Mathematical models of glacier sliding and drumlin formation

One of the central difficulties in many models of glacier and ice sheet flow lies in the prescription of boundary conditions at the bed. Often, processes which occur there dominate the evolution of the ice mass as they control the speed at which the ice is able to slide over the bed. In part I of this thesis, we study two complications to classical models of glacier and ice sheet sliding. First, we focus on the effect of cavity formation on the sliding of a glacier over an undeformable, impermeable bed. Our results do not support the widely used sliding law $u_b = C\tau_b^pN^{-q}$, but indicate that $\tau_b/N$ actually decreases with $u_b/N$ at high values of the latter, as suggested previously for simple periodic beds by Fowler (1986). The second problem studied is that of an ice stream whose motion is controlled by bed obstacles with wavelengths comparable to the thickness of ice. By contrast with classical sliding theory for ice of constant viscosity,the bulk flow velocity does not depend linearly on the driving stress. Indeed, the bulk flow velocity may even be a multi-valued function of driving stress and ice thickness. In the second part of the thesis, attention is turned to the formation of drumlins. The viscous till model of Hindmarsh (1998) and Fowler (2000) is analysed in some detail. It is shown that the model does not predict the formation of three-dimensional drumlins, but only that of two-dimensional features, which may be interpreted as Rogen moraines. A non-linear model allows the simulation of the predicted bedforms at finite amplitude. Results obtained indicate that the growth of bedforms invariably leads to cavitation. A model for travelling waves in the presence of cavitation is also developed, which shows that such travelling waves can indeed exist. Their shape is, however, unlike that of real bedforms, with a steep downstream face and no internal stratification. These results indicate that Hindmarsh and Fowler's model is probably not successful at describing the processes which lead to the formation of streamlined subglacial bedforms

Plastic bed beneath Hofsjökull Ice Cap, central Iceland, and the sensitivity of ice flow to surface meltwater flux

The mechanical properties of glacier beds play a fundamental role in regulating the sensitivity of glaciers to environmental forcing across a wide range of timescales. Glaciers are commonly underlain by deformable till whose mechanical properties and influence on ice flow are not well understood but are critical for reliable projections of future glacier states. Using synoptic-scale observations of glacier motion in different seasons to constrain numerical ice flow models, we study the mechanics of the bed beneath Hofsjökull, a land-terminating ice cap in central Iceland. Our results indicate that the bed deforms plastically and weakens following incipient summertime surface melt. Combining the inferred basal shear traction fields with a Coulomb-plastic bed model, we estimate the spatially distributed effective basal water pressure and show that changes in basal water pressure and glacier accelerations are non-local and non-linear. These results motivate an idealized physical model relating mean basal water pressure and basal slip rate wherein the sensitivity of glacier flow to changes in basal water pressure is inversely related to the ice surface slope.This research was conducted at the California Institute of Technology and the University of Iceland with funding provided by the NASA Crysopherice Sciences Program (Award NNX14AH80G). B. M. was partially funded by a NASA Earth and Space Sciences Fellowship and an Achievement Rewards for College Students (ARCS) fellowship. InSAR data are freely available from the Alaska Satellite Facility via the UAVSAR website (http://uavsar.jpl.nasa.gov).Peer Reviewe

Application of constitutive friction laws to glacier seismicity

While analysis of glacial seismicity continues to be a widely used method for interpreting glacial processes, the underlying mechanics controlling glacial stick-slip seismicity remain speculative. Here, we report on laboratory shear experiments of debris-laden ice slid over a bedrock asperity under carefully controlled conditions. By modifying the elastic loading stiffness, we generated the first laboratory icequakes. Our work represents the first comprehensive lab observations of unstable ice-slip events and replicates several seismological field observations of glacier slip, such as slip velocity, stress drop, and the relationship between stress drop and recurrence interval. We also observe that stick-slips initiate above a critical driving velocity and that stress drop magnitude decreases with further increases in velocity, consistent with friction theory and rock-on-rock friction laboratory experiments. Our results demonstrate that glacier slip behavior can be accurately predicted by the constitutive rate-and-state friction laws that were developed for rock friction

Glacier Surge Mechanism Based on Linked Cavity Configuration of the Basal Water Conduit System

Based on observations of the 1982–1983 surge of Variegated Glacier, Alaska, a model of the surge mechanism is developed in terms of a transition from the normal tunnel configuration of the basal water conduit system to a linked cavity configuration that tends to restrict the flow of water, resulting in increased basal water pressures that cause rapid basal sliding. The linked cavity system consists of basal cavities formed by ice-bedrock separation (cavitation), ∼1 m high and ∼10 m in horizontal dimensions, widely scattered over the glacier bed, and hydraulically linked by narrow connections where separation is minimal (separation gap ≲ 0.1 m). The narrow connections, called orifices, control the water flow through the conduit system; by throttling the flow through the large cavities, the orifices keep the water flux transmitted by the basal water system at normal levels even though the total cavity cross-sectional area (∼200 m^2) is much larger than that of a tunnel system (∼10 m^2). A physical model of the linked cavity system is formulated in terms of the dimensions of the “typical” cavity and orifice and the numbers of these across the glacier width. The model concentrates on the detailed configuration of the typical orifice and its response to basal water pressure and basal sliding, which determines the water flux carried by the system under given conditions. Configurations are worked out for two idealized orifice types, step orifices that form in the lee of downglacier-facing bedrock steps, and wave orifices that form on the lee slopes of quasisinusoidal bedrock waves and are similar to transverse “N channels.” The orifice configurations are obtained from the results of solutions of the basal-sliding-with-separation problem for an ice mass constituting of linear half-space of linear rheology, with nonlinearity introduced by making the viscosity stress-dependent on an intuitive basis. Modification of the orifice shapes by melting of the ice roof due to viscous heat dissipation in the flow of water through the orifices is treated in detail under the assumption of local heat transfer, which guarantees that the heating effects are not underestimated. This treatment brings to light a melting-stability parameter Ξ that provides a measure of the influence of viscous heating on orifice cavitation, similar but distinct for step and wave orifices. Orifice shapes and the amounts of roof meltback are determined by Ξ. When Ξ ≳ 1, so that the system is “viscous-heating-dominated,” the orifices are unstable against rapid growth in response to a modest increase in water pressure or in orifice size over their steady state values. This growth instability is somewhat similar to the jökulhlaup-type instability of tunnels, which are likewise heating-dominated. When Ξ ≲ 1, the orifices are stable against perturbations of modest to even large size. Stabilization is promoted by high sliding velocity ν, expressed in terms of a ν^(−½) and ν^(−1) dependence of Ξ for step and wave cavities. The relationships between basal water pressure and water flux transmitted by linked cavity models of step and wave orifice type are calculated for an empirical relation between water pressure and sliding velocity and for a particular, reasonable choice of system parameters. In all cases the flux is an increasing function of the water pressure, in contrast to the inverse flux-versus-pressure relation for tunnels. In consequence, a linked cavity system can exist stably as a system of many interconnected conduits distributed across the glacier bed, in contrast to a tunnel system, which must condense to one or at most a few main tunnels. The linked cavity model gives basal water pressures much higher than the tunnel model at water fluxes ≳1 m^(3/s) if the bed roughness features that generate the orifices have step heights or wave amplitudes less than about 0.1 m. The calculated basal water pressure of the particular linked cavity models evaluated is about 2 to 5 bars below ice overburden pressure for water fluxes in the range from about 2 to 20 m^(3/s), which matches reasonably the observed conditions in Variegated Glacier in surge; in contrast, the calculated water pressure for a single-tunnel model is about 14 to 17 bars below overburden over the same flux range. The contrast in water pressures for the two types of basal conduit system furnishes the basis for a surge mechanism involving transition from a tunnel system at low pressure to a linked cavity system at high pressure. The parameter Ξ is about 0.2 for the linked cavity models evaluated, meaning that they are stable but that a modest change in system parameters could produce instability. Unstable orifice growth results in the generation of tunnel segments, which may connect up in a cooperative fashion, leading to conversion of the linked cavity system to a tunnel system, with large decrease in water pressure and sliding velocity. This is what probably happens in surge termination. Glaciers for which Ξ ≲ 1 can go into surge, while those for which Ξ ≳ 1 cannot. Because Ξ varies as α^(3/2) (where α is surface slope), low values of Ξ are more probable for glaciers of low slope, and because slope correlates inversely with glacier length in general, the model predicts a direct correlation between glacier length and probability of surging; such a correlation is observed (Clarke et al., 1986). Because Ξ varies inversely with the basal shear stress τ, the increase of τ that takes place in the reservoir area in the buildup between surges causes a decrease in Ξ there, which, by reducing Ξ below the critical value ∼1, can allow surge initiation and the start of a new surge cycle. Transition to a linked cavity system without tunnels should occur spontaneously at low enough water flux, in agreement with observed surge initiation in winter

Sliding of temperate basal ice on a rough, hard bed: creep mechanisms, pressure melting, and implications for ice streaming

Basal ice motion is crucial to ice dynamics of ice sheets. The classic Weertman model for basal sliding over bedrock obstacles proposes that sliding velocity is controlled by pressure melting and/or ductile flow, whichever is the fastest; it further assumes that pressure melting is limited by heat flow through the obstacle and ductile flow is controlled by standard power-law creep. These last two assumptions, however, are not applicable if a substantial basal layer of temperate (T � Tmelt/ ice is present. In that case, frictional melting can produce excess basal meltwater and efficient water flow, leading to near-thermal equilibrium. High-temperature ice creep experiments have shown a sharp weakening of a factor 5–10 close to Tmelt, suggesting standard power-law creep does not operate due to a switch to melt-assisted creep with a possible component of grain boundary melting. Pressure melting is controlled by meltwater production, heat advection by flowing meltwater to the next obstacle and heat conduction through ice/rock over half the obstacle height. No heat flow through the obstacle is required. Ice streaming over a rough, hard bed, as possibly in the Northeast Greenland Ice Stream, may be explained by enhanced basal motion in a thick temperate ice layer

A double continuum hydrological model for glacier applications

Peer reviewedPublisher PD

Rapid basal melting of the Greenland Ice Sheet from surface meltwater drainage

Subglacial hydrologic systems regulate ice sheet flow, causing acceleration or deceleration, depending on hydraulic efficiency and the rate at which surface meltwater is delivered to the bed. Because these systems are rarely observed, ice sheet basal drainage represents a poorly integrated and uncertain component of models used to predict sea level changes. Here, we report radar-derived basal melt rates and unexpectedly warm subglacial conditions beneath a large Greenlandic outlet glacier. The basal melt rates averaged 14 mm ⋅d−1 over 4 months, peaking at 57 mm ⋅d−1 when basal water temperature reached +0.88 ∘C in a nearby borehole. We attribute both observations to the conversion of potential energy of surface water to heat in the basal drainage system, which peaked during a period of rainfall and intense surface melting. Our findings reveal limitations in the theory of channel formation, and we show that viscous dissipation far surpasses other basal heat sources, even in a distributed, high-pressure system

Subglacial roughness of the Greenland Ice Sheet: relationship with contemporary ice velocity and geology

The subglacial environment of the Greenland Ice Sheet (GrIS) is poorly constrained both in its bulk properties, for example geology, the presence of sediment, and the presence of water, and interfacial conditions, such as roughness and bed rheology. There is, therefore, limited understanding of how spatially heterogeneous subglacial properties relate to ice-sheet motion. Here, via analysis of 2 decades of radio-echo sounding data, we present a new systematic analysis of subglacial roughness beneath the GrIS. We use two independent methods to quantify subglacial roughness: first, the variability in along-track topography – enabling an assessment of roughness anisotropy from pairs of orthogonal transects aligned perpendicular and parallel to ice flow and, second, from bed-echo scattering – enabling assessment of fine-scale bed characteristics. We establish the spatial distribution of subglacial roughness and quantify its relationship with ice flow speed and direction. Overall, the beds of fast-flowing regions are observed to be rougher than the slow-flowing interior. Topographic roughness exhibits an exponential scaling relationship with ice surface velocity parallel, but not perpendicular, to flow direction in fast-flowing regions, and the degree of anisotropy is correlated with ice surface speed. In many slow-flowing regions both roughness methods indicate spatially coherent regions of smooth beds, which, through combination with analyses of underlying geology, we conclude is likely due to the presence of a hard flat bed. Consequently, the study provides scope for a spatially variable hard- or soft-bed boundary constraint for ice-sheet models

How dynamic are ice-stream beds?

Projections of sea-level rise contributions from West Antarctica's dynamically thinning ice streams contain high uncertainty because some of the key processes involved are extremely challenging to observe. An especially poorly observed parameter is sub-decadal stability of ice-stream beds, which may be important for subglacial traction, till continuity and landform development. Only two previous studies have made repeated geophysical measurements of ice-stream beds at the same locations in different years, but both studies were limited in spatial extent. Here, we present the results from repeat radar measurements of the bed of Pine Island Glacier, West Antarctica, conducted 3–6 years apart, along a cumulative ∼ 60 km of profiles. Analysis of the correlation of bed picks between repeat surveys shows that 90 % of the bed displays no significant change despite the glacier increasing in speed by up to 40 % over the last decade. We attribute the negligible detection of morphological change at the bed of Pine Island Glacier to the ubiquitous presence of a deforming till layer, wherein sediment transport is in steady state, such that sediment is transported along the basal interface without inducing morphological change to the radar-sounded basal interface. Given the precision of our measurements, the upper limit of subglacial erosion observed here is 500 mm a‾¹, far exceeding erosion rates reported for glacial settings from proglacial sediment yields, but substantially below subglacial erosion rates of 1.0 m a‾¹ previously reported from repeat geophysical surveys in West Antarctica

Variability in ice motion at a land-terminating Greenlandic outlet glacier: the role of channelized and distributed drainage systems

We use a combination of field observations and hydrological modelling to examine the mechanisms through which variability in meltwater input affects ice motion at a land-terminating Greenlandic outlet glacier. We find a close agreement between horizontal ice velocity, vertical ice velocity and modelled subglacial water pressure over the course of a melt season. On this basis, we argue that variation in horizontal and vertical ice velocity primarily reflects the displacement of basal ice during periods of cavity expansion and contraction, a process itself driven by fluctuations in basal water pressure originating in subglacial channels. This process is not captured by traditional sliding laws linking water pressure and basal velocity, which may hinder the simulation of realistic diurnal to seasonal variability in ice velocity in coupled models of glacial hydrology and dynamics