Isothermal Ice Research Articles

Impact of thermomechanical ice sheet coupling on a model of the 100 kyr ice age cycle

Simulations of the most recent 100 kyr glaciation‐deglaciation cycle of the late Pleistocene Ice Age are presented which feature a newly constructed thermomechanically coupled three‐dimensional ice sheet model which is itself coupled to a previously employed global energy balance climate model with realistic geography. The model incorporates both orbital insolation forcing due to the slow time evolution of orbital geometry (arising from many body effects in the solar system) as well as the forcing due to varying atmospheric concentrations of greenhouse gases. Simulations of the Greenland ice sheet are presented with which we are able to investigate the extent to which the ice flow law employed in the ice dynamics component of the model is constrained. The good agreement that we are able to achieve between the model‐generated ice sheet topography and the observed Greenland topography provides a clear demonstration of the quality of the ice dynamical model we have developed. The incorporation of full thermomechanical coupling using the standard Glen flow law in the ice dynamics component of the model is shown to increase the difficulty of achieving complete termination of the 100 kyr cycle that the model delivers. Furthermore, a 20 fold flow parameter enhancement relative to that used for Greenland is required to match the aspect ratio of the ICE‐4G reconstruction [Peltier, 1994]. Analyses presented herein suggest that basal processes are unlikely to account for this need for flow parameter retuning. However, the new thermomechanically coupled model now provides a clear separation of the Cordilleran and Hudson Bay domes at Last Glacial Maximum in contradistinction to analyses previously performed with isothermal ice sheet models. This innovation therefore leads to a major improvement of the generated ice sheet topography in relation to geological inferences. Aside from this important difference, the overall results regarding the ability of the model to fully account for the most recent 100 kyr cycle of glaciation and deglaciation without the necessity of introducing additional ad hoc feedbacks confirms the validity of this conclusion, reached previously on the basis of isothermal model integrations.

The shallow ice approximation for anisotropic ice: Formulation and limits

Ice sheet flow modeling is generally based on the shallow ice approximation (SIA) developed for isotropic ice. We extend this approximation to anisotropic ice and check its validity for both the isotropic and anisotropic cases by comparing the results of up to second‐order SIA with those obtained from a model solving the full set of mechanical equations. The theory is developed for a constitutive relation describing Newtonian and non‐Newtonian behavior, but numerical results are shown only for a Newtonian behavior. The results are compared for plane flow of isothermal ice under steady state conditions. SIA gives an excellent representation of the Newtonian flow of an isotropic ice sheet and a good description for the anisotropic case. The zero‐order approximation is sufficient to describe ice flow precisely over flat bedrock, for the anisotropic and isotropic cases, even close to an ice divide. For uneven bedrock, the second‐order SIA gives excellent results for isotropic ice and acceptable results for anisotropic ice. Finally, we quantify the error introduced by using an enhancement factor to represent the anisotropy when longitudinal stresses are taken into account.

Isothermal flow of an anisotropic ice sheet in the vicinity of an ice divide

Simulations of glacier flow are commonly based on the assumption that ice has an isotropic viscosity. Here we examine the plane flow of ice in the special region of an ice divide using a constitutive relation for an anisotropic, incompressible viscous body that is orthotropic and transversally isotropic. Ice is assumed to be isotropic at the ice sheet surface, with the continuous development of a vertical single maximum c axis fabric with increasing depth. We consider the theoretical case of an isothermal ice sheet over a horizontal bedrock, with no slip at the ice‐bedrock interface. The ice sheet surface elevation is imposed, and the flow corresponding to the steady state is calculated, using a two‐dimensional finite difference model based on the resolution of a pressure‐Poisson equation. In this model, all components of the stress and strain rate tensor are calculated. The main conclusion is that for a fixed surface elevation, the general flow pattern accelerates when the anisotropic behavior of the ice is taken into account due to the greater fluidity with respect to shear stress. The downward motion of the ice is faster, despite a higher resistance to vertical deformation. As a result, the dominance of shear strain rate in the flow of polar ice is stronger in the anisotropic case than in the isotropic case. The shear stresses are slightly relaxed, while the longitudinal stresses are significantly increased in the anisotropic case.

Comparison of isothermal and non-isothermal TTT-curves in the system 2,4-pentanediol in water

The system 2,4-pentanediol-water is shown to form glasses for concentrations above 30% by weight for cooling rates above 80°C min −1. Below this concentration, ice crystallizes during the initial cooling. Ice crystallization is calorimetrically recorded in isothermal and non-isothermal conditions during either the initial cooling from the melt or the subsequent warming from the glassy state for glass-forming concentrations. Time-Temperature-Transformation (TTT) curves for a 0.1 crystallization fraction have been constructed. For 35% w/w 2,4-pentanediol in water, two “noses” are recorded for isothermal conditions. They are identified with the thermal ranges as being the consequences of homogeneous and heterogeneous ice nucleation. A comparison of isothermal and non-isothermal conditions is also presented to underline the difficulty of applying the usual “nose method” for the determination of the critical cooling and warming rates. The Johnson-Mehl-Avrami model is used for the analysis of the isothermal ice crystallization. comparison of the parameters calculated after cooling or after warming from the glassy state shows that the homogeneous nucleation occurs at a constant rate and that the heterogeneous nucleation occurs as approximately the inverse square root of the time rate.

Potential For Basal Melting Under The Summit Coring Site, Central Greenland

The climatic record from Greenland boreholes is likely to extend well beyond the last interglacial only if the basal ice near the drilling sites has never reached its pressure melting point (−2°C). A simplified one-dimensional analysis (Paterson and Waddington, 1986) suggested that this would be true at Crête, Greenland, if the geothermal flux was less than 48 mW m−2. In that study, the vertical velocity pattern for an isothermal ice sheet was used. We have repeated the Crête calculations using the vertical velocity pattern derived by a finite element analysis. Using this temperature-dependent velocity pattern lowered the basal temperature by about 3°C.We have carried out a similar analysis for the Summit coring site further north on the Greenland ice divide. Here we find that the basal ice does not melt if the geothermal flux is less than 54 mW m−2, using the same mass balance and surface temperature histories as the previous study. We are repeating these one-dimensional calculations with more recently compiled histories and plan to present results from a full two-dimensional temperature model that includes processes only parameterized in the one-dimensional models. Using two dimensions, we will more realistically incorporate the special ice-flow patterns found at divides (e.g. Raymond, 1983; Dahl-Jensen, 1989). In steady-state flow models these patterns lead to significant horizontal temperature gradients and a “hot spot” beneath an ice divide (Paterson and Waddington, 1986). In addition, we will more accurately determine the transient effects on basal temperature resulting from the interaction of these flow patterns and the changing climate. Our discussion will include sensitivity to geothermal heat flux, ice thickness and paleoenvironmental history.

Potential For Basal Melting Under The Summit Coring Site, Central Greenland

The climatic record from Greenland boreholes is likely to extend well beyond the last interglacial only if the basal ice near the drilling sites has never reached its pressure melting point (−2°C). A simplified one-dimensional analysis (Paterson and Waddington, 1986) suggested that this would be true at Crête, Greenland, if the geothermal flux was less than 48 mW m−2. In that study, the vertical velocity pattern for an isothermal ice sheet was used. We have repeated the Crête calculations using the vertical velocity pattern derived by a finite element analysis. Using this temperature-dependent velocity pattern lowered the basal temperature by about 3°C. We have carried out a similar analysis for the Summit coring site further north on the Greenland ice divide. Here we find that the basal ice does not melt if the geothermal flux is less than 54 mW m−2, using the same mass balance and surface temperature histories as the previous study. We are repeating these one-dimensional calculations with more recently compiled histories and plan to present results from a full two-dimensional temperature model that includes processes only parameterized in the one-dimensional models. Using two dimensions, we will more realistically incorporate the special ice-flow patterns found at divides (e.g. Raymond, 1983; Dahl-Jensen, 1989). In steady-state flow models these patterns lead to significant horizontal temperature gradients and a “hot spot” beneath an ice divide (Paterson and Waddington, 1986). In addition, we will more accurately determine the transient effects on basal temperature resulting from the interaction of these flow patterns and the changing climate. Our discussion will include sensitivity to geothermal heat flux, ice thickness and paleoenvironmental history.

A Flow-line Model for Calculating the Surface Profile and the Velocity, Strain-rate, and Stress Fields in an Ice Sheet

Abstract A flow-line model is presented for calculating the surface profile and the velocity, strain-rate, and stress fields in an ice sheet with given base-elevation profile, ice thickness at the dome (divide), flow-law parameters, mass-balance distribution, and convergence/divergence conditions along the flow line. The model, which is based on a “quasi-similarity” hypothesis as regards the horizontal velocity-depth profiles, accounts for changes along the flow line in the depth distributions of temperature, normal stress deviators, and possible enhanced flow of deep ice of Wisconsin origin. A curvilinear coordinate system is applied with horizontal axes along flow lines and surface-elevation contours, respectively. The flow equations are reduced to two differential equations, one for the surface-elevation profile, and the other for a profile function that determines the depth distributions of velocities and strain-rates. The two equations are coupled through a profile parameter that communicates the influence of velocity-profile changes to the surface-profile equation. It is shown that the variation along the flow line of this parameter should also be considered when deriving flow-law parameters from ice-sheet flow-line data. For a symmetric dome, explicit expressions are derived for the depth distributions of the vertical velocity, strain-rates, and stresses. The strain-rate profiles display an inflection about half-way down the ice sheet, and, in the case of isothermal ice, have surface values 2.2 times their depth-averaged values. The depth distribution of the vertical velocity indicates that a relatively thick layer of almost stagnant ice is present at the ice-sheet base below a dome.

A Flow-line Model for Calculating the Surface Profile and the Velocity, Strain-rate, and Stress Fields in an Ice Sheet

AbstractA flow-line model is presented for calculating the surface profile and the velocity, strain-rate, and stress fields in an ice sheet with given base-elevation profile, ice thickness at the dome (divide), flow-law parameters, mass-balance distribution, and convergence/divergence conditions along the flow line. The model, which is based on a “quasi-similarity” hypothesis as regards the horizontal velocity-depth profiles, accounts for changes along the flow line in the depth distributions of temperature, normal stress deviators, and possible enhanced flow of deep ice of Wisconsin origin. A curvilinear coordinate system is applied with horizontal axes along flow lines and surface-elevation contours, respectively. The flow equations are reduced to two differential equations, one for the surface-elevation profile, and the other for a profile function that determines the depth distributions of velocities and strain-rates. The two equations are coupled through a profile parameter that communicates the influence of velocity-profile changes to the surface-profile equation. It is shown that the variation along the flow line of this parameter should also be considered when deriving flow-law parameters from ice-sheet flow-line data. For a symmetric dome, explicit expressions are derived for the depth distributions of the vertical velocity, strain-rates, and stresses. The strain-rate profiles display an inflection about half-way down the ice sheet, and, in the case of isothermal ice, have surface values 2.2 times their depth-averaged values. The depth distribution of the vertical velocity indicates that a relatively thick layer of almost stagnant ice is present at the ice-sheet base below a dome.