scholarly journals The Steady Profile of an Axisymmetric Ice Sheet

1981 ◽  
Vol 27 (95) ◽  
pp. 25-37 ◽  
Author(s):  
I. R. Johnson
Keyword(s):  
Aspect Ratio ◽  
Power Law ◽  
Viscous Incompressible Fluid ◽  
Ice Sheet ◽  
Plane Flow ◽  
The Third ◽  
Basal Sliding ◽  
Steady Plane ◽  
Third Dimension ◽  
Special Case

AbstractSteady plane flow under gravity of an axisymmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding is treated according to a power law between shear traction and velocity. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, and temperature variation through the ice sheet is neglected. Illustrations are presented for Glen’s power law (including the special case of a Newtonian fluid), and the polynomial law of Colbeck and Evans. The analysis follows that of Morland and Johnson (1980) where the analogous problem for an ice sheet deforming under plane flow was considered. Comparisons are made between the two models and it is found that the effect of the third dimension is to reduce (or leave unchanged) the aspect ratio for the cases considered, although no general formula can be obtained. This reduction is seen to depend on both the surface accumulation and the sliding law.

scholarly journals The Steady Profile of an Axisymmetric Ice Sheet

1981 ◽  
Vol 27 (95) ◽  
pp. 25-37 ◽  
Author(s):  
I. R. Johnson
Keyword(s):  
Aspect Ratio ◽  
Power Law ◽  
Viscous Incompressible Fluid ◽  
Ice Sheet ◽  
Plane Flow ◽  
The Third ◽  
Basal Sliding ◽  
Steady Plane ◽  
Third Dimension ◽  
Special Case

AbstractSteady plane flow under gravity of an axisymmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding is treated according to a power law between shear traction and velocity. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, and temperature variation through the ice sheet is neglected. Illustrations are presented for Glen’s power law (including the special case of a Newtonian fluid), and the polynomial law of Colbeck and Evans. The analysis follows that of Morland and Johnson (1980) where the analogous problem for an ice sheet deforming under plane flow was considered. Comparisons are made between the two models and it is found that the effect of the third dimension is to reduce (or leave unchanged) the aspect ratio for the cases considered, although no general formula can be obtained. This reduction is seen to depend on both the surface accumulation and the sliding law.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramterνoccurs, but theν= 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factorε(ν) is required, so that the aspect ratioεof a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion inεgives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

scholarly journals Computational Ice-Divide Analysis of a Cold Plane Ice Sheet Under Steady Conditions

1989 ◽  
Vol 12 ◽  
pp. 170-177 ◽  
Author(s):  
F. Szidarovszky ◽  
K. Hutter ◽  
S. Yakowitz
Keyword(s):  
Ice Sheet ◽  
Reduced Model ◽  
Stress Distributions ◽  
Field Equations ◽  
Geothermal Heat ◽  
Local Flow ◽  
Plane Flow ◽  
Basal Sliding ◽  
Flow Features ◽  
Simplifying Assumptions

The dimensionless form of the field equations and boundary conditions governing plane flow of a grounded cold ice sheet emerge from balance statements of mass, momentum, and energy. They constitute an amended version of a reduced model of ice-sheet flow, due to Morland (1984) and Hutter (1983), and circumvent the restrictions imposed by the reduced model, namely the neglect of the longitudinal stretching effects. The amended version permits satisfaction of mass balance at the ice divide for arbitrary basal sliding conditions and gives a better reproduction of the local flow features. Under very mild simplifying assumptions, namely that horizontal thermal conduction can be ignored close to the divide, we present a numerical analysis of the ice divide which has second-order accuracy. This analysis permits determination of the temperature profile, velocity, and stress distributions in a symmetric ice divide, provided that the ice-divide height, the local behavior of the accumulation and surface-temperature functions, and the geothermal heat flow are prescribed.

scholarly journals Effects Of Bed Inclination And Topography On Steady Isothermal Ice Sheets

1982 ◽  
Vol 28 (98) ◽  
pp. 71-90 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Flow Problem ◽  
Ice Sheet ◽  
Lead Order ◽  
Plane Motion ◽  
Order Unity ◽  
Plane Flow ◽  
The Mean ◽  
Steady Plane ◽  
Bed Slope ◽  
Coordinate Scaling

AbstractThe small slope magnitude ε. or aspect ratio. of an ice sheet in steady plane motion under gravity user a horizontal plane bed, subject to balancing surface accumulation and ablation. and basal drainage. is determined by the aceunualation magnitude, maximum depth. and the viscous properties of the ice. Horizontal coordinate scaling by a factor ε allows. series expansions in ε for which the leading-order solution is valid ecenschee under some weak restrictions on the ice law and sliding faw. This procedure is now extended to the plane flow problem when the mean bed line is inclined at angle χ to the horizontal and the bed is not flat. The lead order problems for χ of order unity and χ of order t: are distinct. and both are treated, fiir an isothermal sheet. The present analysis is valid only for a maximum bed slope relative to the mean line of order ε or less. The amplitude of it bed profile is ill, wavelength that of’ the ice sheet may therefore be of the same site as the ice sheet depth which allows treatment of a typical isostatic bed shape.

scholarly journals Sea-level response to melting of Antarctic ice shelves on multi-centennial time scales with the fast Elementary Thermomechanical Ice Sheet model (f.ETISh v1.0)

2017 ◽  
Author(s):  
Frank Pattyn
Keyword(s):  
Sea Level Rise ◽  
Sea Level ◽  
Time Scales ◽  
Power Law ◽  
Coulomb Friction ◽  
Ice Sheet ◽  
Ice Shelf ◽  
Ice Shelves ◽  
Grounding Line ◽  
Basal Sliding

Abstract. The magnitude of the Antarctic ice sheet's contribution to global sea-level rise is dominated by the potential of its marine sectors to become unstable and collapse as a response to ocean (and atmospheric) forcing. This paper presents Antarctic sea-level response to sudden atmospheric and oceanic forcings on multi-centennial time scales with the newly developed fast Elementary Thermomechanical Ice Sheet (f.ETISh) model. The f.ETISh model is a vertically integrated hybrid ice sheet/ice shelf model with an approximate implementation of ice sheet thermomechanics, making the model two-dimensional. Its marine boundary is represented by two different flux conditions, coherent with power-law basal sliding and Coulomb basal friction. The model has been compared to a series of existing benchmarks. Modelled Antarctic ice sheet response to forcing is dominated by sub-ice shelf melt and the sensitivity is highly dependent on basal conditions at the grounding line. Coulomb friction in the grounding-line transition zone leads to significantly higher mass loss in both West and East Antarctica on centennial time scales, leading to 2 m sea level rise after 500 years for a moderate melt scenario of 20 m a−1 under freely-floating ice shelves, up to 6 m for a 50 m a−1 scenario. The higher sensitivity is attributed to higher driving stresses upstream from the grounding line. Removing the ice shelves altogether results in a disintegration of the West Antarctic ice sheet and (partially) marine basins in East Antarctica. After 500 years, this leads to a 4.5 m and a 12.2 m sea level rise for the power-law basal sliding and Coulomb friction conditions at the grounding line, respectively. The latter value agrees with simulations by DeConto and Pollard (2016) over a similar period (but with different forcing and including processes of hydro-fracturing and cliff failure). The chosen parametrizations make model results largely independent of spatial resolution, so that f.ETISh can potentially be integrated in large-scale Earth system models.

scholarly journals Sea-level response to melting of Antarctic ice shelves on multi-centennial timescales with the fast Elementary Thermomechanical Ice Sheet model (f.ETISh v1.0)

2017 ◽  
Vol 11 (4) ◽  
pp. 1851-1878 ◽  
Author(s):  
Frank Pattyn
Keyword(s):  
Sea Level Rise ◽  
Sea Level ◽  
Power Law ◽  
Coulomb Friction ◽  
Ice Sheet ◽  
East Antarctica ◽  
Ice Shelf ◽  
Ice Shelves ◽  
Grounding Line ◽  
Basal Sliding

Abstract. The magnitude of the Antarctic ice sheet's contribution to global sea-level rise is dominated by the potential of its marine sectors to become unstable and collapse as a response to ocean (and atmospheric) forcing. This paper presents Antarctic sea-level response to sudden atmospheric and oceanic forcings on multi-centennial timescales with the newly developed fast Elementary Thermomechanical Ice Sheet (f.ETISh) model. The f.ETISh model is a vertically integrated hybrid ice sheet–ice shelf model with vertically integrated thermomechanical coupling, making the model two-dimensional. Its marine boundary is represented by two different flux conditions, coherent with power-law basal sliding and Coulomb basal friction. The model has been compared to existing benchmarks. Modelled Antarctic ice sheet response to forcing is dominated by sub-ice shelf melt and the sensitivity is highly dependent on basal conditions at the grounding line. Coulomb friction in the grounding-line transition zone leads to significantly higher mass loss in both West and East Antarctica on centennial timescales, leading to 1.5 m sea-level rise after 500 years for a limited melt scenario of 10 m a−1 under freely floating ice shelves, up to 6 m for a 50 m a−1 scenario. The higher sensitivity is attributed to higher ice fluxes at the grounding line due to vanishing effective pressure. Removing the ice shelves altogether results in a disintegration of the West Antarctic ice sheet and (partially) marine basins in East Antarctica. After 500 years, this leads to a 5 m and a 16 m sea-level rise for the power-law basal sliding and Coulomb friction conditions at the grounding line, respectively. The latter value agrees with simulations by DeConto and Pollard (2016) over a similar period (but with different forcing and including processes of hydrofracturing and cliff failure). The chosen parametrizations make model results largely independent of spatial resolution so that f.ETISh can potentially be integrated in large-scale Earth system models.

scholarly journals Effects Of Bed Inclination And Topography On Steady Isothermal Ice Sheets

1982 ◽  
Vol 28 (98) ◽  
pp. 71-90 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Flow Problem ◽  
Ice Sheet ◽  
Lead Order ◽  
Plane Motion ◽  
Order Unity ◽  
Plane Flow ◽  
The Mean ◽  
Steady Plane ◽  
Bed Slope ◽  
Coordinate Scaling

AbstractThe small slope magnitude ε. or aspect ratio. of an ice sheet in steady plane motion under gravity user a horizontal plane bed, subject to balancing surface accumulation and ablation. and basal drainage. is determined by the aceunualation magnitude, maximum depth. and the viscous properties of the ice. Horizontal coordinate scaling by a factor ε allows. series expansions in ε for which the leading-order solution is valid ecenschee under some weak restrictions on the ice law and sliding faw. This procedure is now extended to the plane flow problem when the mean bed line is inclined at angle χ to the horizontal and the bed is not flat. The lead order problems for χ of order unity and χ of order t: are distinct. and both are treated, fiir an isothermal sheet. The present analysis is valid only for a maximum bed slope relative to the mean line of order ε or less. The amplitude of it bed profile is ill, wavelength that of’ the ice sheet may therefore be of the same site as the ice sheet depth, which allows treatment of a typical isostatic bed shape.

Computational Ice-Divide Analysis of a Cold Plane Ice Sheet Under Steady Conditions

1989 ◽  
Vol 12 ◽  
pp. 170-177 ◽  
Author(s):  
F. Szidarovszky ◽  
K. Hutter ◽  
S. Yakowitz
Keyword(s):  
Ice Sheet ◽  
Reduced Model ◽  
Stress Distributions ◽  
Field Equations ◽  
Geothermal Heat ◽  
Local Flow ◽  
Plane Flow ◽  
Basal Sliding ◽  
Flow Features ◽  
Simplifying Assumptions

The dimensionless form of the field equations and boundary conditions governing plane flow of a grounded cold ice sheet emerge from balance statements of mass, momentum, and energy. They constitute an amended version of a reduced model of ice-sheet flow, due to Morland (1984) and Hutter (1983), and circumvent the restrictions imposed by the reduced model, namely the neglect of the longitudinal stretching effects. The amended version permits satisfaction of mass balance at the ice divide for arbitrary basal sliding conditions and gives a better reproduction of the local flow features.Under very mild simplifying assumptions, namely that horizontal thermal conduction can be ignored close to the divide, we present a numerical analysis of the ice divide which has second-order accuracy. This analysis permits determination of the temperature profile, velocity, and stress distributions in a symmetric ice divide, provided that the ice-divide height, the local behavior of the accumulation and surface-temperature functions, and the geothermal heat flow are prescribed.

Sign in / Sign up

Export Citation Format

Share Document