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Recent observations of glacier sliding at temperatures below the melting point are discussed. It is pointed out that these observations can be simply explained by including solid friction in the sliding law. Furthermore, we re-emphasize the point that such sub-temperate sliding has an important effect on the basal boundary conditions which should be applied in model studies of ice sheets and glaciers.
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.
During the austral summers of 1976–77 and 1978–79, several ice cores were taken from the McMurdo Ice Shelf brine zone to investigate its thermal, physical, and chemical properties. This brine zone consists of a series of superimposed brine layers (waves) that originate at the seaward edge of the ice shelf and migrate at various rates, depending on their age and position in the ice shelf. The brine in these layers becomes increasingly concentrated as the waves migrate inland through the permeable ice-shelf firn. Chemical analyses of brine samples from the youngest (uppermost) brine wave show that, except for the advancing front, it contains sea salts in normal sea-water proportions. Further inland, deeper and older brine layers, though highly saline (S > 200°/00), are severely depleted in SO42-, with the SO42-/Na+ ratio being an order of magnitude less than that of normal sea-water. Consideration of the solubility of alternative salts, together with analyses of Na+, K+, Ca2+, Mg2+, SO42-, and Cl- concentrations, shows that the sulfate depletion is probably due to selective precipitation of mirabilite, Na2SO4·10H2O. The location of the inland boundary of brine penetration is closely related to the depth at which the brine encounters the firn/ice transition. However, a small but measureable migration of brine is still occurring in otherwise impermeable ice; this is attributed to eutectic dissolution of the ice by concentrated brine as it moves into deeper and warmer parts of the McMurdo Ice Shelf.
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.
The time-scale for the onset of the explosive growth of a finite-amplitude shear-heating instability in the down-slope creep of a thick ice sheet is determined by integrating the equation for the temporal evolution of the temperature-depth profile subsequent to a sudden change in ice thickness. All instabilities eventually grow explosively after a prolonged period of simmering or relatively slow monotonic growth. Though times for explosive growth depend on initial and final ice thicknesses, surface temperature, accumulation rate, basal heat flux, and ice rheological parameters, the explosion times are extremely sensitive to the activation energy and the pre-exponential constant of the ice-creep law. Sudden increases in ice-sheet thickness of 1–2 km due to a rapid climatic deterioration can lead to explosive instability and melting of the basal shear layer in only thousands of years if ice-creep activation energies are lower than about 60 kJ mol-1.
Ice accretion on a non-rotating cylinder was studied under icing conditions involving a wet-growth (glaze) process. Experiments were performed in an outdoor wind tunnel designed for icing studies. In this paper, the experimental method is introduced and the characteristics of form, size, and time development of deposits are given. In terms of freezing conditions, these characteristics were found to be very complex, in which all the external effects: air temperature, wind conditions, liquid water content in the air, and accretion time, are of the same order of importance. In a wet-growth process there exists above the icing surface a water film, the behaviour and dynamics of which are affected by various variables. The water film seems to play an important role in the wet-growth icing.
The rate of ice growth and heat transfer during a stationary wet-growth icing was studied, based on wind-tunnel experiments of Reynolds numbers of I04 to 105, and air temperatures of 0°C to –13°C. The convective heat-transfer coefficient, a quantity of primary importance entering the heat-balance equation of a freezing surface, was found to depend strongly on the liquid water content in the air (or rather, on the impinging water flow on the surface). The convective heat-transfer coefficient was considered theoretically and the dependence is thought to be caused by an increase in the surface roughness and, especially, by an effect of the impinging water droplets on viscous sub-layers and on turbulent heat transfer near the icing boundary. The above evaluation allows us to calculate an accretion efficiency for each icing condition.
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.
Data from experimental tests of snow-block impact against vertical barriers are used to establish values of parameters in order to computer-model the impact mechanics. The results show that total impulse, impact force, and duration of impact can be modeled by accurate specification of the kinematic viscosity in the fluid representation. In modeling the highly transient impact, kinematic viscosity of the material is determined to vary linearly with the impact velocity. This non-physical condition is attributed to lack of accountability of compressibility effects in the computer model, and reduces modeling to an empirical approach. A biviscous modeling of the impact process is in near correspondence to linear viscous modeling, due to dominant importance of block momentum on impact rather than fluidity of material in the impact region.
Subglacial hydrology is investigated for an ice sheet where the substrate consists of a deformable aquifer resting on an aquitard. If sliding velocities are low or absent, subglacial melt-water drainage is dominated by drainage through the aquifer to water channels. Drainage along the bed is negligible. Efficient melt-water drainage requires that a system of subglacial water channels exists; otherwise, pore-water pressures will exceed the overburden pressure. In general, aquifer deformation near (away from) the terminus is most likely to occur during the winter (summer). The effect of short-term high channel pressures is, in general, not critical to aquifer deformation because the pressure pulse does not propagate far into the aquifer. (For aquifers of high permeability, short periods of high channel pressures constitute the most critical condition.) Aquifer deformation at the terminus is very likely to occur if the terminus ice slope exceeds tan ϕ, where ϕ is the Coulomb friction angle of the aquifer material. Upwelling of basal melt water near the terminus will normally cause soil dilation if the aquifer has a low permeability (e.g. till). Maximal profiles are computed corresponding to various aquifer materials using channel spacings which provide efficient drainage. (A maximal profile is the highest ice profile which the aquifer can sustain without deformation.) In general, maximal profiles lie well above observed profiles (such as h(x) = 3x1/2 (m)) except near the terminus. However, if channel spacings are sufficiently large, pore-water pressures are increased and maximal profiles can lie well below h(x) = 3x1/2.
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.
Wave ogives arise in a solution of the continuity equation by the method of characteristics. Steady ice flow is assumed. Ice velocity, channel width, and mass-balance functions combine to form a wave-excitation potential that yields the forcing function for wave ogives. This linear-systems formulation extends the ogive theory of Nye. Convolution of the temporal cumulative mass balance and spatial forcing functions gives the total wave pattern below an ice fall. Many ice falls do not generate ogives because the wave amplitude is modulated by a factor related to ice-fall length. The wave ogives at Austerdalsbreen, Norway, are due almost entirely to ice acceleration at the top of the ice-fall, i.e. the same zone that King and Lewis showed was responsible for forming Forbes bands.
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.
Recent mapping of ice-surface and bedrock topography from airborne radio-echo sounding has shown that the ice caps of Nordaustlandet, Svalbard, are divided into a series of well-defined drainage basins. Three lines of evidence indicate that several distinctive modes of ice-flow regime characterize these basins: (1) comparison between observed and theoretical ice-surface profiles; (2) analysis of driving stresses; and (3) observations of ice-surface features on satellite imagery and air photographs. The drainage basins are inferred to behave in the following ways. First, basins with low driving stresses and surface profiles, some of them clearly stagnant, are associated with the quiescent phase between glacier surges. Secondly, the ice streams draining southern Vestfonna have low surface profiles, relatively low driving stresses, and marked shear zones at their margins. They are interpreted to be flowing continuously at a relatively faster rate than the ridges between them. Basal melting, perhaps combined with substrate deformation, is probably responsible for the regime of these glaciers. Thirdly, the remaining basins studied on Nordaustlandet have relatively high marginal driving stresses and high surface profiles. They are interpreted to be frozen to their beds, at least near their margins. Some of these basins may also surge, particularly those where a part of the basin is below sea-level, and therefore is probably underlain by considerable thicknesses of deformable sediments.
The “vertically” integrated, exact longitudinal stress-equilibrium equation of Budd (1970) is developed further in such a way as to yield an equation that gives explicitly and exactly the contributions to the basal shear stress made by surface and bed slope, surface curvature, longitudinal stress deviators, and longitudinal stress gradients in a glacier flowing in plane strain over a bed of longitudinally varying slope. With this exact equation, questions raised by various approximate forms of the longitudinal equilibrium equation can be answered decisively, and the magnitude of errors in the approximations can be estimated. To first order, in the angle δ that describes fluctuations in the surface slope α from its mean value, the exact equilibrium equation reduces to
where G and T are the well-known stress-deviator-gradient and “variational stress” terms, K is a “longitudinal curvature” term, and B is a “basal drag” term that contributes a resistance to sliding across basal hills and valleys. Except for T, these terms are expressed in simple form and evaluated for practical situations. The bed slope θ (relative to the mean slope) is not assumed to be small, which allows the effects of bedrock topography to be determined, particularly through their appearance in the B term.
Storglaciären (lat. 67.5° N., long. 17.5° E.) is a sub-polar glacier which has been the object of detailed study for many years. It responds in a sensitive way to annual and long-term changes in climate but it does not surge. Conditions at the bed and the distribution of englacial water are of considerable interest. In this paper we discuss an attempt to learn something about these matters by radio-echo soundings at metric and decametric wavelengths. We analyse radio-echo records mainly using a scalar-wave theory of the diffraction of pulses. The pulse shapes of echoes are useful because they help us to recognize the types of target and processes responsible for the echoes. We then use simple statistical measurements of radio echoes to provide estimates of certain average properties of the targets. We estimate, for example, the roughness of the glacier bed and the distribution and orientations of scatterers within the ice.
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 “T term” in the longitudinal stress-equilibrium equation for glacier mechanics, a double y-integral of ∂2τxy/∂x2 where x is a longitudinal coordinate and y is roughly normal to the ice surface, can be evaluated within the framework of longitudinal flow-coupling theory by linking the local shear stress τxy at any depth to the local shear stress τB at the base, which is determined by the theory. This approach leads to a modified longitudinal flow-coupling equation, in which the modifications deriving from the T term are as follows: 1. The longitudinal coupling length is increased by about 5%. 2. The asymmetry parameter σ is altered by a variable but small amount depending on longitudinal gradients in ice thickness h and surface slope α. 3. There is a significant direct modification of the influence of local h and α on flow, which represents a distinct “driving force” in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂τxy/∂x. For longitudinal variations in h, the “T force” varies as d2h/dx2 and results in an in-phase enhancement of the flow response to the variations in h, describable (for sinusoidal variations) by a wavelength-dependent enhancement factor. For longitudinal variations in α, the “force” varies as dα/dx and gives a phase-shifted flow response. Although the “T force” is not negligible, its actual effect on τB and on ice flow proves to be small, because it is attenuated by longitudinal stress coupling. The greatest effect is at shortest wavelengths (λ 2.5h), where the flow response to variations in h does not tend to zero as it would otherwise do because of longitudinal coupling, but instead, because of the effect of the “T force”, tends to a response about 4% of what would occur in the absence of longitudinal coupling. If an effect of this small size can be considered negligible, then the influence of the T term can be disregarded. It is then unnecessary to distinguish in glacier mechanics between two length scales for longitudinal averaging of τb, one over which the T term is negligible and one over which it is not.
Longitudinal flow-coupling theory also provides a basis for evaluating the additional datum-state effects of the T term on the flow perturbations Δu that result from perturbations Δh and Δα from a datum state with longitudinal stress gradients. Although there are many small effects at the ~1% level, none of them seems to stand out significantly, and at the 10% level all can be neglected.
The foregoing conclusions apply for long wavelengths λh. For short wavelengths (λ h), effects of the T term become important in longitudinal coupling, as will be shown in a later paper in this series.
“Avalanche-type” medial moraines are described on four Torngat Mountains glaciers with single, rather than compound, firn basins. They form from debris avalanching down couloirs and are exposed by ice melt to form extensive debris covers on glacier snouts in the Torngat Mountains. The debris may play an important part in helping the glacier survive periods with warm summers and/or dry winters since, in the Torngat Mountains, average ice melt on debris-covered sites is approximately one-third that on exposed ice.
Variations of area-averaged annual balance can be calculated from sparse stake networks, if the specific balances satisfy a simple version of the linear model proposed by Lliboutry. The linear model is tested with 4 years of data at 13 stakes on Qamanârssûp sermia, West Greenland, and is satisfied with a model error of ±0.34 m water equivalent. From analyses of data from three stakes a few metres apart, the random error in the measurement of annual balance is estimated to be greater than ±0.23 m water equivalent. The variation in balance for the 4 years is ±0.52 m water equivalent which reflects the year–to–year variation in climate. The shortness of the present data series is admitted and its continuation is recommended.