effects of spatial discretization in ice sheet modelling using the shallow ice approximation this paper assesses a two dimensional vertically integrated ice model for its numerical properties in the calculation of ice sheet evolution on a sloping bed using the shallow ice approxi mation we discuss the influence of initial conditions and individual model parameters on the model s numerical behaviour with emphasis on varying spatial discretizations the modelling results suffer badly from numerical problems they show a strong dependence on gridcell size and we conclude that the widely used gridcell spacing of 20 km is too coarse the numerical errors are small in each single time step but increase non linearly over time and with volume change as a result of feedback of the mass balance with height we propose a new method for the calculation of the surface gradient near the margin which improves the results significantly furthermore we show that we may use dimension analysis as a tool to explain in which situations numerical problems are to be expected

grounding line migration in an adaptive mesh ice sheet model 1 grounding line migration is a key process affecting the stability of marine ice sheets such as the west antarctic ice sheet recent studies have shown that ice sheet models employing a fixed spatial grid such as are commonly used for whole ice sheet simulations cannot be used to solve this problem in a robust manner we have developed a one dimensional vertically integrated shelfy stream ice sheet model that employs the adaptive mesh refinement amr technique to bring higher resolution to spatially and temporally evolving subregions of the model domain a higher order solver the piecewise parabolic method ppm is used to compute the thickness evolution both amr and ppm extend readily to greater than one dimension and could be used in full ice sheet simulations we demonstrate that this approach can bring improvements in terms of accuracy and consistency in both grounded ice sheet and ice stream ice shelf simulations given the appropriate choice of refinement criteria in particular we demonstrate that amr in conjunction with a parameterization for subgrid scale grounding line position can produce predictions of grounding line migration

irreversible mass loss of canadian arctic archipelago glaciers 1 the canadian arctic archipelago caa contains the largest volume of glacier ice on earth outside of antarctica and greenland in the absence of significant calving caa glacier mass balance is governed by the difference between surface snow accumulation and meltwater runoff surface mass balance here we use a coupled atmosphere snow model to simulate present day and 21st century caa glacier surface mass balance through comparison with gravity recovery and climate experiment mass anomalies and in situ observations we show that the model is capable of representing present day caa glacier mass loss as well as the dynamics of the seasonal snow cover on the caa tundra next we force this model until 2100 with a moderate climate warming scenario ar5 rcp4 5 we show that enhanced meltwater runoff from caa glaciers is not sufficiently compensated by increased snowfall extrapolation of these results toward an ar5 multimodel ensemble results in sustained 21st century caa glacier mass loss in the vast majority 99 of the 7000 temperature realizations

dynamical processes involved in the retreat of marine ice sheets marine ice sheets with mechanics described by the shallow ice approximation by definition do not couple mechanically with the shelf such ice sheets are known to have neutral equilibria we consider the implications of this for their dynamics and in particular for mechanisms which promote marine ice sheet retreat the removal of ice shelf buttressing leading to enhanced flow in grounded ice is discounted as a significant influence on mechanical grounds sea level rise leading to reduced effective pressures under ice streams is shown to be a feasible mechanism for producing postglacial west antarctic ice sheet retreat but is inconsistent with borehole evidence warming thins the ice sheet by reducing the average viscosity but does not lead to grounding line retreat internal oscillations either specified or generated via a macayeal payne thermal mechanism promote migration this is a noise induced drift phenomenon stemming from the neutral equilibrium property of marine ice sheets this migration occurs at quite slow rates but these are sufficiently large to have possibly played a role in the dynamics of the west antarctic ice sheet after the glacial maximum numerical experiments suggest that it is generally true that while significant changes in thickness can be caused by spatially uniform changes spatial variability coupled with dynamical variability is needed to cause margin movement

interaction and variability of ice streams under a triple valued sliding law and non newtonian rheology 1 ice streams are regions of fast flowing glacier ice that transport a significant portion of the total ice flux from present ice sheets the flow pattern of ice streams can vary both temporally and spatially in particular ice streams can become stagnant and change their path we study the dynamics of ice streams using an idealized model of an isothermal and power law viscous ice flow that includes horizontal lateral shear stresses the basal sliding law is assumed to be triple valued we investigate the spatiotemporal patterns formed because of the flow over a flat bed fed from an upstream mass source the ice flows from the mass source region through one or two gaps in a prescribed upstream topographic ridge which restricts the flow leading to the formation of one or two ice streams we find a relation between the parameters of the ice rheology and the width of the ice stream shear margins and show how these parameters can affect the minimum width of an ice stream we also find that complex asymmetric spatiotemporal patterns can result from the interaction of two ice streams sharing a common mass source the rich spatiotemporal variability is found to mostly be a result of the triple valued sliding law but non newtonian effects are found to play a significant role in setting a more realistic shear margin width and allowing for relevant time scales of the variability

investigating the stability of subglacial lakes with a full stokes ice sheet model despite the large amount of subglacial lakes present underneath the east antarctic ice sheet and the melt processes involved the hydrology beneath the ice sheet is poorly understood changes in subglacial potential gradients may lead to subglacial lake outbursts discharging excess water through a subglacial drainage system underneath the ice sheet such processes can eventually lead to an increase in ice flow in this paper a full stokes numerical ice sheet model was employed which takes into account the ice flow over subglacial water bodies in hydrostatic equilibrium with the overlying ice sensitivity experiments were carried out for small perturbations in ice flow and basal melt rate as a function of ice thickness general surface slope ice viscosity and lake size in order to investigate their influence on the subglacial potential gradient and the impact on subglacial lake drainage experiments clearly demonstrate that small changes in surface slope are sufficient to start and sustain episodic subglacial drainage events lake drainage can therefore be regarded as a common feature of the subglacial hydrological system and may influence to a large extent the present and future behavior of large ice sheets

results from the eismint model intercomparison the effects of thermomechanical coupling this paper discusses results from the second phase of the european ice sheet modelling initiative eismint it reports the intercompartison of ten operational ice sheet models and uses a series of experiments to examine the implications of thermomechanical coupling for model behaviour a schematic circular ice sheet is used in the work which investigates both steady states and the response to stepped changes in climate the major finding is that radial symmetry implied in the experimental design can under certain circumstances break down with the formation of distinct regularly spaced spokes of cold ice which extended from the interior of the ice sheet outward to the surrounding zone of basal melt these features also manifest themselves in the thickness and velocity distributions predicted by the models they appear to be a common feature to all of the models which took part in the intercomparison and may stem from interactions between ice temperature flow and surface form the exact nature of these features varies between models and their existence appears to be controlled by the overall thermal regimne of the ice sheet a second result is that there is considerable agreement between the models in their predictions of global scale response to imposed climate change

transient glacier response with a higher order numerical ice flow model in this paper a higher order numerical flowline model is presented which is numerically stable and fast and can cope with very small horizontal grid sizes 10 m the model is compared with the results from blatter and others 1998 on haut glacier d arolla switzerland and with the european ice sheet modelling initiative benchmarks huybrechts and others 1996 results demonstrate that the significant difference between calculated basal drag and driving stress profiles in a fixed geometry disappears when the glacier profile is allowed to react to the surface mass balance conditions and reaches a steady state dynamic experiments show that the mass transfer in higher order models occurs at a different speed in the accumulation and ablation areas and that the front position is more sensitive to migration compared to the shallow ice approximation

on estimating length fluctuations of glaciers caused by changes in climatic forcing the reaction of alpine glaciers to shifts in the equilibrium line altitude ela is calculated by using a two dimensional numerical model to solve the full equations for the velocity and stress fields full system model in the absence of basal motion rates of advance and retreat of the snout of typically sized alpine glaciers are found to be insensitive to the details of the flow at the snout even when the glaciers are far from steady state a comparison of results obtained with a full system model and a shallow ice approximation sia model yields no significant differences in advance or retreat rates this assumption has been implicitly made in numerous previous climatic studies and is here shown to be well justified using a realistic mass balance altitude feedback only slight model dependent changes in steady state lengths are found the relative importance of mass balance and glacier dynamics for the transient response of alpine glaciers to changes in the ela is given a precise meaning by determining the model dependent additional shifts in ela needed for the sia and the full system models to produce identical changes in length for alpine glaciers these additional shifts in ela are on the order of 10 m which is within the error range of ela estimates it follows that at least in the absence of significant basal motion there is no need to include the effects of horizontal stresses when calculating the reaction of alpine glaciers to climatic changes attention should focus on accurate determination of the mass balance distribution and model tuning to give realistic ice thickness distributions

improvement of a 2 d sia ice flow model application to glacier de saint sorlin france a number of improvements have been made to an existing two dimensional ice flow model applied to an alpine glacier analysis of the results of the existing model revealed several shortcomings the first concerns the lack of mass conservation of the applied alternating direction implicit adi scheme a semi implicit si scheme is therefore proposed and the effects on mass conservation assessed by a comparison with the adi scheme the comparison is first carried out with a simple theoretical glacier for which the improvement is significant concerning the real case of glacier de saint sorlin france the initial deviation in mass conservation was much less pronounced such that the new scheme although improving mass conservation does not significantly change the modelled dynamics however other shortcomings that have a more profound impact on the modelling of glacier behaviour have been identified the ice thickness may become negative over some gridpoints leading to an inconsistency the problem is partly resolved by incorporating extra checks on critical gridpoints at the glacier border finally with the help of ice particle tracking unrealistic ice settlement above the bergschrund has been identified as the main reason for spurious dynamic effects and has been corrected

a thermomechanical model of ice flow in west antarctica this study uses a three dimensional thermo mechanical model to investigate the internal flow dynamics of the west antarctic ice sheet wais the model allows ice thickness flow and temperature to interact freely its domain is prescribed as that of the present day grounded wais realistic present day climatic and topographical boundary conditions are employed the analysis of a series of experiments pays particular attention to the location and dynamics of concentrations of ice flow ice streams underlying topographic troughs are crucial in determining the strength and location of these concentrations of flow the flow pattern generated by subglacial troughs is made more distinct by the inclusion of ice flow temperature coupling the inclusion of sliding leads to the generation of limit cycles in the ice flow they are concentrated around the present day ice streams b and c of the siple coast and have a period of 5 to 10 ky there appears to be competition between several preferred ice flow pathways in this area the two end members of the flow regime are a strong ice stream c with a weakened ice stream a b complex and strong ice streams a and b with a dormant ice stream c ice streams appear to require ice discharges above a certain threshold in order to maintain frictional heat generation and fast flow individual ice streams can therefore interact through changes in catchment area size a reduction in catchment area reduces the volume of ice entering a stream and can cause stagnation as the amount of frictional heating falls

consistent generation of ice streams via thermo viscous instabilities modulated by membrane stresses accurate computation of ice stream location and dynamics is a key aspiration for theoretical glaciology ice sheet models with thermo viscous coupling have been shown to exhibit stream like instabilities using shallow ice approximation mechanics but the location and width of these streams depends on the numerical implementation and are not unique we present results from thermo viscously coupled ice sheet models incorporating membrane stresses spontaneous generation of fast flowing linear features still occurs under certain parameter regimes with computed stream widths between 20 km to 100 km comparable with observations these features are maintained as the grid size is decreased the thermo viscous feedback mechanism that generates ice streams under the shallow ice approximation still operates now selecting a unique stream size computations of thermo viscous ice flows should include membrane stresses when the bed is approximately flat e g parts of antarctica and former ice sheets of the northern hemisphere previous calculations of spontaneous ice stream generation using the shallow ice approximation should be reassessed citation hindmarsh r c a 2009 consistent generation of ice streams via thermo viscous instabilities modulated by membrane stresses geophys res lett 36 l06502 doi 10 1029 2008gl036877

a new three dimensional higher order thermomechanical ice sheet model basic sensitivity ice stream development and ice flow across subglacial lakes 1 a new three dimensional thermomechanically coupled ice sheet model is developed contrary to the majority of three dimensional ice sheet models shallow ice approximation higher order stress gradients longitudinal and transverse are accounted for in the force balance equations the horizontal velocity field is determined from the force balance equations in their derivative form elliptical equation using the method presented by pattyn 2000 2002a the model is solved on a regular grid in the horizontal and an irregular grid in the vertical and is numerically stable basic experiments include the european ice sheet modeling initiative eismint benchmarks for largescale ice sheet models and a comparison with the saito blatter ice sheet model including higher order stress gradients saito et al 2003 detailed calculations of ice flow over three dimensional bedrock perturbations showed the validity of the higher order solution the model is capable of simulating the evolution of an ice stream within the ice sheet and shows important aspects of observed ice stream features such as the surface flattening and the importance of side drag the simulation of the ice flow over a subglacial lake results in a flattening of the surface a local velocity increase over the lake and a deviation of the ice flow from the main flow direction features which are also observed at lake vostok antarctica index terms 1827 hydrology glaciology 1863 1863 hydrology snow and ice 1827 3230 mathematical geophysics numerical solutions keywords higher order ice sheet model thermomechanical ice stream subglacial lake citation pattyn f a new three dimensional higher order thermomechanical ice sheet model basic sensitivity ice stream development and ice flow across subglacial lakes j geophys res 108 b8 2382 doi 10 1029 2002jb002329 2003

european ice sheet modelling initiative eismint model intercomparison experiments with first order mechanics 1 the european ice sheet modelling initiative eismint intercomparison experiments with thermomechanical coupling are repeated with an ice sheet model that applies the first order approximation for computing the flow field the experiments impose radially symmetric boundary conditions most of the previous results have shown the loss of implied radial symmetry i e the formation of distinct regularly spaced spokes of cold ice in the simulated basal temperatures a similar result is also observed in the presented first order model results the computed velocity components scatter widely along the marginal grid points this indicates that the spokes may be triggered by the poor representation of the margin with a regular grid where the steep gradients in the surface enhance the numerical errors additionally the generally applied second order discretization scheme tends to decouple even and odd numbered grid points thus leading to wavy solutions with a wavelength of two grid cells these patterns strongly suggest that the loss in radial symmetry is a numerical artifact

stress gradient damping of thermoviscous ice flow instabilities 1 calculations of thermomechanically coupled ice sheet evolution using the shallow ice approximation exhibit the development of fingering instabilities in velocity temperature and thickness which have been argued to resemble and therefore explain ice stream formation however workers have had difficulty in obtaining comparable results it has been suggested that this is due to ill posing of the evolution problem when the shallow ice approximation is used the influence of additional mechanical terms are examined using both the full system and the hybrid lubrication membrane stress approximation the three dimensional equivalent of the longitudinal stress approximation linearization techniques are used to formulate an analytical problem and a numerical eigenvalue problem the analysis shows that the shallow ice approximation gives highly erroneous stability maps for the combination of shorter transverse wavelengths and longer along flow wavelengths i e ice stream dimensions while the lubrication membrane stress approximation and full systems give different but consistent results which remove the ill posing calculations in two horizontal dimensions of thermoviscous coupling using the shallow ice approximation which comprise a large proportion of ice sheet modeling results need reexamination

time step limits for stable solutions of the ice sheet equation various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions the computational efficiency of various iterated and non iterated marching schemes is compared the stability properties of different marching schemes with and without iterations on the non linear equations are compared newton raphson techniques permit the largest time steps a new technique which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies is shown to improve substantially the performance of picard iteration at a negligible computational cost

results from the ice sheet model intercomparison project heinrich event intercomparison ismip heino results from the heinrich event intercomparison heino topic of the ice sheet model intercomparison project ismip are presented ismip heino was designed to explore internal large scale ice sheet instabilities in different contemporary ice sheet models these instabilities are of interest because they are a possible cause of heinrich events a simplified geometry experiment reproduces the main characteristics of the laurentide ice sheet including the sedimented region over hudson bay and hudson strait the model experiments include a standard run plus seven variations nine dynamic thermodynamic ice sheet models were investigated one of these models contains a combination of the shallow shelf ssa and shallow ice approximation sia while the remaining eight models are of sia type only seven models including the sia ssa model exhibit oscillatory surges with a period of 1000 years for a broad range of parameters while two models remain in a permanent state of streaming for most parameter settings in a number of models the oscillations disappear for high surface temperatures strong snowfall and small sediment sliding parameters in turn low surface temperatures and low snowfall are favourable for the ice surge cycles we conclude that further improvement of ice sheet models is crucial for adequate robust simulations of cyclic large scale instabilities

the validation and sensitivity of a model of the icelandic ice sheet abstract the derivation and implementation of a three dimensional model used to investigate the last glacial maximum ice sheet across iceland by hubbard et al 2006 a modelling insight into the icelandic last glacial maximum ice sheet quaternary science reviews is described it is applied at 2 km resolution and requires boundary distributions of topography geothermal heat flux surface air temperature and mass balance calculated using a temperature index approach based on reference distributions of annual temperature and precipitation the model enables the variables of ice thickness stress strain and temperature to evolve freely through time and caters for the coupling of thermally triggered basal sliding with non local dynamics through the computation of longitudinal stresses it is driven through perturbations in sea level and annual precipitation and temperature a series of contemporary experiments are initiated to validate the model against the present ice cover across iceland forcing the model from ice free conditions with the 1961 1990 reference climatology yields a good simulation of all the ice masses except for vatnajokull where the model falls well short of its present margins however an experiment forced from ice free conditions for 1000 years with a 2 c cooling perturbation followed by 100 years of reference climatology yields a good simulation of vatnajokull in addition to other ice masses implying that it is a remnant icecap inherited from the little ice age and perpetuated through strong ice elevation mass balance coupling an ensemble of experiments are initiated to investigate the sensitivity of the optimum lgm model isolated in hubbard et al 2006 ice sheet volume and aspect ratio but not area are found to be sensitive to basal boundary conditions in particular to the choice of sliding parameter and the applied geothermal conditions due to strong topographic control in particular the configuration of offshore bathymetry and shelf break ice sheet volume and area is sensitive to the calving parameter and sea level change however an asymmetrical response indicates that the ice sheet is effectively decoupled from further climatic deterioration once it advances to the continental shelf break these experiments imply that there is little latitude in the selection of model parameters which yields an ice sheet compatible with the available evidence and that the optimum lgm experiment represents a sound result by inference at least 63 of the optimum lgm ice sheet was grounded below sea level implying potential instability with the onset of deglaciation

dynamics of thermally induced ice streams simulated with a higher order flow model we use a new discretization technique to solve the higher order thermomechanically coupled equations of glacier evolution we find that under radially symmetric continuum equations small perturbations in symmetry due to the discretization are sufficient to produce the initiation of nonsymmetric thermomechanical instabilities which we interpret as ice streams in good agreement with previous studieswhich have indicated a similar instability we find that the inclusion of membrane stresses regularizes the size of predicted streams eliminating the ill posedness evident in previous investigations of ice stream generation through thermomechanical instability ice streams exhibit strongly irregular periodicity which is influenced by neighboring ice streams and the synoptic state of the ice stream ice streams are not always the same size but instead appear to follow a temperature dependent distribution of widths that is robust to grid refinement the morphology of the predicted ice streams corresponds reasonably well to extant ice streams in physically similar environments

exact solutions to the thermomechanically coupled shallow ice approximation effective tools for verification we describe exact solutions to the thermomechanically coupled shallow ice approximation in three spatial dimensions although artificially constructed these solutions are very useful for testing numerical methods in fact they allow us to verify a finite difference scheme that is to show that the results of our numerical scheme converge to the correct continuum values as the grid is refined in three dimensions comparison of numerical results with exact solutions has helped us to precisely quantify and understand some of the numerical errors we are making our verified numerical scheme shows the basal temperature spokes which arose in the eismint european ice sheet modelling initiative ii intercomparison payne and others 2000 a careful analysis describes these warm spokes as numerical errors which occur when the derivative of the strain heating term with respect to the temperature is large on the other hand the appearance of basal temperature spokes in a verified numerical scheme strongly suggests that they are a feature of the eismint ii experiment f continuum problem in fact they are clear evidence of an unstable equilibrium point of the continuum problem this paper is a sequel to bueler and others 2005 which addresses exact solutions and verification in the isothermal case

the eismint benchmarks for testing ice sheet models we present a series of benchmark experiments designed for testing and comparing numerical ice sheet models following the outcome of two eismintworkshops organized to intercompare large scale ice sheet models currently in operation model benchmark experiments are described for ice sheets underfixed and moving margin conditions these address both steady state and time dependent behavior under schematic boundary conditions and with prescribedphysics a comparison was made of each model s prediction of basic geophysical variables such as ice thickness velocity and temperature consensusachieved in the model intercomparison provides reference solutions against which the accuracy and consistency of ice sheet modeling codes can be assessed

back arc rifting initiated with a hot and wet continental lithosphere abstract how back arc rifting is initiated is studied by inferring the tectonic force and strain rate regimes from published subsidence data for nine back arc basins in the european tethyan belt and in the western pacific the results show that rifting only begins after some particular magnitude of tectonic force is reached this implies that subduction has already progressed to a point such that the total negative buoyancy of the slab becomes so large that the relevant tectonic force exceeds its bending strength once back arc rifting has been initiated the force required to maintain it gradually decays this indicates that the processes governing back arc extension are fast enough that lithosphere weakening due to an increase in geothermal gradient exceeds strengthening due to the thermal relaxation and crustal thinning this is achieved given currently accepted estimates for the maximum force 7 tn m if the thickness of the thermal lithosphere is significantly less than 125 km and if the lithosphere has a wet rheology meaning that a weak continental lithosphere is required for the initiation of back arc rifting the presence of significant strength in the uppermost mantle is not precluded but it is not required

on the numerical computation of temperature in an ice sheet quadrature solutions for temperature fields in the central regions of ice sheets where flow is by internal deformation are presented and compared with numerical computations and the european ice sheet modelling initiative eismint benchmark standards the solution appears to be more accurate than the eismint benchmark standards for thermomechanically uncoupled temperature distributions the ability of finite difference and pseudo spectral methods to reproduce this solution is considered errors in the eismint benchmark are larger than expected the possibility that they could arise from inaccurate evaluation of the vertical velocity is considered and found to be unlikely formulae for computing the vertical velocity in jenssen s coordinate are complicated and may lead to programming errors a simple form of the heat transport equation in the coordinate for flow by sliding and internal deformation is derived arid presented and it is shown how this form is particularly suitable for finite element computations it is also shown that only two quadratures are necessary to compute the vertical and horizontal heat transport and the ice flux