Linear Viscous Rheology Research Articles

Clast collision frequency as an indicator of glacier sliding rate

AbstractThe mechanical interaction between a glacier and subglacial sediment can be observed using an instrumented rod that we refer to as the UBC ploughmeter. For clast-rich sediments, the rate of collision between clasts and a rod dragged through these sediments should be related to the glacier sliding rate. By assuming that proglacial measurements of sediment granulometry represent the subglacial granulometry of Trapridge Glacier, Yukon Territory, Canada, we have used ploughmeter results to obtain an estimated sliding rate of ∼45 mm d−1, in good agreement with known rates. In addition, for a subglacial material treated as a solid-liquid dispersion having a linear viscous rheology, the force of collision experienced by the rod should be proportional to the effective sediment viscosity. Our estimate of ∼2.0 × 1010Pa s agrees well with previously, derived values.

Sensitivity analysis of ice motion near Adams Island

The time-averaged motion of shore-fast ice near a small Arctic island is analysed through the solution of the momentum equation for the ice cover using the Galerkin finite element method coupled with an adjoint sensitivity formulation. A linear-viscous rheology is used. Shore boundaries were modeled by prescribing ice velocity components such that the ice velocity vectors at shore boundaries were directed towards the ice cover, normal to the shoreline. A sensitivity analysis using the adjoint methodology showed that the velocity of the ice near the island is strongly influenced by the magnitude of the prescribed boundary velocities. The surface shear stresses from wind and ocean currents and the ice viscosities over part of the modeled domain also have a significant influence. The influence of Coriolis acceleration and the mass of the ice are not significant. Key words : ice mechanics, momentum, viscous rheology, finite elements.

A crack model of afterslip events on shallow faults

SUMMARY Post-seismic evolution of surface displacement on a strike slip fault is studied employing a linear viscous rheology for the shallow fault zone. A mathematical model is built in which the singular integral equation governing the equilibrium of cracks is transformed in an infinite-dimensional linear symmetric system whose positive eigenvalues define a sequence of characteristic relaxation times. The model lends to model vertical heterogeneities in the rheological parameters and the equations can be efficiently solved by truncation. Several specific problems are then considered in order to assess which parameters may govern the fast early relaxation observed in afterslip events in California. Neither stress heterogeneities, nor the spectral distribution of characteristic relaxation times have significant effects on surface displacement if a uniform effective viscosity is employed. Viscosity heterogeneities however, can provide fast early afterslip and, perhaps, precursory events. Post-seismic creep data following the 1966 Parkfield, the 1979 Imperial Valley and the 1987 Superstition Hill earthquakes can be fitted well with this model without resorting necessarily to non-linear rheological relations.

Transfer of Basal Sliding Variations to the Surface of a Linearly Viscous Glacier

AbstractThe transfer of basal velocity anomalies to the surface of a glacier is investigated using a model of a planar parallel-sided slab (thickness H) of linear viscous rheology. Surface velocity parallel (us) and normal (vs) to the surface is calculated for various spatial distributions of basal velocity anomalies with components parallel (ub) and normal (vb) to the surface. Four scales of differing behavior can be identified depending on the spatial length L of the basal anomalies. At very short scales (L ≤ 1H) there is essentially no response at the surface. At short scales (1H ≤ L ≤ 5H), a basal anomaly ub induces a response in both us and vs. The spatial pattern of us is such that velocity peaks in us can be shifted from peaks in ub, and may differ in number. The amplitude of us is up to about 0.3|ub|. The amplitude of the cross-component effect vs may be greater than the amplitude of us. A basal anomaly vb induces a response in both vs and us. The pattern of vs is the same as the pattern of vb, and the amplitude of vs is up to about 0.7 |vb|. The amplitude of the cross-component effect us is less than the amplitude of vs. At intermediate scales (5H ≤ L ≤ 10H), results differ from the short scale in two respects; velocity peaks in us correspond with peaks in ub; and surface amplitudes are increased, except for cross-component effects for which surface amplitudes are of the same order as at the short scale. These cross-component effects at the short and intermediate scales show in particular that substantial anomalous surface-normal motions can be induced by deformation, even though the basal velocity anomaly is parallel to the surface. At long scales (10H ≤ L), the velocity anomaly at the surface is essentially the same as the anomaly at the bed. For all scales, the longitudinal strain-rate averaged over depth is larger in magnitude than the longitudinal strain-rate at the surface and, at the short scale, it may differ in sign, so that vs cannot be easily estimated from surface strain-rate. Although the simplifications of the model do not allow its rigorous quantitative application to field measurements, the results indicate the need for caution in interpreting surface-velocity variations in terms of basal velocity anomalies. It is important to establish the spatial pattern of surface motions for any chance of a confident interpretation in terms of basal motions.

Transfer of Basal Sliding Variations to the Surface of a Linearly Viscous Glacier

AbstractThe transfer of basal velocity anomalies to the surface of a glacier is investigated using a model of a planar parallel-sided slab (thicknessH) of linear viscous rheology. Surface velocity parallel (us) and normal (vs) to the surface is calculated for various spatial distributions of basal velocity anomalies with components parallel (ub) and normal (vb) to the surface. Four scales of differing behavior can be identified depending on the spatial lengthLof the basal anomalies. At very short scales (L≤ 1H) there is essentially no response at the surface. At short scales (1H≤L≤ 5H), a basal anomalyubinduces a response in bothusandvs. The spatial pattern ofusis such that velocity peaks inuscan be shifted from peaks inub, and may differ in number. The amplitude ofusis up to about 0.3|ub|. The amplitude of the cross-component effectvsmay be greater than the amplitude ofus. A basal anomalyvbinduces a response in bothvsandus. The pattern ofvsis the same as the pattern ofvb, and the amplitude ofvsis up to about 0.7 |vb|. The amplitude of the cross-component effectusis less than the amplitude ofvs. At intermediate scales (5H≤L≤ 10H), results differ from the short scale in two respects; velocity peaks inuscorrespond with peaks inub; and surface amplitudes are increased, except for cross-component effects for which surface amplitudes are of the same order as at the short scale. These cross-component effects at the short and intermediate scales show in particular that substantial anomalous surface-normal motions can be induced by deformation, even though the basal velocity anomaly is parallel to the surface. At long scales (10H≤L), the velocity anomaly at the surface is essentially the same as the anomaly at the bed. For all scales, the longitudinal strain-rate averaged over depth is larger in magnitude than the longitudinal strain-rate at the surface and, at the short scale, it may differ in sign, so thatvscannot be easily estimated from surface strain-rate. Although the simplifications of the model do not allow its rigorous quantitative application to field measurements, the results indicate the need for caution in interpreting surface-velocity variations in terms of basal velocity anomalies. It is important to establish the spatial pattern of surface motions for any chance of a confident interpretation in terms of basal motions.

Longitudinal Variations in Glacial Flow: Theory and Test Using Data from the Byrd Station Strain Network, Antarctica

AbstractA theory advanced by Budd (1970) for ice-flow variation due to bed effects is improved and applied to the flow of the ice sheet leading to Byrd Station, Antarctica. Linear viscous rheology is used and a biharmonic equation for stress and strain-rate variation is solved. Basal boundary conditions include an undulating bottom topography and longitudinal variations in basal sliding and shear stress. Near Byrd Station it is found that variations in basal sliding and shear stress have more important direct effects than does the bed topography. This result is important to considerations on the stability of glacial flow. Distortions of internal layers are calculated and these closely match the distortions observed in the internal layers detected by the NSF-SPRI-TUD radar. The phase relationship between internal layer distortion and surface topography provides a positive test for the theory. The theory does not predict a spectral power maximum in surface topographic relief at a wavelength of three ice thicknesses. Such wavelengths however dominate and we speculate that this may be due to the dynamics of glacial erosion and deposition or with subglacial water motion.

Longitudinal Variations in Glacial Flow: Theory and Test Using Data from the Byrd Station Strain Network, Antarctica

AbstractA theory advanced by Budd (1970) for ice-flow variation due to bed effects is improved and applied to the flow of the ice sheet leading to Byrd Station, Antarctica. Linear viscous rheology is used and a biharmonic equation for stress and strain-rate variation is solved. Basal boundary conditions include an undulating bottom topography and longitudinal variations in basal sliding and shear stress. Near Byrd Station it is found that variations in basal sliding and shear stress have more important direct effects than does the bed topography. This result is important to considerations on the stability of glacial flow. Distortions of internal layers are calculated and these closely match the distortions observed in the internal layers detected by the NSF-SPRI-TUD radar. The phase relationship between internal layer distortion and surface topography provides a positive test for the theory. The theory does not predict a spectral power maximum in surface topographic relief at a wavelength of three ice thicknesses. Such wavelengths however dominate and we speculate that this may be due to the dynamics of glacial erosion and deposition or with subglacial water motion.

Coupling of diffusional mass transport and deformation in a tight rock

Simple governing equations coupling diffusional transport at a macroscopic scale and deformation in a tight rock are formulated. The deformation alone is described by a linear viscous rheology. Several deformations of geological interest are analyzed, including the folding of an embedded layer. A basic feature is the length-scale dependence of diffusional effects. Interpretation of natural structures permits the estimation of parameters governing the diffusional transport and rate of deformation.

Lithospheric loading by the 1896 Riku‐u Earthquake, northern Japan: Implications for plate flexure and asthenospheric rheology

Under favorable circumstances the time‐dependent aseismic deformation resulting from the loading of the lithosphere by the stress drop of large dip slip earthquakes can be used to determine both the effective elastic plate thickness and the asthenospheric viscosity. The deformation has several similarities with the deflection of the lithosphere by surface loads and with movements due to postglacial rebound. Level changes obtained in the 80 years since the M = 7.5, 1896 Riku‐u earthquake, an intraplate thrust event in northern Honshu, provide convincing evidence that asthenospheric readjustments are responsible for the observed movements. Leveling surveys crossing the zone of surface faulting have been repeated five times since 1900 and delineate a localized depression that has subsided at a continually decreasing rate. The depression is centered close to the 1896 faulting, and its shape and width, about 75 km, are matched by our model using a plate thickness of 30 km. The decaying subsidence rate constrains the viscosity of the uppermost asthenosphere to be 1×1020 P. A linear viscous rheology matches the observed decay quite well, although measurements are sparse during the several decades following the earthquake.