com

Abstract

The Antarctic Ice Sheet is losing mass as a result of increased ocean-driven melting of its fringing ice shelves. Efforts to represent the effects of basal melting in sea level projections are undermined by poor understanding of the turbulent ice shelf–ocean boundary layer (ISOBL), a meters-thick layer of ocean that regulates heat and salt transfer between the ocean and ice. To address this shortcoming, we perform large-eddy simulations of the ISOBL formed by a steady, geostrophic flow beneath horizontal ice. We investigate melting and ISOBL structure and properties over a range of free-stream velocities and ocean temperatures. We find that the melting response to changes in thermal and current forcing is highly nonlinear due to the effects of meltwater on ISOBL turbulence. Three distinct ISOBL regimes emerge depending on the relative strength of current shear and buoyancy forcing: “well-mixed,” “stratified,” or “diffusive-convective.” We present expressions for mixing-layer depth for each regime and show that the transitions between regimes can be predicted with simple nondimensional parameters. We use these results to develop a novel regime diagram for the ISOBL which provides insight into the varied melting responses expected around Antarctica and highlights the need to include stratified and diffusive-convective dynamics in future basal melting parameterizations. We emphasize that melting in the diffusive-convective regime is time dependent and is therefore inherently difficult to parameterize.

Significance Statement

The purpose of this study is to investigate the processes that control ocean-driven melting of Antarctic ice shelves (100–1000-m-thick floating extensions of the Antarctic ice sheet). Currently, these processes are poorly understood due to the difficulty of accessing the ocean beneath ice shelves. Using an ocean model, we determine the melting response to different ocean conditions, including feedbacks whereby cold, fresh meltwater can enhance or suppress turbulent eddies beneath the ice, depending on the ocean state. Our results point the way to improvements in the representation of ocean-driven melting in ocean/climate models, which will allow more accurate predictions of future climate and sea level.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Madelaine G. Rosevear, madi.rosevear@gmail.com

1. Introduction

Ocean-driven basal melting comprises more than half of the mass loss from Antarctica’s ice shelves (Depoorter et al. 2013; Rignot et al. 2013). Currently, increased basal melting is thinning the Antarctic Ice Sheet at an accelerating rate (Paolo et al. 2015; Shepherd et al. 2018), presenting a major threat to coastal regions (Cazenave and Llovel 2010). Melting of ice shelves also contributes a freshwater flux to the ocean. This flux has a major impact on the production of Antarctic Bottom Water which supplies the lower limb of the global thermohaline circulation (Purkey and Johnson 2013; De Lavergne et al. 2014).

Understanding how the ocean drives basal melting is critical for quantifying the rates of present mass loss and developing the capability to project future change. Despite extremely limited spatial and temporal coverage, observations from within ice shelf cavities have revealed a range of different cavity environments and melting behaviors. In cold, energetic conditions such as those observed below the Filchner–Ronne (Jenkins et al. 2010) and Larsen C (Davis and Nicholls 2019) ice shelves, melting is strongly controlled by current speed. Under these conditions, buoyancy flux due to meltwater has little effect on the turbulent boundary layer beneath the ice (Davis and Nicholls 2019). However, observations suggest that buoyant meltwater plays an important role in ice shelf–ocean interactions in warmer ice shelf cavities. For example, meltwater has been observed to drive an energetic plume along the sloping base of Pine Island Ice Shelf (Stanton et al. 2013), to increase water column stratification beneath the Ross Ice Shelf (Stewart 2018; Begeman et al. 2018), and to drive diffusive convection (DC), a type of convection enabled by the differing diffusivities of heat and salt, beneath George VI Ice Shelf (Kimura et al. 2015).

Recent laboratory and modeling studies using large-eddy simulation (LES) and direct numerical simulation have focused on understanding the effects of buoyant meltwater on the ice shelf–ocean boundary layer (ISOBL). For sloping or vertical ice, buoyant instability can control the fluxes of heat and salt to the ice. In this scenario, the melt rate is a function of ocean temperature, but does not depend directly on the speed of the plume or current adjacent to the ice (Kerr and McConnochie 2015; McConnochie and Kerr 2016; Gayen et al. 2016; Mondal et al. 2019). Beneath horizontal ice, Vreugdenhil and Taylor (2019) investigated effects of meltwater on a turbulent, shear-dominated ISOBL, showing that buoyancy inhibits the turbulent transfer of heat and salt and suppresses melting. However, when DC occurs, increased buoyant meltwater fluxes will enhance—rather than suppress—turbulence, and DC mixing can form a thermohaline staircase beneath the ice (Rosevear et al. 2021). In the DC regime, melt rates are inherently transient due to the presence of a growing diffusive sublayer adjacent to the ice (Keitzl et al. 2016b; Middleton et al. 2021). Furthermore, melting is insensitive to changes in the far field current strength (Rosevear et al. 2021), contrary to existing parameterizations of ice–ocean interactions (e.g., McPhee et al. 1987; Jenkins 1991; Jenkins et al. 2010).

Ice–ocean parameterizations should represent the state of our current understanding of the ISOBL and provide reliable melt rate estimates in most environmental conditions beneath Antarctic ice shelves. Instead, existing parameterizations match observations only under the coldest, most energetic conditions (Rosevear et al. 2022). Existing parameterizations are also subject to issues of implementation as a result of the assumption of a water column structure in which boundary layer turbulence forms a well-mixed layer adjacent to the ice. In reality, this layer is not always well mixed (e.g., Stanton et al. 2013), consequently, we will refer to it as a mixing layer. The mixing-layer temperature (T), salinity (S), and friction velocity ( u*) constitute the driving parameters for typical basal melting parameterizations (e.g., McPhee et al. 1987; Jenkins 1991; Jenkins et al. 2010). In this paradigm, T and S must be sampled within the mixing layer. In practice, the depth at which these properties are sampled depends either on the vertical resolution of the model or an arbitrary choice of sampling depth, resulting in an unphysical dependence of predicted melt rates on ocean model resolution (Gwyther et al. 2020). Similar issues arise when the velocity u, which is required to estimate friction velocity u*, is taken from outside of the relevant region of the boundary layer (Davis and Nicholls 2019).

These issues highlight the need for a holistic picture of the ice–ocean boundary layer, including melting, ocean properties, velocity structure, boundary layer depth, and mixing-layer depth, especially under warm conditions where stratification becomes important. To address this need, we performed high-resolution LES of the ice shelf–ocean boundary layer, forced with a steady, geostrophic current covering wide ranges of cavity conditions. We varied ocean current speed and temperature and analyzed these simulations to determine the principal variables affecting the melt rate and mixing beneath the ice. We classified these results based on the dominant physics observed and developed criteria for the transition between these regimes as a function of commonly observed/resolved ocean variables. In this study, we considered melting of hydraulically smooth, horizontal ice. However, we note that Antarctic ice shelves are commonly sloping and hydraulically rough. Further study will be needed to extend these results to more varied topography and basal conditions.

This paper is structured as follows. Section 2 provides detailed background on oceanic boundary layers, including those with a stabilizing surface buoyancy flux, as well as currently used parameterizations of ice–ocean interactions. The governing equations, numerical implementation, and forcing for the numerical model are detailed in section 3. We then present the results of our simulations in section 4, focusing first on the boundary layer and mixing-layer characteristics, including the classification of boundary layer types based on the observed processes, followed by analysis of the melt rates. In section 5 we synthesize our results into a regime diagram for the ISOBL which we compare to observed sub-ice shelf conditions. Conclusions and future directions are found in section 6.

2. Background and scaling

a. The ice shelf–ocean boundary layer

The ISOBL typically refers to the frictional boundary layer that forms adjacent to an ice shelf due to the presence of a mean flow in the ocean (e.g., Fig. 1). Friction acts to reduce velocity in the vicinity of the ice, creating velocity shear and driving turbulence. Due to Coriolis, the viscous stresses associated with vertical shear result in rotation of the current vector with depth. Rotation gives the boundary layer, also known as an Ekman layer, a natural depth scale (δf) which depends on the Coriolis frequency (f) and the friction velocity ( u*=τb/ρ0), where τb is the shear stress at the ice–ocean interface and ρ0 is a reference density. Commonly used variables and model parameters are defined in Table 1. The vertical scale of the Ekman layer is
δfu*f,
where the turbulent Ekman boundary layer extends to ∼0.4δf (Pope 2001). Stratification will tend to decrease the Ekman layer depth relative to δf.
Fig. 1.
Fig. 1.

Schematic of the model domain, which consists of a horizontal ice–ocean interface with cold, salty ocean below. In the interior of the domain the ocean is in geostrophic balance and has velocity (u, υ) = (U0, 0). Near the ice an Ekman boundary layer forms.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

The ISOBL can be divided into three parts: the viscous sublayer, the surface layer, and the outer layer. The viscous sublayer is the region of laminar flow adjacent to the ice, which is O(10−2) m thick. Within the viscous sublayer, velocity scales with distance from the ice as
u+z+,
where the distance z and velocity u are expressed nondimensionally as
u+=uu*,z+=zu*ν,
where ν is molecular viscosity. Adjacent to the viscous sublayer is the surface or “log” layer, which occupies ∼30% of the total boundary layer depth (Pope 2001). Within the log layer, turbulence is affected by the presence of the solid boundary, in this case the ice–ocean interface. Vertical shear is inversely proportional to the distance from the interface (z) and proportional to the strength of the turbulence. In an unstratified flow, the dimensionless current shear (ϕ) is given by
ϕ=kzu*(uz)=1,
where k = 0.41 is von Kármán’s constant. Integrating ϕ gives a logarithmic current profile in height,
u+=1kln(z+)+C,
applying the identities in (3). This logarithmic scaling of velocity with depth is often called the law of the wall.
For flow that is strongly affected by stratification, the velocity profile is expected to depart from this logarithmic scaling. The departure of the velocity profile from (5) is commonly taken to be a linear function ϕ(ξ) = 1 + βmξ of the normalized distance from the ice ξ = z/L, where L is the Obukhov length,
L=u*3kBb,
and Bb (m2 s−3) is the surface buoyancy flux. Integrating (6) yields the velocity profile
U+=1kln(z+)+βmkξ+C,
where constant C = 5.0 (Bradshaw and Huang 1995) and βm = 4.7 (Businger et al. 1971). This expression implies that stratification is affecting the mixing length, or the maximum vertical distance over which eddies can diffuse momentum (McPhee 1994). For a positive (stabilizing) buoyancy flux, the Obukhov length scale is an estimate of the distance from the ice where stratification effects are felt by ISOBL turbulence. Another key parameter is the ratio of the Obukhov length scale and the viscous length scale (δν), which we will refer to as the viscous Obukhov scale (L+):
L+=Lδν,
where δν=ν/u*. A small L+ indicates there is no region of the flow free from the effects of either stratification or viscosity, both of which suppress turbulence.
Stratification also influences the overall boundary layer depth. Stratification is characterized by the buoyancy frequency N = [−g/ρ0(∂ρ/∂z)]1/2 and can be imposed in the form of both a stratified far-field environment and through a stabilizing buoyancy flux at the boundary, where both are relevant to the ISOBL. External stratification decreases the Ekman layer depth relative to the well-mixed case (Weatherly and Martin 1978; Taylor and Sarkar 2008a; McWilliams et al. 2009). McWilliams et al. (2009) showed that the stratified Ekman layer depth varies as a function of the stratification parameter γ = N2/f2. Typically in the ocean N > f and therefore γ ≫ 1. The mixing-layer depth (h) in the presence of external stratification is given by
h=Cγu*fγ1/4=Cγu*Nf.
This expression is consistent with the scaling of Weatherly and Martin (1978) for γ ≫ 1. Based on the height at which the TKE or turbulent mixing goes to zero Weatherly and Martin (1978) found Cγ = 1.3, while Fer and Sundfjord (2007) found Cγ = 1.5 for the mixing layer beneath sea ice. Zulberti et al. (2022) showed that this expression can also be used to estimate the mixing-layer depth for tidal flow by using the tidal frequency in place of f.
A stabilizing surface buoyancy flux has been shown to limit the depth of an Ekman boundary layer and increase its cross-stream velocity (Coleman et al. 1992; Shah and Bou-Zeid 2014). The vertical scales of turbulence are reduced under stabilizing surface buoyancy forcing, causing the mixing-layer depth to shoal (McPhee 2008). Under a stabilizing buoyancy flux, the mixing-layer depth has been shown to depend on the stability parameter μ:
h=Cμu*fμ1/2=Cμu*Lf
(e.g., Zilitinkevich 1972; Garratt 1982; Deusebio et al. 2014), where μ=u*/(fL) is the ratio of the Ekman scale to the Obukhov length scale.
In situations where DC dominates mixing beneath the ice, the mixing-layer depth no longer depends on u* (Rosevear et al. 2021). Mixing layer depth scaling for this case has not yet been established. However, in a dynamically similar problem in which a stable salinity gradient was heated from below Fernando (1987) found that the mixing-layer depth was controlled by the surface buoyancy flux due to heat only ( BbT) and the far-field stratification as
h=CDC(BbTN03)1/2,
with constant CDC = 41.

b. Models of ice–ocean interaction

Thermodynamic models of ice–ocean interaction aim to determine the ice shelf melt rate (m), interface temperature (Tb), and salinity (Sb) as a function of the ocean mixing-layer and ice temperatures. The interface is assumed to be at the freezing temperature, which is a weakly nonlinear function of salinity and a linear function of pressure. A linearized version is typically used, such that
Tb=Tf(Sb, pb)=λ1Sb+λ2pb+λ3,
where λ1 and λ2 are the liquidus slopes in salinity and pressure, λ3 is an offset, and pb is the interface pressure. The melt rate of the ice is determined by balancing heat and salt fluxes at the ice–ocean interface. In the following, the subscripts i, ML, and b denote ice shelf, mixing layer, and interface properties, respectively. At the interface, conservation of heat allows us to balance the latent heat of melting with divergence between the oceanic and ice heat fluxes. The latent heat of melting is given by ρiceLfm, where ρice is the density of the ice shelf, Lf is the latent heat of freezing, and m is the melt rate. The oceanic heat flux is given by ρcpκT(∂T/∂z)b, where ρ, cp, and κT are the density, specific heat capacity, and thermal diffusivity of the ocean and (∂T/∂z)b is the oceanic temperature gradient at the ice–ocean interface. In the following, the conductive heat flux into the ice is set to zero based on the assumption that it is much smaller than the latent heat term, following, e.g., Gayen et al. (2016), Mondal et al. (2019), and Vreugdenhil and Taylor (2019). For a discussion on the influence of conductive heat flux term, see Holland and Jenkins (1999). The interfacial heat transfer balance is given by
ρiceLfm=ρcpκT(Tz)b.
We can formulate a similar expression for salinity budget by balancing the brine flux due to melting ρiceSbm with the oceanic salt flux ρκS(∂T/∂z)b, where κS is the diffusivity of salt. The ice salinity and the salt flux into the ice are taken to be zero (Oerter et al. 1992). Equating the remaining terms yields:
ρiceSbm=ρκS(Sz)b.
The oceanic heat and salt fluxes in Eqs. (13) and (14) are not resolved in regional or circum-Antarctic ocean models and must instead be parameterized (e.g., Holland and Jenkins 1999). The commonly used three-equation parameterization represents these fluxes in terms of the temperature and salinity difference between the mixing layer and the interface. The surface and sublayer dynamics are encapsulated by the product of friction velocity u* and heat and salt transfer coefficients ΓT and ΓS:
ΓT=κT(Tz)bu*(TMLTb),
ΓS=κS(Sz)bu*(SMLSb).
Typically the friction velocity is not known and is instead modeled as a function of the free-stream velocity U0 and the drag coefficient Cd as u*=Cd1/2U0 (Table 2). The drag coefficient is not well constrained for this problem, since very few observational studies have succeeded in measuring u* beneath an ice shelf. Some exceptions are an estimate of Cd = 0.0022 from Larsen C Ice Shelf and an estimate of Cd ∼ 0.004 from Pine Island Glacier Ice Shelf (inferred from Stanton et al. 2013). However, measurements of Cd are desperately needed, since it is expected to vary spatially based on the roughness of the ice base (Gwyther et al. 2015).
Table 1

Notation.

Table 1

Equations (15) and (16) assume that the thermal and saline diffusive sublayers (diffusive regions next to the ice) are controlled by current shear and thin with increasing friction velocity (e.g., Wells and Worster 2008; McConnochie and Kerr 2017). However, there are many conditions pertinent to the Antarctic ISOBL where shear does not control the thickness of the diffusive sublayer. In the case of a sloping ice shelf, fresh meltwater drives convection adjacent to the ice and it is convective instability, rather than shear, that controls the diffusive sublayer thickness (Kerr and McConnochie 2015; McConnochie and Kerr 2018; Mondal et al. 2019) and sets the melt rate. Under flat ice, in cases with relatively weak currents, differential diffusion of heat and salt allows a type of convection known as diffusive convection to occur. In this instance, it has been shown that the diffusive sublayer grows in time, resulting in a transient melt rate which depends on time as t−1/2 (Martin and Kauffman 1977; Keitzl et al. 2016a; Middleton et al. 2021; Rosevear et al. 2021).

3. Methods

The steady flow and turbulent boundary layer beneath a horizontal ice shelf is modeled using Diablo (Taylor 2008) in a computational domain of height LZ and horizontal dimensions LX = LY. A schematic of the model domain is shown in Fig. 1. The flow is periodic in both horizontal directions, while the upper and lower boundaries are impenetrable. The boundary conditions at the upper and lower boundaries are no-slip and free-slip, respectively. The flow velocities in x, y, and z directions are u, υ, and w, respectively, and the Coriolis parameter f is constant within the domain. We model the flow using large-eddy simulation (LES), in which the larger turbulent scales (which contain most of the energy) are explicitly resolved, while the effects of the small-scale isotropic turbulence are parameterized.

a. Governing equations

The simulations solve the incompressible, nonhydrostatic Navier–Stokes momentum equation under the Boussinesq approximation along with the conservation of mass, heat and salt, and a linear equation of state. In dimensional form, these equations are
u=0,
DuDt=1ρ0p*+fk×u+ν2uρ*ρ0gkτ,
DT*Dt=κT2T*λTwdTbgdz,
DS*Dt=κS2S*λSwdSbgdz,
ρ*=ρ0(βS*αT*).
Here u = (u, υ, w) is the flow velocity and p*, T*, S*, and ρ* are the deviations from background hydrostatic pressure pbg, background temperature Tbg, salinity Sbg, and density ρbg profiles, respectively, e.g., S*=SSbg. The cold, saline water has molecular viscosity ν = 2 × 10−6 m2 s−1, thermal diffusivity κT = 1.4 × 10−7 m2 s−1, salt diffusivity κS = 1.3 × 10−9 m2 s−1, thermal expansion coefficient α = 3.8 × 10−5 °C−1 and haline contraction coefficient β = 7.8 × 10−4 kg g−1, where these values are chosen to be appropriate for the polar ocean. The quantities τ, λT, and λS are the subgrid-scale stress tensor and temperature and salinity flux vectors, respectively. The subgrid-scale stress tensor is represented with a dynamic eddy viscosity model and the subgrid salinity/temperature fluxes with a dynamic eddy diffusivity model. The expressions for the subgrid-scale models are as follows:
τij=2νTMij¯,νT=CΔ¯2|M¯|,
λjT=KTT¯xj,KT=CTΔ¯2|M¯|,
λjS=KSS¯xj,KS=CSΔ¯2|M¯|,
where Mij is the strain tensor and Δ is the filter width. Dynamic Smagorinsky coefficients C, CT, and CS are implemented following Taylor (2008), where our coefficients for temperature and salinity are determined equivalently to the coefficient for buoyancy in Taylor (2008). The coefficients are evaluated through a dynamic procedure introduced by Germano et al. (1991).

b. Numerical method

The simulations use a mixed spectral/finite difference algorithm (see Taylor 2008 for details). Spanwise derivatives are treated with a pseudospectral method, and the wall normal spatial derivatives are computed with second-order finite differences. A third-order Runge–Kutta method is used for time stepping, and viscous terms are treated implicitly with the Crank–Nicolson method. Variable time stepping with a fixed CFL number of 0.9 is used, with typical time steps on the order of 1 s for U0 = 0.014 m s−1 runs. At the upper boundary temperature and salinity are given by the melting boundary condition [Eqs. (13) and (14)], while at the lower boundary they are relaxed back to initial profiles in a 3-m-thick sponge region, to minimize reflection of internal waves back into the domain. Within the sponge layer all fluctuations of velocity, salinity and temperature are suppressed. Detailed implementation of the sponge layer can be found in Gayen et al. (2010).

The melt rate is determined by solving Eqs. (13), (14), and a modified version of (12):
Tb=Tf(Sb)=λ1Sb0.311,
where λ1 = −0.057°C kg g−1 and constant −0.311 accounts for the depression of the freezing point due to a pressure of 500 dbar. The ice–water interface is assumed to remain planar and fixed at z = 0.

The ambient flow, with amplitude U0, is achieved through forcing with a constant pressure gradient in the y direction. A geostrophic balance of 1/ρ0(∂p/∂y) = −fU0 holds in the far field. As the model domain is periodic in both horizontal directions, the mean vertical motion is constrained to zero. Therefore, the features of oceanic boundary layers that would result from Ekman pumping/suction driven by large-scale horizontal gradients in the ambient flow are not present in these simulations. A stable ambient stratification with buoyancy frequency N0 = 1.75 × 10−3 s−1 is imposed using the background salinity gradient Sbg = S0 − 4 × 10−4 z, where S0 = 34.5 g kg−1. The stratification is based on observed stratification beneath the Amery Ice Shelf (Rosevear et al. 2022). The initial temperature (T0) is constant with depth, and chosen to obtain a specific elevation above freezing, i.e., T0*=T0Tf(S0).

c. Domain and resolution

The model domain size is based upon δf, which is estimated using the expression δf=u0*/f, where u0* (m s−1) is the friction velocity, estimated a priori using the drag law relationship u0*=Cd1/2U0 taking Cd = 2.5 × 10−3. The domain sizes LX × LY × LZ, which are roughly 1.4δf × 1.4δf × 1.8δf, can be found in Table 1. The LES is performed using a grid that is uniform in both the x and y directions and stretched in the z direction to achieve higher resolution near the boundary. Grid resolution is based upon viscous length scale δν=ν/u0* where ν = 2 × 10−6 m2 s−1 is the molecular viscosity. The grid resolution for each experiment is expressed nondimensionally as ΔX+ = ΔX/δν. We are performing “resolved” LES, which means that resolution near the ice is sufficient to resolve the viscous stresses, and no wall model is needed. The grid resolution is chosen to be consistent with the results of Salon et al. (2007) for resolved LES. The horizontal resolution is typically ΔX+ = 20, with the exception of the 0.007 m s−1 case, where it is increased to ΔX+ = 10 in order to satisfy horizontal-to-vertical grid aspect ratio and vertical resolution constraints simultaneously. We require extremely high resolution near the boundary to capture the near-ice diffusive sublayers, which are thin due to the realistic κT and κS values. The grid stretching is performed using a tanh function with resolution ΔZ+ < 1 at the wall. The aspect ratio of grid cells at the edge of the viscous sublayer (ΔZ+ ∼ 50) has also been shown to be important, with A = ΔZX = 1/8 being recommended in a previous study (Vreugdenhil and Taylor 2019). We found no dependence on aspect ratio for 1/30 < A < 1/7, and most of our runs are performed with A = 1/11.

Table 2

Grid specifications for different model runs. Free-stream velocity U0 (cm s−1); Coriolis frequency f (s−1); domain horizontal dimension LX (=LY); domain height LZ; nondimensional horizontal resolution ΔX+ (=ΔY+); vertical resolution at the ice–ocean interface ΔZ+.

Table 2

To ease computational constraints for our highest velocity (U0 = 2.8 cm s−1) cases we double the Coriolis parameter from the physically relevant value at a latitude of 70°S of f = 1.37 × 10−4 to f = 2.75 × 10−4 s−1, decreasing the expected boundary layer depth based on δf and allowing us to decrease the domain size. In appendix A we show that this does not affect the near-ice dynamics or heat transfer.

For each case, a neutral (unstratified) spinup run is performed to obtain the velocity field of a steady-state turbulent Ekman boundary layer. At model time t = 0 h the initial temperature and salinity profiles are applied, after which time the temperature, salinity and velocity fields coevolve. The simulations are run for 2–3 inertial periods ti = 2π/f (s). The temperature and velocity forcing for the simulations span T0*=0.0025°0.05°C and U0 = 0.7–2.8 cm s−1 (Table 3). For context, observed current speeds beneath ice shelves span two orders of magnitude from 1 to 50 cm s−1 (Begeman et al. 2018; Nicholls et al. 2004) and temperature measurements range from 0.001° to 2.5°C above the local freezing point (Stevens et al. 2020; Kimura et al. 2015). As we are focusing on melting, we only consider conditions with temperatures above freezing, although “supercooled” conditions are also possible.

Table 3

Key simulation parameters and results. Free-stream velocity U0; initial thermal driving T0*; friction velocity u*; drag coefficient Cd; temperature difference between the mixing-layer and the interface TMLTb; melt rate m; Obukhov length scale L; viscous Obukhov scale L+. All values are computed at t = 3ti, except runs A3, B3, and C3, which are at t = 2ti. Unsteady melt rates are denoted by an asterisk. Mixing-layer classifications are WM (well-mixed), STR (stratified), and DC (diffusive-convective). DC/STR indicates a transition from DC to STR during the simulation, and the transition time tDC is shown in the final column.

Table 3

4. Results

a. Boundary layer evolution and velocity characteristics

Flow beneath a stationary ice shelf results in vertical shear and forms a turbulent boundary layer beneath the ice. This boundary layer determines the melt rate by regulating the transfer of heat and salt from the far-field ocean to the ice. Figure 2 shows the ice–ocean boundary layer and melt rate at the ice–ocean interface for run A3 (Table 3) with free-stream velocity U0 = 2.8 cm s−1. Turbulence beneath the ice, visible as fluctuations in the vertical velocity field, mixes cold meltwater over the upper ∼2.5 m of the water column and supplies heat to the ice–ocean interface. The resulting melt rate is spatially heterogeneous due to the presence of elongated turbulent structures near the ice, which increase heat flux locally (Vreugdenhil and Taylor 2019). These structures are aligned with the wall stress which has a cross-stream component due to Coriolis.

Figure 3 shows the evolution and characteristics of this boundary layer under cold conditions over three inertial periods. The difference between the mixing-layer temperature and the in situ freezing temperature (Tf) at mixing-layer salinity and interface pressure [ T*=TMLTf(SML)], often termed “thermal driving,” is a useful measure of the heat available for melting. For run A2, T0*0.0025°C at the beginning of the simulation. The steady far-field flow with magnitude U0 = 1.4 cm s−1 exerts a shear stress on the stationary ice–ocean interface, which we characterize by the friction velocity u*, and creates a turbulent boundary layer. Plane-averaged profiles of turbulent kinetic energy (TKE), defined as
TKE=12(uu¯+υυ¯+ww¯),
where u′, υ′, and w′ are deviations from the horizontal mean velocity, show active turbulence over the upper 2 m of the water column, which is most intense near the ice (Fig. 3b). Both u* and TKE are quasi-steady over for the duration of the simulation. The boundary layer turbulence rapidly homogenizes the initial scalar profiles forming a layer that is well mixed in temperature (Fig. 3c) and salinity (not shown) beneath the ice. The squared buoyancy frequency N2 = −g/ρ0(∂ρ/∂z) (Fig. 3d) characterizes the stratification of the water column. As the flow develops, a strong density gradient (pycnocline) forms at the base of the mixing layer. The pycnocline deepens gradually over the course of the experiment due to entrainment of ambient fluid into the mixing layer. The melt rate (Fig. 3a) adjusts rapidly at the beginning of the experiment as diffusion sets up thermal and saline boundary layers adjacent to the ice. After ∼1ti the melt rate achieves a quasi-steady value, evolving only in response to the gradual cooling of the mixing layer.
Fig. 2.
Fig. 2.

Snapshot of output for case A3 at t = ti. Horizontal (x–y) plane shows instantaneous melt rate m at the ice–ocean boundary. Vertical planes show vertical velocity w (z–y plane) and temperature T (x–z plane). Only the upper 3.6 m of the domain is shown.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

Fig. 3.
Fig. 3.

Temporal evolution of key quantities for (left) case A2 and (right) case C2 over three inertial periods: (a),(e) melt rate [m (cm yr−1)] and friction velocity [ u* (cm s−1)] [note melt rate is multiplied by a factor of 20 in (a)]; (b),(f) turbulent kinetic energy [TKE (m2 s−2)]; (c),(g) temperature [T (°C)]; and (d),(h) squared buoyancy frequency [N2 (s−2)].

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

ISOBL characteristics and evolution are strongly influenced by ocean temperature. Higher T* results in an increased melt rate and higher surface buoyancy forcing, which suppresses boundary layer turbulence. Figures 3e–h show the evolution of the boundary layer in case C2, which has the same current forcing (U0 = 1.4 cm s−1) as case A2 but a much higher thermal driving ( T*0.05°C). For the warmer case the melt rate is much higher, while the boundary layer turbulence is weaker. After ∼1ti the TKE drops off at depth and turbulent mixing is confined to the upper ∼1 m of the water column. The change in mixing depth causes a new pycnocline to form above the first from 2ti. The friction velocity also decreases by ∼30% from its original value over the simulation. Melting, which both cools and freshens the water near the interface, constitutes a stabilizing buoyancy flux which tends to stratify the water column. The increased melting, and therefore stabilizing buoyancy flux, results in suppression of turbulence and reduction of the mixing depth compared with the cooler case presented in Fig. 3b.

1) The surface velocity layer

The velocity structure of the boundary layer varies significantly across simulations, leading to variation in the wall stress and drag coefficient (Fig. 4). Immediately below the ice, in the viscous sublayer, flow is laminar and velocity scales with distance from the ice (u+z+). This is shown for velocities of U0 = 0.7, 1.4 and 2.8 cm s−1 at T*0.0025°,0.008°, and 0.05°C, in Figs. 4a–c, where the velocity profiles all follow the viscous scaling for z+ < 5. Away from the interface we compare the velocity profiles to the classic log law (5) and Monin–Obukhov (MO) similarity scaling (7). At warmer conditions, we find that the boundary layer velocity profile deviates from the classic logarithmic velocity profile, with the profile appearing more “laminar” in the near-wall region.

Fig. 4.
Fig. 4.

(a)–(c) Velocity profiles in wall units. Overplotted are the viscous boundary layer scaling u+ = z+ (solid black line), logarithmic boundary layer scaling [Eq. (5); solid gray line], and MO similarity scaling [Eq. (7); dashed lines, where colors correspond to labeled experiments]. (d)–(f) Profiles of Ozmidov scale Loz. The black dashed line shows Loz = −z. All profiles are from t = 3ti, except runs A3, B3, and C3, which are taken at t = 2ti.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

At low velocity (Fig. 4a) the boundary layer profile is poorly explained by the MO similarity scaling, regardless of the thermal driving. However, this may be attributed to the low Reynolds number of the flow. At higher free-stream velocities (Figs. 4b,c), the MO similarity scaling describes the simulated velocity profile over 50 ≤ z+ ≤ 500 relatively well, but only at low thermal driving. Under these conditions, which correspond to large L, the MO similarity scaling does not deviate much from the classic log-layer scaling. At higher temperatures, the fit between the velocity profiles and (7) is poor. While the scaling captures the tendency for the shear to be concentrated closer to the boundary, it does not accurately represent the shape of the velocity profile. For case C2, the velocity profile remains closer to the viscous scaling than the log-layer scaling over z+ < 100.

The velocity profiles in Figs. 4a–c are scaled as u+=u/u*, thus the far field u+ is related to the drag coefficient as Cd = (1/u+)2. We observe a tendency for Cd to decrease with increasing thermal driving. Drag coefficients for each experiment can be found in Table 3. A decrease in drag due to increasing ocean temperatures in a melting scenario has been reported by Rosevear et al. (2021) and Vreugdenhil and Taylor (2019), and is consistent with results from stabilized Ekman boundary layers, where increasing static stability decreases drag (Deusebio et al. 2014; Shah and Bou-Zeid 2014).

Under warm, high melt conditions, turbulent motions increasingly feel the effect of buoyancy. This effect can be quantified using the Ozmidov scale Loz = (ϵ/N3)1/2, where ϵ is turbulence dissipation. The Ozmidov scale is an estimate of the smallest spatial scale that is influenced by stratification (e.g., Smyth and Moum 2000). Figures 4d–f show that increased thermal driving leads to a reduced Loz for a fixed free-stream velocity. For example, comparing runs A3 and C3 with T0*=0.0025 and T0*=0.05, respectively, Loz is considerably reduced at higher T0*.

Within the log layer, the scale of a turbulent eddy is approximately proportional to the distance from the interface (Pope 2001). For some simulations we observe that Loz is smaller than the distance to the ice (i.e., Loz < |z|) within the boundary layer. This indicates that buoyancy is affecting turbulence within the region of the flow where a log layer is expected and may explain the large deviation of the velocity profile from the logarithmic scaling in, for example, run C2 (Fig. 4b).

b. Mixing-layer structure

Boundary layer turbulence homogenizes temperature and salinity profiles, forming a mixing layer adjacent to the ice. In our experiments we observe three distinct mixing-layer structures which we term “well-mixed,” “stratified,” or “diffusive-convective,” depending on the dominant boundary layer processes. In this section we categorize our experiments and use existing theory to model the mixing-layer depth in each regime.

1) Well-mixed regime: Effects of current shear and stratification

Under low thermal driving, the depth of the mixing layer can be predicted using a simple scaling law based on the friction velocity, Coriolis frequency and far field stratification. Figure 5 shows profiles of xy-averaged buoyancy frequency (N2) for all experiments, where depth is normalized by the Ekman scale δf. All A and B experiments have a similar density structure, with a single, well-mixed layer and pycnocline beneath the ice. We consider these cases, which we classify as “well-mixed,” first. The A and B cases evolve in a qualitatively similar manner to that shown in Fig. 3 for case A2. However, the pycnocline depths do not collapse with the scaled vertical coordinate. Rather than the f−1 dependence suggested by the Ekman scale, we find that the mixing-layer depth scales with f−1/2, consistent with the stratified Ekman boundary layer scale (9). See appendix A for an extension of this result to higher f. The stratification parameter (γ) scaling explains the mixing-layer depth well for all A and B experiments (Fig. 6a) with constant A = 1.3, as suggested by Weatherly and Martin (1978). This result is consistent with the work of McWilliams et al. (2009) and Taylor and Sarkar (2008b), who showed that increasing external stratification thins the Ekman boundary layer and mixing layer.

Fig. 5.
Fig. 5.

(top) Profiles of squared buoyancy frequency (N2) for all cases (labeled) at t = 2ti. Profiles are offset by N2 = 2 × 10−5 s−2. (bottom) Schematic showing typical profiles of N2 for the three observed ISOBL regimes at t = ti and t = 2ti. In each profile the pycnocline depth, which we use as a proxy for mixing-layer depth, is shown with an arrow. Qualitatively, the DC and well-mixed regimes look similar, except for a small region with N2 < 0 beneath the diffusive sublayer and a deeper pycnocline for the DC case.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

Fig. 6.
Fig. 6.

(a) Observed mixing-layer depth (hobs) scaled by δf as a function of γ. Dashed line is hobs/δf = 1.4γ−1/4. (b) hobs/δf as a function of μ; the dashed line is hobs/δf = 0.31μ−1/2. (c) hobs as a function of BbT/N03; the dashed line is hobs=11(BbT/N03)1/2. In all figures, markers denote well-mixed (circle), stratified (square), and diffusive-convective (triangle) boundary layers, respectively.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

2) Stratified regime: Effects of current shear and surface buoyancy flux

At warmer ocean conditions (C cases) the mixing-layer depth and structure departs from the stratified Ekman boundary layer scaling. For cases C2 and C3, with current speeds of U0 = 1.4 and 2.8 cm s−1, respectively, we observe a shoaling of the mixing depth over the course of the experiment, indicating that the boundary layer turbulence is being strongly suppressed by the surface meltwater flux. While these cases loosely retain a density structure with a mixing layer and pycnocline beneath the ice, the mixing layer is thin and quite strongly stratified. We classify these cases as “stratified.” Whether or not the boundary layer will stratify can be determined by comparing L with δf (McPhee 2008). The Obukhov length scale (6) gives an estimate of the depth at which turbulence will feel the effects of stratification due to surface buoyancy forcing. If the Obukhov length scale is much larger than the Ekman scale [ μ=u*/(fL)1] then the mixing layer is unlikely to be affected by the surface buoyancy forcing. However, if the two are of similar magnitude (μ ∼ 1) then the boundary layer will be strongly affected by stabilizing buoyancy. We compare L with δf rather than Eq. (9) based on the observation that the stratified boundary layer forms over the top of an initially well-mixed region (as seen in Fig. 3f) indicating that far field stratification N0 is unlikely to influence the boundary layer turbulence. For case C2 μ = 2.3, while for case C3 μ = 0.9. Drawing in a case from Rosevear et al. (2021), we find that hobs/δf decreases as μ increases for the stratified cases (Fig. 6b).

We find that hobs/δfμ−1/2, in agreement with previous numerical (Deusebio et al. 2014) and theoretical (Zilitinkevich 1972) studies. The expression of best fit to our data is given by hobs/δf = 0.31 μ−1/2. In addition we find that the drag coefficient depends on μ; Fig. 7 shows that u*/U0 scales with μ−1/5, also consistent with the findings of Deusebio et al. (2014).

Fig. 7.
Fig. 7.

Friction velocity ( u*) normalized by U0 as a function of μ, dashed line is u*/U0=0.043μ1/5. Markers are as for Fig. 6.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

3) Diffusive-convective regime: Effects of surface buoyancy flux and ambient stratification

For high temperatures and weak currents, DC sets the mixing-layer depth. Following Middleton et al. (2021) and Rosevear et al. (2021) we use the sign of the buoyancy flux B = wT′〉 − wS′〉 beneath the diffusive sublayers to determine when DC is occurring, where B > 0 indicates that convection is active. For case C1, which is in the DC regime, the mixing layer is much deeper than expected from the Ekman scale (Fig. 5). Beneath the strongly stratified diffusive sublayer, N2 becomes negative over a short distance, further evidence of DC. Cold, fresh meltwater has a stabilizing effect, resulting in strong stratification near the ice. However, in diffusing away from the interface, heat outpaces salt by a factor of O(100), and can drive convection some distance from the ice. In this diffusive-convective case the mixing-layer depth depends on T*, and the melt rate is inherently transient (Middleton et al. 2021; Rosevear et al. 2021). Mixing-layer depth in the DC regime depends on the thermal buoyancy flux from meltwater and the ambient stratification, and is well predicted by (11) (Fig. 6c), although we find CDC ∼ 11, rather than the 41 suggested by Fernando (1987). One key feature of the DC regime is that the melt rate, and therefore BbT, are inherently transient. We evaluate the surface buoyancy flux due to heat as the mean over the first 5 h of simulation time, as the majority of the mixing-layer growth occurs over that period.

4) Implications of the departure from well-mixed conditions

In the stratified runs the velocity boundary layer and mixing layer are much thinner than in the well-mixed cases. Enhanced near-wall stratification and the double pycnocline structure beneath the mixing layer support large gradients in T and S over the upper ∼2 m of the water column. There is a much larger disconnect between the interface, mixing layer, and far field conditions and as a result it will be much more challenging to sample or resolve (either observationally or numerically) the mixing-layer properties required to determine the ice shelf melt rate using a parameterization such as the three-equation parameterization. In situ ice shelf–ocean observations are collected in a variety of ways, such as autonomous underwater vehicles (e.g., Dutrieux et al. 2014) and profiling/installation of moorings via boreholes drilled through the ice shelf (e.g., Davis and Nicholls 2019; Stewart et al. 2019; Stevens et al. 2020). A key issue with the majority of these studies is identifying and sampling within the mixing layer (e.g., Stewart 2018), if one is even present. Similar issues exist with sampling the velocity boundary layer in order to estimate the friction velocity (Davis and Nicholls 2019). The reduction in drag we observe under stratified conditions will also have implications for the accuracy of basal melting predictions in large-scale models. Regional and global ocean models typically take the drag coefficient to be Cd = 2.5 × 10−3 (Gwyther et al. 2015). While this is a reasonable approximation for our results in the well-mixed regime (e.g., case A3 with Cd = 2.3 × 10−3), we found that Cd depends on stability parameter μ. For example, at μ = 2.3 we find Cd = 1.4 × 10−3 (case C2). As a result, in the stratified regime, friction velocity will be overestimated in melting parameterizations, contributing to an overestimation of melt rates.

Diffusive convection results in the formation of a unique thermohaline structure known as a “diffusive staircase” with well-mixed layers in temperature and salinity separated by sharp interfaces. These convecting layers likely affect the boundary layer and heat supply to the ice. However, while Rosevear et al. (2021) showed that a diffusive staircase can emerge in LES, layer formation away from the ice will not be well resolved, since grid resolution decreases with distance from the ice. Consequently, the present simulations might underestimate the potential effects of DC on the ISOBL.

c. Melting

Determining how melting depends on key drivers such as ocean temperature and current speed is critical for the prediction of melting at cavity and circum-Antarctic scales. We find that simulated melt rates fall into two categories: simulations where the melt rate is steady in time and depends on u*, i.e., melting is shear controlled (e.g., Fig. 3a), and simulations in which melting is unsteady (e.g., Fig. 3e). To investigate these behaviors, we consider time series of heat and salt transfer coefficients ΓT (15) and ΓS (16), which describe the efficiency of heat and salt transport to the ice. The mixing-layer temperature [TML(t)] is taken as the temperature 0.5 m below the ice, with an equivalent definition for SML. For all simulations, this location is within the mixing layer. Figure 8 shows the temporal evolution of ΓT and ΓS for four selected simulations, where a constant transfer coefficient indicates steady-state melting. One of the selected cases exhibits transient melting, indicating that current shear is not controlling the width of the laminar sublayer across which heat and salt must diffuse. We find that the magnitude of the viscous Obukhov scale (L+) is a good predictor of whether melt rates will be shear controlled. Transfer coefficients ΓT and ΓS collapse to the same value for experiments B2 and C3 with the same L+, despite having different thermal and current forcing (Figs. 8a,b). This observation provides further support for the hypothesis that L+ determines near-ice properties, as found in Vreugdenhil and Taylor (2019).

Fig. 8.
Fig. 8.

(a) Heat transfer coefficient (ΓT) and (b) salt transfer coefficient (ΓS) for selected cases.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

Here we compare melt rates estimated using the three-equation parameterization (m3eq) with our simulated melt rates, where m3eq is calculated using transfer coefficients ΓT = 0.011, ΓS = 3.9 × 10−4, and simulation-derived inputs u*, TML, and SML. The ratio m3eq/m is plotted as a function of the viscous Obukhov scale L+ (Fig. 9). For high L+ we find that the three equation parameterization reproduces simulated melt rates quite well, however, as L+ decreases the fit worsens. For L+2500 we find that melt rates are inherently transient, and consequently the ratio m3eq/m also evolves in time. Transfer coefficients ΓT and ΓS depend on L+ over L+5×104 (Figs. 9b,c). We determine empirical logarithmic fits to the heat and salt transfer coefficients for L+ > 2500, using the maximum transfer coefficients found in our simulations as an upper bound for the functions, which are given by
ΓT(L+)=min[0.00165log(8.30L+)0.00837,0.0113],L+>2500,
ΓS(L+)=min[0.00010log(199L+)0.00117,3.8×104],L+>2500.

The high L+ limits of these expressions are motivated by the observation that both ΓT and ΓS appear to asymptote to constant values at high L+, although further data at higher L+ are required to confirm this. Vreugdenhil and Taylor (2019) performed a “passive scalar” experiment where T and S fields did not influence the flow through buoyancy. For this case, intended to determine the heat and salt transfer at infinite L+, they found ΓT = 0.012 and ΓS = 3.9 × 10−4, close to the values used as upper bounds in Eqs. (27) and (28).

5. Discussion

a. Transfer coefficients

The transfer coefficients found in this study at high L+ add to a growing body of evidence suggesting that constant values of ΓT and ΓS are appropriate for calculating melt rates for cold and/or energetic ice shelf cavities. Our highest transfer coefficients agree extremely well with the transfer coefficients found by Vreugdenhil and Taylor (2019) at high L+ (Table 4), as well as with observationally determined coefficients from beneath the Filchner Ronne Ice Shelf (Jenkins et al. 2010) and sea ice in the arctic (Sirevaag 2009). Beneath the Larsen C Ice Shelf, Davis and Nicholls (2019) found extremely good agreement between their observed melt rates and the three-equation parameterization using Cd1/2ΓT=0.011 with ΓTS = 35. Combining this with the local drag coefficient, determined from measurements of boundary layer turbulence to be Cd = 0.0022, they propose a heat transfer coefficient of ΓT = 0.024, roughly double that found in the other observational and numerical studies presented in Table 4. However, this value of ΓT may be an overestimate, since they determine that Cd = 0.0022 for flow speeds 0.1ms1 only. For slower speeds, which are more frequently observed, Davis and Nicholls (2019) find Cd = 0.0022, which would suggest ΓT < 0.024. Alternatively—or additionally—higher ΓT at Larsen C may be due to tidal currents or other processes, which are not represented in the present study or that of Vreugdenhil and Taylor (2019). This latter point warrants further investigation using numerical models with tidal forcing. Despite this discrepancy, the observations and simulations in Table 4 collectively support the hypothesis that the three-equation parameterization can perform well in cold and/or high velocity (L+ > 104) ice shelf cavities.

Table 4

Comparison of transfer coefficients from observational and numerical studies. Thermal Stanton number ( Cd1/2ΓT), drag coefficient (Cd), heat transfer coefficient (ΓT), and ratio of heat and salt transfer coefficients (ΓTS). An asterisk indicates that the ratio is assumed, not measured.

Table 4

Consistent with the results of Vreugdenhil and Taylor (2019), we have found that stratification effects on the efficiency of heat and salt transfer (i.e., ΓT and ΓS) are best described by L+. However, it is interesting to note that, in our simulations, stratification effects on the boundary layer are better described by μ. This result is consistent with the findings of Deusebio et al. (2014) for an Ekman boundary layer under a stabilizing heat flux, who found that the Nusselt number (the ratio of convective to conductive heat flux) varied with L+, while the boundary layer depth and friction velocity depended on μ.

b. Regimes and transitions in ISOBL dynamics

In this section, we present a regime diagram that maps the DC, stratified, and well-mixed ISOBL regimes as a function of the elevation of mixing-layer temperature above freezing and friction velocity. This diagram draws on results from section 4 to determine the transition from well-mixed to stratified conditions and builds on theory and results from Martin and Kauffman (1977) and Rosevear et al. (2021) to estimate the transition between DC and stratified conditions.

Diffusive convection is not expected to coexist with strong turbulence (Radko 2013; Shibley et al. 2017). Simulations of DC beneath ice shelves with forced turbulence from a background current or other mechanical forcing show that DC is long-lived for weakly forced turbulence and/or high temperatures, and short-lived or inhibited completely when background turbulence is strong and/or temperatures are low (Rosevear et al. 2021; Middleton et al. 2021). In their melting simulations, which were forced by a level of “background” turbulence rather than a turbulent shear flow, Middleton et al. (2021) determined a criterion for the transition from DC to non-DC conditions based on the magnitude of the buoyancy Reynolds number Reb = ϵ/(νN2). They showed that if Reb > 1 at a critical depth within the diffusive region beneath the ice, then DC is suppressed. However, a model for the vertical structure of temperature, salinity, and dissipation is needed in order to apply this criterion.

We observed DC in several of our simulations with high T* and low u*, where DC-dominated mixing is defined by a positive buoyancy flux (B > 0) within the mixing layer. For some simulations, a transition from DC to non-DC conditions was observed over the course of the simulation, and the time at which this transition occurred (tDC) is listed in Table 3. We find that the Obukhov length scale (L) is a relatively good predictor of whether DC will occur or not, and the duration of time for which it will be active. DC is observed when L0.5m (Table 3), and as L decreases we find that tDC increases. To classify the transition between DC and shear-dominated conditions using L, a model of the surface buoyancy flux under DC conditions (Bb) is needed. We propose a simple model for melting and Bb under DC conditions based on the “conductive turbulent” melting model for DC from Martin and Kauffman (1977), with some simplifying assumptions based on results from Rosevear et al. (2021) and the present simulations. This model is described in appendix B. Using Eq. (30) and the criterion L=u*3/[kBb(tDC)]=0.5, we can predict the time scale (tDC) at which a transition from DC to stratified turbulence will occur. We find that DC is short lived for cold and/or energetic conditions, and long lived for conditions that are warm and/or quiescent, and that the predicted transition time scales agree well with tDC from our simulations (Fig. 10). However, since our simulations only cover a relatively small range of tDC and u*, we were not able rigorously test this criterion.

Fig. 9.
Fig. 9.

(a) Scatterplot showing ratio of simulated to parameterized melt rates as a function of the viscous Obukhov scale (L+), where the predicted melt rates are obtained from the three-equation parameterization with ΓT = 0.011 and ΓS = 3.9 × 10−4. Circles and triangles denote non-steady-state and steady-state experiments, respectively. For the steady-state experiments we plot (b) ΓT and (c) ΓS as a function of L+. The pale blue lines in (b) and (c) are Eqs. (27) and (28), respectively.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

Fig. 10.
Fig. 10.

Transition time scale (tDC) as a function of T* and u*. Simulations are overplotted, markers indicate whether DC is observed (circles) or not observed (squares). Where relevant, the observed tDC is indicated by the color map. Note that the three simulations in the top-left corner all have tDC = 100 h, since the simulations were only run for 100 h and a transition from DC to shear-dominated mixing was not observed during that period.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

Physically, the Obukhov length scale can be understood as comparing the interfacial buoyancy flux due to DC melting to the dissipation ϵ at a depth z = L, since ϵu*3/z within the log layer for a wall-bounded shear flow (Davidson 2015). For our simulations, we found L ∼ 0.5 m to be a good choice for distinguishing DC and non-DC conditions. We note that this criterion is somewhat related to the critical Reb criterion proposed by Middleton et al. (2021), since the interfacial buoyancy flux and the near-ice stratification are linked, and that our Fig. 10 qualitatively resembles Fig. 11 from Middleton et al. (2021). Further work on DC melting over a broad range of conditions is needed to thoroughly investigate the transition from DC to non-DC conditions in the presence of a shear flow.

In Antarctic conditions, DC results in transient melting due to growing thermal and saline diffusive sublayers adjacent to the ice [steady-state melting is possible in the DC regime, but only for a sublayer that is nearly fresh (Middleton et al. 2021)]. However, while transient melting is always observed when DC is occurring, the reverse is not always true: we have observed that transient melting can persist after convection has ceased. In this scenario, shear is sufficiently strong to inhibit DC fluxes at the margin of the diffusive sublayer/s, but not sufficiently strong to control the sublayer thickness and drive steady melting. Consequently, as well as the DC, stratified, and well-mixed ISOBL regimes, we depict two melting regimes in Fig. 11; transient melting for L+ < 2500 and steady melting for L+ > 2500. Using the empirical transfer coefficients [Eqs. (27) and (28)], we solve (12)(14) iteratively to obtain ΓT as a function of T* and u* (Fig. 11). Inherently transient melting is expected for a large portion of relevant parameter space. Within the steady melting regime we also delineate an area of reduced heat transfer for 2500 < L+ < 25000, in which heat and salt transfer have a functional dependence on L+.

Fig. 11.
Fig. 11.

Regime diagram for the ISOBL showing the DC, stratified, and well-mixed regimes in T*u* parameter space. Conditions associated with transient (L+ < 2500) and steady (L+ > 2500) melting are overlaid, as are field observations. For the observations, if u* is not measured directly, it is estimated from the free streamflow using u*=CdU with Cd = 0.0025. Error bars (bottom right) indicate the range of u* due to choosing Cd = 0.001 or 0.01 instead. Observational data are described in appendix C. Data sources are WGZ (Begeman et al. 2018), RIS (Stewart 2018), LCIS (Davis and Nicholls 2019), HWD2 (Stevens et al. 2020), PIIS (Stanton et al. 2013), and AIS (Rosevear et al. 2022).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

To characterize the well-mixed to stratified transition, we solve for the stratification parameter (μ), where stratified conditions are expected for μ ≥ 1. The buoyancy flux at the interface is given by Bb = (∂T/∂z)b(∂S/∂z)b, where (∂T/∂z)b and (∂S/∂z)b are predicted using the three-equation parameterization with empirical transfer coefficients (27) and (28).

Figure 11 depicts the DC, stratified, and well-mixed regimes as a function of T* and u*. Thermal driving and friction velocity span 0.001°–4°C and 0.03–1.25 cm s−1, respectively, where these friction velocity values correspond to free-stream velocities in the range 0.6–25 cm s−1, assuming a quadratic drag relationship and a constant drag coefficient of 0.0025. For reference, our simulations span a subset of these conditions, with T0*=0.0025°0.05°C and U0 = 0.7–2.8 cm s−1 ( u*=0.030.13 cm s1).

The regimes in Fig. 11 are supported by field observations beneath Antarctic ice shelves. For example, the Larsen C Ice Shelf (LCIS) falls within the well-mixed regime, consistent with the observations of a turbulent, homogeneous boundary layer (Davis and Nicholls 2019), as does the central Ross Ice Shelf Site (HWD2), where thermal driving is low and a well-mixed ISOBL was observed (Stevens et al. 2020). Near the front of the Ross Ice Shelf (RIS), seasonality in thermal and current forcing result in summertime (RIS S) and wintertime (RIS W) conditions moving between the well-mixed and stratified regimes (Fig. 11). Stewart (2018) showed that when conditions beneath RIS were less well mixed, the fit between the three-equation parameterization and the observed melt rates was worse and the dependence of melt rate on thermal driving was sublinear. These observations suggest strong buoyancy effects on both melting and ISOBL dynamics, consistent with both a transition from well-mixed to stratified conditions and a transition from constant to reduced heat transfer as shown in our regime diagram. Other sites are also affected by stratification; Pine Island Ice Shelf (PIIS) falls within the stratified regime, consistent with the observation of a strongly stratified water column to within 1 m of the ice shelf base (Stanton et al. 2013), as does the Amery Ice Shelf (AIS). Observations beneath the AIS revealed a temperature-stratified boundary layer and low melt rates relative to the thermal and current forcing, indicating inefficient heat and salt transfer due to stratification effects on near-ice dynamics, consistent with its position on our regime diagram (Rosevear et al. 2022). Finally, our diagram indicates that transient melting and short-lived DC are expected at the Wissard Grounding Zone (WGZ) of the Ross Ice Shelf, consistent with the low melt rates and staircase-like temperature and salinity structure observed at the site (Begeman et al. 2018).

While Fig. 11 does not include all sub-ice shelf observations, it does encompass a broad range of ice shelf sites. Crucially, it shows that stratification effects are very important to Antarctic ice–ocean interactions generally and that ice shelves must be very cold, energetic, or rough—to create enhanced interfacial drag—in order to be unaffected by buoyancy. It is important to note that the observational data included in Fig. 11 come from individual sites (point measurements) beneath ice shelves and should by no means be considered representative of their entire respective ice shelf cavities. Some combination of the three ISOBL regimes, as well as regimes such as the natural convection case for sloping ice, will likely be present across each ice shelf to varying degrees.

6. Summary and conclusions

We have presented results from a suite of large-eddy simulations of the ISOBL forced by a steady, geostrophic flow under melting conditions, with free-stream velocity and thermal driving ranging from 0.7 to 2.8 cm s−1 and 0.0025°–0.05°C, respectively. Over this range of conditions we observed three distinct boundary layer regimes: well-mixed, stratified, and diffusive-convective. In the well-mixed regime, the boundary layer was not strongly affected by meltwater, and the depth of the mixing layer could be predicted using a simple scaling law based on the friction velocity, Coriolis frequency and far field stratification. In the stratified regime, which occurs when the Obukhov length scale is the same or less than the Ekman scale (μ > 1), we showed that the boundary layer depth and interfacial drag coefficient decrease as the stability parameter μ increases. In the diffusive-convective regime, the mixing-layer depth depends upon the thermal component of the surface buoyancy flux and the far field stratification. The current speeds that we simulate with the LES are small compared to many of the observed currents beneath ice shelves (e.g., Davis and Nicholls 2019; Stanton et al. 2013). However, dynamic similarity using the scaling parameters L+ and μ allows us to apply our results to more energetic ice shelf conditions.

We have shown that the viscous Obukhov scale can be used to determine the efficiency of heat and salt transfer to the ice–ocean interface in agreement with the findings of Vreugdenhil and Taylor (2019). For L+2500 we found that current shear controlled the thickness of the diffusive sublayer adjacent to the ice, yielding a steady melt rate, and for L+25000 the heat and salt transfer coefficients tended to a constant value. At lower L+ the diffusive sublayer was found to grow in time, resulting in time-dependent melting much like in the diffusive-convective melting regime (Rosevear et al. 2021; Middleton et al. 2021). However, we note that transient melting does not necessarily mean diffusive convection will occur.

Our results have important implications for both the parameterization and the observation of the ISOBL. First, the transient melt rates we observe for L+2500 are inherently difficult to parameterize and cannot be accounted for by current ice–ocean parameterizations, which assume steady-state melting. Second, the L+ dependence of heat and salt transfer coefficients for 2500L+25000 is not currently accounted for in basal melting parameterizations, and as a result melt rates will be overpredicted over a decade of L+. For example, for case B2 with L+ ∼ 3000, the Jenkins et al. (2010) parameterization would overestimate melting by ∼70%. Third, in the stratified boundary layer regime, the elevated stratification and thin boundary layer will increase the difficulty of sampling or resolving the mixing layer in both field observations and ocean model simulations, and result in a (relatively) cooler mixing layer. Melt parameterizations, which do not take stratification effects into account, will likely overestimate melt rates in this regime through overestimating the mixing-layer temperature.

Melt parameterizations can be easily adjusted to include the functional dependence of transfer coefficients on L+ identified in the present study. This adjustment will improve the accuracy of melt predictions under warm and/or quiescent—but still shear-controlled—conditions. However, the effects of buoyancy on the mixing layer and interfacial drag will also need to be included for accurate melting predictions. To include these effects a more sophisticated parameterization with knowledge of the boundary and mixing-layer structure is needed, with the stability parameter μ offering a useful starting point. This would allow thermal driving and friction velocity to be accurately estimated under stratified conditions.

Parameterizing diffusive-convective melting accurately will be far more challenging, since the melt rate is a function of time. To represent diffusive-convective melting in large-scale ocean models, it will be important to understand how it behaves in more realistic flow settings where many processes are occurring simultaneously. For example, in the presence of a tidal current, we might see the boundary layer switch between shear-controlled and diffusive-convective melting, resulting in a quasi-steady melt rate. Further work is required in this area. Other aspects of Antarctic ice shelf–ocean interactions such as internal wave activity and the effects of a sloped or rough ice–ocean interface also warrant investigation using high-resolution simulations in the future.

Acknowledgments.

This research was supported by the Australian Research Council through the Special Research Initiative for Antarctic Gateway Partnership (project ID SR140300001), and also by the Australian Government through the Antarctic Science Collaboration Initiative Australian Antarctic Program Partnership (project ID ASCI000002). During the preparation of this manuscript, B.G. received support from the Australian Research Council through a Future Fellowship (Grant ID FT180100037). Numerical simulations were conducted on the Australian National Computational Infrastructure at the Australian National University, which is supported by the Australian Government.

Data availability statement.

The model output is available at https://doi.org/10.5281/zenodo.5912262.

APPENDIX A

Effect of Varying Coriolis Parameter f

To achieve our highest velocity (U0 = 2.8 cm s−1) cases, we set the Coriolis parameter to f = 2f0 in cases A3, B3, and C3, where f0 = −1.37 × 10−4 s−1 is the Coriolis parameter at a latitude of 70°S. This yields a thinner boundary layer and allows us to use a slightly smaller domain size than would be required at f = f0. To ensure that this does not affect our results we performed two supplementary cases, S1 and S2, with the Coriolis parameter set to f = 2f0 and 6f0, respectively. These cases were both run with a thermal forcing of T0*=0.0025°C and a free-stream velocity of U0 = 1.4 cm s−1 allowing a direct comparison with (main text) case A2, isolating the effect of f.

Figure A1 shows that the melt rate is not affected by varying f in the range f0f ≤ 6f0, despite a slight increase in u* with increasing f. Figure A2 shows that the stratified Ekman boundary layer scaling (9) also holds over the f0f ≤ 6f0. Based on these results we do not expect that employing f = 2f0 for cases A3, B3, and C3 will influence our results relating to heat transfer. Increasing f simply results in a thinner boundary and mixing layer, as already described in the main text.

Fig. A1.
Fig. A1.

Effect of Coriolis parameter (f) on (top) melt rate; (middle) melt rate normalized by TMLTb; (bottom) friction velocity u* for cases A2 (main text) supplementary cases S1 and S2. Thermal driving and free-stream velocity are constant between cases.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

Fig. A2.
Fig. A2.

Profiles of squared buoyancy frequency (N2) for cases A2, S1, and S2. Depth axis is scaled by γ−1/4.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

APPENDIX B

Model for Melting and Buoyancy Flux under DC Conditions

Here we describe a simple empirical model for melting and surface buoyancy flux under DC conditions. This model has strong similarities to the “conductive turbulent” melting model for DC from Martin and Kauffman (1977), with some simplifying assumptions based on results from Rosevear et al. (2021) and the present simulations. We follow Martin and Kauffman (1977) in assuming that the thickness of the diffusive (nonconvecting) region adjacent to the ice (δd) is proportional to the salt diffusion length scale 4κSt, i.e., δd(t)=a4κSt. Following Martin and Kauffman (1977) we assume that the temperature profile across the diffusive region is linear, and that the temperature is constant at the interface (Tb) and the lower edge of the diffusive layer ( Tδd).

We also make several additional assumptions, diverging from the approach in Martin and Kauffman (1977). First, we assume that the interface is fixed in space, consistent with our model setup. Second, we assume that all the temperature variation occurs over the diffusive region, i.e., TδdTML. Third, based on results from Rosevear et al. (2022), we approximate the temperature difference across the diffusive region as a fixed fraction of the thermal driving TMLTb=0.5T* (Fig. B1) and set a = 3. Combining these results with Eq. (13) we can express the melt rate as
m=ρcpκTLfρice(0.5T*34κSt).
This expression compares extremely well to the simulation melt rates (Fig. B2). Using the fact that the interface heat and salt fluxes [Eqs. (13) and (14)] are related by the melt rate we can write an expression for the buoyancy flux at the interface (Bb):
Bb=gβκS(Sz)bgακT(Tz)bgκT0.5T*34κSt(βcpSbLfα),
where Sb can be calculated using Eq. (25).
Fig. B1.
Fig. B1.

Ratio of the temperature difference between the mixed layer and interface (TMLTb) and the thermal driving ( T*) for runs 1, 2, 3, 3*, and C1 over time.

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

Fig. B2.
Fig. B2.

Evolution of the melt rate (m) for runs 1, 2, 3, 3*, and C1 (solid lines) compared to the modeled melt rate from Eq. (B1) (dashed lines).

Citation: Journal of Physical Oceanography 52, 10; 10.1175/JPO-D-21-0317.1

APPENDIX C

Observational Data

Data from Davis and Nicholls (2019): U¯ calculated from their Fig. 3 (upper instrument); u* calculated from U¯ using their reported Cd = 0.0022; T* estimated from their mixing-layer temperature (TML = −2.06° to −2.04°C; from temperature profile in Fig. 2b and text), salinity (SML = 34.54 psu), and interface pressure (pb = 304 dbar) as Tf = −0.0573SML + 0.0832 − 7.53 × 10−4pb, yielding T*=0.04°0.06°C (Table C1).

Table C1

Observational data included in Fig. 11. An asterisk indicates that values should be considered approximate only.

Table C1

Data from Stevens et al. (2020): U¯ from visual inspection of their Fig. 3.

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