The plane steady flow of a grounded ice sheet is numerically analysed using the approximate model of Morland or Hutter. In this, the ice behaves as a non-linear viscous fluid with a strongly temperature-dependent rate factor, and ice sheets are assumed to be long and shallow. The climate is assumed to be prescribed via the accumulation/ablation distribution and the surface temperature, both of which are functions of position and unknown height. The rigid base exerts external forcings via the normal heat flow, the geothermal heat, and a given basal sliding condition connecting the tangential velocity, tangential traction, and normal traction. The functional relations are those of Morland (1984) or motivated by his work. We use equations in his notation.

The governing equations and boundary conditions in dimensionless form are briefly stated and dimensionless variables are related to their physical counterparts. The thermo-mechanical parabolic boundary-value problem, found to depend on physical scales, constitutive properties, and external forcing functions, has been numerically solved. For reasons of stability, the numerical integration must proceed from the ice divide towards the margin, which requires a special analysis of the ice divide. We present this analysis and then describe the versatility and limitations of the constructed computer code.

Results of extensive computations are shown. In particular, we prove that the Morland–Hutter model for ice sheets is only applicable when sliding is sufficiently large (satisfying inequality (30)). In the range of the validity of this inequality, it is then demonstrated that of all physical scaling parameters only a single π-product influences the geometry and the flow within the ice sheet. We analyse the role played by advection, diffusion, and dissipation in the temperature distribution, and discuss the significance of the rheological non-linearities. Variations of the external forcings, such as accumulation/ablation conditions, free surface temperature, and geothermal heat, demonstrate the sensitivity of the ice-sheet geometry to accumulation conditions and the robustness of the flow to variations in the thermal state. We end with a summary of results and a critical review of the model.

This paper examines the deterioration of an iceberg grounded outside St. John’s Harbour, Newfoundland, Canada, in terms of its initial velocity prior to grounding. Theoretical expressions for the lifting of the iceberg and hence its buoyancy loss during grounding are derived as a function of initial iceberg velocity and ocean-bottom slope. Wave erosion and calving are two of the most significant mechanisms for iceberg deterioration. With wave erosion occurring on the seaward side of the grounded iceberg, model simulations are compared with observational data from a field study conducted on the grounded iceberg between 10 and 17 June 1983. Model–simulated time of re-flotation of the iceberg agrees with observations, for initial iceberg velocity of 0.3–0.5 m/s. Model simulations of the deterioration of the iceberg beyond the point of re-flotation are also compared with observations. Allowing for a 10% error in the observed above-water volume of the iceberg on 10 June, the model-simulated mass losses are in good agreement with observations. Best results are obtained for the model initialized with data observed on 14 June 1983, the first day for which detailed observational data are available following re-flotation of the iceberg.

Based on observations of naleds (also called aufeis or icings) grown in a refrigerated laboratory flume, a detailed, composite description of the processes associated with naled ice growth is presented. A spread length is derived that represents a mass balance between the water supplied to a naled’s surface and the ice that freezes on to the surface. Herein, this spread length is termed an equilibrium length. Guided by this concept of equilibrium length and the laboratory data, length- and time-scales applicable to the growth of two-dimensional naleds are proposed. These scales proved useful for normalizing the times and streamwise lengths that correspond to distinct phases of naled ice growth. It is observed that, even in a laboratory flume, naleds spread and thicken in a complex, layer-by-layer manner.

To the best of the authors’ knowledge, this is the first reported laboratory study on naled ice growth. The descriptions and concepts presented herein should be useful to engineers concerned with the effects of naleds on engineering works, and of interest to those who are planning experiments on naleds.

Wire strain meters and seismometers spaced longitudinally along the upper part of Variegated Glacier, Alaska, showed quasi-periodic episodes of increased velocity (mini-surges), which lasted about 1 day and recurred at intervals of a few days to 2 weeks during the early part of the melt seasons of 1979, 1980, and 1981. The zone affected by these mini-surges corresponds to the zone of highest velocity and basal stress increase over the previous decade, and the initiation of the most recent surge in 1982. Mini-surges initiate locally; as a single melt season progresses, the later mini-surges start at higher locations and show a distinct down-glacier propagation of a characteristic strain pattern and associated zone of acoustic emissions at speeds of 0.1–0.6 km h−1. During mini-surges, extensile and compressive strain-rates exceed 10 × 10−4 d−1 and 40 × 10−4 d−1, respectively; typical strain-rates between mini-surges were less than 2 × 10−4d−1 in magnitude. Seismic activity jumped by two orders of magnitude and was distinctly audible during a mini-surge. Maximum strain-rate during mini-surges decreased from year to year. The high time resolution of the strain allows short time-scale structure of velocity variations to be deduced. As a propagating mini-surge passes, the velocity anomaly at a fixed location is characterized by a rapid initial rise over a few hours to two distinct peaks separated by a few hours, followed by a slower return to normal velocity taking up to a day. The double peak in velocity may arise from a single, very sharp, transient peak in the basal slip velocity associated with the initial opening of cavities at the base in response to a sudden rise in basal water pressure (observed by Kamb and Engelhardt). This supports an important role for basal cavitation in the mini-surge mechanism.

Radio-echo signals travel faster in firn than in ice, which affects the analysis of ice-mass thickness. If this effect is neglected, then the calculated thickness of an ice mass may be in error by an amount of the order of one-tenth of the firn thickness. An exact formulation is derived for the path of the signal through the firn and the ice. Explicit solutions are given for constant, linear, and quadratic profiles of the refractive index in the firn layer.

Discharges of water, sediment, and dissolved impurities from Variegated Glacier, Alaska, were monitored in the early summers of 1980 and 1981 during the occurrence of mini-surges. Seasonal trends, weather-related events, and diurnal variations similar to behavior of other temperate glacier streams were found. The principal effect in the stream associated with mini-surge occurrence was a brief discharge of extremely turbid water. The turbidity is assumed to be introduced into the basal hydraulic system by initiation of the fast motion of a mini-surge at a time and location on the upper glacier known from other measurements. The mean water velocity in the hydraulic system over the intervening distance is thereby determined (0.3 ms−1). The mean water velocity, together with the water discharge (≈16 m3 s−1 at the terminus), places constraints on the distribution of water velocity u and total cross-sectional area AT of the flow paths along the glacier base. This leads to the conclusion that within the zone of mini-surge occurrence in its unperturbed state: u is about 0.1 ms−1 or possibly less; AT is about 102 m 2 or possibly more, and it must be divided into a very large number of small passageways, be blocked by constrictions, or both. The total water cross-section corresponds to a layer 0.1–0.2 m thick when spread uniformly over the glacier width. The water velocity is close to or less than the propagation velocity of the mini-surges. Between the zone affected by mini-surges and the stream, a dynamically less active lower section of the glacier is probably underlain by a small number of conduits, in which the water velocity may be very high (≥ 2 m s−1). Water discharge following the mini-surges puts an upper limit on water-storage changes associated with the anomalous ice motion.

The net mass balance of four small cirque glaciers (0.7–1.4 km2) in the Torngat Mountains of northern Labrador was measured for 1981–84, allowing three complete mass-balance years to be calculated. The two largest glaciers experienced positive mass-balance conditions in 1982 while all the glaciers were negative in 1983. The temporal pattern relates directly to general climatic conditions, in particular winter snowfall. Spatial variations of mass balance on the glaciers are the result of several factors including altitude, extent of supraglacial debris cover, slope, proximity to side and backwalls of the enclosing cirque, and the height of the backwall above the ice surface. Abraham Glacier, the smallest studied and with consistently the largest negative mass balance (–1.28 m in 1983), re-advanced an average of 1.2 m each year between 1981 and 1984. Mean equilibrium-line altitude (ELA) for the four glaciers is 1050 m, varying substantially from one glacier to another (+240 to –140 m) and from year to year (+60 to –30 m).

Large ice masses contain a wealth of information regarding past climates and atmospheric chemistry. To interpret properly information from ice cores obtained from glaciers, a time-scale for the ice core must be established. A procedure based on the finite-element method, using velocity-pressure and stream-function formulations to establish particle paths and hence isochrones, is outlined. Examples are presented which demonstrate the ability of the procedure to predict particle paths and isochrones which can be used to determine the time-scale or to confirm dates established by other methods, of ice cores obtained from large ice masses.

In order to determine the effect of basal debris drag as a component in a sliding law, it is necessary to know the basal debris concentration. Does, for example, this debris concentration exhibit uniformity, in an average sense, across a valley? What effect does a localized region of high quarryability have on basal debris concentration down-stream? The analysis presented here suggests that, for the case of sparse debris, debris concentration tends to be uniform over the entire bed. Consequently, the debris-drag term in a sliding law will also tend to be uniform.

In order to reach the above conclusions, it is necessary to consider debris balance, the balance of the quarrying, abrasion, and debris-flushing rates. This entails proposing and testing quarrying and flushing laws, since these laws have not been previously proposed. It is concluded that the quarrying, flushing, and erosion rates depend weakly on the sliding velocity. Furthermore, the abrasion rate is negligible compared to the quarrying rate. (This conclusion depends partially on the definition of abrasion.) The quarrying rate also depends on other factors such as the effective pressure_pressure-fluctuation mechanism; the cross-valley dependency of this mechanism is investigated. The flushing rate is found to depend on the ratio, N-channel area/bed area, and on average N-channel diameter.

Conduit and regelation water are inferred to drain as ground water from up to 130 km2 of Columbia Icefield. Subglacial conduits appear to be generally occupied by free surface streams. Ground-water flow will allow exchange between the regelation film and conduits. Present-day discharge from beneath the accumulation zone carries little sediment, and past injections of sediment appear to correspond to ancient interglacials. Transport of sediment through cave passages may be analogous to transport through basal conduits.

The late Wisconsin Shelbyville till was deposited in southern Illinois c. 20 000–21 000 year B.P. and records the maximum southern advance of the Lake Michigan lobe of the Laurentide ice sheet. The yield strength calculated for a representative till debris flow found just south of the ice margin is 8 kPa (0.08 bar), and probably approximates yield strength of basal Shelbyville till. An ice-profile model assuming plastic behavior in basal till suggests the southern Lake Michigan lobe may have been unusually thin. Reconstructed Laurentide glacier profiles from the south-west and western Great Plains (South Dakota, Alberta, Minnesota, and Montana), and the MacKenzie Delta, N.W.T., are similar to those inferred for the southern Great Lakes area, and much thinner than those of most modern ice sheets. The Pleistocene Laurentide ice sheet may have been asymmetric: thicker in the east than in the west. Glaciers resting on weak sediments can move both by subglacial sediment deformation (creep) and sliding at the sediment–ice interface. Till rheology is complex; shearing of till by over-riding glaciers would increase porosity and further reduce yield strength.

Shipboard observations (in accordance with Norsk Polar-Institutt guidelines) from 6 years of Australian National Antarctic Research Expeditions (ANARE) voyages have provided data giving a detailed knowledge of iceberg sizes and concentrations in the Southern Ocean between long. 60° and 140° E. The resulting size–frequency distributions are examined in conjunction with a knowledge of water movement along known drift tracks in a selected study area (between lat. 59° and 64° S., and long. 90° and 120° E.) to determine iceberg-dissolution rates. The “median life” (before breaking) of icebergs less than 1000 m in horizontal dimension is estimated to be 0.2 a, which is significantly lower than was previously thought. The mean melt rate is estimated to be 0.12 m d–1, which agrees broadly with previous laboratory studies. The relative contributions of melt, calving, and breakage, plus the enhancement effect of roll-over, are examined in estimating the natural dissolution rate. Breakage appears to be the dominant mechanism for larger icebergs with melt and calving able to explain the disappearance of icebergs in the smallest categories only (within the mean “median-life” period). Examination of the historical records of Captain Cook indicates that iceberg concentrations, as well as the northerly extent in this region 200 years ago, were compatible with the present data.

Satellite images recorded in 1973, 1974, and 1985 of the Brunt Ice Shelf are compared. There are sufficient identifiable features moving with the ice shelf to show flow patterns over an area of ~10 000 km2. Velocities vary from 500 m a−1 in the west, near the Dawson-Lambton Ice Stream, to 1300 m a−1 in the east, within the Stancomb-Wills Ice Stream.

A mathematical model is presented of non-stationary melting processes of ice including particles of morainic material. The problem is treated as a Stephen-type one with the phase boundary of ice melting being located under the debris cover. The main terms of the heat-balance equation for a glacier surface are solar radiation and convective heat transfer. The quantitative relationships characterizing the effect of glacier run-off augmentation from under a thin layer of debris cover are obtained for different bulk moraine concentrations inside the ice. The concept of equivalent time is introduced. It is defined as the time elapsed until the moment the sub-moraine ice-ablation rate becomes equal to the ablation rate of clean ice. This moment signifies the beginning of the shielding stage. Thus, a glacier can be considered as a self-controlling system with respect to its summer run-off. A series of numerical tests for Djankuat glacier, Central Caucasus, has been carried out. The dynamics of moraine-cover growth and alterations of seasonal ablation rate under debris show perfect agreement between the computed data and the results of 14 years of direct observations. Some practical recommendations concerning artificial blackening of a glacier surface for augmentation of liquid run-off are presented. Conditions promoting increase of run-off are: relatively high albedo, relatively low summer air temperature, and relatively small convective heat transfer between the air and the ice surface. The method of artificially blackening a glacier surface is by means of a durable thin dark polymer film. In conclusion, some further aspects of the problem are discussed.

For a glacier flowing over a bed of longitudinally varying slope, the influence of longitudinal stress gradients on the flow is analyzed by means of a longitudinal flow-coupling equation derived from the “vertically” (cross-sectionally) integrated longitudinal stress equilibrium equation, by an extension of an approach originally developed by Budd (1968). Linearization of the flow-coupling equation, by treating the flow velocity u (“vertically” averaged), ice thickness h, and surface slope α in terms of small deviations Δu, Δh, and ∆α from overall average (datum) values uo, h0, and α0, results in a differential equation that can be solved by Green’s function methods, giving Δu(x) as a function of ∆h(x) and ∆α(x), x being the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of local h(x) and α(x) on the flow u(x): where the integration is over the length L of the glacier. The ∆ operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variable x′, represents the influence of local h(x′), α(x′), and channel-shape factor f(x′), at longitudinal coordinate x′, on the flow u at coordinate x, the influence being weighted by the “influence transfer function” exp (−|x′ − x|/ℓ) in the integral.

The quantity ℓ that appears as the scale length in the exponential weighting function is called the longitudinal coupling length. It is determined by rheological parameters via the relationship , where n is the flow-law exponent, η the effective longitudinal viscosity, and η the effective shear viscosity of the ice profile, η is an average of the local effective viscosity η over the ice cross-section, and (η)–1 is an average of η−1 that gives strongly increased weight to values near the base. Theoretically, the coupling length ℓ is generally in the range one to three times the ice thickness for valley glaciers and four to ten times for ice sheets; for a glacier in surge, it is even longer, ℓ ~ 12h. It is distinctly longer for non-linear (n = 3) than for linear rheology, so that the flow-coupling effects of longitudinal stress gradients are markedly greater for non-linear flow.

The averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations in h or α, with wavelength λ, are attenuated by the factor 1/(1 + (2πℓ/λ)2) relative to what they would be without longitudinal coupling. The short, intermediate, and long scales of glacier motion (Raymond, 1980), over which the longitudinal flow variations are strongly, partially, and little attenuated, are for λ ≲ 2ℓ , 2ℓ ≲ λ ≲ 20ℓ, and λ ≳ 20ℓ.

For practical glacier-flow calculations, the exponential weighting function can be approximated by a symmetrical triangular averaging window of length 4ℓ, called the longitudinal averaging length. The traditional rectangular window is a poor approximation. Because of the exponential weighting, the local surface slope has an appreciable though muted effect on the local flow, which is clearly seen in field examples, contrary to what would result from a rectangular averaging window.

Tested with field data for Variegated Glacier, Alaska, and Blue Glacier, Washington, the longitudinal averaging theory is able to account semi-quantitatively for the observed longitudinal variations in flow of these glaciers and for the representation of flow in terms of “effective surface slope” values. Exceptions occur where the flow is augmented by large contributions from basal sliding in the ice fall and terminal zone of Blue Glacier and in the reach of surge initiation in Variegated Glacier. The averaging length 4l that gives the best agreement between calculated and observed flow pattern is 2.5 km for Variegated Glacier and 1.8 km for Blue Glacier, corresponding to ℓ/h ≈ 2 in both cases.

If ℓ varies with x, but not too rapidly, the exponential weighting function remains a fairly good approximation to the exact Green’s function of the differential equation for longitudinal flow coupling; in this approximation, ℓ in the averaging integral is ℓ(x) but is not a function of x′. Effects of longitudinal variation of J are probably important near the glacier terminus and head, and near ice falls.

The longitudinal averaging formulation can also be used to express the local basal shear stress in terms of longitudinal variations in the local “slope stress” with the mediation of longitudinal stress gradients.

As a result of the coupling effects of longitudinal stress gradients, the perturbations ∆u in glacier-flow velocity that result from longitudinally varying perturbations in ice thickness ∆h and surface slope ∆α are determined by a weighted longitudinal average of ϕh∆h and ϕα∆α, where ϕh and ϕα are “influence coefficients” that control the size of the contributions made by local ∆h and ∆α to the flow increment in the longitudinal average. The values of ϕh and ϕα depend on effects of longitudinal stress and velocity gradients in the unperturbed datum state. If the datum state is an inclined slab in simple-shear flow, the longitudinal averaging solution for the flow perturbation is essentially that obtained previously (Kamb and Echelmeyer, 1985) with equivalent values for the longitudinal coupling length l and with ϕh = n + 1 and ϕα + n, where n is the flow-law exponent. Calculation of the influence coefficients from flow data for Blue Glacier, Washington, indicates that in practice ϕα differs little from n, whereas ϕh can differ considerably from n + 1. The weighting function in the longitudinal averaging integral, which is the Green’s function for the longitudinal coupling equation for flow perturbations, can be approximated by an asymmetric exponential, whose asymmetry depends on two “asymmetry parameters” μ and σ, where μ is the longitudinal gradient of ℓ(= dℓ/dx). The asymmetric exponential has different coupling lengths ℓ+ and ℓ− for the influences from up-stream and from down-stream on a given point of observation. If σ/μ is in the range 1.5–2.2, as expected for flow perturbations in glaciers or ice sheets in which the ice flux is not a strongly varying function of the longitudinal coordinate x, then, when dℓ/dx > 0, the down-stream coupling length ℓ+ is longer than the up-stream coupling length ℓ−, and vice versa when dℓ/dx < 0. Flow-, thickness- and slope-perturbation data for Blue Glacier, obtained by comparing the glacier in 1957–58 and 1977–78, require longitudinal averaging for reasonable interpretation. Analyzed on the basis of the longitudinal coupling theory, with 4ℓ + 1.6 km up-stream, decreasing toward the terminus, the data indicate n to be about 2.5, if interpreted on the basis of a response factor Ѱ + 0.85 derived theoretically by Echelmeyer (unpublished) for the flow response to thickness perturbations in a channel of finite width. The data contain an apparent indication that the flow response to slope perturbations is distinctly smaller, in relation to the response to thickness perturbations, than is expected on a theoretical basis (i.e. ϕα/ ϕh + (n/n + 1) for a slab). This probably indicates that the effective ℓ is longer than can be tested directly with the available data set owing to its limited range in x.

The Digital Thermo-Resistograph is a portable microprocessor-based data-probe system for quick and accurate field collection of snow-cover strength. This was accomplished by constructing a probe with a load cell, a small snow platform for probe-position information, and a Z-80 microprocessor-based data acquisition system. A 64 × 240 dot matrix LCD graphic display unit is used to show immediately complete strength profiles in the field. Sufficient memory for the storage of 25 profiles is provided. Temperature and temperature-gradient collection is also planned as a part of the instrument but as yet this work has not been completed.

The results of winter 1984 field tests are presented. The Digital Thermo-Resistograph proved to be fast and reliable in collecting snow-strength information. Comparisons with the ram penetrometer are shown and suggestions for future developments are discussed.

A simple technique is presented for growing large single crystals of sea ice. Using this technique, crystals with dimensions of 20 cm × 20 cm × 10 cm or more can be readily obtained within an orientational accuracy of 3°. Such crystals can then be used to investigate the physical properties of sea ice.