Structure of ice III

[No abstract

Stress-gradient coupling in glacier flow: IV. Effects of the "T" term

The "T term" in the longitudinal stress equilibrium equation for glacier mechanics, a double y-integral of ∂^2T_(xy)/∂x^2 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 T_(xy) at any depth to the local shear stress T_B 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 I 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 ɑ. 3. There is a significant direct modification of the influence of local h and ɑ on flow, which represents a distinct "driving force" in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂T_(xy)/∂_x. For longitudinal variations in h, the "T force" varies as d^2h/dx^2 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 ɑ, the "force" varies as dɑ/dx and gives a phase-shifted flow response. Although the "T force" is not negligible, its actual effect on T_B 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 T_B, 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 Δɑ 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 λ ≳ h. 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

Stress-gradient coupling in glacier flow: II. Longitudinal averaging of the flow response to small perturbations in ice thickness and surface slope

As a result of the coupling effects of longitudinal stress gradients, the perturbations Δu in glacier-flow velocity that result from longitudinally varying perturbations in ice thickness Δh and surface slope Δɑ are determined by a weighted longitudinal average of ФhΔh and Ф_ɑΔɑ, where Фh and Ф_ɑ are "influence coefficients" that control the size of the contributions made by local Δh and Δɑ to the flow increment in the longitudinal average. The values of Ф_h and Ф_ɑ 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 Ф_h = n + l and Ф_ɑ + n, where n is the flow-law exponent. Calculation of the influence coefficients from flow data for Blue Glacier, Washington, indicates that in practice Ф_ɑ differs little from n, whereas Ф_h 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 ψ + 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. Ф_ɑ/Ф_h + 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

Stress-gradient coupling in glacier flow: I. Longitudinal averaging of the influence of ice thickness and surface slope

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 ɑ in terms of small deviations Δu, Δh, and Aɑ from overall average (datum) values u)0, h_0, and ɑ_0 results in a differential equation that can be solved by Green's function methods, giving Δu(x) as a function of Δh(x) and Δɑ(x), x being the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of local h(x) and ɑ(x) on the flow u(x): Δu(x)= u0/2ℓ ∫^L_0 Δln(ɑ^nƒ^nh^(n+l)exp(-│x' -x│/ℓ)dx' where the integration is over the length L of the glacier. The Δ 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' ), ɑ(x'), and channel-shape factor ƒ(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 ℓ = 2h√n ƒn/3n, where n is the flowlaw exponent, n the effective longitudinal viscosity, and n the effective shear viscosity of the ice profile. n is an average of the local effective viscosity n over the ice cross-section, and (n^(-1) is an average of n^(-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 ɑ, 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 ℓ 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 "stope stress" with the mediation of longitudinal stress gradients

A direct optical method for the study of grain boundary melting

The structure and evolution of grain boundaries underlies the nature of polycrystalline materials. Here we describe an experimental apparatus and light reflection technique for measuring disorder at grain boundaries in optically clear material, in thermodynamic equilibrium. The approach is demonstrated on ice bicrystals. Crystallographic orientation is measured for each ice sample. The type and concentration of impurity in the liquid can be controlled and the temperature can be continuously recorded and controlled over a range near the melting point. The general methodology is appropriate for a wide variety of materials.Comment: 8 pages, 8 figures, updated with minor changes made to published versio

The thermodynamics and roughening of solid-solid interfaces

The dynamics of sharp interfaces separating two non-hydrostatically stressed solids is analyzed using the idea that the rate of mass transport across the interface is proportional to the thermodynamic potential difference across the interface. The solids are allowed to exchange mass by transforming one solid into the other, thermodynamic relations for the transformation of a mass element are derived and a linear stability analysis of the interface is carried out. The stability is shown to depend on the order of the phase transition occurring at the interface. Numerical simulations are performed in the non-linear regime to investigate the evolution and roughening of the interface. It is shown that even small contrasts in the referential densities of the solids may lead to the formation of finger like structures aligned with the principal direction of the far field stress.Comment: (24 pages, 8 figures; V2: added figures, text revisions

Structure of ice III

[No abstract

Short-term variability in Greenland Ice Sheet motion forced by time-varying meltwater inputs: implications for the relationship between subglacial drainage system behavior and ice velocity.

High resolution measurements of ice motion along a -120 km transect in a land-terminating section of the GrIS reveal short-term velocity variations (<1 day), which are forced by rapid variations in meltwater input to the subglacial drainage system from the ice sheet surface. The seasonal changes in ice velocity at low elevations (<1000 m) are dominated by events lasting from 1 day to 1 week, although daily cycles are largely absent at higher elevations, reflecting different patterns of meltwater input. Using a simple model of subglacial conduit behavior we show that the seasonal record of ice velocity can be understood in terms of a time-varying water input to a channelized subglacial drainage system. Our investigation substantiates arguments that variability in the duration and rate, rather than absolute volume, of meltwater delivery to the subglacial drainage system are important controls on seasonal patterns of subglacial water pressure, and therefore ice velocity. We suggest that interpretations of hydro-dynamic behavior in land-terminating sections of the GrIS margin which rely on steady state drainage theories are unsuitable for making predictions about the effect of increased summer ablation on future rates of ice motion. © 2012. American Geophysical Union