A Numerical Study of Plane Ice-Sheet Flow

AbstractThe plane steady flow of a grounded ice sheet is numerically analysed using the approximate model of Morland or Hutter. In this, the ice behaves as a non-linear viscous fluid with a strongly temperature-dependent rate factor, and ice sheets are assumed to be long and shallow. The climate is assumed to be prescribed via the accumulation/ablation distribution and the surface temperature, both of which are functions of position and unknown height. The rigid base exerts external forcings via the normal heat flow, the geothermal heat, and a given basal sliding condition connecting the tangential velocity, tangential traction, and normal traction. The functional relations are those of Morland (1984) or motivated by his work. We use equations in his notation.The governing equations and boundary conditions in dimensionless form are briefly stated and dimensionless variables are related to their physical counterparts. The thermo-mechanical parabolic boundary-value problem, found to depend on physical scales, constitutive properties, and external forcing functions, has been numerically solved. For reasons of stability, the numerical integration must proceed from the ice divide towards the margin, which requires a special analysis of the ice divide. We present this analysis and then describe the versatility and limitations of the constructed computer code.Results of extensive computations are shown. In particular, we prove that the Morland–Hutter model for ice sheets is only applicable when sliding is sufficiently large (satisfying inequality (30)). In the range of the validity of this inequality, it is then demonstrated that of all physical scaling parameters only a single π-product influences the geometry and the flow within the ice sheet. We analyse the role played by advection, diffusion, and dissipation in the temperature distribution, and discuss the significance of the rheological non-linearities. Variations of the external forcings, such as accumulation/ablation conditions, free surface temperature, and geothermal heat, demonstrate the sensitivity of the ice-sheet geometry to accumulation conditions and the robustness of the flow to variations in the thermal state. We end with a summary of results and a critical review of the model.

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The effect of geothermal heat flux on the deglacial evolution of the Greenland ice sheet

10.5194/egusphere-egu21-13647 ◽

2021 ◽

Author(s):

Parviz Ajourlou ◽

François PH Lapointe ◽

Glenn A Milne ◽

Yasmina Martos

Keyword(s):

Boundary Condition ◽

Heat Flux ◽

Thermal State ◽

Ice Sheet ◽

Greenland Ice Sheet ◽

Sampling Procedure ◽

Geophysical Research ◽

Geothermal Heat ◽

Input Field ◽

Uncertainty Estimates

<p>Geothermal heat flux (GHF) is known to be an important control on the basal thermal state of an ice sheet which, in turn, is a key factor in governing how the ice sheet will evolve in response to a given climate forcing. In recent years, several studies have estimated GHF beneath the Greenland ice sheet using different approaches (e.g. Rezvanbehbahani et al., Geophysical Research Letters, 2017; Martos et al., Geophysical Research Letters, 2018; Greve, Polar Data Journal, 2019). Comparing these different estimates indicates poor agreement and thus large uncertainty in our knowledge of this important boundary condition for modelling the ice sheet. The primary aim of this study is to quantify the influence of this uncertainty on modelling the past evolution of the ice sheet with a focus on the most recent deglaciation. We build on past work that considered three GHF models (Rogozhina et al., 2011) by considering over 100 different realizations of this input field. We use the uncertainty estimates from Martos et al. (Geophysical Research Letters, 2018) to generate GHF realisations via a statistical sampling procedure. A sensitivity analysis using these realisations and the Parallel Ice Sheet Model (PISM, Bueler and Brown, Journal of Geophysical Research, 2009) indicates that uncertainty in GHF has a dramatic impact on both the volume and spatial distribution of ice since the last glacial maximum, indicating that more precise constraints on this boundary condition are required to improve our understanding of past ice sheet evolution and, consequently, reduce uncertainty in future projections.</p>

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Last Interglacial climate and sea-level evolution from a coupled ice sheet–climate model

Climate of the Past ◽

10.5194/cp-12-2195-2016 ◽

2016 ◽

Vol 12 (12) ◽

pp. 2195-2213 ◽

Cited By ~ 22

Author(s):

Heiko Goelzer ◽

Philippe Huybrechts ◽

Marie-France Loutre ◽

Thierry Fichefet

Keyword(s):

Surface Temperature ◽

Sea Level ◽

Ice Sheets ◽

Ice Sheet ◽

Greenland Ice Sheet ◽

Last Interglacial ◽

A Minor ◽

The Antarctic ◽

The Impact ◽

And Sea Level

Abstract. As the most recent warm period in Earth's history with a sea-level stand higher than present, the Last Interglacial (LIG, ∼  130 to 115 kyr BP) is often considered a prime example to study the impact of a warmer climate on the two polar ice sheets remaining today. Here we simulate the Last Interglacial climate, ice sheet, and sea-level evolution with the Earth system model of intermediate complexity LOVECLIM v.1.3, which includes dynamic and fully coupled components representing the atmosphere, the ocean and sea ice, the terrestrial biosphere, and the Greenland and Antarctic ice sheets. In this setup, sea-level evolution and climate–ice sheet interactions are modelled in a consistent framework.Surface mass balance change governed by changes in surface meltwater runoff is the dominant forcing for the Greenland ice sheet, which shows a peak sea-level contribution of 1.4 m at 123 kyr BP in the reference experiment. Our results indicate that ice sheet–climate feedbacks play an important role to amplify climate and sea-level changes in the Northern Hemisphere. The sensitivity of the Greenland ice sheet to surface temperature changes considerably increases when interactive albedo changes are considered. Southern Hemisphere polar and sub-polar ocean warming is limited throughout the Last Interglacial, and surface and sub-shelf melting exerts only a minor control on the Antarctic sea-level contribution with a peak of 4.4 m at 125 kyr BP. Retreat of the Antarctic ice sheet at the onset of the LIG is mainly forced by rising sea level and to a lesser extent by reduced ice shelf viscosity as the surface temperature increases. Global sea level shows a peak of 5.3 m at 124.5 kyr BP, which includes a minor contribution of 0.35 m from oceanic thermal expansion. Neither the individual contributions nor the total modelled sea-level stand show fast multi-millennial timescale variations as indicated by some reconstructions.

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Modes of Operation of Thermo-Mechanically Coupled Ice Sheets

Annals of Glaciology ◽

10.3189/s0260305500006960 ◽

1989 ◽

Vol 12 ◽

pp. 57-69 ◽

Cited By ~ 9

Author(s):

Richard C.A. Hindmarsh ◽

Geoffrey S. Boulton ◽

Kolumban Hutter

Keyword(s):

Temperature Field ◽

Time Scale ◽

Ice Sheets ◽

Ice Sheet ◽

Thermal Inertia ◽

Coupled System ◽

Thermal Processes ◽

Temperature Dependent ◽

Geothermal Heat ◽

Diffusive Time

A dimensionless model of thermo-mechanically coupled ice sheets is used to analyse the operation of the system. Three thermal processes are identified: (i) dissipation, having a maximum time-scale of thousands of years; (ii) advection, having a time-scale of tens of thousands of years; and (iii) conduction, having a time-scale of 100000 years. Kinematical processes occur on two time-scales: (i) a marginal advective time-scale of thousands of years; and (ii) a diffusive time-scale of tens of thousands of years dominant in the divide area.The coupling with the temperature field in the bed produces fluctuations to the depth of a few kilometres, which means that horizontal conduction in the bed can be ignored except perhaps in the marginal area. The thermal inertia of the bed could produce significant fluctuations in the geothermal heat gradient.The operation of the thermo-mechanically coupled system is explored with a time-dependent thermo-mechanically coupled numerical algorithm. Dependence of the basal friction on temperature is introduced heuristically, and an enthalpy method is used to represent the effect of latent heat. The marginal area is shown to be dissipation-driven, and always reaches melting point. The divide area can show two modes of behaviour: a warm-based mode where the ice sheet is thin, and a cold-based mode where the ice sheet is thick. Which mode operates depends upon the applied temperature field and the amount of heat conducted from the bed.Calculations where sliding is limited were not found to be possible owing to problems with the reduced model which resulted in a violation of the approximation conditions at the margin. Cases which did work required a substantial sliding component; as a result, a significant coupling between geometry and temperature can only be demonstrated when sliding is made temperature-dependent.

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The Dynamics of the Antarctic Ice Sheet

Annals of Glaciology ◽

10.3189/s026030550000690x ◽

1989 ◽

Vol 12 ◽

pp. 16-22 ◽

Cited By ~ 5

Author(s):

W.F. Budd ◽

D. Jenssen

Keyword(s):

Heat Flux ◽

Surface Temperature ◽

Accumulation Rate ◽

Three Dimensional ◽

Ice Sheet ◽

Surface Velocity ◽

Geothermal Heat ◽

Antarctic Ice Sheet ◽

The Past ◽

The Antarctic

A three-dimensional dynamic, thermodynamic ice-sheet model has been developed to simulate the past, present, and future behaviour of the Antarctic ice sheet. The present ice velocities depend on the deep ice temperatures which in turn depend on the past changes of the ice sheet, including surface temperature, accumulation rate, and ice thickness. The basal temperatures are also strongly dependent on the geothermal heat flux. The model has therefore been used to study the effect on the basal temperatures, of changes to the geothermal heat flux, as well as the past changes of surface temperature and accumulation rate based on results obtained from the Vostok deep ice core. The model is also used to compute the distribution of surface velocity required to balance the present accumulation rate and the dynamics velocity based on the stress, temperature, and flow properties of ice, for the internal deformation, plus a component due to ice sliding. These velocities are compared to observed surface velocities in East Antarctica to assess the state of balance and the performance of the dynamics formulation.

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Thermomechanical modelling of Northern Hemisphere ice sheets with a two-level mass-balance parameterization

Annals of Glaciology ◽

10.1017/s0260305500015688 ◽

1995 ◽

Vol 21 ◽

pp. 111-116 ◽

Cited By ~ 59

Author(s):

Philippe Huybrechts ◽

Stephen T’ Siobbel

Keyword(s):

Surface Temperature ◽

Mass Balance ◽

Northern Hemisphere ◽

Three Dimensional ◽

Ice Sheets ◽

Ice Sheet ◽

Snow Accumulation ◽

Summer Temperature ◽

Balance Model ◽

Geological Evidence

A three-dimensional time-dependent thermomechanical ice-sheet model was used together with a two-level (snow-accumulation/runoff) mass-balance model to investigate the Quaternary ice sheets of the Northern Hemisphere. The model freely generates the ice-sheet geometry in response to specified changes in surface temperature and mass balance, and includes bedrock adjustment, basal sliding and a full temperature calculation within the ice. The mass-balance parameterization makes a distinction between snowfall and melting. Yearly snowfall rates depend on the present precipitation distribution, and are varied proportionally to changes in surface temperature and the moisture content of the air. The ablation model is based on the positive-degree-day method, and distinguishes between ice and snow melting. This paper discusses steady-slate characteristics, conditions for growth and retreat, and response time-scales of ice sheets as a function of a prescribed lowering of summer temperature. Most notably, the modelled extents of the Eurasian ice sheet for a summer temperature lowering of 6–7 K and of the Laurentide ice sheet for a cooling of 9–10 K are in reasonable agreement with most reconstructions based on geological evidence, except for the presence of a large ice sheet stretching from Alaska across the Bering Strait to most of eastern Siberia. In addition, wet basal conditions turned out to be always confined to the margin, whereas central areas in these reconstructions remained always cold-based. This is of relevance for processes involving reduced basal traction.

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The ice-rock interface for the climap 18000 years b.p. and 120000 years b.p. experiments

Journal of Glaciology ◽

10.1017/s0022143000030185 ◽

1979 ◽

Vol 23 (89) ◽

pp. 425-428

Author(s):

T. J. Hughes

Keyword(s):

Water Distribution ◽

Ice Sheets ◽

Ice Sheet ◽

Thermal Conditions ◽

Constant Thickness ◽

Shear Stresses ◽

Glacial Erosion ◽

Geothermal Heat ◽

Rock Interface ◽

Basal Shear

Abstract Most numerical models of present ice-sheet dynamics predict basal thermal conditions for an assumed geothermal heat flux and measured ice thickness, surface temperature, and snow precipitation. These models are not ideally suited for reconstructing former ice sheets because what is known for present ice sheets is unknown for former ones, and vice versa. In particular, geothermal heat fluxes are immeasurable at an ice-sheet bed but can be measured after the ice sheet is gone, and the thermal conditions predicted at an ice-sheet bed can be inferred from the glacial-geological–topographic record after the ice sheet is gone. The Maine CLIMAP ice-sheet reconstruction model uses these inferred basal thermal conditions to compute ice thicknesses from basal shear stresses. Basal shear stress is assumed to reflect the degree of ice–bed coupling which, in turn, is assumed to reflect the amount and distribution of basal water under the ice sheet. Under the ice-sheet interior, basal water exists in a thin film of constant thickness covering the low places on the bed. This film expands for a melting bed and contracts for a freezing bed. Along the ice-sheet margin, basal water exists in narrow channels of varying thickness corresponding to troughs on the bed. These water channels become deeper for a melting bed and shallower for a freezing bed. In areas covered by the Laurentide and Scandinavian ice sheets, myriads of interconnected lakes in regions of greatest postglacial rebound are interpreted as evidence suggesting the interior basal water distribution, whereas eskers pointed toward terminal moraines and troughs across continental shelves are interpreted as evidence suggesting the basal water distribution toward the margins. Continental-shelf troughs were assumed to correspond to former ice streams, by analogy with observations in Greenland and Antarctica. Three modes of glacial erosion are considered to be responsible for the lakes, eskers, troughs, and associated topography. Quarrying is by a freeze-thaw mechanism which occurs where the melting-point isotherm intersects bedrock, so it is important only for freezing or melting beds because high places on the bed are frozen, low places are melted, and minor basal temperature fluctuations shift the isotherm separating them. Crushing results when rocks at the ice-bed interface are ground against each other and the bed by glacial sliding, so it occurs where the bed is melted and is most important when the entire bed is melted. Abrasion of bedrock occurs when rock cutting tools imbedded in the ice at the ice–rock interface are moved across the interface by glacial sliding, so it is also most important when the entire bed is melted. If basal melting continued after the entire bed is melted, abrasion-rates drop because the basal water layer thickens and drowns bedrock projections otherwise subjected to abrasion. Basal freezing reduces both crushing and abrasion-rates by coating quarried rocks with a sheath of relatively soft ice and transporting them upward from the ice–rock interface. An initially flat subglacial topography will develop depressions where glacial erosion is greatest and deposition is least, and ridges where the opposite conditions prevail. We interpret the central depressions represented today by Hudson Bay and the Gulf of Bothnia as caused by erosion on a melting bed under the Laurentide and Scandinavian ice sheets, respectively. The arc of lakes, gulfs, and shallow seas surrounding these depressions are interpreted as resulting from a freezing bed under the former ice sheets. The present watershed separating the depressions from the arcs marks the approximate former basal equilibrium line where the bed was melted. The Canadian and Baltic continental shields beyond these arcs are blanketed by material eroded from within the arcs, and represent areas having a frozen bed where evidence for abrasion is missing and a second zone having a melting bed where evidence for abrasion is present. This basic pattern was assumed to be imprinted on the bed during the steady-state period of maximum ice-sheet extent, and maintained in varying degrees during growth and shrinkage of these ice sheets.

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The ice-rock interface for the climap 18000 years b.p. and 120000 years b.p. experiments

Journal of Glaciology ◽

10.3189/s0022143000030185 ◽

1979 ◽

Vol 23 (89) ◽

pp. 425-428

Author(s):

T. J. Hughes

Keyword(s):

Water Distribution ◽

Ice Sheets ◽

Ice Sheet ◽

Thermal Conditions ◽

Constant Thickness ◽

Shear Stresses ◽

Glacial Erosion ◽

Geothermal Heat ◽

Rock Interface ◽

Basal Shear

AbstractMost numerical models of present ice-sheet dynamics predict basal thermal conditions for an assumed geothermal heat flux and measured ice thickness, surface temperature, and snow precipitation. These models are not ideally suited for reconstructing former ice sheets because what is known for present ice sheets is unknown for former ones, and vice versa. In particular, geothermal heat fluxes are immeasurable at an ice-sheet bed but can be measured after the ice sheet is gone, and the thermal conditions predicted at an ice-sheet bed can be inferred from the glacial-geological–topographic record after the ice sheet is gone. The Maine CLIMAP ice-sheet reconstruction model uses these inferred basal thermal conditions to compute ice thicknesses from basal shear stresses.Basal shear stress is assumed to reflect the degree of ice–bed coupling which, in turn, is assumed to reflect the amount and distribution of basal water under the ice sheet. Under the ice-sheet interior, basal water exists in a thin film of constant thickness covering the low places on the bed. This film expands for a melting bed and contracts for a freezing bed. Along the ice-sheet margin, basal water exists in narrow channels of varying thickness corresponding to troughs on the bed. These water channels become deeper for a melting bed and shallower for a freezing bed. In areas covered by the Laurentide and Scandinavian ice sheets, myriads of interconnected lakes in regions of greatest postglacial rebound are interpreted as evidence suggesting the interior basal water distribution, whereas eskers pointed toward terminal moraines and troughs across continental shelves are interpreted as evidence suggesting the basal water distribution toward the margins. Continental-shelf troughs were assumed to correspond to former ice streams, by analogy with observations in Greenland and Antarctica.Three modes of glacial erosion are considered to be responsible for the lakes, eskers, troughs, and associated topography. Quarrying is by a freeze-thaw mechanism which occurs where the melting-point isotherm intersects bedrock, so it is important only for freezing or melting beds because high places on the bed are frozen, low places are melted, and minor basal temperature fluctuations shift the isotherm separating them. Crushing results when rocks at the ice-bed interface are ground against each other and the bed by glacial sliding, so it occurs where the bed is melted and is most important when the entire bed is melted. Abrasion of bedrock occurs when rock cutting tools imbedded in the ice at the ice–rock interface are moved across the interface by glacial sliding, so it is also most important when the entire bed is melted. If basal melting continued after the entire bed is melted, abrasion-rates drop because the basal water layer thickens and drowns bedrock projections otherwise subjected to abrasion. Basal freezing reduces both crushing and abrasion-rates by coating quarried rocks with a sheath of relatively soft ice and transporting them upward from the ice–rock interface.An initially flat subglacial topography will develop depressions where glacial erosion is greatest and deposition is least, and ridges where the opposite conditions prevail. We interpret the central depressions represented today by Hudson Bay and the Gulf of Bothnia as caused by erosion on a melting bed under the Laurentide and Scandinavian ice sheets, respectively. The arc of lakes, gulfs, and shallow seas surrounding these depressions are interpreted as resulting from a freezing bed under the former ice sheets. The present watershed separating the depressions from the arcs marks the approximate former basal equilibrium line where the bed was melted. The Canadian and Baltic continental shields beyond these arcs are blanketed by material eroded from within the arcs, and represent areas having a frozen bed where evidence for abrasion is missing and a second zone having a melting bed where evidence for abrasion is present. This basic pattern was assumed to be imprinted on the bed during the steady-state period of maximum ice-sheet extent, and maintained in varying degrees during growth and shrinkage of these ice sheets.

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The Dynamics of the Antarctic Ice Sheet

Annals of Glaciology ◽

10.1017/s026030550000690x ◽

1989 ◽

Vol 12 ◽

pp. 16-22 ◽

Cited By ~ 24

Author(s):

W.F. Budd ◽

D. Jenssen

Keyword(s):

Heat Flux ◽

Surface Temperature ◽

Accumulation Rate ◽

Three Dimensional ◽

Ice Sheet ◽

Surface Velocity ◽

Geothermal Heat ◽

Antarctic Ice Sheet ◽

The Past ◽

The Antarctic

A three-dimensional dynamic, thermodynamic ice-sheet model has been developed to simulate the past, present, and future behaviour of the Antarctic ice sheet. The present ice velocities depend on the deep ice temperatures which in turn depend on the past changes of the ice sheet, including surface temperature, accumulation rate, and ice thickness. The basal temperatures are also strongly dependent on the geothermal heat flux. The model has therefore been used to study the effect on the basal temperatures, of changes to the geothermal heat flux, as well as the past changes of surface temperature and accumulation rate based on results obtained from the Vostok deep ice core. The model is also used to compute the distribution of surface velocity required to balance the present accumulation rate and the dynamics velocity based on the stress, temperature, and flow properties of ice, for the internal deformation, plus a component due to ice sliding. These velocities are compared to observed surface velocities in East Antarctica to assess the state of balance and the performance of the dynamics formulation.

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Modes of Operation of Thermo-Mechanically Coupled Ice Sheets

Annals of Glaciology ◽

10.1017/s0260305500006960 ◽

1989 ◽

Vol 12 ◽

pp. 57-69 ◽

Cited By ~ 50

Author(s):

Richard C.A. Hindmarsh ◽

Geoffrey S. Boulton ◽

Kolumban Hutter

Keyword(s):

Temperature Field ◽

Time Scale ◽

Ice Sheets ◽

Ice Sheet ◽

Thermal Inertia ◽

Coupled System ◽

Thermal Processes ◽

Temperature Dependent ◽

Geothermal Heat ◽

Diffusive Time

A dimensionless model of thermo-mechanically coupled ice sheets is used to analyse the operation of the system. Three thermal processes are identified: (i) dissipation, having a maximum time-scale of thousands of years; (ii) advection, having a time-scale of tens of thousands of years; and (iii) conduction, having a time-scale of 100000 years. Kinematical processes occur on two time-scales: (i) a marginal advective time-scale of thousands of years; and (ii) a diffusive time-scale of tens of thousands of years dominant in the divide area. The coupling with the temperature field in the bed produces fluctuations to the depth of a few kilometres, which means that horizontal conduction in the bed can be ignored except perhaps in the marginal area. The thermal inertia of the bed could produce significant fluctuations in the geothermal heat gradient. The operation of the thermo-mechanically coupled system is explored with a time-dependent thermo-mechanically coupled numerical algorithm. Dependence of the basal friction on temperature is introduced heuristically, and an enthalpy method is used to represent the effect of latent heat. The marginal area is shown to be dissipation-driven, and always reaches melting point. The divide area can show two modes of behaviour: a warm-based mode where the ice sheet is thin, and a cold-based mode where the ice sheet is thick. Which mode operates depends upon the applied temperature field and the amount of heat conducted from the bed. Calculations where sliding is limited were not found to be possible owing to problems with the reduced model which resulted in a violation of the approximation conditions at the margin. Cases which did work required a substantial sliding component; as a result, a significant coupling between geometry and temperature can only be demonstrated when sliding is made temperature-dependent.

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