regionally differentiated contribution of mountain glaciers and ice caps to future sea level rise the contribution to sea level rise from mountain glaciers and ice caps has grown over the past decades a projection of their melting during the twenty first century based on temperature and precipitation projections from ten climate models suggests that by 2100 these glaciers will lose about 21 of their total global volume

transfer of basal sliding variations to the surface of a linearly viscous glacier on etudie pour une plaque a faces paralleles d epaisseur h et de viscosite lineaire l effet en surface des fluctuations de la vitesse de derapage

reconstruction of basal properties in ice sheets using iterative inverse methods inverse problems are used to estimate model parameters from observations many inverse problems are ill posed because they lack stability meaning it is not possible to find solutions that are stable with respect to small changes in input data regularization techniques are necessary to stabilize the problem for nonlinear inverse problems iterative inverse methods can be used as a regularization method these methods start with an initial estimate of the model parameters update the parameters to match observation in an iterative process that adjusts large scale spatial features first and use a stopping criterion to prevent the overfitting of data this criterion determines the smoothness of the solution and thus the degree of regularization here iterative inverse methods are implemented for the specific problem of reconstructing basal stickiness of an ice sheet by using the shallow shelf approximation as a forward model and synthetically derived surface velocities as input data the incomplete gauss newton ign method is introduced and compared to the commonly used steepest descent and nonlinear conjugate gradient methods two different stopping criteria the discrepancy principle and a recent improvement threshold are compared the ign method is favored because it is rapidly converging and it incorporates the discrepancy principle which leads to optimally resolved solutions

theoretical limitations to englacial velocity calculations to study the dynamics of ice sheets and glaciers velocities at the bed of a glacier must be measured directly or calculated using data gathered from boreholes and surface surveys boreholes to the bed are expensive and time consuming to drill so the determination of basal velocity is almost exclusively by numerical inversion of velocities observed at the surface for non linearly viscous glaciers a perturbation analysis demonstrates that inversions for englacial velocities will magnify measurement errors at an exponential rate with depth the rate at which calculation errors grow is proportional to a lyapunov exponent a measure of information loss which is shown to be a simple linear function of spatial frequency with a coefficient depending on glen s flow law exponent n the coefficient decreases with increasing non linearity demonstrating that inversions with non linearly viscous ice have smaller calculation errors than inversions with linearly viscous ice in both the linear and nonlinear cases the lyapunov exponent and rate of error growth increases with decreasing wavelength which limits velocity calculations at the bed to wavelengths on the order of one ice thickness or greater this limitation is theoretical and cannot be countered by more accurate survey data or special numerical techniques

the evolution of the patagonian ice sheet from 35 ka to the present day patice abstract we present patice a gis database of patagonian glacial geomorphology and recalibrated chronological data patice includes 58 823 landforms and 1 669 geochronological ages and extends from 38 s to 55 s in southern south america we use these data to generate new empirical reconstructions of the patagonian ice sheet pis and subsequent ice masses and ice dammed palaeolakes at 35 ka 30 ka 25 ka 20 ka 15 ka 13 ka synchronous with the antarctic cold reversal 10 ka 5 ka 0 2 ka and 2011 ad at 35 ka the pis covered of 492 6 x103 km2 had a sea level equivalent of 1 496 mm was 350 km wide and 2090 km long and was grounded on the pacific continental shelf edge outlet glacier lobes remained topographically confined and the largest generated the suites of subglacial streamlined bedforms characteristic of ice streams the pis reached its maximum extent by 33 28 ka from 38 s to 48 s and earlier around 47 ka from 48 s southwards net retreat from maximum positions began by 25 ka with ice marginal stabilisation then at 21 18 ka which was then followed by rapid irreversible deglaciation by 15 ka the pis had separated into disparate ice masses draining into large ice dammed lakes along the eastern margin which strongly influenced rates of recession glacial readvances or stabilisations occurred at least at 14 13 ka 11 ka 6 5 ka 2 1 ka and 0 5 0 2 ka we suggest that 20th century glacial recession a 1 is occurring faster than at any time documented during the holocene

a tutorial on the use of control methods in ice sheet modeling control methods are recommended as an efficient means to estimate undetermined physical parameters and boundary conditions and in so doing to improve the fidelity of a given ice sheet flow model to specific ice sheet velocity observations to accomplish this task the underlying dynamics of the model are inverted this permits model tuning adjustments to be represented explicitly in terms of model observation misfit

force budget i theory and numerical methods a practical method is developed for calculating stresses and velocities at depth using field measurements of the geometry and surface velocity of glaciers to do this it is convenient to partition full stresses into lithostatic and resistive components the horizontal gradient in vertically integrated lithostatic stress is the driving stress and it describes the horizontal action of gravity the horizontal resistive stress gradients describe the reactions resistive stresses are simply related to deviatoric stresses and hence to strain rates through a constitutive relation a numerical scheme can be used to calculate stresses and velocities from surface velocities and slope and from ice thickness there is no mathematical requirement that the variations in these quantities be small

volume area scaling parameterisation of norwegian ice caps a comparison of different approaches over the recent decades glaciers have in general continued to lose mass causing surface lowering volume reduction and frontal retreat thus contributing to global sea level rise when making assessments of present and future sea level change and management of water resources in glaciated catchments precise estimates of glacier volume are important the glacier volume cannot be measured on every single glacier therefore the global glacier volume must be estimated from models or scaling approaches volume area scaling is mostly applied for estimating volumes of glaciers and ice caps on a regional and global scale by using a statistical theoretical relationship between glacier volume v and area a v ca for explanation of the parameters c and see eq 1 in this paper a two dimensional 2d glacier model has been applied on four norwegian ice caps hardangerjokulen nordre folgefonna sporteggbreen and vestre svartisen in order to obtain values for the volume area relationship on ice caps

stress gradient coupling in glacier flow i longitudinal averaging of the influence of ice thickness and surface slope for a glacier flowing over a bed of longitudinally varying slope the influence of longitudinal stress gradients on the flow is analyzed by means of a longitudinal flow coupling equation derived from the vertically cross sectionally integrated longitudinal stress equilibrium equation by an extension of an approach originally developed by budd 1968 linearization of the flow coupling equation by treating the flow velocity u vertically averaged ice thickness h and surface slope in terms of small deviations u h and a from overall average datum values u 0 h 0 and 0 results in a differential equation that can be solved by green s function methods giving u x as a function of h x and x x being the longitudinal coordinate the result has the form of a longitudinal averaging integral of the influence of local h x and x on the flow u x u x u0 2l l 0 ln n nh n l exp x x l dx where the integration is over the length l of the glacier the operator specified deviations from the datum state and the term on which it operates which is a function of the integration variable x represents the influence of local h x x and channel shape factor x at longitudinal coordinate x on the flow u at coordinate x the influence being weighted by the influence transfer function exp x x l in the integral the quantity l that appears as the scale length in the exponential weighting function is called the longitudinal coupling length it is determined by rheological parameters via the relationship l 2h n n 3n where n is the flowlaw exponent n the effective longitudinal viscosity and n the effective shear viscosity of the ice profile n is an average of the local effective viscosity n over the ice cross section and n 1 is an average of n 1 that gives strongly increased weight to values near the base theoretically the coupling length l is generally in the range one to three times the ice thickness for valley glaciers and four to ten times for ice sheets for a glacier in surge it is even longer l 12h it is distinctly longer for non linear n 3 than for linear rheology so that the flow coupling effects of longitudinal stress gradients are markedly greater for non linear flow the averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations in h or with wavelength are attenuated by the factor 1 1 2 l 2 relative to what they would be without longitudinal coupling the short intermediate and long scales of glacier motion raymond 1980 over which the longitudinal flow variations are strongly partially and little attenuated are for 2l 2l 20l and 20l for practical glacier flow calculations the exponential weighting function can be approximated by a symmetrical triangular averaging window of length 4l called the longitudinal averaging length the traditional rectangular window is a poor approximation because of the exponential weighting the local surface slope has an appreciable though muted effect on the local flow which is clearly seen in field examples contrary to what would result from a rectangular averaging window tested with field data for variegated glacier alaska and blue glacier washington the longitudinal averaging theory is able to account semi quantitatively for the observed longitudinal variations in flow of these glaciers and for the representation of flow in terms of effective surface slope values exceptions occur where the flow is augmented by large contributions from basal sliding in the ice fall and terminal zone of blue glacier and in the reach of surge initiation in variegated glacier the averaging length 4l that gives the best agreement between calculated and observed flow pattern is 2 5 km for variegated glacier and 1 8 km for blue glacier corresponding to l h 2 in both cases if l varies with x but not too rapidly the exponential weighting function remains a fairly good approximation to the exact green s function of the differential equation for longitudinal flow coupling in this approximation l in the averaging integral is l x but is not a function of x effects of longitudinal variation of l are probably important near the glacier terminus and head and near ice falls the longitudinal averaging formulation can also be used to express the local basal shear stress in terms of longitudinal variations in the local stope stress with the mediation of longitudinal stress gradients

global distributions of glacier properties a stochastic scaling paradigm many problems in global climate and earth systems science require knowledge of regional or global scale distributions of glacier properties which includes mass balance ice velocity flux thickness volume and surface area among others with roughly 160 000 glaciers worldwide obtaining information on the global probability distributions of most ice properties is expensive and often infeasible only surface area distributions are relatively easy to measure either by direct observation or remote sensing while other properties are difficult to observe this work shows that scaling relationships from the continuum dynamics of ice can link the distribution of surface areas to the global and regional distributions of any other continuum property some data inventories already exist for constructing reasonable distributions of glacier sizes and this analysis presents theoretical arguments based on glacier network topologies similar in concept to river networks to suggest a power law times an exponential distribution of surface areas in agreement with the data therefore by using existing data or the theoretical distributions of surface areas specific distributions for other glacier properties in any region of the world can be constructed as an example predictions up to a scaling constant are made for the distribution of glacier volumes characteristic thicknesses characteristic velocities and characteristic response times in the alps

a consensus estimate for the ice thickness distribution of all glaciers on earth knowledge of the ice thickness distribution of the world s glaciers is a fundamental prerequisite for a range of studies projections of future glacier change estimates of the available freshwater resources or assessments of potential sea level rise all need glacier ice thickness to be accurately constrained previous estimates of global glacier volumes are mostly based on scaling relations between glacier area and volume and only one study provides global scale information on the ice thickness distribution of individual glaciers here we use an ensemble of up to five models to provide a consensus estimate for the ice thickness distribution of all the about 215 000 glaciers outside the greenland and antarctic ice sheets the models use principles of ice flow dynamics to invert for ice thickness from surface characteristics we find a total volume of 158 41 103 km3 which is equivalent to 0 32 0 08 m of sea level change when the fraction of ice located below present day sea level roughly 15 is subtracted our results indicate that high mountain asia hosts about 27 less glacier ice than previously suggested and imply that the timing by which the region is expected to lose half of its present day glacier area has to be moved forward by about one decade the ice volume of glaciers outside the greenland and antarctic ice sheets totals about 158 000 km3 with about 27 less ice in high mountain asia than thought according to multiple models that estimate ice thickness from surface characteristics

glaciers dominate eustatic sea level rise in the 21st century ice loss to the sea currently accounts for virtually all of the sea level rise that is not attributable to ocean warming and about 60 of the ice loss is from glaciers and ice caps rather than from the two ice sheets the contribution of these smaller glaciers has accelerated over the past decade in part due to marked thinning and retreat of marine terminating glaciers associated with a dynamic instability that is generally not considered in mass balance and climate modeling this acceleration of glacier melt may cause 0 1 to 0 25 meter of additional sea level rise by 2100

sea level rise from glaciers and ice caps a lower bound 1 one of the most easily measured dimensions of a glacier the accumulation area is linked to future changes in glacier volume and consequent changes in sea level currently observed accumulation areas are too small forcing glaciers to lose 27 of their volume to attain equilibrium with current climate as a result at least 184 33 mm of sea level rise are necessitated by mass wastage of the world s mountain glaciers and ice caps even if the climate does not continue to warm if the climate continues to warm along current trends a minimum of 373 21 mm of sea level rise over the next 100 years is expected from glaciers and ice caps when compared to recent estimates from all other sources melt water from glaciers must be considered as a particularly important fraction of the total sea level rise expected this century

basal shear stress of the ross ice streams from control method inversions 1 we used control method inversions to determine the basal shear stress beneath the ross ice streams where new high resolution velocity data sets have recently become available the inversion algorithm was adapted from an earlier viscous bed algorithm to allow solution for the basal shear stress corresponding to a weak plastic bed we performed several experiments using synthetic data to determine the quality of the inversions these experiments indicate that with high quality surface elevation data e g errors 5 m the inversions are relatively robust with respect to errors in ice flow velocity and bed topography the inversions are consistent with seismic and borehole observations and indicate that the ross ice streams lie atop a bed that is nearly everywhere weak in contrast the tributaries feeding these ice streams overlie alternating patches of strong and weak bed

regional and global volumes of glaciers derived from statistical upscaling of glacier inventory data 1 very few global scale ice volume estimates are available for mountain glaciers and ice caps although such estimates are crucial for any attempts to project their contribution to sea level rise in the future we present a statistical method for deriving regional and global ice volumes from regional glacier area distributions and volume area scaling using glacier area data from 123 000 glaciers from a recently extended world glacier inventory we compute glacier volumes and their sea level equivalent sle for 19 glacierized regions containing all mountain glaciers and ice caps on earth on the basis of total glacierized area of 741 103 68 103 km2 we estimate a total ice volume of 241 103 29 103 km3 corresponding to 0 60 0 07 m sle of which 32 is due to glaciers in greenland and antarctica apart from the ice sheets however our estimate is sensitive to assumptions on volume area scaling coefficients and glacier area distributions in the regions that are poorly inventoried i e antarctica north america greenland and patagonia this emphasizes the need for more volume observations especially of large glaciers and a more complete world glacier inventory in order to reduce uncertainties and to arrive at firmer volume estimates for all mountain glaciers and ice caps

estimate of the total volume of svalbard glaciers and their potential contribution to sea level rise using new regionally based scaling relationships we present a set of new volume scaling relationships specific to svalbard glaciers derived from a sample of 60 volume area pairs glacier volumes are computed from ground penetrating radar gpr retrieved ice thickness measurements which have been compiled from different sources for this study the most precise scaling models in terms of lowest cross validation errors are obtained using a multivariate approach where in addition to glacier area glacier length and elevation range are also used as predictors using this multivariate scaling approach together with the randolph glacier inventory v3 2 for svalbard and jan mayen we obtain a regional volume estimate of 6700 835 km3 or 17 2 mm of sea level equivalent sle this result lies in the mid to low range of recently published estimates which show values as varied as 13 and 24 mm sle we assess the sensitivity of the scaling exponents to glacier characteristics such as size aspect ratio and average slope and find that the volume of steep slope and cirque type glaciers is not very sensitive to changes in glacier area

theory of lattice boltzmann simulations of glacier flow a lattice boltzmann technique for modeling navier stokes fluid flow is extended to allow steady state simulations of glaciers and other slow flowing solids the technique is based on a statistical mechanical representation of flowing ice as a set of particles populations which translate and collide on a face centered cubic lattice the average trajectories of the populations give the velocities of the ice at any point in the glacier the method has considerable advantages over other techniques including its ability to handle complex realistic geometries without additional complications to the code examples are presented for two dimensional simulations

the physical basis of glacier volume area scaling ice volumes are known for only a few of the roughly 160 000 glaciers worldwide but are important components of many climate and sea level studies which require water flux estimates a scaling analysis of the mass and momentum conservation equations shows that glacier volumes can be related by a power law to more easily observed glacier surface areas the relationship requires four closure choices for the scaling behavior of glacier widths slopes side drag and mass balance reasonable closures predict a volume area scaling exponent which is consistent with observations giving a physical and practical basis for estimating ice volumes glacier volume is insensitive to perturbations in the mass balance scaling but changes in average accumulation area ratios reflect significant changes in the scaling of both mass balance and ice volume

a neoproterozoic seamount in the paleoasian ocean evidence from zircon u pb geochronology and geochemistry of the mayile ophiolitic melange in west junggar nw china abstract the mayile ophiolitic melange mom is located in the southwestern part of the west junggar nw china and forms part of the southern altaids the mom comprises ultramafic rocks gabbro pillow and massive lavas abyssal radiolarian cherts and volcaniclastic rocks zircons with magmatic crystallization features including oscillatory zoning and high th u values from the isotropic gabbro within the mom yield la icp ms u pb age of 572 9 ma mswd 1 0 marking the timing of crystallization of these rocks as late neoproterozoic geochemically the basalts of the corresponding gabbros from mom display oib type alkali basalt and e morb type tholeiitic basalt features both of these groups are characterized by lile and lree enrichment and hree depletion very weak or no eu anomalies eu eu 0 9 1 and no obvious nb ta and ti negative anomalies suggesting a typical oib affinity we propose that these volcanic rocks were derived from a mantle plume related magmatism associated with the evolution of the paleoasian oceanic system with the mantle source containing 2 5 garnet 2 spinel and 2 amphibole the basalts show within plate affinity marked geochemical similarities with those from hawaii and xigaze seamount suggestive of their intra oceanic setting subduction of the oceanic lithosphere commenced during late cambrian to early ordovician with the eventual accretion of the seamounts in the fore arc together with oceanic fragments forming the mayile ophiolitic melange

ice volume and subglacial topography for western canadian glaciers from mass balance fields thinning rates and a bed stress model abstracta method is described to estimate the thickness of glacier ice using information derived from the measured ice extent surface topography surface mass balance and rate of thinning or thickening of the ice column shear stress beneath an ice column is assumed to be simply related to ice thickness and surface slope as for an inclined slab but this calculation is cast as a linear optimization problem so that a smoothness regularization can be applied assignment of bed stress is based on the flow law for ice and a mass balance calculation but must be preceded by delineation of the ice flow drainage basin validation of the method is accomplished by comparing thickness estimates to the known thickness generated by a numerical ice dynamics model once validated the method is used to estimate the subglacial topography for all glaciers in western canada that lie south of 60 n adding the present ice volume of each glacier gives the estimated total volume as 2320 km3 equivalent to 5 8 mm of sea leve

stress gradient coupling in glacier flow iii exact longitudinal equilibrium equation the vertically integrated exact longitudinal stress equilibrium equation of budd 1970 is developed further in such a way as to yield an equation that gives explicitly and exactly the contributions to the basal shear stress made by surface and bed slope surface curvature longitudinal stress deviators and longitudinal stress gradients in a glacier flowing in plane strain over a bed of longitudinally varying slope with this exact equation questions raised by various approximate forms of the longitudinal equilibrium equation can be answered decisively and the magnitude of errors in the approximations can be estimated to first order in the angle that describes fluctuations in the surface slope from its mean value the exact equilibrium equation reduces to 1 2sin 2 b pghsin 2g t b k where g and t are the well known stress deviator gradient and variational stress terms k is a longitudinal curvature term and b is a basal drag term that contributes a resistance to sliding across basal hills and valleys except for t these terms are expressed in simple form and evaluated for practical situations the bed slope relative to the mean slope is not assumed to be small which allows the effects of bedrock topography to be determined particularly through their appearance in the b term

distributed ice thickness and volume of all glaciers around the globe 1 a new physically based approach for calculating glacier ice thickness distribution and volume is presented and applied to all glaciers and ice caps worldwide combining glacier outlines of the globally complete randolph glacier inventory with terrain elevation models shuttle radar topography mission advanced spaceborne thermal emission and reflection radiometer we use a simple dynamic model to obtain spatially distributed thickness of individual glaciers by inverting their surface topography results are validated against a comprehensive set of thickness observations for 300 glaciers from most glacierized regions of the world for all mountain glaciers and ice caps outside of the antarctic and greenland ice sheets we find a total ice volume of 170 103 21 103 km3 or 0 43 0 06 m of potential sea level rise

glacier volume estimation as an ill posed inversion estimating a glacier s volume by inferring properties at depth e g bed topography or basal slip from properties observed at the surface e g area and slope creates a calculation instability that grows exponentially with the size of the glacier random errors from this inversion instability can overwhelm all other sources of error and can corrupt thickness and volume calculations unless problematic short spatial wavelengths are specifically excluded volume area scaling inherently filters these short wavelengths and automatically eliminates the instability while numerical inversions can also give stable solutions by filtering the correct wavelengths explicitly as is frequently done when regularizing a model each of the scaling and numerical techniques has applications to which it is better suited and there are trade offs in resolution and accuracy but when calculating volume neither the modeling nor the scaling approach offers a fundamental advantage over the other both are significantly limited by the inherently ill posed inversion and even though both provide stable volume solutions neither can give unique solutions

an iterative scheme for determining glacier velocities and stresses there are no direct methods to measure the boundary condition at the base of a glacier here we propose a method that works by iteratively changing boundary conditions until a satisfactory fit to surface observations is obtained the method is an accelerated version of one known as kozlov maz ya iteration we apply it to the problem of ice flow through a transverse cross section and show it to be effective by solving the inverse problem for a sequence of synthetic data we also apply the method to two real glaciers athabasca and perito moreno one of which has a known basal velocity distribution

the basal speed of valley glaciers an inverse approach geophysical inverse methods are used to calculate the basal motion of a glacier they are applied to a one dimensional forward model that can be linearized to make the analysis simpler the inverse method finds a solution that fits the data within a given error it selects for smooth solutions to discriminate against unrealistic oscillations the method is applied to a simple model glacier of uniform shape and thickness to test how well a given basal motion field can be reconstructed it shows as expected that optimizing for smoothness lowers maxima and increases minima of the solution a step change in basal velocity is drawn out in the inversion over a distance that is given by the half width of a resolving function this is typically about three times the ice thickness but is also affected by the sampling rate of the data the method is then applied to two glaciers where suitable data are available brown glacier on heard island southern indian ocean and mccall glacier in the brooks range alaska u s a the mccall results agree well with earlier estimates of basal motion