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.

The average three-dimensional coordination number, n3, is an important measure of firn structure. The value of n3 can be estimated from n2, the average measured two-dimensional coordination number, and from a function, Γ, that depends only on the ratio of average bond radius to grain radius in the sample. This method is easy to apply and does not require the use of unknown shape factors or tunable parameters.

Observations of surging glaciers indicate that the flow regime is one dominated by extensional flow. The stress state has substantial longitudinal deviatoric stress. This regime is very different from the conventional model for glacier dynamics which is dominated by shearing flow. In addition, the initiation of surging often involves a compression front which travels down the glacier. The compression front seems to divide an up-stream region of high drag at the base of the glacier from one of low drag which allows the rapid sliding. We develop a framework for the mechanics of glaciers undergoing surging. Relevant issues are the extensional and compression flows, high longitudinal deviatoric stress, and the stress state near the basal discontinuity. We find that some of the down-slope component of glacier weight is borne by longitudinal stress in the rapidly sliding region. This stress thrusts against the slowly moving parts of the glacier. We hypothesize that this effect causes the rapidly sliding part to spread and causes the compression front to travel down the glacier. A criterion for spreading of the rapidly sliding part is developed. The mechanics outlined above are used to develop a highly idealized model for glacier surging. We propose that regions of low drag are relatively common features of glaciers. The surge initiates when conditions are met which allow the surge nucleus to spread. The rapidly sliding region of low drag spreads to a large part of the glacier. Surging ends when the low-drag conditions terminate. Because of the changed state of the glacier, surge nuclei are now stable against spreading. Several years of rebuilding must occur before nuclei are once more unstable. Calculations are performed for the evolution of the shape of Medvezhy Glacier during the surge of 1963. We find a remarkable similarity between the data and our computations.

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.

Series of experiments were conducted with the aim of determining the influences of the following factors on freeze-bonding between contacting ice blocks in floating ice rubble: pressure normal to the contact plane, period and area of contact, and salinity of the water in which freeze-bonding occurred. Freeze-bonding between ice blocks in air was also investigated. The experiments were conducted with water and air temperatures of about 0°C and normal pressures, between ice blocks, up to 4 kPa. This range of normal pressures may occur hydrostatically between ice blocks in layers of floating ice rubble up to about 10 m thick, or in 2-3 m thick layers which are in a passive Rankine state of pressure. The experiments show that stronger freeze-bonds develop between ice blocks in distilled water, tap water, and water from the Iowa River than develop between ice blocks contacting in air at 0°C. However, stronger freeze-bonds developed in air at 0°C than developed between ice blocks in 0°C saline (NaCl) solutions with salinities in excess of 12.5% by weight. The strength of freeze-bonding increased linearly with contact period for ice blocks in distilled, tap, and river waters, but did not increase with contact period for ice blocks contacting in saline solutions or in air. The results of the experiments are useful contributions to explanations of the shear-strength behavior of a layer of floating ice rubble. For example, thicker layers of ice rubble may show greater cohesive behavior, because normal pressures and thus freeze-bond strengths increase with layer thickness.

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.

Existence of a very thin layer of adsorbed water adjacent to particles embedded in ice allows relative motion between ice and particles even at sub-freezing temperatures if there are either applied stresses or macroscopic temperature gradients. Theoretical analysis of such motion involving a single sphere demonstrates that such motion is dominantly due either to “viscous” deformation in the ice or to mass transport in the liquid layer at temperatures below the nominal pressure melting-point, depending on the ratio of the sphere’s radius to a temperature-dependent “transition radius”. This result should also hold for motion of a cylinder (for which the creeping flow problem has no known analytical solution). Reviewing data on wire regelation at sub-freezing temperatures in the context of this analysis suggests that all “anomalous” data correspond to cases in which wire radii were greater than the transition radius, leading to dominance of ice-deformation effects. Ice motion past very small particles, on the other hand, is essentially accommodated entirely by mass transfer through the liquid layer. This result lends support to the “rigid-ice” model of frost heaving as proposed by R.D. Miller and co-workers, and permits approximate analysis of ice movement through a porous soil. In all cases involving relative motion between ice and particles at sub-freezing temperatures, the existence of macroscopic temperature gradients plays an important role.

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.

Many observations regarding grain growth in ice sheets are glaciologically interesting but imperfectly understood. Here we develop the theory of grain growth in ice that is not deforming rapidly, and in the succeeding paper we use this theory to explain observations from glacial ice. In the absence of significant strain energy, the driving force for grain growth arises from grain-boundary curvature. Grain growth is slowed by the interaction of grain boundaries with extrinsic materials (microparticles, bubbles, and dissolved impurities). If the driving force for growth is not large enough to cause boundaries to separate from an extrinsic material, then the grain-boundary velocity is determined by the velocity characteristic of the extrinsic material (low-velocity regime). If the driving force is large enough to cause separation, then boundaries migrate more rapidly than the extrinsic material (high-velocity regime) but the net driving force is reduced through transient pinning by the extrinsic material. Polar ice is typically in the low-velocity regime relative to dissolved impurities and the high-velocity regime relative to microparticles and bubbles. Cross-sectional area of grains is predicted to increase linearly with time under most but not all circumstances.

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.

Grain growth observed in polar ice that is not deforming rapidly can be accounted for if concentrations and distributions of extrinsic materials (microparticles, bubbles, and dissolved impurities) are characterized fully. Dissolved impurities segregate to grain boundaries and slow grain growth in all cold glacial ice. The high concentration of soluble impurities in Wisconsinan ice from the Dome C (Antarctica) ice core (and perhaps other ice cores) probably causes the small grain-sizes observed in that ice. Microparticles have little effect on grain growth in ordinary ice. In ice layers that appear dirty owing to concentrations of volcanic tephra (such as in the Byrd Station (Antarctica) ice core) or of morainal material, micro particles reduce grain-growth rates significantly. The relatively high vapor pressure of ice allows rapid growth and high mobility of intergranular necks, so grain growth in firn is limited by boundary migration rather than by neck growth. Bubbles formed by pore close-off at the firn-ice transition are less mobile than grain boundaries, causing bubble-boundary separation whenever geometric constraints are satisfied; however, such separation reduces grain-growth rates by only about 10%. The observed linear increase of grain area with time is thus predicted by theory, but the growth rate depends on soluble-impurity concentrations as well as on temperature.

A statistical model characterizing the granular structure of snow is developed using quantitative stereology. The model is based on specific parameters (e.g. bond radius, grain-size, etc.) which take the form of internal-state variables in a constitutive theory for high-rate deformation of snow. In addition to parameters developed by other authors in previous investigations, a new parameter characterizing the mean bond length is developed. More significantly, general relations are derived for the mean number of bonds per grain and mean number of grains per unit volume without making any assumptions regarding the shape or size of the ice grains, or their respective distributions.

A theoretical model is developed to describe the steady-state behavior of interconnected, water-filled cavities at the glacier bed. Physically plausible cavities should contain constrictions along the flow path, with flow in the wider sections being relatively sluggish. Mean flow rates in cavities may be at least one order of magnitude less than in channels incised into the basal ice (R channels). Melting due to viscous dissipation - the process that allows R channels to exist - probably plays a minor or negligible role, as compared to glacier sliding, in determining the size of cavities. Furthermore, a system of subglacial cavities should not show a tendency for localization of flow in a few main conduits, as does an R-channel system. If water pressure rises to within several bars of overburden pressure, the rate of cavity closure by creep falls below the rate of cavity opening by sliding and melting, with cavities then becoming unstable. Subsequent evolution of the drainage system should depend upon the total melt-water flux. Circumstances may arise in which cavities and channels act as conduits for melt water; such a configuration would probably show unusual transient behavior.

The optimum continuation of series of mass-balance measurements and their extension to unmonitored glaciers are important problems in contemporary glaciology. For this purpose, two new practical survey methods are proposed, based on the linear-balance variations model of Lliboutry (1974). The first method is a simplified application of the linear model that uses only a data set limited to selected fixed-measurement sites. It was developed to obtain the mass-balance variation in cases where data are too scarce to obtain the global mass balance or to apply the Lliboutry algorithm. This simplified linear model is used with the 8 years’ of surveys on glacier d’Argentière. The second method uses the continuity equation to derive the mass balance of a glacier sector delimited by two cross-profiles where the surface velocities, surface altitudes, and depths are known. By using this continuity method, the entire mass-balance series is established for a sector of glacier de Gébroulaz (Vanoise area, France) from 1908 to 1950, as well as for two sectors of Unteraargletscher (Oberland, Switzerland) from 1924 to 1981.

Expressions are found for the water pressure in channels and cavities under a temperate glacier in a steady state. By steady state is meant that not only is the sliding velocity constant but also the basal water flow as well as the basal water production. The calculations are based on a simplified bimodal obstacle bed. One-size obstacles are of the magnitude of the conventional “controlling” obstacles size. The other obstacle size is several orders of magnitude larger than the controlling size. The large obstacles are introduced in order to investigate the storage of large amounts of water at the bed. The effect of the missing obstacles is mimicked by adding a friction stress whose values decrease linearly with ice–rock separation. It is found that water flow is primarily in a water film if the water flux is below a critical limit. The physical reason for this result is that a minimum water flow must be exceeded in order to keep a water channel open in high ice-pressure regions in front of obstacles. The new ingredient in our analysis of water flow is the explicit account taken of the effect of excess ice pressure (at the up-stream side of obstacles) on the closing of water channels. The excess ice-pressure regions play a critical role in the development of water films. The recognition of the possible importance of high-pressure basal ice regions to the regulation of water flow is the primary result of this paper. It is found that, in steady state, the water pressure is close to the overburden ice pressure provided that water transported through Nye channels is only a minor part of the total. The water flows primarily in a water film. If a large flux of upper-surface melt water descends to the bed, this water could cause a reduction of the water pressure and a channelization of the water flow.

The seaward edge of the Ross Ice Shelf advanced northward at a minimum average velocity of 0.8 km a–1 between 1962 and 1985. That advance approximated velocities that have been obtained from glaciological data, indicating little recent wastage by iceberg calving. West of long. 178° E., the ice shelf has attained its most northerly position in the past 145 years, and has not experienced a major calving episode for at least 75 years. Since 1841 the ice-front position has advanced and retreated within a zone from about lat. 77° 10’S. (near long. 171° E.) to lat. 78° 40’ S. (near long. 164° W.). The central ice front is now farthest south but has the highest advance rate. Calving may occur at more frequent intervals in that sector, which also overlies the warmest ocean currents that flow into the sub-ice-shelf cavity. Available information on ice-shelf advance, thickness, spreading rate, and surface accumulation indicates a basal melting rate around 3 m a–1 near the ice front. These data and independent estimates imply that basal melting is nearly as large a factor as iceberg calving in maintaining the ice-shelf mass balance. In recent years, the Ross, Ronne, and Filchner Ice Shelves have contributed few icebergs to the Southern Ocean, while projections from a contemporaneous iceberg census are that circumpolar calving alone may exceed accumulation on the ice sheet. Large-scale ice-shelf calving may have preceded historical sightings of increased numbers of icebergs at sea.