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Corresponding author: R. Sayag, Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK. (rs620@cam.ac.uk)

Abstract

[1] We combine the dynamics of ice, bed, and ocean in a new elastic model for the tidal-timescale migration of grounding lines on deformable foundations. Previous interpretations of tidal flexure using models of elastic ice shelves with fixed grounding lines were found to be inconsistent, suggesting an elasticity of ice that varies spatially and temporally and that is significantly smaller than measured experimentally. We argue here that with our model, a consistent, purely elastic interpretation can be made. Combining this new approach with remote-sensing measurements, we show that the grounding line migrates several kilometers during a tidal cycle, that we can infer the effective elastic properties of the bed, and that the elastic pressure of the ice leads to a hydrological barrier near the grounding line that controls subglacial hydrology. Our findings imply that subglacial lubrication and melting induced by the ocean thermal forcing can increase substantially during high tide.

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[2] Grounding zones of marine ice sheets are interfaces between grounded and floating ice. The response of such regions to tidal forcing correlates with ice-flow acceleration [Bindschadler et al., 2003; Anandakrishnan et al., 2003; Gudmundsson, 2006] and is an indicator of the grounding line (GL) position [Rignot et al., 2011], knowledge of which is required to determine the ice flux to the ocean and the consequent contribution to sea level rise.

[3] Previous interpretations of tidal flexure using elastic models included only the floating shelves, clamped at a fixed GL over a stiff bed (stiff-fixed models, Figures 1a and 1b) [Holdsworth, 1977]. Such models predicted ice elasticity that varies significantly between different glaciers [Vaughan, 1995; Rignot, 1996; Sykes et al., 2009] that varies with tidal phases in the same glacier [Schmeltz et al., 2002] and is significantly smaller than measured experimentally [Gammon et al., 1983]. These discrepancies have led to the inclusion of viscous deformation in the modeling of tidal flexure [Reeh et al., 2003; Gudmundsson, 2011; Walker et al., 2013].

[4] Alternatively, such discrepancies may have resulted from misinterpreted data due to the absence of grounded-ice dynamics in past elastic models. This argument is supported by observational evidence of tidal migration of GLs [Brunt et al., 2011] and of substantial bending that extends several kilometers upstream from the GL [Jacobel et al., 1994], in addition to correlation with periodic accelerations of grounded ice. These observations together with evidence of thick layers of deformable sediments near GLs [Anandakrishnan et al., 2007] indicate that the coupling of the shelf to the grounded sheet can be important to the interpretation of tidal flexure and subglacial processes near GLs.

2 Grounding Line Dynamics Under Tidal Forcing

[5] Scaling analysis of the forces involved in viscous bending of an ice sheet under tidal forcing implies that the flexure of ice on tidal timescales is dominantly elastic (see supporting information). We therefore consider the instantaneous deformation under tidal forcing of an elastic sheet of density ρ_{i} supported by an ocean of density ρ_{w}>ρ_{i} and an elastically deformable bed with effective stiffness k_{0}, and a free GL at x_{g}(the soft-free model, Figures 1a and 1b). The sheet is assumed to have uniform thickness H and bending stiffness D=EH^{3}/12(1−ν^{2}), where E and νare Young's modulus and Poisson ratio, respectively. Under small deflections, the vertical force balance in the floating and grounded parts is given by the Euler-Bernoulli equation

where y is the midsurface deflection of the sheet with respect to sea level along the horizontal coordinate x, y_{b}(x)=−Sx is the undeformed shape of the bed with slope S, and T is the sea level relative to its mean. Solutions of (1) have undulations that are damped exponentially away from the GL with relaxation lengths proportional to ℓ≡(D/ρ_{w}g)^{1/4} in the floating shelf and to ℓ/K in the grounded sheet, where K=(k_{0}/ρ_{w}g)^{1/4} (supporting information). In the stiff bed limit (k_{0}→∞), the solution has been found to be in good agreement with laboratory experiments [Sayag and Worster, 2011].

[6] The capacity of the bed to deform results in the GL position shifting by a distance of about ρ_{i}H/(ρ_{w}SK^{2}) relative to the GL position on a stiff bed, which can be several kilometers (supporting information). The stiffness k_{0}, which we use as a fitting parameter, represents an effective spring constant of the underlying bedrock plus the water-till mixture, having stiffnesses k_{b} and k_{t}, respectively, whence 1/k_{0}=1/k_{t}+1/k_{b}. We can estimate each stiffness using Hooke's law as σ/δ, where σ is the normal pressure and δis the vertical displacement of the bed. The displacement of a till having thickness h and Young's modulus E_{t} is δ=σh/E_{t}. The bedrock can be modeled as an elastic medium with Young's modulus E_{b} that occupies a half space, implying δ≃σR/E_{b}, where R is the horizontal range along which pressure is applied [Landau and Lifshitz, 1970]. The bed and the till stiffnesses are therefore k_{t}=E_{t}/h and k_{b}=E_{b}/R, respectively. The till can be of order 1 m thick with E_{t} ranging between 1 MPa for soft till [Walker et al., 2013] and 1.5 GPa for overconsolidated till [Bell, 2000], implying k_{t}≈1 MPa/m for thick, dilatant till or 10 GPa/m for thin, dense till. For the bedrock, E_{b}≈500 GPa [Turcotte and Schubert, 2002], while R≈10^{2}−10^{4}m represents the range over which the GL migrates, implying that k_{b}≈10−1000 MPa/m. Therefore, k_{0} ranges from about 1 MPa/m to about 1 GPa/m.

[7] Testing the consistency of elastic theories with observations requires the removal of long-time-scale footprints of viscous deformation in the observational data. This is achieved by considering the tidal curve Δy, defined as the difference between two deflection curves measured at tidal phases with amplitudes T_{1} and T_{0}, and corresponding GL positions x_{g}(T_{1}) and x_{g}(T_{0}) (Figures 1a and 1c and supporting information).

3 Analysis and Implications

3.1 Consistency With the Young's Modulus of Ice

[8] We now consider the tidal curve measured across the grounding zone of Rutford Ice Stream, Antarctica (Figure 2), starting above a steep knoll in the center of this 30 km wide stream of nearly uniform thickness. Regression to a stiff-fixed model with tidal difference ΔT=T_{1}−T_{0}, GL position xgfix and Young's modulus Eas free parameters implies E≈1.8 GPa – about 5 times less than the value found by laboratory measurements. In contrast, the regression to the soft-free model is performed with a constant Young's modulus of 9.33 GPa given by the laboratory measurements and with the same number of free parameters as in the stiff-fixed case, given by ΔT,x_{g}(T_{1}) and the bed stiffness k_{0}. The result (Figure 2) shows excellent agreement with the data, indicating k_{0}≈1 MPa/m and a GL position nearly 75 m upstream from that set by the stiff-fixed model. This agreement is possible within the soft-free model, despite the substantially larger elastic modulus, owing to the capacity of the bed to deform and the consequent unconstrained ice slope at the GL. In addition, the fact that ℓ∝E^{1/4}implies that a factor five increase of Young's modulus results in ℓgrowing by only about 50%. Such consistency of the purely elastic, soft-free model both with the flexure data and with the known elastic properties of ice provides an alternative interpretation of the dominant underlying processes, suggesting that viscous interpretations may not always be relevant.

[9] The resultant bed stiffness, implying 1 m vertical displacement, and the small range of GL migration indicate the presence of a thick layer of subglacial material having the capacity to become highly dilatant, which may be consistent with some observations beneath Rutford Ice Stream [King et al., 2009]. It is conceivable that ℓ is effectively smaller, owing to nonuniform ice density and surface fractures that imply smaller effective thickness or Young's modulus or larger Poisson ratio [Jenkins et al., 2006], which would lead to a larger bed stiffness and smaller displacement. In addition, the knoll beneath the stream is a significant disruption to the bed slope, which requires a more detailed analysis. We note that similar bed stiffness values in a soft-bed and fixed-GL model [Walker et al., 2013] imply smaller bed compression since the grounded ice is pinned at both ends.

3.2 Grounding Line Migration Between High and Low Tides

[10] The explicit coupling of the GL position and the tidal amplitude in the soft-free model becomes important when resolving GL migration on shallow slopes such as ice plains. We use altimetry measurements taken in tidal phases T_{0},T_{1}and T_{2} at two locations in the Filchner-Ronne grounding zone and compute regressions to two tidal curves at each location (supporting information). Regression to the stiff-fixed model is computed separately for each of the curves (as in §3.1), since the GL and the tidal phase are not coupled. Using ΔT, x_{g} and Young's modulus as fitting parameters for each curve (six parameters in all), we find that a good agreement between this model and the data (Figure 3) is not obtained unless Young's modulus is dependent on phase [Schmeltz et al., 2002]. Regressions to the soft-free model are computed to both tidal curves simultaneously using fewer (five) fitting parameters, namely T_{0,1,2}, k_{0} and the GL position at T_{0}. The results provide better agreement between the theory and the data than the stiff-fixed model, despite the fewer fitting parameters. They indicate that k_{0}≈7.6 GPa/m (δ∼1.3 mm), consistent with GL migration of 4.1±0.1 km at one track (Figure 3a) and k_{0}≈6.3 GPa/m (δ=1.4 mm) with GL migration of 13.3±0.2 km at another (Figure 3b). The dimensional analysis in §2 suggests that such GL migrations should correspond to softer beds (about 10–100 MPa/m). However, it is conceivable that the large ice rise a few tens of kilometers downstream of the GL along these tracks, a distance comparable to ℓ≈32 km, pins the shelf even at high tide so that the tidal response at this location involves substantial tension, which the present theory does not account for.

[11] As the GL migrates at high tide, the contact area between the ice base and the ocean increases. Although this increase can be small compared with the length of an ice shelf, the impact on basal melting can be substantial, because the maximum melt rate that is driven by the ocean thermal forcing occurs near the GL in a small region compared with the shelf length [Jenkins and Doake, 1991; Rignot et al., 2013].

3.3 The Impact of Basal Deformation on Subglacial Hydraulics

[12] The deformation of the bed under the elastic pressure of the ice has significant implications for subglacial hydrology and lubrication near the GL. A subglacial hydraulic potential that accounts for the ice elastic pressure is φ=ρ_{w}gy+ρ_{i}gH+D∇^{4}y, which together with equation (1) implies a gradient of the subglacial hydraulic potential, ∇φ=ρ_{w}g∇y+k_{0}∇(H/2−y+y_{b}). Far from the GL, the elastic pressure in the ice diminishes, leading to y≈y_{b}+H/2 and therefore to ∇φ≈−ρwgS≈O(1−10Pa/m). However, bending of the ice near the GL enhances the elastic pressure substantially, leading to ∇φ≈−k0S+∇y≈O(103Pa/m) over a distance of the order of ℓ, in which the gradient of the hydraulic potential is negative close to the GL and positive further upstream (Figure 4a). Consequently, this elastically controlled region acts as a barrier that prevents leakage of subglacial melt water toward the ocean and transfer of ocean water landward.

[13] A large tidal migration of the GL on shallower slopes combined with the action of the elastically dominated hydraulic barrier can potentially control subglacial lubrication (Figure 4b). At high tide, the landward migration of the hydraulic barrier can also force the subglacial water system to migrate, increasing the area of subglacial lubrication that could consequently trigger a slip action [Bindschadler et al., 2003; Anandakrishnan et al., 2003]. At low tide, the hydraulic barrier migrates toward the ocean, thereby reducing the lubricated area while pushing ocean water downstream. The larger ΔT/S/ℓ is, the larger the hydraulic barrier migration and the potential for extended slip conditions at high tide. Therefore, ice located on wet beds and shallow bed slopes (e.g., ice streams) is more likely to undergo slip events under this mechanism. This outcome is in contrast with the case of a pinned GL [Walker et al., 2013], in which the elastic pressure gradient changes sign at low tide, implying suction of ocean water into the subglacial domain. These considerations indicate a mechanism by which subglacial water can be redistributed but much more detailed modeling of the subglacial hydrology is required to assess this potential.

4 Conclusions

[14] Our analysis coupling the grounded sheet to the floating shelf demonstrates that the grounding line can migrate several kilometers during a tidal cycle (estimated at 13 km at one location of Filchner-Ronne ice shelf, for example), implying that at high tide the area of ice that undergoes basal melting at maximum rate is larger. It also shows that at high tide subglacial lubrication is enhanced, which provides an explanation for the observed slip action associated with tides. Consequently, these processes can combine to accelerate the transfer of mass from ice sheets to the ocean, making it essential to account for such processes on tidal timescales in order to improve the prediction of the mass balance of ice sheets and ultimately of sea level rise.

Acknowledgments

[15] We thank D. Vaughan and J. Neufeld for commenting on this manuscript and J. Lister for discussions. We also thank D. Vaughan for providing the data in Figure 2 and to K. Brunt, M. Siegfried, and H. A. Fricker for providing the data in Figure 3.

[16] The Editor thanks two anonymous reviewers for assistance evaluating this paper.