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Euromech colloquium 172: Mechanics of glaciers, Interlaken, 19–23 September, 1983
CoM Regions Science and Technology, 9 (1984) 7 7 - 8 6
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Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
REVIEW EUROMECH COLLOQUIUM 172: MECHANICS OF GLACIERS, INTERLAKEN, 19-.23 SEPTEMBER, 1983 Organi$ing Committee: K. Hutter* and U. Spring** Report by K. Hutter and L.W. Morland*** *Laboratory o f Hydraulics, Hydrology and Glaciology, ETH, Zurich (Switzerland) * *Electrowatt Engineering Services Ltd., Zurich (Switzerland) * **School o f Mathematics and Physics, University o f East Anglia, Norwich (U.K.)
(Received January 11,1984; accepted January 11,1984)
INTRODUCTION The objective of the Colloquium was to promote the interaction of scientists from a multidiscipline group comprising geologists, physicists, engineers and mathematicians in a forum focussing on snow and glacier mechanics. A format of formal lectures, typically of 45 minutes duration, accompanied by ample discussion periods provided a presentation of both practical and theoretical approaches, and, most importantly, a vigorous interchange of views from the wide spectrum of interests represented. Papers describing recently completed or current research were presented under eight broad headings, and here we attempt to summarize the subject matter covered in each section, incorporating some of the discussion points. These are, of course, personal interpretations which may not accurately reflect the lecture and discussion intentions. A list of lecturers, in alphabetical order, is appended, with their institutions, lecture titles, and co-workers when noted. Cited references are distinguished by author and publication year. In addition, a 12 hour field excursion was organised, starting with a trip to the Research Station on the Jungfraujoch, then a hike alongside the Grindelwald Oberergletscher.
CREEP FLOW OF ICE, SIMPLECONFIGURATIONS The crucial ingredient of all flow predictions is the constitutive model adopted for the long time
0165-232X/84/$03.00 © 1984 ElsevierSciencePublishers B.V.
creep response of ice. Lliboutry emphasized that natural glacier ice is a "metamorphic rock ice" with properties depending on its history. Laboratory tests on cold ice show, at constant stress, decelerating primary creep to a minimum (secondary or steady) strain rate followed by accelerating tertiary creep on recrystallisation, but the long time tertiary response is not realised. Deductions about the secondary response are often confused by primary and tertiary effects. He suggested that an ice sheet flow description requires a model reflecting the anisotropic nature of the fabric after a long stress history, and proposed a model based on a dissipation potential. Returning to the conventional incompressible non-linearly viscous fluid model (necessarily isotropic), Lliboutry discussed the flow down an inclined cylindrical channel and over an inclined plane bed. When the velocities and wall drag are uniform down the channel, and free surface accumulation is absent, the surface is necessarily plane (Lliboutry, 1981) and there is no transverse circulation. It follows for a Newtonian fluid that singularities (non-physical) arise at the intersection of the surface with a nonvertical side wall when no-slip is assumed, and also at the point of maximum velocity on the free surface, though such results have not been confirmed for a non-linearly viscous fluid. It was noted that the no-slip condition is appropriate to a cold glacier in which non-uniform temperature would modify the creep response, and that a temperate glacier at uniform temperature would exhibit sliding, casting doubt on the relevance of Nye's (1965a) calculations.
78 In the uniform depth plane flow with stresses depending only on depth (Nye, 1957), the basal shear stress is necessarily uniform along the length while any non-zero sliding velocity must vary when surface accumulation/ablation arises. This conflicts with a conventional sliding law relating basal shear stress and velocity. Lliboutry proposed a generalisation of Nye's solution for uniform (non-zero) accumulation and presented an illustration which exhibits an increase of differential velocity between surface and base by a factor of 1.65 when zero accumulation is replaced by 0.1 m/yr. The uniformity of the longitudinal velocity through the thickness of a floating ice shelf away from the front or any grounded region allows a simplifying integration of the longitudinal momentum balance over the thickness even when a significant temperature profile is considered (Morland and Shoemaker, 1982). Reeh (1968, 1978) has used the integrated equations of a small strain, small rotation beam theory to study non-steady wave forms which develop at the front of a shelf, and here discussed their application to steady wave forms in the grounding transition zone. First a Newtonian fluid model is adopted to allow analytic progress and an estimation of the errors arising from the beam theory approximation. The resulting stress compatibility failure is then improved by an additional longitudinal stress contribution depending on load so that compatibility can be satisfied in mean over the thickness. This modified beam theory is extended to temperature dependent non-linearly viscous models and less simple geometries. Such engineering approximations may yield useful results for complex flow regions, but must be compared carefully with the underlying system of equations and boundary conditions to assess the possibility of serious error.
LARGE SCALE PHENOMENA
Hapgood and Campbell's (1958) theory of the Earth's crust movement is based on supposed stresses induced by the addition of antarctic ice on time scales too short for isostasy to be attained. Spinelli introduced a model of the crust as a two-dimensional non-developable surface (one which cannot be
mapped continuously onto a plane), or shel|, subjected to forces in the tangent plane which would represent the ice effects, the shell resting on a rigid ellipsoid. The analysis is not complete, and stress results are not yet available. The application of an isothermal steady plane flow solution for a non-linearly viscous ice sheet (Morland and Johnson, 1980, 1982) to investigate the influence of net mass balance pattern, bed topography, and basal sliding conditions on the size and shape of the ice sheet was described by Smith. Relationships between span, accumulation pattern, and equilibrium line altitude (ELA) are obtained for a wide range of basal conditions, showing that only sliding near the margin has a significant influence, and that both pattern and ELA influence possible expansion or contraction (Boulton, Smith and Morland, 1984). An objective is to correlate former ice sheet extents with palaeoclimate. The model proposed by Nye (1965b, 1965c) to estimate the response of a glacier to change of accumulation was applied by Reeh to reconstruct the West Greenland ice margin movement around 69.5°N over the last 1400 years. A set of parallel steady plane flows covering the width of a sub-glacier is matched to EGIG and Dye 3 ice core data, but with estimations of missing information required. The calculations give good agreement with recorded margin fluctuations over the last 100 years, and reasonable agreement with advanced positions around AD 800 and AD 1700 and retreated positions over AD 11001450. The model then predicts that the present margin recession will continue for several decades if climatic conditions do not change dramatically.
GLACIER AND ICE SHEET G E O M E T R Y
The significant temperature variation through large ice sheets and shelves, and the strongly temperature-dependent viscisoty of ice, imply a strong thermomechanical coupling of the physical balances. A dimensionless analysis, and coordinate stretching which reflects the long aspect ratio or small surface slope of natural ice masses, displays the roles of diffusion, horizontal and vertical advection, and dissipation in the energy balance (Morland, 1984). On the assumption of a homogeneous non-linearly
79 viscous fluid response, Morland showed that a weak thermal basal boundary layer may arise in a thick grounded sheet, but not otherwise, and that dissipation should not be neglected in basal regions. Discussions pointed out that the older ice in basal regions of large sheets could be less viscous and could have anisotropic structure, when the homogeneous fluid model would not be satisfactory and larger strain rate gradients could arise, enhancing the boundary layer effect. It was also shown that seasonal thermal variations are restricted to a surface boundary layer. The perturbation analysis (Morland and Johnson, 1980, 1982) developed for an isothermal theory extends to the coupled theory, and reduces the steady plane flow problem to a simpler parabolic system of leading order partial differential equations offering a useful starting point for numerical solution. A simplification of the unsteady problem is also obtained. However, solutions of the steady plane coupled flow system have yet to be obtained, and these are essential to a convincing description of the flow and geometry of the conventional ice sheet model. Hurter (1982b, 1983) has already proposed an alternative iteration scheme, so we have two distinct computational methods to compare for accuracy and efficiency. The influence of prescribed temperature variation can be estimated for the temperature dependent viscosity model from the mass and momentum balances alone. Smith described a series of illustrations of steady plane flows under different plausible temperature patterns, obtained by an extension of the isothermal perturbation analysis which again reduces to a second order ordinary differential equation for the surface profile (Morland and Smith, 1983). These illustrations confirm that no mean temperature isothermal solution is satisfactory even for temperature variation with depth alone, and also that very small variation of temperature along the span can have significant effect, emphasising the need for a valid coupled solution. The regular perturbation analysis is strictly valid only for a viscosity bounded at zero shear stress, and while a leading order solution for steady plane flow with a power law of exponent n > 1, reflecting unbounded viscosity as stress approaches zero, was obtained (Morland and Johnson, 1980, 1982), restrictions on n were imposed by physical validity at a
margin and a divide. It was also noted that n > 1 leads to an unbounded longitudinal deviatoric stress at the free surface. A valid analysis for n > 1 using a singular perturbation method (Johnson and McMeeking, 1982) exhibits a surface boundary layer in which the longitudinal deviatoric stress is significant, but remains bounded. The important conclusion emphasised by Johnson is that the leading order velocity components, pressure, shear stress, and profile determined by the regular perturbation analysis are indeed correct, and the boundary layer is a passive correction of the higher order longitudinal stress behaviour. Perturbation analyses have also been used to 'investigate the effects of bed undulations on a steady plane flow solution, in particular the effects on the free surface geometry (Hutter, 1981; Hutter et al., 1981; Morland and Johnson, 1982). Each scheme requires that the magnitude O of the undulation slope relative to a mean bed plane is small, with further restrictions on undulation wavelength compared to ice thickness. The Hurter et al. (1981) analysis allows wavelengths of order thickness, but fails when the mean bed plane is nearly horizontal. It is essentially a central zone solution constructed as an iteration on a plane surface (uniform thickness) flow. Reeh described an extension of this iteration scheme for undulations in both longitudinal and transverse directions in the simplest case of an incompressible linearly viscous fluid, but considered a nearly horizontal bed plane for illustration. -'
BASAL FEATURES AND GLACIER SLIDING
Crescent shaped fractures and gouges of mean length between 30 and 1000 mm in the granite rock of former glacier beds can be used as indicators of former basal ice flow direction, which is perpendicular to the main chord of the fractured arches. Wintges presented arguments for basal mechanisms which could be responsible for the fractured features. The basic hypothesis is that only the flatter of the two bounding fractured arches is caused by basal flow processes, the remaining fracture zones being the result of thawing and refreezing processes under basically ice free conditions after retreat. Using experimental results on ultimate strength under trta~ial
8() stress conditions for the granite under consideration. and assuming that basal debris would cause the basal pressure and shear stress to peak wherever a boulder touches the base, computational and photoelastic methods were applied to model ice, boulder and rock by various artificial resins (Ficker and Weber, 1981). These investigations made the initiation of the crescent shaped fractures and gouges by the indicated processes a very likely one. Further work is in progress. Basal Friction with and without cavity formation is still a central question; it was approached from several different angles. Lliboutry presented Weertman's (1957, 1964) 'tombstone'-model and his own alternative consisting of a regular array of hemispheres mounted on a flat base. When cavities form, autonomous and non-autonomous regimes must be differentiated; in the former the water pockets are interconnected and drainage may occur, in the latter they are not. The model presented is very much in the realm of Lliboutry's (1979) review, but the tenor seems to be that basal shear traction T, sliding velocity V and effective pressure N are functionally related by a statement T = T (V, N). While this is commonly accepted in glaciology the explicit functional form of this relation is debatable. Meyssonnier applied the finite element method to obtain the friction law on a sinusoidal bed; melting, refreezing and cavity formation are ignored, and by contrast to classical treatments, non-infinitesimal bed inclinations are assumed. Extremum principles for velocities and stresses are used to obtain upper and lower bounds for the drag of a polycrystalline ice obeying Glen's flow law with n = 3. The computations indicate that for a given basal drag, sliding velocities are somewhat larger than those of Lliboutry (1975) and smaller than those of Kamb (1970). Meyssonnier also calculates the drag for a profile of several superimposed sine curves. These and the previous results will be made available in a forthcoming dissertation. When an incompressible linearly viscous fluid is assumed, the steady plane flow reduces to a biharmonic theory for the velocities and stresses, and harmonic theory for temperature when regelation is considered. For small amplitude, small slope periodic undulations of the bed, series expansions have been constructed by using Fourier transforms
(Nye, 1969, 1970; Kamb, 1970) and by using Hilbert transforms in a complex variable representation (Morland, 1976a, 1976b), when there is no cavitation. Except for very small mean bed inclination, the solutions predict that cavitation must occur. Fowler adopted the complex variable representation to fornmlate the boundary-value problem, ignoring regelation, when cavities are present. Cavities are zones across which the pressure of the basal ice is at a uniform cavitation pressure, but their end points are to be determined as part of the flow solution. Boundary conditions are distinct in the contact and cavity zones, giving rise to a mixed Hilbert problem. Periodicity allows the leading order plane boundary to be mapped onto a circle, and the leading order solution for a sinusoidal undulation was presented. This exhibits a monotonic basal shear stress sliding velocity relation which depends on the cavitation pressure, with the shear stress approaching an asymptotic limit with increase of sliding velocity while cavitation extends over the entire bed. Fowler asks if this has implications for a surge mechanism. If the boundary of a semi-infinite complex domain can be mapped conformally onto the real axis by a rational function, then a standard boundary-value problem for the plane biharmonic equation can be solved in terms of Cauchy integral representations (Morland and Boulton, 1975). By formulating the linearly viscous plane flow over a periodic bedform with humps of finite slope as the flow over an isolated hump (or humps) with conditions outside the hump region imposed to reflect the effects of the ignored periodic extension, rational mappings can be found for interesting shapes. The Cauchy integral method first reduces the mixed boundaryvalue problem to a singular integral equation, which in turn can be formulated as a non-singular integrodifferential equation. Jones described numerical solutions for the pressure and velocity variations over two different hump profiles, as part of" a continuing investigation of the influence of profile shape and hump separation on the drag-sliding velocity relation. These solutions will also be used to check finite element programs constructed for the non-linearly viscous fluid problem. Longitudinal strain rates of order 10 -7 s-i, corresponding to longitudinal deviatoric stresses of order l0 s Nm -2, are observed during glacier surges, which
81 are comparable with the basal shear stress in a steep glacier. The successful perturbation analysis for a small surface slope glacier (relative to the bed plane) is based on a co-ordinate stretching which implies that the longitudinal deviatoric stress is an order of magnitude smaller than the shear stress, with their roles reversed in a floating shelf perturbation analysis. No such rational scheme has been developed when both stresses are comparable, so that surge conditions still lack a complete description. McMeeking proposed that loss of adhesion in a region of basal rupture, which would be a consequence of particular sliding conditions, must transfer the effects of local glacier weight to the downstream ice, inducing higher longitudinal stress and possible spread of the rupture region. As illustration he starts with a rupture region of order the ice thickness, on which the basal shear stress is zero, and deduces the leading order velocity field to be matched (crudely) with the flows upstream and downstream of the rupture region. The possibilities of rupture spread and of adhesion recovery were discussed.
WATER IN GLACIERS
Today no-one denies the importance of water in the dynamics of glaciers. However, there are two different points of view: either one looks at the effect of the water on the behavior of the ice, or that of the glacier on the water within it. Iken presented measurements of glacier movement and subglacial water pressure on Findelengletscher from May/June 1982. Surface ice velocities were measured several times a day at 4 transverse stake profiles and, in the same area, bore holes were drilled to the glacier bed, connecting to the subglacial drainage system, and water pressures were recorded. Velocity variations correlate well with the variations in water pressure, even for very low water pressure, and are thought to be explicable by a sliding mechanism with cavity formation. Since the data suggest a low separation pressure and small bed roughness, the observed conditions would cause the glacier to become unstable, which is obviously not the case, so friction between bedrock and debris
is also included in the sliding mechanism and is shown to stabilize the glacier sliding. If correct it has both a considerable stabilizing effect on the glacier flow (sliding velocity) and a dramatic effect on the hydrology (water table) of the glacier. Spring presented model calculations on turbulent flow in intraglacial channels in which an evolution equation for the temperature was taken into account. The original ideas date back to Shreve (1972), Roethlisberger (1972) and Nye (1976) but the model equations were based on the Spring and Hutter (1981, 1982) equations. The importance of temperature was emphasized and supported by a collection of measurements of water temperature of the Gornersee at the subglacial outlet cross section, indicating a surprisingly large range of temperature variation of the draining lake water. Spring's steady state equations differ from those of Roethlisberger (Rchannels!). Numerical solutions for simple ice configurations subject to different temperature and water pressure conditions at the boundaries of the conduit demonstrate that, to each given set of external parameters, either two steady conduits exist, or none. Of the two, only one is stable. Polythermal glaciers consist of cold and temperate zones separated by a discontinuity surface. To properly describe the interaction, the transport of ice plus moisture must be described together with associated boundary and transition conditions at the cold-temperate interface. Fowler and Larson's (1978) and Hutter's (1982a) models take the moisture transport into account by a diffusion equation. Fowler presented a two-phase concept formulating mass and momentum balances for water and ice separately and energy balance for the mixture. In this model the water flows in accordance with Darcy's law and the water pressure is given by Nye's (1976) vein-closure condition, complemented by a new term due to melting. Boundary conditions at the cold-temperate transition surface follow from the jump conditions of the balance laws but open additional questions regarding a proper formulation. A typical feature of the model is that temperate ice and water are both at melting temperature, avoiding water temperature as an additional field variable.
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SNOW AVALANCHI::~
Snow avalanches exist in various forms, two limiting cases being the wet flow avalanche and the airborne powder snow avalanche. Classically, hydraulic concepts are used for their description (VoellmySalm theory) and both are treated alike. More recently, theoretical and experimental studies on the flow of cohesionless granular materials have led to a new and more detailed understanding of flow avalanches (Lun et al., 1984; Scheiwiller and Hutter, 1982) and turbulent two-phase models have been suggested to describe powder snow avalanches (Scheiwiller and Hutter, 1982). Savage presented extensions of the granular flow kinetic theories of Savage and Jeffrey (1981) and Jenkins and Savage (1983). In the evaluation of the statistical averages he regarded the particles as finite spheres and included both the collision and kinetic (advective) transports in the evaluation of the 'granular' stresses, energy flux and energy dissipation. The model incorporates two phenomenological collisional coefficients which permit non-conservation of translational and rotational energy in binary collisions. The complete polar theory is not yet published but its non-polar counterpart is submitted for publication (Lun et al., 1984). The potential applicability of the model was made plausible by demonstrating agreement of the theoretical model with rapid shear flow of granules down a chute. The most important mechanism a model of powder snow avalanches must incorporate is snow settlement. Both Savage and Scheiwiller presented such models. Savage extended Hopfinger's (1983) 'inclined thermal model' by looking at a cloud of particles moving down a slope, in which the sedimentation of the particles was accounted for by a mass balance of the settling particles. Using this, an entrainment hypothesis for the ambient fluid and the momentum equation, predicted flows compared favourably with laboratory experiments. The more complete two-phase model of powder snow avalanches presented by Scheiwiller permits (in general) the prediction of mean flow and density profiles by air and particles, and allows incorporation of air and snow entrainment from above and below respectively. Snow settlement is incorporated ab initio by the two phase nature of the model. A boundary layer approximation for plane flow down
an inclined rough plane was presented, and two numerical procedures (Kantorovich and FD-methods) were discussed, by which steady plane flow in an avalanche tail could be obtained. Results look promising but must still be verified experimentally. The physical mechanisms that are responsible for the formation of powder snow avalanches from flow avalanches are still not fully understood. Hutter suggested that under a certain flow state, characteristic of the transition, air entrainment into the avalanche will be enhanced and cause tile rapid 'granular' shear flow to become unstable. A binary mixture model appropriate for flow avalanches is presented in which the snow particles obey the Jenkins-Savage (1983) theory while the air is modelled as a linearly viscous fluid. At the free surface (this is a discontinuity surface at which the jump conditions of mass and momentum nmst be satisfied) air is entrained, and this entrainment is functionally related to a surface Richardson number; the functional relationship is selected to enforce the transition from flow to powder snow avalanche. It is demonstrated how simple shear flow of a strictly parallel-sided flow can be analysed and how its stability can be studied. Field observations of snow avalanches are important in order that danger zones can be classified. Depending on severeness, three different subdomains are distinguished and forecasted using essentially either hydraulic or hydrodynamic channel flow models. To improve the classification, flow properties in actual avalanches must be studied. Gubler described microwave radar systems developed at the Swiss Institute for Snow and Avalanches Research capable of measuring speed (profiles) and flow heights of a dense flow avalanche. The equipment, mounted on oversnow vehicles and able to perform several measuring cycles in one avalanche event, has a radar range of 2 km and resolutions of 10 m, 0.5 ms -1, 0.2 s for in-plane distances, speeds and time, respectively. The system, described in Gubler (1981, 1984), has been tested for small size, dry snow avalanches, and results have been compared with the classical models (Voellmy-Salm-Perla). The ultimate goal is to obtain detailed information on in-situ flow properties which allow extension of existing models and corroborate or disprove the new theoretical models. In a field program on gliding snow avalanches, in
83 Tirol (Austria), described by Lackinger, causes of avalanche initiation and, after the fact, avalanche extent are sought. Indications of avalanche formation are, among others, distinct fractures of the snow pack and variations of seismicity, which was recorded by piezoelectric acceleration gauges. Gliding velocities and run-out distances were also measured as were meteorological and topographic routine data. Avalanche initiation was explained by means of static force balances, familiar in soil mechanics.
CREEPING SNOW
Snow 'at rest' was studied in two contributions. Ambach described the deformation of a 20 m deep firn pit in the accumulation area of the Kesselwandferner (Oetztal, Austria). The originally circular (2 m diameter) pit translated approximately 200 m over the 11 year long observation period, and exhibited vertical compaction and shear deformation. The pit axis was bent and its 14 observed cross sections deformed nearly into ellipses. Longitudinal and shear strain rates (engineering strains) were measured at various points along the circumference of the pit, as were all geometric data pertaining to compaction and local geometry. Linear and non-linear viscous constitutive models with shear and bulk viscosities were suggested as a suitable constitutive behavior between stress and strain rate but a theoretical reconstruction of the pit deformation was not given. The description of the flow o f mass and energy through a snow pack should be modeled by a fourphase mixture concept of snow, meltwater, vapour and air, but current understanding, inferred from both thermodynamic and experimental considerations is not yet sufficiently developed. Instead, Morris presented a hierarchy of reduced snowmelt forecasting models, one based on a two-phase concept using snow and meltwater as the constituents, the other being a 'classical' energy budget model. The boundary conditions of the two-phase model are based on empirical equations describing the physical processes in the atmospheric boundary layer and the underlying soil. This model, prepared for the Syst~me Europ~en Hydrologique, is appropriate for ripe snow in the funicular r~gime, and appears to be superior to the energy budget model.
BRIEF PRESENTATIONS
Veteran glaciologist Henri Bader drew attention to the texture (orientation pattern of grain axes) in ice, and the question of whether this was satisfactorily represented by specifying the distribution of grain sizes. He recalled an early experiment in which identical plasticene spheres with lubricated surfaces were placed loosely in a rubber bag, and then the air was evacuated to produce a space-filling aggregate of polyhedra. Sectioning then showed that the most frequent number of faces was 11, and the most frequent number of edges per face was 5. He asked if a theoretical correlation between plane section properties and an initial volume distribution of unequal spheres is possible, so that, by analogy, plane ice section properties could be correlated with grain size distribution. Fast sea ice dynamics in the open ocean have been studied by AIDJEX. Ito reported that observations of fast sea ice in the Greenland-Iceland-Spitzbergen area, however, indicate that mesoscale ice flow behavior is chiefly boundary dominated, a situation apparently not adequately covered by the theoretical formulations arising from AIDJEX. MIZEX will observe sea ice motion in the Greenland-IcelandSpitzbergen area and plans to develop new constitutive models that are also appropriate in near boundary zones. In a short film Savage demonstrated granular chute flow from a steady source. It indicated that, depending on discharge, chute inclination and downstream conditions, oscillatory flow conditions may develop. Inhibiting the downstream flux resulted in a granular jump, similar to a hydraulic bore. Variegated Glacier is currently very active, going through a surging phase, and observations from the 1983 campaign were shown and interpreted by Iken.
CONCLUDING REMARKS (LWM)
This Colloquium encouraged a wide audience participation, and demonstrated that the views held by practical glaciologists and the recently arrived mathematical enthusiasts do have common ground, which augurs well for the development
84 of glacier mechanics. The success of the forum was in no small part due to the efficient organisation and constant attention of Drs. Hurter and Spring, and the excellent venue they arranged in collaboration with the Swiss Government. There was a unanimous call for a repeat performance in the not too distant future (setting aside a British comment about lecture programmes from 8.30 a.m. to 6 p.m.).
APPENDIX Lecturers (in alphabetic order), titles of their presentations and coworkers
Prof. Dr. W. Ambach, University of Innsbruck, Innsbruck (Austria). - Viscosity parameters of high density firn from field experiments (with H. Eisner). Dr. H. Bader, Pompano Beach, Florida (U.S.A.). Texture in ice. Dr. E. Ficker, University of Munich (Germany). An ansatz to interpret mechanical processes at the ice rock interface of glaciers using remains of microforms. Prof. Dr. A.C. Fowler, MIT, Cambridge (U.S.A.). - Glacier sliding with cavitation; On the transport of moisture in polythermal glaciers. Dr. H. Gubler, Weissfluhjoch-Davos (Switzerland). - M e a s u r e m e n t s of speed distributions and flow heights in dense flow snow avalanches. Dr. K. Hutter, ETH, Zurich (Switzerland). - S o m e thoughts on the transition flow/powder snow avalanches (with T. Scheiwiller). Dr. A. Iken, ETH, Zurich (Switzerland). Measurements of glacier movement and subglacial water pressure, Findelengletscher, May/June 1982. Conclusions on sliding mechanism and bed roughness; Observations on Variegated Glacier. Dr. H. Ito, Alfred Wegener Institute, Bremerhafen (Germany). Mesoscale, boundary affected sea ice dynamics. Prof. Dr. R.E. Johnson, University of Illinois, UrbanaChampaign (U.S.A.). Near surface flow in glaciers obeying Glen's law (with R.M. McMeeking). -
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Dr. A.S. Jones (visiting University of East Anglia. Norwich (U.K.)), University of Queensland. Brisbane (Australia). - Pressure distribution over a finite hump beneath a glacier (with L.W. Morland). Dr. B. Lackinger, University of Innsbruck, Innsbruck (Austria). - Snow gliding and gliding snow avalanches. Prof. Dr. L.A. Lliboutry, Lab. de Glaciologie, Grenoble (France). - Creep law of ice (with P. Duval); Flow in a cylindrical channel; - Nye flow; - Friction laws for glaciers (with J. Meysonnier). Prof. Dr. R. McMeeking (visiting Cambridge University, Cambridge (U.K.)), University of Illinois, Urbana-Champaign (U.S.A.). - A model for glacier surge due to basal rupture (with R.E. Johnson). Dr. J. Meyssonnier, Lab. de Glaciologie, Grenoble (France). - Friction of ice over a sine profile derived from FEM computations. Dr. L.W. Morland, University of East Anglia, Norwich (U.K.). Thermomechanical balances of ice sheet flows. Dr. E.M. Morris, Institute of Hydrology, Wallingford (U.K3. - Modelling the flow of mass and energy through snow. Dr. N. Reeh, University of Copenhagen, Copenhagen (Denmark). Steady state wave forms on floating glaciers and ice shelves treated by beam theory methods; Response of the West Greenland ice sheet margin to a 2600 year long mass balance history deduced from a Greenland ice core record; Steady state three-dimensional ice flow over an undulating base. First order theory with linear ice rheology. Prof. Dr. S.B. Savage, McGill University, Montreal (Canada). Granular flow mechanics and snow avalanches. Mr. T. Scheiwiller, ETH Zurich (Switzerland). of powder snow avalanches (with K. Hurter). -
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Dr. G.D. Smith, University o f East Angiia, N o r w i c h (O.K.). Influence o f n o n - u n i f o r m t e m p e r a t u r e distribution on the steady m o t i o n o f ice sheets (with L.W. Morland); - The reconstruction o f former ice sheets and their mass balance characteristics using a non-linear viscous flow m o d e l (with G.S. B o u l t o n and L.W. Morland). Prof. Dr. G. Spinelli, Politecnico di Milano, Milano (Italy). - A first order m o d e l o f the crust o f the earth for calculating stresses due to a tangential force. Dr. U. Spring, E l e c t r o w a t t Engineering Services Ltd., Zurich (Switzerland). - T u r b u l e n t flow in intraglacial conduits. Temperature-induced instabilities. Dr. T. Wintges, Techn. University Munich (Germany). - An a t t e m p t to interpret mechanical processes at the ice/rock interface o f glaciers using remains o f m i c r o f o r m s (with E. Ficker and S. Jecic). -
REFERENCES Boulton, G.S., Smith, G.D. and Morland, L.W. (1984). The reconstruction of former ice sheets and their mass balance characteristics using a non-linearly viscous flow method. J. Glaciology (to appear). Ficker, E. and Weber, E. (1981). Auf den Spuren der Gletscher oder Untersuchungen zum Problem dot Rissentstehung an Felsoberflffchen. VDI-Berieht, 399: 89-96. Fowler, A.C. and Larson, D.A. (1978). On the flow of polythermal glaciers - I. Model and preliminary analyses. Prec. R. Soc., A 36: 217-242. Gubler, H.U. (1981). Messungen an Fliesslawinen. Int. Ber. 600, EISLF, Dares. Gubler, H.U. (1984). The use of microwave FMCW radar in snow and avalanche research, Cold Reg. Sci. Teeh. (to appear). Hapgood Ch. and Campbell, J.H. (1958). Earth's Shifting Crust; a Key to Some Basic Problems of Earth Science. Foreword by Albert Einstein, New York, Pantheon. Hopflnger, E.J. (1983). Snow avalanche motion and related phenomena. Ann. Rev. Fluid Mech., 15: 47-76. Hurter, K. (1981). The effect of longitudinal strain on the shear stress of an ice sheet: in defence of using stretched coordinates. J. Glaciology, 27: 39- 56. Hutter, K. (1982a). A mathematical model of polythermal glaciers and ice sheets. Geophys. Astrophys. Fluid Dyn., 21 : 201-224. Hutter, K. (1982b). Dynamics of glaciers and large ice masses. Ann. Rev. Fluid Mech., 14: 87-130.
Hutter, K. (1983). Theoretical Glaciology. Reidel, Dordrecht, 510 pp. Hutter, K., Legerer, F. and Spring, U. (1981). First-order stresses and deformations in glaciers and ice sheets. J. Glaciology, 27: 227-270. Jenkins, J.T. and Savage, S.B. (1983). A theory for the rapid flow of identical, smooth, nearly elastic particles. Fluid Mech., 130: 187-202. Johnson, R.E. and McMeeking, R.M. (1982). Near-surface flow in glaciers obeying Glen's Law. Univ. Illinois T & AM Rep. 454. Kamh, W.B. (1970). Sliding motion of glaciers: Theory and observations. Rev. Geophys. and Space Phys., 8: 6 7 3 728. Lliboutry, L.A. (1975). Loi de glissement d'un glacier sans cavitation. Annales de G6ophysique, 31: 207-226. Lliboutry, L.A. (1979). Local friction laws for glaciers: a critical review and new openings. J. Glaciology, 23: 67-95. Lliboutry, L.A. (1981). Ecoulement a surface libre d'un fluide tr6s visqueux dans un chenal eylindrique. Comptesrendus Ac. Sci. Paris, 293: 333-336. Lun, C.K.K., Savage, S.B., Jeffrey, D.J. and Chepurniy, N. (1984). Kinetic theories for granular flow: Inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. (to appear). Morland, L.W. (1976a). Glacier sliding down an inclined wavy bed. J. Glaciology, 17: 447-462. Morland, L.W. (1976b). Glacier sliding down an inclined wavy bed with friction. J. Glaciology, 17: 463-4;/7. Morland, L.W. (1984). Thermomechanical balances of ice sheet flows. Geophys. Astrophys. Fluid Dyn. (to appear). Morland, L.W. and Boulton, G.S. (1975). Stress in an elastic hump: the effects of glacier flow over elastic bedrock. Proc. R. Soc., A344: 157-173. Morland, L.W. and Shoemaker, E.M. (1982). Ice shelf balances. Cold Reg. SCI. Tech., 5: 235-251. Morland, L.W. and Smith, G.D. (1984). Influences of nonuniform temperature distribution on the steady motion of ice sheets. J. Fluid Mech., 140: 113-133. Morland, L.W. and Johnson, I.R. (1980). Steady motion of ice sheets. J. Glaciology, 25 : 229-246. Morland, L.W. and Johnson, I.R. (1982). Effects of bed inclination and topography on steady isothermal ice sheets. J. Glaciology, 28: 71-90. Nye, J.F. (1957). The distribution of stress and velocity in glaciers and ice sheets. Proc. R. Sot., A239: 113-133. Nye, J.F. (1959). The motion of ice sheets and glaciers. J. Glaciology, 3: 493-507. Nye, J.F. (1965a). The flow of a glacier in a channel of rectangular, elliptic or parabolic cross-section. J. Glaciology, 5 : 6 6 1 - 6 9 0 . Nye, J.F. (1965b). The frequency response of glaciers. J. Glaciology, 5 : 567-588. Nye, J.F. (1965c). A numerical method of inferring the budget history of a glacier from its advance and retreat. J. Glaciology, 5: 589-607.
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Nye, J.F. (1969). A calculation of the sliding of ice over a wavy surface using a Newtonian viscous approximation. Proc. R. Soc., A311: 4 4 5 - 4 6 7 . Nye, J.F. (1970). Glacier sliding without cavitation in a linear viscous approximation. Proc. R. Soc., A315: 381-403. Nye, J.F. (1976). Water flow in glaciers: J~ikulhlaups, tunnels and veins. J. Glaciology, 17: 181-207. Reeh, N. (1968). On the calving of ice from floating glaciers and ice shelves. J. Glaciology, 7: 2 1 5 - 2 3 2 . Reeh, N. (1978). On the Application of Beam Theory to Glacier Mechanics. University of Copenhagen Geophysical Isotope Report, 217 pp. Roethlisberger, H. (1972). Water pressure in intra and subglacial channels. J. Glaciology, 11: 177-203. Savage, S.B. and Jeffrey, D.J. (1981). The stress tensor in a granular flow at high shear rates. J. Fluid Mech., 110: 255-272.
Scheiwiller, T. and Hutter, K. (1982). Lawincndynamik, Uebersicht fiber Experimente und theoretische Modelle yon Fliess- und Staublawinen. Mitteilung No. 58 tier Versuchsanstalt fiJr Wasserbau, Hydrologic und Glaciologie. Shreve, R.L. (1972). Movement of water in glaciers..1. Glaciology, 11 : 205 214. Spring, U. and Hutter, K. (1981). Numerical studies of J~Jkulhlaups. Cold Reg. Sci. Tech., 4: 227-244. Spring, U. and Hutter, K. (1982). Conduit flow of a fluid through its solid phase and its application to intraglacial channel flow. Int. J. Engr. Sci., 20: 327-363. Weertman, J. (1957). On the sliding of glaciers. J. Glaciology, 3: 3 3 - 3 8 . Weertman, J. (1964). The theory of glacier sliding. J. Glaciology, 5: 287-303.
77
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
REVIEW EUROMECH COLLOQUIUM 172: MECHANICS OF GLACIERS, INTERLAKEN, 19-.23 SEPTEMBER, 1983 Organi$ing Committee: K. Hutter* and U. Spring** Report by K. Hutter and L.W. Morland*** *Laboratory o f Hydraulics, Hydrology and Glaciology, ETH, Zurich (Switzerland) * *Electrowatt Engineering Services Ltd., Zurich (Switzerland) * **School o f Mathematics and Physics, University o f East Anglia, Norwich (U.K.)
(Received January 11,1984; accepted January 11,1984)
INTRODUCTION The objective of the Colloquium was to promote the interaction of scientists from a multidiscipline group comprising geologists, physicists, engineers and mathematicians in a forum focussing on snow and glacier mechanics. A format of formal lectures, typically of 45 minutes duration, accompanied by ample discussion periods provided a presentation of both practical and theoretical approaches, and, most importantly, a vigorous interchange of views from the wide spectrum of interests represented. Papers describing recently completed or current research were presented under eight broad headings, and here we attempt to summarize the subject matter covered in each section, incorporating some of the discussion points. These are, of course, personal interpretations which may not accurately reflect the lecture and discussion intentions. A list of lecturers, in alphabetical order, is appended, with their institutions, lecture titles, and co-workers when noted. Cited references are distinguished by author and publication year. In addition, a 12 hour field excursion was organised, starting with a trip to the Research Station on the Jungfraujoch, then a hike alongside the Grindelwald Oberergletscher.
CREEP FLOW OF ICE, SIMPLECONFIGURATIONS The crucial ingredient of all flow predictions is the constitutive model adopted for the long time
0165-232X/84/$03.00 © 1984 ElsevierSciencePublishers B.V.
creep response of ice. Lliboutry emphasized that natural glacier ice is a "metamorphic rock ice" with properties depending on its history. Laboratory tests on cold ice show, at constant stress, decelerating primary creep to a minimum (secondary or steady) strain rate followed by accelerating tertiary creep on recrystallisation, but the long time tertiary response is not realised. Deductions about the secondary response are often confused by primary and tertiary effects. He suggested that an ice sheet flow description requires a model reflecting the anisotropic nature of the fabric after a long stress history, and proposed a model based on a dissipation potential. Returning to the conventional incompressible non-linearly viscous fluid model (necessarily isotropic), Lliboutry discussed the flow down an inclined cylindrical channel and over an inclined plane bed. When the velocities and wall drag are uniform down the channel, and free surface accumulation is absent, the surface is necessarily plane (Lliboutry, 1981) and there is no transverse circulation. It follows for a Newtonian fluid that singularities (non-physical) arise at the intersection of the surface with a nonvertical side wall when no-slip is assumed, and also at the point of maximum velocity on the free surface, though such results have not been confirmed for a non-linearly viscous fluid. It was noted that the no-slip condition is appropriate to a cold glacier in which non-uniform temperature would modify the creep response, and that a temperate glacier at uniform temperature would exhibit sliding, casting doubt on the relevance of Nye's (1965a) calculations.
78 In the uniform depth plane flow with stresses depending only on depth (Nye, 1957), the basal shear stress is necessarily uniform along the length while any non-zero sliding velocity must vary when surface accumulation/ablation arises. This conflicts with a conventional sliding law relating basal shear stress and velocity. Lliboutry proposed a generalisation of Nye's solution for uniform (non-zero) accumulation and presented an illustration which exhibits an increase of differential velocity between surface and base by a factor of 1.65 when zero accumulation is replaced by 0.1 m/yr. The uniformity of the longitudinal velocity through the thickness of a floating ice shelf away from the front or any grounded region allows a simplifying integration of the longitudinal momentum balance over the thickness even when a significant temperature profile is considered (Morland and Shoemaker, 1982). Reeh (1968, 1978) has used the integrated equations of a small strain, small rotation beam theory to study non-steady wave forms which develop at the front of a shelf, and here discussed their application to steady wave forms in the grounding transition zone. First a Newtonian fluid model is adopted to allow analytic progress and an estimation of the errors arising from the beam theory approximation. The resulting stress compatibility failure is then improved by an additional longitudinal stress contribution depending on load so that compatibility can be satisfied in mean over the thickness. This modified beam theory is extended to temperature dependent non-linearly viscous models and less simple geometries. Such engineering approximations may yield useful results for complex flow regions, but must be compared carefully with the underlying system of equations and boundary conditions to assess the possibility of serious error.
LARGE SCALE PHENOMENA
Hapgood and Campbell's (1958) theory of the Earth's crust movement is based on supposed stresses induced by the addition of antarctic ice on time scales too short for isostasy to be attained. Spinelli introduced a model of the crust as a two-dimensional non-developable surface (one which cannot be
mapped continuously onto a plane), or shel|, subjected to forces in the tangent plane which would represent the ice effects, the shell resting on a rigid ellipsoid. The analysis is not complete, and stress results are not yet available. The application of an isothermal steady plane flow solution for a non-linearly viscous ice sheet (Morland and Johnson, 1980, 1982) to investigate the influence of net mass balance pattern, bed topography, and basal sliding conditions on the size and shape of the ice sheet was described by Smith. Relationships between span, accumulation pattern, and equilibrium line altitude (ELA) are obtained for a wide range of basal conditions, showing that only sliding near the margin has a significant influence, and that both pattern and ELA influence possible expansion or contraction (Boulton, Smith and Morland, 1984). An objective is to correlate former ice sheet extents with palaeoclimate. The model proposed by Nye (1965b, 1965c) to estimate the response of a glacier to change of accumulation was applied by Reeh to reconstruct the West Greenland ice margin movement around 69.5°N over the last 1400 years. A set of parallel steady plane flows covering the width of a sub-glacier is matched to EGIG and Dye 3 ice core data, but with estimations of missing information required. The calculations give good agreement with recorded margin fluctuations over the last 100 years, and reasonable agreement with advanced positions around AD 800 and AD 1700 and retreated positions over AD 11001450. The model then predicts that the present margin recession will continue for several decades if climatic conditions do not change dramatically.
GLACIER AND ICE SHEET G E O M E T R Y
The significant temperature variation through large ice sheets and shelves, and the strongly temperature-dependent viscisoty of ice, imply a strong thermomechanical coupling of the physical balances. A dimensionless analysis, and coordinate stretching which reflects the long aspect ratio or small surface slope of natural ice masses, displays the roles of diffusion, horizontal and vertical advection, and dissipation in the energy balance (Morland, 1984). On the assumption of a homogeneous non-linearly
79 viscous fluid response, Morland showed that a weak thermal basal boundary layer may arise in a thick grounded sheet, but not otherwise, and that dissipation should not be neglected in basal regions. Discussions pointed out that the older ice in basal regions of large sheets could be less viscous and could have anisotropic structure, when the homogeneous fluid model would not be satisfactory and larger strain rate gradients could arise, enhancing the boundary layer effect. It was also shown that seasonal thermal variations are restricted to a surface boundary layer. The perturbation analysis (Morland and Johnson, 1980, 1982) developed for an isothermal theory extends to the coupled theory, and reduces the steady plane flow problem to a simpler parabolic system of leading order partial differential equations offering a useful starting point for numerical solution. A simplification of the unsteady problem is also obtained. However, solutions of the steady plane coupled flow system have yet to be obtained, and these are essential to a convincing description of the flow and geometry of the conventional ice sheet model. Hurter (1982b, 1983) has already proposed an alternative iteration scheme, so we have two distinct computational methods to compare for accuracy and efficiency. The influence of prescribed temperature variation can be estimated for the temperature dependent viscosity model from the mass and momentum balances alone. Smith described a series of illustrations of steady plane flows under different plausible temperature patterns, obtained by an extension of the isothermal perturbation analysis which again reduces to a second order ordinary differential equation for the surface profile (Morland and Smith, 1983). These illustrations confirm that no mean temperature isothermal solution is satisfactory even for temperature variation with depth alone, and also that very small variation of temperature along the span can have significant effect, emphasising the need for a valid coupled solution. The regular perturbation analysis is strictly valid only for a viscosity bounded at zero shear stress, and while a leading order solution for steady plane flow with a power law of exponent n > 1, reflecting unbounded viscosity as stress approaches zero, was obtained (Morland and Johnson, 1980, 1982), restrictions on n were imposed by physical validity at a
margin and a divide. It was also noted that n > 1 leads to an unbounded longitudinal deviatoric stress at the free surface. A valid analysis for n > 1 using a singular perturbation method (Johnson and McMeeking, 1982) exhibits a surface boundary layer in which the longitudinal deviatoric stress is significant, but remains bounded. The important conclusion emphasised by Johnson is that the leading order velocity components, pressure, shear stress, and profile determined by the regular perturbation analysis are indeed correct, and the boundary layer is a passive correction of the higher order longitudinal stress behaviour. Perturbation analyses have also been used to 'investigate the effects of bed undulations on a steady plane flow solution, in particular the effects on the free surface geometry (Hutter, 1981; Hutter et al., 1981; Morland and Johnson, 1982). Each scheme requires that the magnitude O of the undulation slope relative to a mean bed plane is small, with further restrictions on undulation wavelength compared to ice thickness. The Hurter et al. (1981) analysis allows wavelengths of order thickness, but fails when the mean bed plane is nearly horizontal. It is essentially a central zone solution constructed as an iteration on a plane surface (uniform thickness) flow. Reeh described an extension of this iteration scheme for undulations in both longitudinal and transverse directions in the simplest case of an incompressible linearly viscous fluid, but considered a nearly horizontal bed plane for illustration. -'
BASAL FEATURES AND GLACIER SLIDING
Crescent shaped fractures and gouges of mean length between 30 and 1000 mm in the granite rock of former glacier beds can be used as indicators of former basal ice flow direction, which is perpendicular to the main chord of the fractured arches. Wintges presented arguments for basal mechanisms which could be responsible for the fractured features. The basic hypothesis is that only the flatter of the two bounding fractured arches is caused by basal flow processes, the remaining fracture zones being the result of thawing and refreezing processes under basically ice free conditions after retreat. Using experimental results on ultimate strength under trta~ial
8() stress conditions for the granite under consideration. and assuming that basal debris would cause the basal pressure and shear stress to peak wherever a boulder touches the base, computational and photoelastic methods were applied to model ice, boulder and rock by various artificial resins (Ficker and Weber, 1981). These investigations made the initiation of the crescent shaped fractures and gouges by the indicated processes a very likely one. Further work is in progress. Basal Friction with and without cavity formation is still a central question; it was approached from several different angles. Lliboutry presented Weertman's (1957, 1964) 'tombstone'-model and his own alternative consisting of a regular array of hemispheres mounted on a flat base. When cavities form, autonomous and non-autonomous regimes must be differentiated; in the former the water pockets are interconnected and drainage may occur, in the latter they are not. The model presented is very much in the realm of Lliboutry's (1979) review, but the tenor seems to be that basal shear traction T, sliding velocity V and effective pressure N are functionally related by a statement T = T (V, N). While this is commonly accepted in glaciology the explicit functional form of this relation is debatable. Meyssonnier applied the finite element method to obtain the friction law on a sinusoidal bed; melting, refreezing and cavity formation are ignored, and by contrast to classical treatments, non-infinitesimal bed inclinations are assumed. Extremum principles for velocities and stresses are used to obtain upper and lower bounds for the drag of a polycrystalline ice obeying Glen's flow law with n = 3. The computations indicate that for a given basal drag, sliding velocities are somewhat larger than those of Lliboutry (1975) and smaller than those of Kamb (1970). Meyssonnier also calculates the drag for a profile of several superimposed sine curves. These and the previous results will be made available in a forthcoming dissertation. When an incompressible linearly viscous fluid is assumed, the steady plane flow reduces to a biharmonic theory for the velocities and stresses, and harmonic theory for temperature when regelation is considered. For small amplitude, small slope periodic undulations of the bed, series expansions have been constructed by using Fourier transforms
(Nye, 1969, 1970; Kamb, 1970) and by using Hilbert transforms in a complex variable representation (Morland, 1976a, 1976b), when there is no cavitation. Except for very small mean bed inclination, the solutions predict that cavitation must occur. Fowler adopted the complex variable representation to fornmlate the boundary-value problem, ignoring regelation, when cavities are present. Cavities are zones across which the pressure of the basal ice is at a uniform cavitation pressure, but their end points are to be determined as part of the flow solution. Boundary conditions are distinct in the contact and cavity zones, giving rise to a mixed Hilbert problem. Periodicity allows the leading order plane boundary to be mapped onto a circle, and the leading order solution for a sinusoidal undulation was presented. This exhibits a monotonic basal shear stress sliding velocity relation which depends on the cavitation pressure, with the shear stress approaching an asymptotic limit with increase of sliding velocity while cavitation extends over the entire bed. Fowler asks if this has implications for a surge mechanism. If the boundary of a semi-infinite complex domain can be mapped conformally onto the real axis by a rational function, then a standard boundary-value problem for the plane biharmonic equation can be solved in terms of Cauchy integral representations (Morland and Boulton, 1975). By formulating the linearly viscous plane flow over a periodic bedform with humps of finite slope as the flow over an isolated hump (or humps) with conditions outside the hump region imposed to reflect the effects of the ignored periodic extension, rational mappings can be found for interesting shapes. The Cauchy integral method first reduces the mixed boundaryvalue problem to a singular integral equation, which in turn can be formulated as a non-singular integrodifferential equation. Jones described numerical solutions for the pressure and velocity variations over two different hump profiles, as part of" a continuing investigation of the influence of profile shape and hump separation on the drag-sliding velocity relation. These solutions will also be used to check finite element programs constructed for the non-linearly viscous fluid problem. Longitudinal strain rates of order 10 -7 s-i, corresponding to longitudinal deviatoric stresses of order l0 s Nm -2, are observed during glacier surges, which
81 are comparable with the basal shear stress in a steep glacier. The successful perturbation analysis for a small surface slope glacier (relative to the bed plane) is based on a co-ordinate stretching which implies that the longitudinal deviatoric stress is an order of magnitude smaller than the shear stress, with their roles reversed in a floating shelf perturbation analysis. No such rational scheme has been developed when both stresses are comparable, so that surge conditions still lack a complete description. McMeeking proposed that loss of adhesion in a region of basal rupture, which would be a consequence of particular sliding conditions, must transfer the effects of local glacier weight to the downstream ice, inducing higher longitudinal stress and possible spread of the rupture region. As illustration he starts with a rupture region of order the ice thickness, on which the basal shear stress is zero, and deduces the leading order velocity field to be matched (crudely) with the flows upstream and downstream of the rupture region. The possibilities of rupture spread and of adhesion recovery were discussed.
WATER IN GLACIERS
Today no-one denies the importance of water in the dynamics of glaciers. However, there are two different points of view: either one looks at the effect of the water on the behavior of the ice, or that of the glacier on the water within it. Iken presented measurements of glacier movement and subglacial water pressure on Findelengletscher from May/June 1982. Surface ice velocities were measured several times a day at 4 transverse stake profiles and, in the same area, bore holes were drilled to the glacier bed, connecting to the subglacial drainage system, and water pressures were recorded. Velocity variations correlate well with the variations in water pressure, even for very low water pressure, and are thought to be explicable by a sliding mechanism with cavity formation. Since the data suggest a low separation pressure and small bed roughness, the observed conditions would cause the glacier to become unstable, which is obviously not the case, so friction between bedrock and debris
is also included in the sliding mechanism and is shown to stabilize the glacier sliding. If correct it has both a considerable stabilizing effect on the glacier flow (sliding velocity) and a dramatic effect on the hydrology (water table) of the glacier. Spring presented model calculations on turbulent flow in intraglacial channels in which an evolution equation for the temperature was taken into account. The original ideas date back to Shreve (1972), Roethlisberger (1972) and Nye (1976) but the model equations were based on the Spring and Hutter (1981, 1982) equations. The importance of temperature was emphasized and supported by a collection of measurements of water temperature of the Gornersee at the subglacial outlet cross section, indicating a surprisingly large range of temperature variation of the draining lake water. Spring's steady state equations differ from those of Roethlisberger (Rchannels!). Numerical solutions for simple ice configurations subject to different temperature and water pressure conditions at the boundaries of the conduit demonstrate that, to each given set of external parameters, either two steady conduits exist, or none. Of the two, only one is stable. Polythermal glaciers consist of cold and temperate zones separated by a discontinuity surface. To properly describe the interaction, the transport of ice plus moisture must be described together with associated boundary and transition conditions at the cold-temperate interface. Fowler and Larson's (1978) and Hutter's (1982a) models take the moisture transport into account by a diffusion equation. Fowler presented a two-phase concept formulating mass and momentum balances for water and ice separately and energy balance for the mixture. In this model the water flows in accordance with Darcy's law and the water pressure is given by Nye's (1976) vein-closure condition, complemented by a new term due to melting. Boundary conditions at the cold-temperate transition surface follow from the jump conditions of the balance laws but open additional questions regarding a proper formulation. A typical feature of the model is that temperate ice and water are both at melting temperature, avoiding water temperature as an additional field variable.
82
SNOW AVALANCHI::~
Snow avalanches exist in various forms, two limiting cases being the wet flow avalanche and the airborne powder snow avalanche. Classically, hydraulic concepts are used for their description (VoellmySalm theory) and both are treated alike. More recently, theoretical and experimental studies on the flow of cohesionless granular materials have led to a new and more detailed understanding of flow avalanches (Lun et al., 1984; Scheiwiller and Hutter, 1982) and turbulent two-phase models have been suggested to describe powder snow avalanches (Scheiwiller and Hutter, 1982). Savage presented extensions of the granular flow kinetic theories of Savage and Jeffrey (1981) and Jenkins and Savage (1983). In the evaluation of the statistical averages he regarded the particles as finite spheres and included both the collision and kinetic (advective) transports in the evaluation of the 'granular' stresses, energy flux and energy dissipation. The model incorporates two phenomenological collisional coefficients which permit non-conservation of translational and rotational energy in binary collisions. The complete polar theory is not yet published but its non-polar counterpart is submitted for publication (Lun et al., 1984). The potential applicability of the model was made plausible by demonstrating agreement of the theoretical model with rapid shear flow of granules down a chute. The most important mechanism a model of powder snow avalanches must incorporate is snow settlement. Both Savage and Scheiwiller presented such models. Savage extended Hopfinger's (1983) 'inclined thermal model' by looking at a cloud of particles moving down a slope, in which the sedimentation of the particles was accounted for by a mass balance of the settling particles. Using this, an entrainment hypothesis for the ambient fluid and the momentum equation, predicted flows compared favourably with laboratory experiments. The more complete two-phase model of powder snow avalanches presented by Scheiwiller permits (in general) the prediction of mean flow and density profiles by air and particles, and allows incorporation of air and snow entrainment from above and below respectively. Snow settlement is incorporated ab initio by the two phase nature of the model. A boundary layer approximation for plane flow down
an inclined rough plane was presented, and two numerical procedures (Kantorovich and FD-methods) were discussed, by which steady plane flow in an avalanche tail could be obtained. Results look promising but must still be verified experimentally. The physical mechanisms that are responsible for the formation of powder snow avalanches from flow avalanches are still not fully understood. Hutter suggested that under a certain flow state, characteristic of the transition, air entrainment into the avalanche will be enhanced and cause tile rapid 'granular' shear flow to become unstable. A binary mixture model appropriate for flow avalanches is presented in which the snow particles obey the Jenkins-Savage (1983) theory while the air is modelled as a linearly viscous fluid. At the free surface (this is a discontinuity surface at which the jump conditions of mass and momentum nmst be satisfied) air is entrained, and this entrainment is functionally related to a surface Richardson number; the functional relationship is selected to enforce the transition from flow to powder snow avalanche. It is demonstrated how simple shear flow of a strictly parallel-sided flow can be analysed and how its stability can be studied. Field observations of snow avalanches are important in order that danger zones can be classified. Depending on severeness, three different subdomains are distinguished and forecasted using essentially either hydraulic or hydrodynamic channel flow models. To improve the classification, flow properties in actual avalanches must be studied. Gubler described microwave radar systems developed at the Swiss Institute for Snow and Avalanches Research capable of measuring speed (profiles) and flow heights of a dense flow avalanche. The equipment, mounted on oversnow vehicles and able to perform several measuring cycles in one avalanche event, has a radar range of 2 km and resolutions of 10 m, 0.5 ms -1, 0.2 s for in-plane distances, speeds and time, respectively. The system, described in Gubler (1981, 1984), has been tested for small size, dry snow avalanches, and results have been compared with the classical models (Voellmy-Salm-Perla). The ultimate goal is to obtain detailed information on in-situ flow properties which allow extension of existing models and corroborate or disprove the new theoretical models. In a field program on gliding snow avalanches, in
83 Tirol (Austria), described by Lackinger, causes of avalanche initiation and, after the fact, avalanche extent are sought. Indications of avalanche formation are, among others, distinct fractures of the snow pack and variations of seismicity, which was recorded by piezoelectric acceleration gauges. Gliding velocities and run-out distances were also measured as were meteorological and topographic routine data. Avalanche initiation was explained by means of static force balances, familiar in soil mechanics.
CREEPING SNOW
Snow 'at rest' was studied in two contributions. Ambach described the deformation of a 20 m deep firn pit in the accumulation area of the Kesselwandferner (Oetztal, Austria). The originally circular (2 m diameter) pit translated approximately 200 m over the 11 year long observation period, and exhibited vertical compaction and shear deformation. The pit axis was bent and its 14 observed cross sections deformed nearly into ellipses. Longitudinal and shear strain rates (engineering strains) were measured at various points along the circumference of the pit, as were all geometric data pertaining to compaction and local geometry. Linear and non-linear viscous constitutive models with shear and bulk viscosities were suggested as a suitable constitutive behavior between stress and strain rate but a theoretical reconstruction of the pit deformation was not given. The description of the flow o f mass and energy through a snow pack should be modeled by a fourphase mixture concept of snow, meltwater, vapour and air, but current understanding, inferred from both thermodynamic and experimental considerations is not yet sufficiently developed. Instead, Morris presented a hierarchy of reduced snowmelt forecasting models, one based on a two-phase concept using snow and meltwater as the constituents, the other being a 'classical' energy budget model. The boundary conditions of the two-phase model are based on empirical equations describing the physical processes in the atmospheric boundary layer and the underlying soil. This model, prepared for the Syst~me Europ~en Hydrologique, is appropriate for ripe snow in the funicular r~gime, and appears to be superior to the energy budget model.
BRIEF PRESENTATIONS
Veteran glaciologist Henri Bader drew attention to the texture (orientation pattern of grain axes) in ice, and the question of whether this was satisfactorily represented by specifying the distribution of grain sizes. He recalled an early experiment in which identical plasticene spheres with lubricated surfaces were placed loosely in a rubber bag, and then the air was evacuated to produce a space-filling aggregate of polyhedra. Sectioning then showed that the most frequent number of faces was 11, and the most frequent number of edges per face was 5. He asked if a theoretical correlation between plane section properties and an initial volume distribution of unequal spheres is possible, so that, by analogy, plane ice section properties could be correlated with grain size distribution. Fast sea ice dynamics in the open ocean have been studied by AIDJEX. Ito reported that observations of fast sea ice in the Greenland-Iceland-Spitzbergen area, however, indicate that mesoscale ice flow behavior is chiefly boundary dominated, a situation apparently not adequately covered by the theoretical formulations arising from AIDJEX. MIZEX will observe sea ice motion in the Greenland-IcelandSpitzbergen area and plans to develop new constitutive models that are also appropriate in near boundary zones. In a short film Savage demonstrated granular chute flow from a steady source. It indicated that, depending on discharge, chute inclination and downstream conditions, oscillatory flow conditions may develop. Inhibiting the downstream flux resulted in a granular jump, similar to a hydraulic bore. Variegated Glacier is currently very active, going through a surging phase, and observations from the 1983 campaign were shown and interpreted by Iken.
CONCLUDING REMARKS (LWM)
This Colloquium encouraged a wide audience participation, and demonstrated that the views held by practical glaciologists and the recently arrived mathematical enthusiasts do have common ground, which augurs well for the development
84 of glacier mechanics. The success of the forum was in no small part due to the efficient organisation and constant attention of Drs. Hurter and Spring, and the excellent venue they arranged in collaboration with the Swiss Government. There was a unanimous call for a repeat performance in the not too distant future (setting aside a British comment about lecture programmes from 8.30 a.m. to 6 p.m.).
APPENDIX Lecturers (in alphabetic order), titles of their presentations and coworkers
Prof. Dr. W. Ambach, University of Innsbruck, Innsbruck (Austria). - Viscosity parameters of high density firn from field experiments (with H. Eisner). Dr. H. Bader, Pompano Beach, Florida (U.S.A.). Texture in ice. Dr. E. Ficker, University of Munich (Germany). An ansatz to interpret mechanical processes at the ice rock interface of glaciers using remains of microforms. Prof. Dr. A.C. Fowler, MIT, Cambridge (U.S.A.). - Glacier sliding with cavitation; On the transport of moisture in polythermal glaciers. Dr. H. Gubler, Weissfluhjoch-Davos (Switzerland). - M e a s u r e m e n t s of speed distributions and flow heights in dense flow snow avalanches. Dr. K. Hutter, ETH, Zurich (Switzerland). - S o m e thoughts on the transition flow/powder snow avalanches (with T. Scheiwiller). Dr. A. Iken, ETH, Zurich (Switzerland). Measurements of glacier movement and subglacial water pressure, Findelengletscher, May/June 1982. Conclusions on sliding mechanism and bed roughness; Observations on Variegated Glacier. Dr. H. Ito, Alfred Wegener Institute, Bremerhafen (Germany). Mesoscale, boundary affected sea ice dynamics. Prof. Dr. R.E. Johnson, University of Illinois, UrbanaChampaign (U.S.A.). Near surface flow in glaciers obeying Glen's law (with R.M. McMeeking). -
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Dr. A.S. Jones (visiting University of East Anglia. Norwich (U.K.)), University of Queensland. Brisbane (Australia). - Pressure distribution over a finite hump beneath a glacier (with L.W. Morland). Dr. B. Lackinger, University of Innsbruck, Innsbruck (Austria). - Snow gliding and gliding snow avalanches. Prof. Dr. L.A. Lliboutry, Lab. de Glaciologie, Grenoble (France). - Creep law of ice (with P. Duval); Flow in a cylindrical channel; - Nye flow; - Friction laws for glaciers (with J. Meysonnier). Prof. Dr. R. McMeeking (visiting Cambridge University, Cambridge (U.K.)), University of Illinois, Urbana-Champaign (U.S.A.). - A model for glacier surge due to basal rupture (with R.E. Johnson). Dr. J. Meyssonnier, Lab. de Glaciologie, Grenoble (France). - Friction of ice over a sine profile derived from FEM computations. Dr. L.W. Morland, University of East Anglia, Norwich (U.K.). Thermomechanical balances of ice sheet flows. Dr. E.M. Morris, Institute of Hydrology, Wallingford (U.K3. - Modelling the flow of mass and energy through snow. Dr. N. Reeh, University of Copenhagen, Copenhagen (Denmark). Steady state wave forms on floating glaciers and ice shelves treated by beam theory methods; Response of the West Greenland ice sheet margin to a 2600 year long mass balance history deduced from a Greenland ice core record; Steady state three-dimensional ice flow over an undulating base. First order theory with linear ice rheology. Prof. Dr. S.B. Savage, McGill University, Montreal (Canada). Granular flow mechanics and snow avalanches. Mr. T. Scheiwiller, ETH Zurich (Switzerland). of powder snow avalanches (with K. Hurter). -
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Dr. G.D. Smith, University o f East Angiia, N o r w i c h (O.K.). Influence o f n o n - u n i f o r m t e m p e r a t u r e distribution on the steady m o t i o n o f ice sheets (with L.W. Morland); - The reconstruction o f former ice sheets and their mass balance characteristics using a non-linear viscous flow m o d e l (with G.S. B o u l t o n and L.W. Morland). Prof. Dr. G. Spinelli, Politecnico di Milano, Milano (Italy). - A first order m o d e l o f the crust o f the earth for calculating stresses due to a tangential force. Dr. U. Spring, E l e c t r o w a t t Engineering Services Ltd., Zurich (Switzerland). - T u r b u l e n t flow in intraglacial conduits. Temperature-induced instabilities. Dr. T. Wintges, Techn. University Munich (Germany). - An a t t e m p t to interpret mechanical processes at the ice/rock interface o f glaciers using remains o f m i c r o f o r m s (with E. Ficker and S. Jecic). -
REFERENCES Boulton, G.S., Smith, G.D. and Morland, L.W. (1984). The reconstruction of former ice sheets and their mass balance characteristics using a non-linearly viscous flow method. J. Glaciology (to appear). Ficker, E. and Weber, E. (1981). Auf den Spuren der Gletscher oder Untersuchungen zum Problem dot Rissentstehung an Felsoberflffchen. VDI-Berieht, 399: 89-96. Fowler, A.C. and Larson, D.A. (1978). On the flow of polythermal glaciers - I. Model and preliminary analyses. Prec. R. Soc., A 36: 217-242. Gubler, H.U. (1981). Messungen an Fliesslawinen. Int. Ber. 600, EISLF, Dares. Gubler, H.U. (1984). The use of microwave FMCW radar in snow and avalanche research, Cold Reg. Sci. Teeh. (to appear). Hapgood Ch. and Campbell, J.H. (1958). Earth's Shifting Crust; a Key to Some Basic Problems of Earth Science. Foreword by Albert Einstein, New York, Pantheon. Hopflnger, E.J. (1983). Snow avalanche motion and related phenomena. Ann. Rev. Fluid Mech., 15: 47-76. Hurter, K. (1981). The effect of longitudinal strain on the shear stress of an ice sheet: in defence of using stretched coordinates. J. Glaciology, 27: 39- 56. Hutter, K. (1982a). A mathematical model of polythermal glaciers and ice sheets. Geophys. Astrophys. Fluid Dyn., 21 : 201-224. Hutter, K. (1982b). Dynamics of glaciers and large ice masses. Ann. Rev. Fluid Mech., 14: 87-130.
Hutter, K. (1983). Theoretical Glaciology. Reidel, Dordrecht, 510 pp. Hutter, K., Legerer, F. and Spring, U. (1981). First-order stresses and deformations in glaciers and ice sheets. J. Glaciology, 27: 227-270. Jenkins, J.T. and Savage, S.B. (1983). A theory for the rapid flow of identical, smooth, nearly elastic particles. Fluid Mech., 130: 187-202. Johnson, R.E. and McMeeking, R.M. (1982). Near-surface flow in glaciers obeying Glen's Law. Univ. Illinois T & AM Rep. 454. Kamh, W.B. (1970). Sliding motion of glaciers: Theory and observations. Rev. Geophys. and Space Phys., 8: 6 7 3 728. Lliboutry, L.A. (1975). Loi de glissement d'un glacier sans cavitation. Annales de G6ophysique, 31: 207-226. Lliboutry, L.A. (1979). Local friction laws for glaciers: a critical review and new openings. J. Glaciology, 23: 67-95. Lliboutry, L.A. (1981). Ecoulement a surface libre d'un fluide tr6s visqueux dans un chenal eylindrique. Comptesrendus Ac. Sci. Paris, 293: 333-336. Lun, C.K.K., Savage, S.B., Jeffrey, D.J. and Chepurniy, N. (1984). Kinetic theories for granular flow: Inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. (to appear). Morland, L.W. (1976a). Glacier sliding down an inclined wavy bed. J. Glaciology, 17: 447-462. Morland, L.W. (1976b). Glacier sliding down an inclined wavy bed with friction. J. Glaciology, 17: 463-4;/7. Morland, L.W. (1984). Thermomechanical balances of ice sheet flows. Geophys. Astrophys. Fluid Dyn. (to appear). Morland, L.W. and Boulton, G.S. (1975). Stress in an elastic hump: the effects of glacier flow over elastic bedrock. Proc. R. Soc., A344: 157-173. Morland, L.W. and Shoemaker, E.M. (1982). Ice shelf balances. Cold Reg. SCI. Tech., 5: 235-251. Morland, L.W. and Smith, G.D. (1984). Influences of nonuniform temperature distribution on the steady motion of ice sheets. J. Fluid Mech., 140: 113-133. Morland, L.W. and Johnson, I.R. (1980). Steady motion of ice sheets. J. Glaciology, 25 : 229-246. Morland, L.W. and Johnson, I.R. (1982). Effects of bed inclination and topography on steady isothermal ice sheets. J. Glaciology, 28: 71-90. Nye, J.F. (1957). The distribution of stress and velocity in glaciers and ice sheets. Proc. R. Sot., A239: 113-133. Nye, J.F. (1959). The motion of ice sheets and glaciers. J. Glaciology, 3: 493-507. Nye, J.F. (1965a). The flow of a glacier in a channel of rectangular, elliptic or parabolic cross-section. J. Glaciology, 5 : 6 6 1 - 6 9 0 . Nye, J.F. (1965b). The frequency response of glaciers. J. Glaciology, 5 : 567-588. Nye, J.F. (1965c). A numerical method of inferring the budget history of a glacier from its advance and retreat. J. Glaciology, 5: 589-607.
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Nye, J.F. (1969). A calculation of the sliding of ice over a wavy surface using a Newtonian viscous approximation. Proc. R. Soc., A311: 4 4 5 - 4 6 7 . Nye, J.F. (1970). Glacier sliding without cavitation in a linear viscous approximation. Proc. R. Soc., A315: 381-403. Nye, J.F. (1976). Water flow in glaciers: J~ikulhlaups, tunnels and veins. J. Glaciology, 17: 181-207. Reeh, N. (1968). On the calving of ice from floating glaciers and ice shelves. J. Glaciology, 7: 2 1 5 - 2 3 2 . Reeh, N. (1978). On the Application of Beam Theory to Glacier Mechanics. University of Copenhagen Geophysical Isotope Report, 217 pp. Roethlisberger, H. (1972). Water pressure in intra and subglacial channels. J. Glaciology, 11: 177-203. Savage, S.B. and Jeffrey, D.J. (1981). The stress tensor in a granular flow at high shear rates. J. Fluid Mech., 110: 255-272.
Scheiwiller, T. and Hutter, K. (1982). Lawincndynamik, Uebersicht fiber Experimente und theoretische Modelle yon Fliess- und Staublawinen. Mitteilung No. 58 tier Versuchsanstalt fiJr Wasserbau, Hydrologic und Glaciologie. Shreve, R.L. (1972). Movement of water in glaciers..1. Glaciology, 11 : 205 214. Spring, U. and Hutter, K. (1981). Numerical studies of J~Jkulhlaups. Cold Reg. Sci. Tech., 4: 227-244. Spring, U. and Hutter, K. (1982). Conduit flow of a fluid through its solid phase and its application to intraglacial channel flow. Int. J. Engr. Sci., 20: 327-363. Weertman, J. (1957). On the sliding of glaciers. J. Glaciology, 3: 3 3 - 3 8 . Weertman, J. (1964). The theory of glacier sliding. J. Glaciology, 5: 287-303.