Search

Acknowledgements. Funding for DIB was provided by NE/R018243/1 REBUS (Resolving Enthalpy Budget to Understand Surging). We present the first general theory of glacier surging that includes both temperate and polythermal glacier surges, based on coupled mass and enthalpy budgets. Enthalpy (in the form of thermal energy and water) is gained at the glacier bed from geothermal heating plus frictional heating (expenditure of potential energy) as a consequence of ice flow. Enthalpy losses occur by conduction and loss of meltwater from the system. Because enthalpy directly impacts flow speeds, mass and enthalpy budgets must simultaneously balance if a glacier is to maintain a steady flow. If not, glaciers undergo out-of-phase mass and enthalpy cycles, manifest as quiescent and surge phases. We illustrate the theory using a lumped element model, which parameterizes key thermodynamic and hydrological processes, including surface-to-bed drainage and distributed and channelized drainage systems. Model output exhibits many of the observed characteristics of polythermal and temperate glacier surges, including the association of surging behaviour with particular combinations of climate (precipitation, temperature), geometry (length, slope) and bed properties (hydraulic conductivity). Enthalpy balance theory explains a broad spectrum of observed surging behaviour in a single framework, and offers an answer to the wider question of why the majority of glaciers do not surge. Publisher PDF

In this paper we undertake a quantitative analysis of the dynamic process by which ice underneath a dry porous debris layer melts. We show that the incorporation of debris-layer airflow into a theoretical model of glacial melting can capture the empirically observed features of the so-called Østrem curve (a plot of the melt rate as a function of debris depth). Specifically, we show that the turning point in the Østrem curve can be caused by two distinct mechanisms: the increase in the proportion of ice that is debris-covered and/or a reduction in the evaporative heat flux as the debris layer thickens. This second effect causes an increased melt rate because the reduction in (latent) energy used for evaporation increases the amount of energy available for melting. Our model provides an explicit prediction for the melt rate and the temperature distribution within the debris layer, and provides insight into the relative importance of the two effects responsible for the maximum in the Østrem curve. We use the data of Nicholson and Benn (2006) to show that our model is consistent with existing empirical measurements.

This paper reviews the seminal contributions (and tussles) of Weertman and Lliboutry to the theory of glacier sliding in the 1950s and 1960s, and summarizes later developments up to the present day.

Of several problems associa ted wit h theoretica l expla nation s of the jokul­ hla ups which emerge from the outlet gla cie r Skeioarar jokull of the ice ca p Vatna jokull in sou theast Iceland, the mechani sm of flood init iation is one that has hi therto defied expl anat ion . We provide such an expl ana tion based on a car efu l analysis of the cla ssica l Nye-Ro thlisbe rger model; near the subglacial lake Grimsvotn, the hyd raulic potenti al gradie nt is towards the lake, and there is theref ore a catchme nt boun dary under the ice, whose location dep ends on the subglacial mel twater drai nage charac terist ics. As the con ­ dit ions for a flood approa ch, we show that the wat er divi de migra tes towards the lake, while at the sa me time the lake pre ssure in creases. When the hydra ulic potential gradient to­ wards the lake is low and the refillin g rate is sl ow, the se al will "break" whe n the cat chment boun dary reaches the lake, while the la ke level is st ill below flotati on pressu re, whereas if refillin g is rapid, flo tation can be achie ved bef ore a flood is initia ted. This theory can thus explain why the seal is normally broken when the lake le vel at Grimsvotn is still some 60 m below flotation level. In addit ion , we are abl e to explain why thej okulhla up following the 1996 eruption did not occur until flo tation level was achie ved, and we show how the cyeli­ cit y and magnitude ofjoku lhla ups can be explained wit hin this theory.

A simplified model of a two-dimensional ice sheet is described. It includes basal ice sliding dependent, on the basal water pressure, which itself is described by a simple theory of basal drainage. We show that this simple but sophisticated model predicts surges of the ice mass in realistic circumstances, and we describe these surges by solving the problem numerically. We also are able to describe some parts of the surge analytically. The numerical solution of the model is a delicate matter, and highlights pitfalls to be avoided if more complicated models are to be solved successfully.

By using a simple parameterized model of thermomechanically coupled flow in cold ice sheets, together with a physically based sliding law which includes a description of basal drainage, we show that relationships between ice flux and ice thickness can realistically be multi-valued, and hence that hydraulically induced surges can occur. We term this mechanism hydraulic run-away, as it relies on the positive feed-back between sliding velocity and basal melt production. For this feedback to operate, it is essential that water pressure increases with water storage. This is consistent with various recent ideas concerning drainage, under ice sheets, be it through a system of canals, a distnbuted film or a subglacial aquifer. For confined flows, such as valley glaciers (e.g. Trapridge Glacier) or topographically constrained ice streams (e.g. Hudson Strait in the Laurentide ice sheet), which are underlain by sufficiently deformable sediment, we can expect thermally regulated surges to occur, while in a laterally unconfined drainage basin (such as that which flows into the Ross Ice Shelf), we might expect ice streams to develop.

We develop theoretically a description of a possible subglacial drainage mechanism for glaciers and ice sheets moving over saturated, deformable till. The model is based on the plausible assumption that flow of water in a thin film at the ice-till interface is unstable to the formation of a channelized drainage system, and is restricted to the case in which meltwater cannot escape through the till to an underlying aquifer. In describing the physics of such channelized drainage, we have generalized and extended Röthlisberger’s model of channels cut into basal ice to include “canals” cut into the till, paying particular attention to the role of sediment properties and the mechanics of sediment transport. We show that sediment-floored Röthlisberger (R) channels can exist for high effective pressures, and wide, shallow, ice-roofed canals cut into the till for low effective pressures. Canals should form a distributed, non-arborescent system, unlike R channels. For steep slopes typical of alpine glaciers, both drainage systems can exist, but with the water pressure lower in the R channels than in the canals; the canal drainage should therefore be unstable in the presence of channels. For small slopes typical of ice sheets, only canals can exist and we therefore predict that, if channelized meltwater flow occurs under ice sheets moving over deformable till, it takes the form of shallow, distributed canals at low effective pressure, similar to that measured at Ice Stream B in West Antarctica. Geologic evidence derived from land forms and deposits left by the Pleistocene ice sheets in North America and Europe is also consistent with predictions of the model.

We here describe the process by which a complex model set of equations and boundary conditions may be rationally reduced to a simpler and more manageable set by the. processes of non-dimensionalization and asymptotic approximation. Such a reduced model (derived elsewhere) is then presented for an incompressible, two-dimensional ice flow. It consists of two coupled equations for the stream function and enthalpy variable, together with a complex set of boundary conditions.The important dimensionless parameters which arise are given, and various limiting values of these are commented on. Nye’s (1960) equation for kinematic waves may be reproduced, and a non-linear analysis of this reveals that disturbances reach the glacier snout in finite time, and are uniformly bounded there: in the particular case considered here, one can also show that the temperature field is stable.It is shown that the effect of introducing a (realistic) sliding law which is continuously dependent on the temperature has a major effect on the bedrock temperature profile.Lastly we consider seasonal waves using a kinematic wave equation based on a plausible form of the sliding law when cavitation is present. The main observed features are qualitatively reproduced.

We present a model for the determination of a sliding law in the presence of subglacial cavitation. This law determines the basal stress at a clean ice‒bedrock interface in terms of the velocity and effective pressure. The method is based on an exact solution of the Nye—Kamb (linearly viscous) sliding problem with cavities, and uses ideas of Lliboutry (1979) to construct, via renormalization methods, an approximate law for general bedrock form. We show that, for a bedrock whose spectrum has a power‒law behaviour, one obtains a sliding law which gives the basal shear stress proportional to a power of the velocity, and to a power of the effective pressure.The effect of subglacial cavitation on the drainage system is examined, using recent ideas of Kamb. For sufficiently high velocities, drainage through a Röthlisberger tunnel system is unstable, and drainage takes place through the linked system of cavities. This leads to a reduction of the effective pressure, and by taking account of this, one can rewrite the sliding law in terms of stress and velocity only.This sliding law can be multi‒valued, and it is suggested that this underlies the dynamic phenomenon of surges.

We present a model for the determination of a sliding law in the presence of subglacial cavitation. This law determines the basal stress at a clean ice‒bedrock interface in terms of the velocity and effective pressure. The method is based on an exact solution of the Nye—Kamb (linearly viscous) sliding problem with cavities, and uses ideas of Lliboutry (1979) to construct, via renormalization methods, an approximate law for general bedrock form. We show that, for a bedrock whose spectrum has a power‒law behaviour, one obtains a sliding law which gives the basal shear stress proportional to a power of the velocity, and to a power of the effective pressure.The effect of subglacial cavitation on the drainage system is examined, using recent ideas of Kamb. For sufficiently high velocities, drainage through a Röthlisberger tunnel system is unstable, and drainage takes place through the linked system of cavities. This leads to a reduction of the effective pressure, and by taking account of this, one can rewrite the sliding law in terms of stress and velocity only.This sliding law can be multi‒valued, and it is suggested that this underlies the dynamic phenomenon of surges.

Previous theories of glacier sliding are subject to the criticism that they are not properly formulated. Here we describe how the basal ice flow may be related to the bulk ice flow by means of the formal mathematical method of matched asymptotic expansions. A complete model of the basal sliding (involving coupled problems in ice, water film, and bedrock) may be rationally reduced by a dimensional analysis to a consideration of the ice flow only, and regelation may be neglected provided roughness is absent on the finest scales (

Recent observations of glacier sliding at temperatures below the melting point are discussed. It is pointed out that these observations can be simply explained by including solid friction in the sliding law. Furthermore, we re-emphasize the point that such sub-temperate sliding has an important effect on the basal boundary conditions which should be applied in model studies of ice sheets and glaciers.

Recent observations of glacier sliding at temperatures below the melting point are discussed. It is pointed out that these observations can be simply explained by including solid friction in the sliding law. Furthermore, we re-emphasize the point that such sub-temperate sliding has an important effect on the basal boundary conditions which should be applied in model studies of ice sheets and glaciers.

We discuss the possibility of the occurrence of thermal instability in ice sheets and glaciers. This may arise from the non-linear viscous heating term, which could provide for the existence of multiple steady states in the flow and temperature fields: a word of warning is given about the applicability of these ideas.