1. Introduction

The largest mass of floating ice on earth is located in the southern Weddell Sea at the margin of the Antarctic ice sheet. The Filchner–Ronne system of ice shelves (FRIS) (Fig. 1) is a floating extension of the Antarctic Ice Sheet, draining 24% of the area of the continent's grounded ice and covering an area slightly less than that of France (Fox and Cooper 1994). Direct oceanographic measurements from beneath FRIS confirm that high salinity shelf water (HSSW), a product of sea ice formation north of the ice shelf, flows into the sub-ice-shelf cavity (Nicholls and Makinson 1998; Nicholls et al. 2001). This relatively warm, dense water occupies the lower portion of the water column in an environment isolated from atmospheric forcing, in which tidal and thermohaline processes are thought to dominate the circulation. MacAyeal (1984) suggested tidal stirring as a probable mechanism for mixing HSSW up through the sub-ice-shelf water column to be transformed, through melting at the underside of the ice shelf, into ice shelf water (ISW). Since ISW is a precursor of Antarctic Bottom Water (Foldvik and Gammelsrød 1988) and Southern Ocean bottom water formation takes place mostly in the Weddell Sea (Ganachaud and Wunsch 2000), mixing processes beneath FRIS are potentially significant for global ocean circulation.

In the cavity beneath an ice shelf frictional stress at both the seabed and the ice shelf base influences the vertical structure of tidal currents, induces vertical mixing, and enables the development of two well-defined mixed boundary layers. These frictional boundary layers are made up of a thin region of high shear and a thicker region of low shear. Beyond this boundary layer is the free stream region where there is no shear and where the current is independent of depth (Soulsby 1983). Where fixed ice cover is present and tidal currents in deep water are weak, the frictional boundary layers occupy a small fraction of the water column. However, in shallow water with strong tidal currents, they can occupy the entire water column, dominating the tidal current structure and associated vertical mixing (Prinsenberg and Bennett 1989). Tidal simulations for the southern Weddell Sea show that the mean barotropic tidal currents beneath FRIS attain speeds of up to 0.2–0.5 m s⁻¹, with peak velocities reaching 1 m s⁻¹ (Makinson and Nicholls 1999; Robertson et al. 1998). Close to the Ronne Ice Front, and along grounding lines of FRIS where the ice first comes afloat, these tidal currents are thought to result in significant levels of vertical mixing, and high basal melt rates.

To determine the influence of vertical tidal mixing on melting beneath Ross Ice Shelf, Scheduikat and Olbers (1990) developed two one-dimensional mixed layer models, thermodynamically coupled to an ice shelf, one with two layers and one with three layers. Both were Kraus and Turner type models (Kraus and Turner 1967), which were forced by a barotropic tidal current. In the two-layer model, the horizontally homogeneous upper mixed layer interacted with the ice shelf while the lower layer remained passive and represented an infinite heat and salt reservoir. In the three-layer model, the bottom layer was treated as a mixed layer and included the advection of heat and salt, while a third layer was introduced between the two mixed layers. With the two-layer model it was found that basal melting or freezing was strongly dependent on the ratio of the major and minor axes of the tidal current ellipse. Flat or degenerate ellipses yielded cyclic currents with the top layer entraining and then detraining water from the layer below. When the current was at its maximum during the tidal cycle, entrainment and melting would occur. With open, circular ellipses there was little or no entrainment from the layer below and heat lost from the upper layer to the ice shelf gave a small amount of freezing, irrespective of the tidal current amplitude. The three-layer model gave similar results, but with reduced melt rates resulting from a better representation of the thermocline. Scheduikat and Olbers (1990), however, did not include the effects of the earth's rotation.

The rotational effect on vertical mixing is accounted for by decomposing the elliptical tidal current into purely anticlockwise (R₊) and clockwise (R₋) components and considering their respective frictional boundary layers. The two constant magnitude rotary velocity components rotate at the tidal frequency, and each has a characteristic frictional boundary layer thickness, δ₊ and δ₋, given by (Prandle 1982):

View Expanded

where K_M is the eddy viscosity, f is the Coriolis parameter, and ω the frequency of the tidal oscillation. Assuming a constant eddy viscosity for semidiurnal tides at northern midlatitudes the thickness of the clockwise frictional boundary layer will be about three times greater than for the anticlockwise frictional boundary layer. An effective combined boundary layer thickness (δ) is suggested by Soulsby (1983)

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Consequently, tidal currents with more negative polarizations in the Northern Hemisphere have larger boundary layers, allowing mixing to penetrate farther into the water column interior and tidal fronts to advance into deeper water.

The spatial variability of the tidal currents beneath FRIS was determined by Makinson and Nicholls (1999) using a two-dimensional depth-averaged barotropic tidal model of the southern Weddell Sea. The M₂ tidal ellipse polarization over this region is shown in Fig. 2. All the semidiurnal tidal constituents exhibit an almost identical pattern of polarization with regions of strongly positive and negative polarization present beneath FRIS, suggesting a wide range of boundary layer depths.

At higher northern latitudes the boundary layer of R₋ is much larger than the R₊ component as |ω − f| tends to zero at the critical latitude (φ_crit). At this latitude the tidal frequency equals the Coriolis or inertial frequency, ω = |f| = 2Ω sin φ_crit, where the earth's angular velocity Ω = 7.2921151 × 10⁻⁵ rad s⁻¹. Furevik and Foldvik (1996) showed that a thick benthic boundary layer develops as the critical latitude is approached. A thicker boundary layer produces less intense shear near boundaries but extends the shear over a larger thickness of the water column, and is likely to enhance vertical mixing near the critical latitude.

In the Southern Hemisphere, the northwest corner of FRIS lies near the critical latitude of the M₂ tidal constituent at 74°28′18″S, while the southernmost areas of the ice shelf are closest to the S₂ critical latitude at 85°45′54″S. In the shallow regions close to the Ronne Ice Front, south of Henry Ice Rise and along grounding lines, the combined effects of strong tidal currents and the proximity of the critical latitude will further thicken the boundary layers. This produces extensive areas over which the boundary layers may occupy the entire water column, and here the free stream region of the tidal current will be absent. At high latitudes the diurnal tides do not exhibit these large differences between the thickness of the boundary layers of R₊ and R₋ as, for these frequencies; |f| is almost twice as large as ω.

Boundary layer thickness is further complicated by the stratifying effect of any buoyancy input. The only source of buoyancy beneath FRIS is the net production of an estimated 1500–3000 m³ s⁻¹ of meltwater (Nicholls and Makinson 1998), which is critical for driving the circulation over large areas of the region and has important implications for water column structure. Buoyancy effects caused by freezing or melting under sea ice have been found to play an important role in modifying turbulent processes (Shirasawa and Ingram 1991). During melting the water column becomes stratified while the tidal forcing remains unchanged. The effect of stratification is to suppress turbulence, inhibiting the transfer of momentum and reducing the eddy viscosity, hence reducing the boundary layer thickness from the homogenous case (Souza and Simpson 1996). Further modification of the vertical tidal current structure occurs in the vicinity of the pycnocline, which becomes decoupled from the mixed layers at the seabed and ice base (Prinsenberg and Bennett 1989). Near the critical latitude, the influence of eddy viscosity on the vertical tidal current profiles is significant (Nøst 1994): a small change results in a large variation in the anticlockwise component in the Southern Hemisphere, while the clockwise component remains unaffected.

Effects of tides are usually ignored within sub-ice-shelf circulation models. For FRIS, this means that the estimated 25 GW of tidal energy dissipation associated with surface friction are also ignored (Makinson and Nicholls 1999; Robertson et al. 1998). Vertical mixing resulting from tidal currents has been parameterized only as a function of average water column speeds derived from two-dimensional barotropic tidal models (Makinson and Nicholls 1999; MacAyeal 1984). This parameterization is poor for high-latitude regions dominated by semidiurnal tides with a wide range of tidal current ellipse polarizations.

In this paper a one-dimensional ocean model, coupled thermodynamically to the ice shelf, is applied to a sub-ice-shelf water column to study vertical profiles of tidal currents, vertical mixing, and basal melting beneath Filchner–Ronne Ice Shelf. The model is described in the next section, together with the parameters used to define the model properties and the initial conditions. Results from the model in section 3 show the proximity to the M₂ and S₂ critical latitudes affects the vertical tidal current structure and how this is further affected by the presence of stratification. Also described are the effects of depth-averaged tidal current ellipse polarization on vertical mixing and basal melting over a wide range of polarizations and current amplitudes. The implications for FRIS of the results from the mixing model are then discussed.

2. Model description

The one-dimensional model uses an implicit finite difference scheme to integrate the horizontal velocity components (u, υ) of the momentum equations

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with x and y positive in the east and north directions and z increasing positively upward, with z = 0 at the ice–ocean boundary and z = −h at the seabed. Here, K_M is the vertical eddy viscosity, f is the Coriolis parameter, g the gravitational acceleration, and ω is the tidal frequency; S_x, S_y and ϕ_x, ϕ_y are the prescribed tidal surface slope amplitudes and phases. A phase difference of π/2 is maintained between ϕ_x and ϕ_y, while the amplitude of the tidal slopes are determined from

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where U determines the maximum speed of the tidal current (semimajor axis) and P, which ranges between ±1.0, controls the polarization of the tidal current ellipse. More details on the derivation of (7) and (8) are given in the appendix. At both the ice–ocean interface and the seabed, a quadratic drag law is applied with a drag coefficient of C_d100 = 0.0025. A smoothly graded vertical mesh is used to give the highest resolution at the boundaries. The thickness of the two layers adjacent to each of the boundaries are set to 1.0 m. The node depth at the center of each layer of the remaining water column (z_rem) is determined from Lynch et al. (1996)

Z_nodez_remz_remβπ(9)

where ε increases linearly with node number from 0 at the bottom to 1 at the top. For a total water column thickness of 200 m with 61 nodes, the constant β is set to give the desired layer resolution of 1.0 m near the boundaries and approximately 6 m in the mid–water column.

Both heat and salt are conserved in the model and their vertical diffusion within the water column are governed in the model by

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where K_H is the vertical eddy diffusivity, T is the water temperature, and S the water salinity. There is no flux of heat or salt through the seabed. At the ice–ocean interface, phase changes during melting or freezing produce heat and freshwater fluxes, which will be considered later in this section.

The effects of turbulence in transferring momentum and mixing heat and salt are determined by the widely used, level 2.5 turbulence closure scheme of Mellor and Yamada (1982) incorporating the modifications of Galperin et al. (1988). This provides vertical viscosity and diffusivity coefficients that depend on the interaction of flow and stratification, which are provided by the relation

K_MK_HqlS_MS_H(11)

where q²/2 is the turbulent kinetic energy (TKE), l is turbulence mixing length scale, and S_M and S_H are stability functions related to the Richardson number. The local evolution of both q and l are given by

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The terms on the right-hand side are, respectively, the vertical turbulent diffusion, shear production, buoyancy production, and dissipation of turbulent kinetic energy; B₁ and E₁ are empirical constants, ρ₀ is the seawater reference density, and K_q is the vertical turbulent diffusivity for q² and q²l. A wall proximity function (w) is used to maintain a log-law near the boundaries. The seawater density (ρ) is related to temperature and salinity by the equation of state (Gill 1982).

At the ice–ocean interface melting is a function of temperature, salinity, and speed of the water adjacent to the interface. As the interface is approached eddies are suppressed and the eddy diffusivity decreases to a point where molecular diffusion becomes the principal transfer mechanism for heat and salt. The parameterization of transfer coefficients for heat (γ_t) and salt (γ_s) follow Kader and Yaglom (1972)

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where Pr and Sc are the molecular Prandtl and Schmidt numbers for seawater, ν is the kinematic viscosity, u∗ is the friction velocity, and D is the layer thickness adjacent to the ice shelf.

Basal melt rate, m (negative for melting) is determined from Nicholls et al. (2001)

View Expanded

where T_b and S_b are the temperature and salinity of the seawater adjacent to the ice–ocean interface; c_w and c_i are the specific heat capacities of seawater and ice; L and T_i are the latent heat of ice fusion and ice shelf surface temperature; T_f is the pressure freezing point for seawater of salinity S_f at the ice–ocean interface and is determined from the linear form of pressure freezing point relationship (Millero 1978)

T_faS_fbcp_io(17)

where a, b, and c are physical constants (−0.0573°C psu⁻¹, 0.0832°C, and 7.61 × 10⁻⁴°C dbar⁻¹), and p_io is the pressure at the ice–ocean interface in dbar.

During the production of melt water, heat is required to warm the ice to the in situ melting point and melt the ice prior to mixing. Assuming the salinity of the ice is zero, the bulk change in temperature (ΔT) and salinity (ΔS) for the 1-m thick layer adjacent to the ice shelf base over the time step (Δt) is prescribed by

View Expanded

where ΔM is the meltwater mass determined from (16) and M_b is the layer water mass.

To examine the vertical structure of the M₂ tidal currents, the resulting u and υ model velocity data from (5) and (6) are analyzed to determine the amplitude (A_u, A_υ) and phase (ϕ_u, ϕ_υ) of the velocity components. These data are used to construct clockwise and anticlockwise rotary components (Souza and Simpson 1996):

View Expanded

When viewed from above, R₋ and R₊ are the purely clockwise and anticlockwise constant magnitude velocity vectors that rotate at the tidal frequency. These two components combine to give the time-varying velocity vector that results in the tidal current ellipse (Prandle 1982) and are used to determine the semimajor (U) and semiminor (V) axes and polarization, from (1), of the tidal ellipse:

View Expanded

In this study, the one-dimensional model simulations are for a 200-m water column beneath an ice shelf with a surface temperature of −27°C, similar to the more tidally active regions close to the Ronne Ice Front, around Korff and Henry Ice Rises, and close to grounding lines. The surface forcing is ramped up over approximately 16 tidal cycles and the model allowed to stabilize. The wintertime conditions found on Berkner Shelf are used to provide the initial water column temperature and salinity of −1.9°C and 34.65 psu and the ice shelf draft of 250 m is similar to that found along the Ronne Ice Front. Although these initial oceanographic conditions and ice shelf draft represent a particular region, the temperature and salinity range is relatively small beneath FRIS (approximately −1.90° to −2.55°C and 34.77 to 34.48 psu) (Nicholls et al. 2001; Nicholls and Makinson 1998) and the model results will provide an insight into the broad pattern of tidal mixing regimes within the sub-ice-shelf cavity.

3. Model result

4. Application to FRIS

The broad range of surface forcings applied to the model generate a wide range of tidal currents that are analyzed to obtain the mean rotary components. Therefore, the effects of latitude on the tidal current structure will be embedded within the mean R₊ and R₋ components. Since FRIS occupies a narrow band of latitudes (79°S ± 4°) the dependence of pycnocline depth and melt rate on mean R₊ and R₋ should vary little with position on the ice shelf. However, the influence of proximity to the critical latitude on the R₊ component results in some differences from the data at 79°S in Figs. 7 and 8. The greatest differences of up to 20% occur for positive polarizations and speeds below 0.25 m s⁻¹, where the contribution of turbulence from R₊ dominates the vertical mixing. Here, the critical latitude strongly influences the R₊ current component (Fig. 4a) with additional turbulence being generated from the increased shear as R₊ attains its free stream velocity around the developing pycnocline (Fig. 5). This results in higher melt rates and slightly shallower pycnocline depths as the critical latitude is approached. For negative polarizations and higher current speeds there is little or no latitudinal influence. Consequently, the model results from 79°S are broadly representative of the range of latitudes occupied by FRIS.

Beneath the ice shelf, the modeled M₂ and S₂ tidal velocities can reach 0.5 m s⁻¹ and 0.3 m s⁻¹, respectively, typically three to six times higher than for the diurnal tides (Makinson and Nicholls 1999). The M₂ ellipse polarization ranges from −0.95 to +0.7 (Fig. 2) with predominantly anticlockwise rotation associated with the shallow water close to the Ronne Ice Front. In the central part of Ronne Ice Shelf and at the mouths of the ice stream and glacier inlets predominantly clockwise rotation occurs, which results from the interaction of a semidiurnal Kelvin wave with basins or inlets of different scales (Carbajal 1997).

The almost complete range of possible polarizations present beneath FRIS suggests that, in light of the model results, a wide range of vertical mixing regimes will exist. The boxes in Figs. 7 and 8 broadly divide the tidal current characteristics beneath FRIS into three main regions. However, the melt rates in Fig. 8 are not predicting basal melt rates beneath FRIS, only the contribution that tidal vertical mixing makes to the heat supply at the ice shelf base. Taking a representative point from each of these three tidal current regions beneath FRIS (indicated by the crosses in Fig. 7), profiles from the model of mixing length, turbulent kinetic energy, eddy viscosity, dissipation, and density change are plotted in Fig. 9. The model was spun up and allowed to stabilize over 80 tidal cycles. The profiles are plotted at hourly intervals over the final M₂ tidal cycle prior to melting and then after a further 10 tidal cycles, with melting, to show the effects of stratification.

Regions where P < −0.3 in water approximately 200 m deep are confined to a few small areas such as the easternmost part of Filchner Ice Shelf and just to north of Doake Ice Rumples. The combination of weak tidal currents and negative polarization (region 1 in Figs. 6 and 7) produce low levels of TKE (Fig. 9a). The turbulent dissipation is greatest close to the boundaries followed by an almost linear decrease on a log₁₀ scale with distance away from the boundaries, characteristic of all the dissipation profiles in Fig. 9. The effect of stratification on the water column is to suppress turbulence, inhibiting the transfer of momentum and reducing the eddy viscosity, particularly in the vicinity of the pycnocline. The pycnocline effectively separates the water column into two parts with TKE generated at the upper boundary being confined to a small upper mixed layer. This small mixed layer cools rapidly toward the in situ freezing point, creating a large density contrast across the pycnocline. The remainder of the water column comes under the influence of the lower boundary with the pycnocline acting effectively as a free surface. Consequently, TKE and dissipation increase in the mid–water column, increasing both the eddy viscosity and mixing length over a large portion of the lower water column (Fig. 9a).

Other regions where P < −0.3 such as the central Ronne Ice Shelf and mouths of ice stream inlets, the water column thickness can be up to 600 m with low speed tidal currents, compounding the effects of low polarization. Although the model does not specifically cover these water thicknesses, it can be inferred that vertical mixing will be confined to the small boundary layers of R₋ and the upper and lower boundary layers will be isolated from each other. In this situation, turbulence generated from tidal currents will not contribute to mid-water-column mixing, and melt resulting from tidal vertical mixing will be close to zero. However, where heat is available from horizontal advection in the upper mixed layer tidal currents will enhance the basal melting.

Most of FRIS lies between P = ±0.3, similar to the region south of Korff and Henry ice rises. This region is characterized by moderate tidal currents and a water column around 200 m thick (Fig. 1). Its M₂ polarization and velocity range occupies region 2 in Figs. 6 and 7, with representative water column properties shown in Fig. 9b. When compared with region 1, the moderate tidal currents provide increased TKE, enhancing the eddy viscosity and leading to a deeper penetration of the pycnocline into the water column. Other water column characteristics develop in the same way as in region 1 (Fig. 9a). The initial development and penetration into the water column of the pycnocline is sensitive to changes in polarization over this range (P = ±0.3) and the increasing polarization can lead to a three- to fourfold increase in the pycnocline depth (Fig. 6). The longer term melt rate is less sensitive where the polarization is positive, however, once the polarization falls below P = 0 the melt rate begins to decline rapidly (Fig. 8). In areas where tidal currents are constrained to flow approximately parallel to grounding lines (R₊ ≈ R₋), such as the Orville coast and south of the ice rises, modification of the ellipse polarization is not possible, therefore the transverse pressure gradient must balance the Coriolis force to maintain flow parallel to the coast. Here, there will be regions extending from grounding lines that are dominated by the interacting R₊ boundary layers and therefore exhibit high levels of mixing.

The Ronne Ice Front region is characterized by shallow water (<200 m), strong tidal currents, positive ellipse polarizations (0.3 < P < 0.7) and occupies region 3 in Figs. 6 and 7. In this high current regime intense mixing takes place, with rapid deepening of the pycnocline and strong melting sustained for as long as the water column remains above the in situ freezing point. These features are shown in Fig. 9c, with high levels of TKE extending into the midwater column, forming a deep pycnocline with the smallest density contrast of the three regions. This small density contrast after 10 tidal cycles is the result of the rapid mixing of meltwater down into the midwater column while maintaining a strong vertical heat flux, keeping the upper water column above the in situ freezing point and maintaining further melting.

5. Conclusions

The application of a 1D turbulent closure model, thermodynamically coupled to an ice shelf, has shown the role played by tidal currents under the influence of the earth's rotation in vertical mixing. The model shows that vertical tidal current structure under ice shelves is dramatically modified by the proximity of critical latitudes and changes in stratification; and both the pycnocline development and basal melt rate are sensitive to tidal current ellipse polarization. Beneath Filchner–Ronne Ice Shelf, which is bounded by the critical latitudes of the M₂ and S₂ semidiurnal tides, several key areas experience the extremes of the polarization range, creating regions of widely differing levels of tidally induced vertical mixing and basal melting. The key results from this study are

1. For predominantly clockwise rotation the model predicts mixing restricted by the small boundary layer of the R₋ rotary component.

2. As the polarization becomes increasingly positive, the thickening boundary layer and associated shear of R₊ causes the mixing to rapidly penetrate deeper into the midwater column.

3. The resultant melting gives over an order of magnitude difference between the lowest melt rates for purely clockwise tidal currents and the highest for purely anticlockwise tidal currents.

4. Changes in stratification can dramatically modify the vertical tidal current structure, particularly near semidiurnal critical latitudes.

As a consequence of these points, simple parameterizations of tidally induced vertical mixing for use in sub-ice-shelf circulation models are unlikely to represent the combined effects of semidiurnal critical latitudes, stratification, and ellipse polarization. Ultimately, a more comprehensive representation of tides and their associated mixing processes will be needed to determine their contribution to sub-ice-shelf circulation.