The research in this thesis shows that our understanding of the first-order short time-scale behavior of natural ice is incomplete and works towards our improvement. Computational modeling of the response of glaciers in Greenland and Antarctica to hydrological forcing for a time shorter than one month is the focus of this thesis.
The Cryosphere
Over long time scales, changes in the climate system can dramatically affect the response of ice streams to ocean conditions. By examining tidal forcing of ice streams (chapters 2 to 4) and rapid drainage of supraglacial lakes (chapter 5), this thesis demonstrates some of the modeling issues of processes that span the gap between very fast (elastic) response to the order. days and more.
Ice Stream Dynamics
At one end of the spectrum are the Siple Coast, Antarctic or Rutford Ice Streams, which are. At the other end of the ice flow spectrum are outlet glaciers in Greenland such as Helheim and Jakobshavn Isbrae.
Tidal Interaction with Grounded Ice
Antarctic Tidal Interactions
Next is a summary of observations where ocean tides do not have an influence on the movement of an ice stream well inland from the grounding line. For the Whillans ice flow and plane, ice slide motion makes it unstable to determine a phase delay in the displacement signal.
Greenland Tidal Interactions
Similarly between recordings there is a slight trend for increasing phase with the best fit phase values, but that this trend is well within the error of the. However, the mean values of the best fit seem to indicate that the grounded ice may have an increased phase lag compared to the floating ice.
Observation Summary
The ice flow appears to filter some of the tidal frequencies so that the detrended GPS records do not exhibit many of the beat frequencies seen from the vertical component of GPS stations on floating ice. Such temporal delays likely provide information about the rheology of the material that transmits the tidal stress inland.
General Finite Element Methods
PyLith solves these equations using the Galerkin spatial equation formulation and the unconditionally stable implicit time step method (following the form of Bathe, 1995). 2012a/b), so that our model results are independent of convergence tolerances to a factor smaller than 1/1000%.
Thesis Outline
Images of the Siple Coast and Rutford Ice Stream come from the Atlas of the Cryosphere, a service provided by the National Snow and Ice Data Center (NSIDC). The resistance at the edges of the ice stream causes an intrinsic, exponential decay of the tidal tension.
Introduction
Such a model is only suitable for ice "far" from the lateral edges of the ice stream. Of the published models following Anandakrishnan and Alley (1997) , the most useful tidal propagation models are those of Gudmundsson (2011) and Walker et al.
Methodology
Model Descriptions
Based on the insight gained from these two-dimensional models, we then move to our three-dimensional models (figure 2.1B), and study the impact of resistance shear at the lateral margins of an ice stream on the upstream transmission of the applied tidal load. Such models investigate the role that the overall geometry of the ice stream (i.e. ice stream width and thickness) has on the transfer of the stress inland from the ground line.
Model Construction
In two-dimensional ice shelf models, the tidal load acts normal to the bottom and vertical edge of the ice shelf. In the two-dimensional shelfless and three-dimensional models, the tidal load acts on the vertical edge of the ice stream at the grounding line.
Results
Two-Dimensional Results
For the model with a frozen bed (figure 2.3), the bending and axial stresses decay exponentially with distance inland from the ground line with similar failure rates. Thus, the same 𝐿𝐿𝑡𝑡𝑡𝑡 in tables 2.3 and 2.4 calculated for stress also represents the behavior of displacements.
Three-Dimensional Results
Such a shear field is akin to the solution for an elastic (Bernoulli–Euler) beam under constant pressure simply supported at both edges (e.g. Turcotte and Schubert, 2002). Also remember that the displacements in our three-dimensional models decay exponentially with inland distance at the same rate as the stress signal decays due to the elastic rheology of the ice.
Transmission of Tidal Stresses
Thus, tidal stresses at a distance inland from the grounding line equal to 2.5 times the width of the ice flow should have no tidal influence on the movement of the ice flow. Note that for the Bindschadler Ice Stream, the grounding line bends along the edge of the ice stream for almost 75 kilometers.
Discussion
Comparison to Previous Models
Comparing our current results with those of Anandakrishnan and Alley (1997) , the results of our two-dimensional model represent the extremes of Anandakrishnan and Alley's model. Our sliding bed model corresponds to the model of Anandakrishnan and Alley with an infinitely weak viscous layer (𝜂 ≈ 0).
Model Shortcomings
The overall effect would be equivalent to having a larger effective ice stream width. We note that in the case of the ice cliff edges of the fjord-bounded Greenland outlet glaciers discussed in.
Summary and Conclusions
In the highlighted case, the standard deviation is large, as the value of 𝛿𝛿𝑥𝑥 drops to zero near the center of the ice stream. Near the top and bottom of the ice flow, the stress drop for 𝛿𝛿𝑥𝑥 is more consistent with the values for the other stress components. The transfer length scale 𝐿𝐿𝑡𝑡𝑡𝑡 is the value found for the equivalent stress decay that matches the value found using the surface displacement magnitude.
Appendix 2A: Importance of the Ice Shelf
The basal condition under the ice stream determines the influence of the ice shelf on the overall magnitude of the stress in the far-field ice stream. Since ice streams have little basal resistance, the finding that the overall stress magnitude is independent of the ice shelf outside the bending zone applies here. Our interest is in the value of stresses many tens of kilometers inland from the ground line, so we can safely neglect the ice shelf in our models without changing the transmission of tidal, non-bending stresses.
Appendix 2B: Flotation Condition for a One-Dimensional Ice Shelf
In addition, we model the linear thinning of the ice shelf (with modification I, using 𝐼𝐼=�12𝑤𝑤� ∙ �[ℎ0−(ℎ0 − ℎ1)]𝑥𝑥𝐿𝐿�3, where the thickness varies linearly from ℎ0 to ℎ1) and find that only a small effect on stress and deflection across the shelf. This simpler condition overestimates the stress and deflection in the model domain compared to the more correct flotation condition. However, since the boundary condition does not depend on and is therefore decoupled from the deflection w, we use this constant load as the "pseudo-flotation" condition of the ice shelf boundary in our final modelling.
Appendix 2C: Two-Dimensional Model Results
Appendix 2D: Three-Dimensional Model Results
Introduction
- Elastic Rheological Effects
- Temperature-Dependent Rheology
- Fabric Dependence
- Enhanced Deformation in the Shear Margins
- Appropriateness of Viscoelasticity
- Simple Rheological Models for Viscoelasticity
- The Maxwell Relaxation Time
Unfortunately, current measurement of the magnitude of the elastic response of ice flows is limited to GPS stations located near short-term perturbations to the background stress of an ice flow. As such, we now explore. the possibility that the viscous component of deformation in ice streams is important on hourly to weekly time scales in the context of the transmission of tidal loads within the grounding line of an ice stream. 2011) explicitly state that viscoelastic effects within the ice flow may play a role in the phase delay of the ice flow response to tidal loading.
Strain-Weakening in the Shear Margins
- Continuum Damage Mechanics Formulation
- Continuous Margin Results
- Discrete Margin Results
- Strain-Weakening Discussion
Note that the value of Ltr is constant over the profile of the model ice stream (stored immediately near the ground line), even though the Young's modulus is not. Unlike the continuous margin models, Ltr is not constant over a cross profile of the ice stream. In the latter, Ltr is elevated in the central ice compared to that of the edges.
Viscoelasticity
- Viscoelastic Model Considerations
- Nonlinearity and the Loss of Superposition
- Time-Dependent Behavior
- Temperature-Dependent Viscosity
- Enhanced Flow in the Margins
- Homogeneous Viscoelastic Modeling Results
- Temperature-Dependent Viscosity Results
- Viscoelasticity Discussion
For reference, the Ross Sea adjoins the ice streams at the Siple Coast, while the Rutford Ice Stream flows into the Weddell Sea. In the context of ice flows under tidal influence, this phase delay is expressed as a time delay in the maximum stress and shear disturbance of the ice flow. In our models, we need to distinguish between the effects of the oscillating loading of seawater and the effects of the static loading due to the gravitational driving stress in the ice flow.
Summary and Conclusions
The ground line is marked with arrows, as the ground line is the location of the applied tidal forcing. Figures on the right show the increase of the relative values of Ltr as a function of shear margin width. The triangles represent GPS stations on the surface of the ice stream and ice shelf.
Appendix 3A: Full Tidal Loading vs. Partial Tidal Loading
Introduction
The previous two chapters showed that ice flows are unlikely to transmit tidal stress through much of the ice flow itself over the extreme distances seen. And along the ice stream, the question of the observed phase lag in the response of the ice stream to ocean tides remains. Our goal is to develop a methodology that uses the multiple time scales of oscillatory tidal loading in conjunction with the observed phase shift in the tidal response to infer constraints on the viscoelastic parameters of ice flow.
Phase Shift in Analytic Models
- One-Dimensional Phase Shift
- Phase Shift for a Nonlinear Maxwell Material
The Deborah number encapsulates the choice of material parameters. shear modulus and viscosity) and the forcing frequency, allowing us to calculate the phase shift with respect to a single non-dimensional quantity. The nonlinear viscosity of ice complicates the understanding of the phase shift in the oscillatory response of a one-dimensional nonlinear material model. The phase shift values for the semi-diurnal, diurnal, fortnightly and combined tides are shown in Figure 4.2 as functions of De and for a linear, Glen and the two.
Two-Dimensional Finite Element Models
- Methodology
- Numerical Results
- Application to Helheim Glacier Data
The model geometry is a simplified version of the lower part of the Helheim glacier (750 meters thick and six kilometers wide). Figures 4.4 and 4.5 show the behavior of the phase for our basal and sidewall models, respectively, as a function of De. To better demonstrate this distance dependence, Figures 4.6 and 4.7 show the phase shift of the centerline ice as a function of inland distance (note that the horizontal length scale varies due to the difference in Ltr between the two models).
Discussion
- Data Constraints and Accuracy
- Survey Requirements
De Juan-Verger (2011) estimated the phase delay in the Helheim Glacier GPS network for. The error in the estimated ice Maxwell time is directly related to the error in the phase estimate. Thus, our two-dimensional models are necessarily oversimplifications of the phase behavior of ice flows.
