scholarly journals On The Use of the Equilibrium Equations and Flow Law in Relating the Surface and Bed Topography of Glaciers and Ice Sheets

1968 ◽  
Vol 7 (50) ◽  
pp. 199-204 ◽  
Author(s):  
I. F. Collins
Keyword(s):  
Surface Topography ◽  
Equilibrium Equation ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Equilibrium Equations ◽  
Bed Topography ◽  
Flow Law ◽  
Mathematical Justification ◽  
Longitudinal Strains

In a recent paper Robin has developed a method of calculating the relation between bed and surface topography of an ice sheet. He found that by including the effect of longitudinal strains in the equilibrium equation the correlation between theory and observation could be much improved. This paper is concerned with the mathematical justification of the assumption made by Robin.

scholarly journals On The Use of the Equilibrium Equations and Flow Law in Relating the Surface and Bed Topography of Glaciers and Ice Sheets

1968 ◽  
Vol 7 (50) ◽  
pp. 199-204 ◽  
Author(s):  
I. F. Collins
Keyword(s):  
Surface Topography ◽  
Equilibrium Equation ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Equilibrium Equations ◽  
Bed Topography ◽  
Flow Law ◽  
Mathematical Justification ◽  
Longitudinal Strains

In a recent paper Robin has developed a method of calculating the relation between bed and surface topography of an ice sheet. He found that by including the effect of longitudinal strains in the equilibrium equation the correlation between theory and observation could be much improved. This paper is concerned with the mathematical justification of the assumption made by Robin.

Relative control of bedrock roughness versus topography on global glacier bed friction

2021 ◽  
Author(s):  
Olivier Gagliardini ◽  
Fabien Gillet-Chaulet ◽  
Florent Gimbert
Keyword(s):  
Sea Level Rise ◽  
Sea Level ◽  
Inverse Method ◽  
Ad Hoc ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Grid Resolution ◽  
Classical Approach ◽  
Bed Topography ◽  
Bed Friction

<p>Friction at the base of ice-sheets has been shown to be one of the largest uncertainty of model projections for the contribution of ice-sheet to future sea level rise. On hard beds, most of the apparent friction is the result of ice flowing over the bumps that have a size smaller than described by the grid resolution of ice-sheet models. To account for this friction, the classical approach is to replace this under resolved roughness by an ad-hoc friction law. In an imaginary world of unlimited computing resource and highly resolved bedrock DEM, one should solve for all bed roughnesses assuming pure sliding at the bedrock-ice interface. If such solutions are not affordable at the scale of an ice-sheet or even at the scale of a glacier, the effect of small bumps can be inferred using synthetical periodic geometry. In this presentation,<span>  </span>beds are constructed using the superposition of up to five bed geometries made of sinusoidal bumps of decreasing wavelength and amplitudes. The contribution to the total friction of all five beds is evaluated by inverse methods using the most resolved solution as observation. It is shown that small features of few meters can contribute up to almost half of the total friction, depending on the wavelengths and amplitudes distribution. This work also confirms that the basal friction inferred using inverse method<span>  </span>is very sensitive to how the bed topography is described by the model grid, and therefore depends on the size of the model grid itself.<span> </span></p>

scholarly journals Reconstruction and Disintegration of Ice Sheets for the Climap 18000 And 125000 Years b.p. Experiments: Results

1979 ◽  
Vol 24 (90) ◽  
pp. 495-496
Author(s):  
G. H. Denton ◽  
T. J. Hughes ◽  
J. L. Fastook ◽  
D. H. Schilling ◽  
C. S. Lingle
Keyword(s):  
Northern Hemisphere ◽  
Land Surface ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Recursive Formula ◽  
Ice Age ◽  
Flow Lines ◽  
Bed Topography ◽  
Surface Areas ◽  
Basal Shear

AbstractLate-Wisconsin ice sheets were reconstructed for the CLIMAP 18000 years b.p. experiment. This experiment modeled the ice-age steady-state climate using boundary conditions that differed from present ones mainly in Earth-surface albedos, sea-surface areas, and land-surface topography. These required determinations of the area, volume, and elevation of Late Wisconsin ice sheets. An initial-value finite-difference numerical model for ice-sheet reconstruction was developed from a recursive formula which gave ice thickness for known variations of bed topography and theoretical variations of basal shear stress. Ice thicknesses were calculated in 50 km to 100 km steps along flow lines from margins to domes of late-Wisconsin ice sheets. We assumed that terrestrial margins were along the furthermost moraines, marine margins were along the present 500 m bathymetric contour, domes were centers of maximum post-glacial isostatic rebound, and flow lines were along glacial lineations (eskers, striations, drumlins, etc.) connecting margins to domes. At various locations ice-sheet margins were verified by dated moraines for terrestrial margins and Egga-type moraines for marine margins. Ice-sheet elevations and thicknesses were contoured from profiles reconstructed for 40 Antarctic flow lines and 137 Northern Hemisphere flow lines for a maximum ice-sheet extent, and 86 Northern Hemisphere flow lines for a minimum ice-sheet extent.

scholarly journals Reconstruction and Disintegration of Ice Sheets for the Climap 18000 and 125000 Years B.P. Experiments: Theory

1979 ◽  
Vol 24 (90) ◽  
pp. 493-495
Author(s):  
T. J. Hughes
Keyword(s):  
Shear Stress ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Frictional Heat ◽  
Ice Streams ◽  
Ice Stream ◽  
Grounding Line ◽  
Flow Law ◽  
Basal Shear ◽  
Stress Variations

AbstractSize, shape, and surface albedo of former ice sheets are needed in order to model atmospheric circulation for the CLIMAP 18000 years B.P. experiment. Both the size and shape of an ice sheet depend on the hardness of ice and its coupling to bedrock. Ice hardness is controlled by ice temperature and fabric, which are not adequately described by any ice flow law. Ice–bed coupling is controlled by bed roughness and basal melt water, which are not adequately described by any ice sliding law. With these inadequacies in mind, we assumed equilibrium ice-sheet conditions 18000 years ago and combined the standard steady-state flow and sliding laws of ice with the equation of mass balance to obtain separate basal shear-stress variations along ice-sheet flow lines for a frozen bed when the flow law dominates and for a melted bed when the sliding law dominates. Theoretical basal shear-stress variations were then derived for freezing and melting beds on the assumption that separate melted areas of the bed had water films of constant thickness which expanded and merged for a melting bed but contracted and separated for a freezing bed. Theoretical basal shear-stress variations were also derived for ice streams along marine ice-sheet margins and ice lobes along terrestrial ice-sheet margins on the assumption that the entire area of their bed was wet so that further melting increased the water-layer thickness, which would then be decreased by freezing. Melting was assumed to continue to the grounding line of an ice stream and the minimum-slope surface inflection line of an ice lobe, where freezing began and continued to the ice-lobe terminus. Ice–bed uncoupling is complete at an ice-stream grounding line and maximized at an ice-lobe minimum-slope inflection line, so ice velocity and consequent generation of frictional heat were assumed to reach maxima across these lines. Theoretical basal shear-stress variations were derived for the zone of converging flow at the heads of ice streams and ice lobes, and from domes to saddles along the ice divide for both frozen and melted beds.

scholarly journals Basal control of supraglacial meltwater catchments on the Greenland Ice Sheet

2018 ◽  
Author(s):  
Josh Crozier ◽  
Leif Karlstrom ◽  
Kang Yang
Keyword(s):  
Surface Topography ◽  
Transfer Functions ◽  
Ice Sheet ◽  
Greenland Ice Sheet ◽  
Spatial Density ◽  
Bed Topography ◽  
Subglacial Environment ◽  
Ice Surface ◽  
Basal Sliding ◽  
Elevation Data

Abstract. Ice surface topography controls the primary routing of surface meltwater on ablation zones of glaciers and ice sheets. Meltwater routing is important for understanding and predicting ice sheet evolution because surface melt can be both a direct source of ice mass loss and an influence on basal sliding and ice advection. Although controls on ice sheet topography at long wavelengths are well known, smaller scale features relevant for meltwater routing are not well understood. Here we examine the effects of two processes that can influence ice sheet surface topography: bed topography transfer and thermal-fluvial incision by supraglacial streams. We implement 2D bed topography and basal sliding transfer functions in seven study regions of the western Greenland Ice Sheet (GIS) ablation zone to study the influence of basal conditions on ice surface topography. Although bed elevation data quality is spatially variable, we find that ∼ 1–10 km scale ice surface features under variable ice thickness, velocity, and surface slope are well predicted by these transfer functions. We then use flow-routing algorithms to extract supraglacial stream networks from 2–5 m resolution digital elevation models, and compare these with synthetic flow networks calculated on ice surfaces predicted by bed topography transfer. Quantitative comparison of these networks reveals that bed topography can explain ∼ 1–10 km surface meltwater routing patterns without significant contributions from thermal-fluvial erosion by streams. We predict how supraglacial internally drained catchment (IDC) patterns on the GIS would change under time-varying ice flow and/or basal sliding regimes. Basal sliding variations exert a significant influence on IDC spatial distribution, and suggest a potential positive feedback between subglacial hydrologic regime to surface IDC patterning. Increased basal sliding will increase IDC spatial density (by decreasing IDC sizes) and cause more disperse meltwater input to the englacial and subglacial environment. This could result in less efficient subglacial channelization and increased basal sliding that would then further increase IDC density.

scholarly journals Reconstruction and Disintegration of Ice Sheets for the Climap 18000 and 125000 Years B.P. Experiments: Theory

1979 ◽  
Vol 24 (90) ◽  
pp. 493-495 ◽  
Author(s):  
T. J. Hughes
Keyword(s):  
Shear Stress ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Frictional Heat ◽  
Ice Streams ◽  
Ice Stream ◽  
Grounding Line ◽  
Flow Law ◽  
Basal Shear ◽  
Stress Variations

Abstract Size, shape, and surface albedo of former ice sheets are needed in order to model atmospheric circulation for the CLIMAP 18000 years B.P. experiment. Both the size and shape of an ice sheet depend on the hardness of ice and its coupling to bedrock. Ice hardness is controlled by ice temperature and fabric, which are not adequately described by any ice flow law. Ice–bed coupling is controlled by bed roughness and basal melt water, which are not adequately described by any ice sliding law. With these inadequacies in mind, we assumed equilibrium ice-sheet conditions 18000 years ago and combined the standard steady-state flow and sliding laws of ice with the equation of mass balance to obtain separate basal shear-stress variations along ice-sheet flow lines for a frozen bed when the flow law dominates and for a melted bed when the sliding law dominates. Theoretical basal shear-stress variations were then derived for freezing and melting beds on the assumption that separate melted areas of the bed had water films of constant thickness which expanded and merged for a melting bed but contracted and separated for a freezing bed. Theoretical basal shear-stress variations were also derived for ice streams along marine ice-sheet margins and ice lobes along terrestrial ice-sheet margins on the assumption that the entire area of their bed was wet so that further melting increased the water-layer thickness, which would then be decreased by freezing. Melting was assumed to continue to the grounding line of an ice stream and the minimum-slope surface inflection line of an ice lobe, where freezing began and continued to the ice-lobe terminus. Ice–bed uncoupling is complete at an ice-stream grounding line and maximized at an ice-lobe minimum-slope inflection line, so ice velocity and consequent generation of frictional heat were assumed to reach maxima across these lines. Theoretical basal shear-stress variations were derived for the zone of converging flow at the heads of ice streams and ice lobes, and from domes to saddles along the ice divide for both frozen and melted beds.

scholarly journals Steady-State Temperatures at the Bottom of Ice Sheets and Computation of the Bottom Ice Flow Law from the Surface Profile

1968 ◽  
Vol 7 (51) ◽  
pp. 363-376 ◽  
Author(s):  
L. Lliboutry
Keyword(s):  
Steady State ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Surface Profile ◽  
Greenland Ice Sheet ◽  
Bottom Temperature ◽  
Ice Flow ◽  
Actual Surface ◽  
Flow Law ◽  
Strain Heating

AbstractA solution for the steady flow of a cold ice sheet is recalled, which takes account of the heat released by deformation. As this strain heating increases the strain velocity, the bottom temperature may be unstable. A set of five equations with five unknowns is written, which allows the surface profile and the bottom temperature to be computed step by step by an iterative process. This has been done by computer for three very different models of ice sheets. and in each case with three distinct values of the constant B in Glen’s ice flow law. It was found in every case that steady-state temperature profiles could not be computed beyond a moderate distance from the ice divide. The correct value of B for bottom ice may be deduced from the actual surface profile. At the bottom of Greenland ice sheet, B ≈ 2.18 bar −3 year−1. This is about thirteen times bigger than for the bulk of the alpine glaciers.

The effects of bed topography and strength on stability of marine ice sheets

2021 ◽  
Author(s):  
Olga Sergienko ◽  
Duncan Wingham
Keyword(s):  
Accumulation Rate ◽  
Ice Sheets ◽  
Ice Sheet ◽  
West Antarctica ◽  
Amundsen Sea ◽  
Bed Topography ◽  
Wide Range ◽  
Grounding Line ◽  
Bed Slope ◽  
Conditions Of Stability

<p>The "marine ice-sheet instability hypothesis", which states that unconfined marine ice sheets are unconditionally unstable on retrograde slopes, was developed under assumptions of negligible bed slopes. Realistic ice sheets, however, flow over beds which topographies have a wide range of bed slopes (for example, Thwaites Glacier in the Amundsen Sea sector, West Antarctica). Reexamining the original model of marine ice sheets proposed by Schoof (2007), and relaxing an assumption of negligible bed slopes, we find that a steady-state ice flux at the grounding line is an implicit function of the grounding-line ice thickness, bed slope and accumulation rate. Depending on the sliding conditions, the magnitudes of the ice flux at the grounding line differ by one-to-three orders of magnitudes from that computed with a power-law expression derived by Schoof (2007) under assumptions of the negligible bed slopes. Non-negligible bed slopes also result in conditions of stability of the grounding line that are significantly more complex than those associated with the "marine ice sheet instability hypothesis". Bed slopes are no longer the sole determinant of whether the grounding line is stable or unstable. We find that the grounding line can be stable on beds with retrograde slopes and unstable on beds with prograde slopes. </p>

Creep and recrystallization of large polycrystalline masses. III. Continuum theory of ice sheets

2006 ◽  
Vol 462 (2073) ◽  
pp. 2797-2816 ◽  
Author(s):  
Sérgio H Faria
Keyword(s):  
Grain Boundary ◽  
Large Scale ◽  
Boundary Migration ◽  
Ice Sheets ◽  
Ice Sheet ◽  
Continuum Theory ◽  
Grain Boundary Migration ◽  
Cauchy Stress ◽  
Flow Law ◽  
Continuous Diversity

This work sets forth the first thermodynamically consistent constitutive theory for ice sheets undergoing strain-induced anisotropy, polygonization and recrystallization effects. It is based on the notion of a mixture with continuous diversity, by picturing the ice sheet as a ‘mixture of lattice orientations’. The fabric (texture) is described by an orientation-dependent field of mass density which is sensitive not only to lattice spin, but also to grain boundary migration. No constraint is imposed on stress or strain of individual crystallites, aside from the assumption that basal slip is the dominant deformation mechanism on the grain scale. In spite of the fact that individual ice crystallites are regarded as micropolar media, it is inferred that couples on distinct grains counteract each other, so that the ice sheet behaves on a large scale as an ordinary (non-polar) continuum. Several concepts from materials science are translated to the language of continuum theory, like, for example, lattice distortion energy , grain boundary mobility and Schmid tensor , as well as some fabric (texture) parameters like the so-called degree of orientation and spherical aperture. After choosing suitable expressions for the stored energy and entropy of dislocations, it is shown that the driving pressure for grain boundary migration can be associated to differences in the dislocation potentials ( viz . the Gibbs free energies due to dislocations) of crystallites with distinct c -axis orientations. Finally, the generic representation derived for the Cauchy stress is compared with former generalizations of Glen's flow law, namely the Svendsen–Gödert–Hutter stress law and the Azuma–Goto–Azuma flow law.

scholarly journals Basal control of supraglacial meltwater catchments on the Greenland Ice Sheet

2018 ◽  
Vol 12 (10) ◽  
pp. 3383-3407 ◽  
Author(s):  
Josh Crozier ◽  
Leif Karlstrom ◽  
Kang Yang
Keyword(s):  
Surface Topography ◽  
Transfer Functions ◽  
Ice Sheet ◽  
Greenland Ice Sheet ◽  
Flow Routing ◽  
Ice Flow ◽  
Fluvial Erosion ◽  
Bed Topography ◽  
Ice Surface ◽  
Basal Sliding

Abstract. Ice surface topography controls the routing of surface meltwater generated in the ablation zones of glaciers and ice sheets. Meltwater routing is a direct source of ice mass loss as well as a primary influence on subglacial hydrology and basal sliding of the ice sheet. Although the processes that determine ice sheet topography at the largest scales are known, controls on the topographic features that influence meltwater routing at supraglacial internally drained catchment (IDC) scales (<10s of km) are less well constrained. Here we examine the effects of two processes on ice sheet surface topography: transfer of bed topography to the surface of flowing ice and thermal–fluvial erosion by supraglacial meltwater streams. We implement 2-D basal transfer functions in seven study regions of the western Greenland Ice Sheet ablation zone using recent data sets for bed elevation, ice surface elevation, and ice surface velocities. We find that ∼1–10 km scale ice surface features can be explained well by bed topography transfer in regions with different multiyear-averaged ice flow conditions. We use flow-routing algorithms to extract supraglacial stream networks from 2 to 5 m resolution digital elevation models and compare these with synthetic flow networks calculated on ice surfaces predicted by bed topography transfer. Multiple geomorphological metrics calculated for these networks suggest that bed topography can explain general ∼1–10 km supraglacial meltwater routing and that thermal–fluvial erosion thus has a lesser role in shaping ice surface topography on these scales. We then use bed topography transfer functions and flow routing to conduct a parameter study predicting how supraglacial IDC configurations and subglacial hydraulic potential would change under varying multiyear-averaged ice flow and basal sliding regimes. Predicted changes to subglacial hydraulic flow pathways directly caused by changing ice surface topography are subtle, but temporal changes in basal sliding or ice thickness have potentially significant influences on IDC spatial distribution. We suggest that changes to IDC size and number density could affect subglacial hydrology primarily by dispersing the englacial–subglacial input of surface meltwater.

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