Bedrock Undulations Research Articles

Distribution Characteristics and Genesis Mechanism of Ground Fissures in Three Northern Counties of the North China Plain

The North China Plain is among the regions most afflicted by ground fissure disasters in China. Recent urbanization has accelerated ground fissure activity in the three counties of the northern North China Plain, posing significant threats to both the natural environment and socioeconomic sustainability. Despite the increased attention, a lack of comprehensive understanding persists due to delayed recognition and limited research. This study conducted field visits and geological surveys across 43 villages and 80 sites to elucidate the spatial distribution patterns of ground fissures in the aforementioned counties. By integrating these findings with regional geological data, we formulated a causative model to explain ground fissure formation. Our analysis reveals a concentration of ground fissures near the Niuxi and Rongxi faults, with the former exhibiting the most extensive distribution. The primary manifestations of ground fissures include linear cracks and patch-shaped collapse pits, predominantly oriented in east-west and north-south directions, indicating tensile failure with minimal vertical displacement. Various factors contribute to ground fissure development, including fault activity, ancient river channel distribution, bedrock undulations, rainfall, and ground settlement. Fault activity establishes a concealed fracture system in shallow geotechnical layers, laying the groundwork for ground fissure formation. Additionally, the distribution of ancient river channels and bedrock undulations modifies regional stress fields, further facilitating ground fissure emergence. Rainfall and differential ground settlement serve as triggering mechanisms, exposing ground fissures at the surface. This research offers new insights into the causes of ground fissures in the northern North China Plain, providing crucial scientific evidence for sustaining both the natural environment and the socio-economic stability of the region.

Assessment of a seismo-neotectonic origin for the New Liskeard–Thornloe scarp, Timiskaming graben, northeastern Ontario

To test an inference that the New Liskeard–Thornloe scarp (NLTS), Timiskaming graben, Ontario, is a deglacial–postglacial seismo-neotectonic fault, we collected shallow geophysical data along lines 7.2, 0.28, and 0.52 km long, located on three roads crossing the middle portion of the scarp. Data revealed a valley subsurface composed of bedrock (seismic unit a), glaciolacustrine–lacustrine deposits (units b to f), mass movement deposits (units ls 1 to ls 3), wave-worked sediments, mass wasting deposits and (or) artificial fill (unit g), and a minor occurrence of roadfill (unit h). The bedrock surface exhibits only minor undulations in the area underlying the scarp, indicating that the scarp morphology is unrelated to the underlying bedrock topography. Parallel reflectors in glaciolacustrine seismic units b and c conformably overlie the minor bedrock undulations and there is an absence of disturbed or offset zones within the reflectors underlying the scarp. This lack of disturbance or offset provides strong evidence that the scarp is not the product of deglacial–postglacial seismo-neotectonic faulting. The erosive truncation of glaciolacustrine seismic units d and e indicate that the scarp is an erosive feature cut into the glaciolacustrine deposits. It is likely a bluff formed by shoreline erosion, as is consistent with a geomorphic setting previously inundated by a large glacial lake and subsequent recessional lake stages. The non-fault origin for the NLTS limits the northern extent of the hypothesized Timiskaming East Shore Fault to within the Lake Timiskaming basin and, hence, constrains estimates of maximum rupture length.

Transmission of basal variability to a glacier surface

Transmission of basal variability to a glacier surface is investigated using analytical models for a linearly viscous medium. The three‐dimensional transient response of the surface to both bedrock undulations and spatial variations in basal slipperiness for perturbations of arbitrary wavelengths is determined using perturbation methods. Both information transfer toward the surface and lateral transmission of horizontal stresses are strongly affected by the slip ratio, that is, the ratio of basal sliding to deformational velocity. For any mean bedrock slope, and above a minimum value of slip ratio, the amplitude transfer of bedrock undulations toward the surface has a local maximum at undulation span corresponding to about 3–8 times the mean ice thickness. The transmission of basal variability to a glacier surface increases quite significantly with increasing slip ratio. This explains why the surfaces of fast flowing ice streams are more undulating than the slower moving bordering areas. At slip ratios higher than about 100, the flow of glaciers and ice sheets becomes nonlocal in the sense that surface velocities and buildup and propagation of surface undulations cannot be calculated accurately on the basis of local thickness and slope. Using linearized long‐wave theories at these slip ratios, instead of the more accurate arbitrary wavelength theory, gives estimates of decay times that are an order‐of‐magnitude too small and phase velocities several times too large. The problem of the propagation and decay of small‐amplitude surface undulations on glaciers in three dimensions is solved. Small‐amplitude surface waves on glaciers are strongly diffusive and dispersive. Redistribution of mass on ice sheets and glaciers is a diffusion process, and it is misleading, albeit not mathematically incorrect, to describe the reaction of glaciers to surface perturbations in terms of a wave propagation.

Scattering of VHF radio waves from within the top 700 m of the Antarctic ice sheet and its relation to the depositional environment: a case-study along the Syowa–Mizuho–Dome Fuji traverse

AbstractRadio-wave scattering is a convenient method to image the properties of large internal regions of ice sheets. We used a ground-based radar system with short pulses of 60 and 179MHz frequencies to scatter off internal strata within 100–700 m of the surface in the ice sheet of East Antarctica. Data were examined along an 1150 kmlong traverse line that was approximately along the ice flowline from inland of Dome Fuji station to the coast. The scattered waves are from strata, and the dominant cause of the scattering was changes in dielectric permittivity across the strata. Therefore, density fluctuations primarily cause the scattering, although variations in ice-crystal fabrics and acidity could also have effects. The power scattered from the same depths varied by > 15 dB from one location to another. These variations correlate with the accumulation rate, changes in the surface slope, and subglacial bedrock undulations. Variations of the scattered power suggest that density contrasts in the strata are highly variable depending on these interdependent local conditions. The distribution of strata along the route allowed estimates of the ice-flow trajectories to depths of about 250 m.

A stacked mixing sequence at the base of the Dye 3 Core, Greenland

This paper documents, by a detailed isotopic and gas composition study of the basal ice from the Dye 3 core, a large degree of flow‐induced mixing at the ice sheet base. This mixing results most probably from circular motion in troughs of bedrock undulations and flow separations with some entrainment of the underlying ice. The measured distribution of isotopes and gases in the basal ice is the consequence of successive mixing events having occurred upglacier from the site in an area of very rough bedrock topography. The deformation produced makes it possible for the basal part of the ice sheet to cross the subglacial valley existing in the Dye 3 area.

Basal-flow characteristics of a linear medium sliding frictionless over small bedrock undulations

AbstractThe basal deformation of a gravity-driven linear creeping flow sliding frictionless over slowly varying bed undulations in two dimensions is analysed analytically, using results from second-order perturbation theory. One of the key results is that, close to sinusoidal bedrock undulations, up to two different spatial regions of local extrusion flow may arise. The offset and onset of extrusion flow is controlled primarily by the amplitude-to-wavelength ratio. Above the crest of a sinusoidal bed line, a local maximum of the surface-parallel velocity develops for ε : =ak< 0.138, whereais the amplitude andkis the wave number. Asεincreases from zerо to this critical value, the vertical position of the velocity maximum moves fromkz= 1 tokz≈ 1.98, wherezis the vertical distance above the mean bed line. Within and above the trough of a sinusoid, a region of local minimum of the surface-parallel velocity component develops, which shifts fromkz= 1 towards the bed line asεincreases front zero to 1/2. Below this velocity minimum, and for some distance above the velocity maximum, the surface-parallel velocity increases with depth. This type of extrusion flow will cause a reversal of borehole-inclination profiles close to the bedrock.

Basal-flow characteristics of a linear medium sliding frictionless over small bedrock undulations

AbstractThe basal deformation of a gravity-driven linear creeping flow sliding frictionless over slowly varying bed undulations in two dimensions is analysed analytically, using results from second-order perturbation theory. One of the key results is that, close to sinusoidal bedrock undulations, up to two different spatial regions of local extrusion flow may arise. The offset and onset of extrusion flow is controlled primarily by the amplitude-to-wavelength ratio. Above the crest of a sinusoidal bed line, a local maximum of the surface-parallel velocity develops for ε : = ak < 0.138, where a is the amplitude and k is the wave number. As ε increases from zerо to this critical value, the vertical position of the velocity maximum moves from kz = 1 to kz ≈ 1.98, where z is the vertical distance above the mean bed line. Within and above the trough of a sinusoid, a region of local minimum of the surface-parallel velocity component develops, which shifts from kz = 1 towards the bed line as ε increases front zero to 1/2. Below this velocity minimum, and for some distance above the velocity maximum, the surface-parallel velocity increases with depth. This type of extrusion flow will cause a reversal of borehole-inclination profiles close to the bedrock.

Basal-flow characteristics of a non-linear flow sliding frictionless over strongly undulating bedrock

AbstractThe flow field of a medium sliding without friction over a strongly undulating surface is calculated numerically. The results are used to elucidate the basal-flow characteristics of glacier flow and they are discussed with reference to known analytical solutions. Extrusion flow is found to become increasingly pronounced as the value of n, where n is a parameter in Glen’s flow law, becomes larger. For sinusoidal bedrock undulations, a flow separation occurs if the amplitude-to-wavelength ratio exceeds a critical value of about 0.28. The main flow then sets up a secondary flow circulation within the trough, and the ice participating in this circular motion theoretically never leaves it. The sliding velocity is calculated numerically as a function of the mean basal shear stress, the amplitude-to-wavelength ratio and the flow parameter n. For moderate and high slope fluctuations, the sliding velocity is significantly different from what would be expected from results based on the small-slope approximation.

Glacier sliding over sinusoidal bed and the characteristics of creeping flow over bedrock undulations : G. Hilmar Gudmundsson, Mitteilungen - Versuchsanstalt fur Wasserbau, Hydrologie und Glaziologie, ETH Zurich, 130, 1994, 102 pp

Glacier sliding over sinusoidal bed and the characteristics of creeping flow over bedrock undulations : G. Hilmar Gudmundsson, Mitteilungen - Versuchsanstalt fur Wasserbau, Hydrologie und Glaziologie, ETH Zurich, 130, 1994, 102 pp

Application of a Flow Model to the Ice-divide Region of Devon Island Ice Cap, Canada

AbstractThe steady-state flow model of Reeh (1988) is applied to a flow line that starts at the highest point of the Devon Island ice cap, follows the surface crest for 7.6 km, and then runs down the slope for a further 3.7 km. The effects of bedrock undulations, divergence of the flow lines, the variation of temperature with depth, and a basal layer of “soft” ice-age ice are taken into account. A flow law with n = 3 and a value of A close to that of Paterson (1981) is used. Longitudinal stress variations are neglected so that shear stress is calculated by the usual formula. It is estimated that these calculated values may be in error by at most 30%. Depth profiles of effective shear stress, and of the components of velocity and normal strain-rate, are presented at selected points along the flow line. These illustrate the large variations that occur near an ice divide and over bedrock undulations of amplitude comparable with the mean ice thickness. The model gives good predictions of the surface profile and of longitudinal and transverse surface strain-rates measured at ten points along the flow line. Predicted depth profiles of horizontal and vertical velocity components are compared with those measured in a bore hole. Comparison is limited by the fact that the model works in ice equivalent, whereas about 20% of the ice column consists of firn with different rheological properties from ice. The vertical velocity prediction is good. However, the model does not reproduce well the shape of the horizontal velocity profile, although measured and calculated fluxes differ only slightly. Predicted annual-layer thicknesses are within 15% of the measured ones in the upper half of the ice column, which consists of ice deposited in the last 1000 years. Predicted thicknesses in older ice are too small and the discrepancy increases with depth. This might indicate increased precipitation or, more likely, a thinner ice cap in the climatic optimum. However, it could also result from the fact that the layer of “soft” ice has been thinning continuously since the end of the ice age, so that the ice cap has never been in a steady state.

Application of a Flow Model to the Ice-divide Region of Devon Island Ice Cap, Canada

AbstractThe steady-state flow model of Reeh (1988) is applied to a flow line that starts at the highest point of the Devon Island ice cap, follows the surface crest for 7.6 km, and then runs down the slope for a further 3.7 km. The effects of bedrock undulations, divergence of the flow lines, the variation of temperature with depth, and a basal layer of “soft” ice-age ice are taken into account. A flow law withn= 3 and a value ofAclose to that of Paterson (1981) is used. Longitudinal stress variations are neglected so that shear stress is calculated by the usual formula. It is estimated that these calculated values may be in error by at most 30%. Depth profiles of effective shear stress, and of the components of velocity and normal strain-rate, are presented at selected points along the flow line. These illustrate the large variations that occur near an ice divide and over bedrock undulations of amplitude comparable with the mean ice thickness. The model gives good predictions of the surface profile and of longitudinal and transverse surface strain-rates measured at ten points along the flow line. Predicted depth profiles of horizontal and vertical velocity components are compared with those measured in a bore hole. Comparison is limited by the fact that the model works in ice equivalent, whereas about 20% of the ice column consists of firn with different rheological properties from ice. The vertical velocity prediction is good. However, the model does not reproduce well the shape of the horizontal velocity profile, although measured and calculated fluxes differ only slightly. Predicted annual-layer thicknesses are within 15% of the measured ones in the upper half of the ice column, which consists of ice deposited in the last 1000 years. Predicted thicknesses in older ice are too small and the discrepancy increases with depth. This might indicate increased precipitation or, more likely, a thinner ice cap in the climatic optimum. However, it could also result from the fact that the layer of “soft” ice has been thinning continuously since the end of the ice age, so that the ice cap has never been in a steady state.

First-Order Stresses and Deformations in Glaciers and Ice Sheets

Abstract In this article the distribution of stress and velocities in glaciers and ice sheets is reinvestigated. We first derive the general equations governing non-linear viscous flow under plane deformations and formulate the relevant boundary conditions, including, in particular, a proper treatment of the accumulation–ablation mechanism. It is then shown how the emerging set of non-linear equations for the established boundary-value problem can be separated into a system covering steady-state problems on the one hand, and transient, time-dependent processes on the other hand. This separation is performed under the assumption that steady-state stresses are larger than the corresponding transient counterparts, suggesting a linearization of the transient equations with regard to the stresses. The steady-state equations are then analysed for the special case of an infinitely long, nearly parallel-sided slab. With the assumption that bottom undulations are small as compared to the glacier thickness it is shown that the original non-linear boundary-value problem can be decomposed into an infinite hierarchy of boundary-value problems defined on the simpler domain of the exactly parallel-sided slab, all of which are linear except for the lowest order one. Since its solution is readily available, the determination of the velocities and stresses due to bedrock protuberances is basically a linear problem, even though the constitutive response may be non-linear. Assuming harmonic bedrock undulations we show for a Navier–Stokes fluid that the transfer of the bedrock undulations to the surface strongly depends on the mean inclination of the slab, but, more importantly, does now show a maximum when plotted as a function of wavelength λ. This result is contradictory to the corresponding results of Budd (1970[a]) and implies serious drawbacks to his calculations of longitudinal stresses and strain-rates in his subsequent article (Budd, 1970[b]). Yet, it is not true that for maximal transfer of bottom protuberances to the surface a distinct wavelength would not exist. The calculations of Budd must rather be extended to include non-linear constitutive behaviour, variations of temperature with depth, and sliding at the bed. It then turns out that under certain circumstances maximal transfer of bottom undulations to the surface in a distinct wavelength domain (3 < λ < 5) may indeed exist. Sliding at the bed and vertical temperature variation thereby play a decisive role. Equally important is the stress distribution at the base, in particular the influence of the longitudinal strain effects on the latter. Rheological non-linearities, vertical temperature variations, and the sliding law at the bed play an important role and are investigated in detail. For non-linear constitutive behaviour and spatially dependent temperature-variation solutions must be sought numerically. The finite-difference scheme used suggests a generalization of Glen’s flow law so as to account for a nearly linear behaviour at low strain-rates. We conclude with a perspective of possible extensions of the general theory to various other time-dependent and time-independent problems.

First-Order Stresses and Deformations in Glaciers and Ice Sheets

AbstractIn this article the distribution of stress and velocities in glaciers and ice sheets is reinvestigated. We first derive the general equations governing non-linear viscous flow under plane deformations and formulate the relevant boundary conditions, including, in particular, a proper treatment of the accumulation–ablation mechanism. It is then shown how the emerging set of non-linear equations for the established boundary-value problem can be separated into a system covering steady-state problems on the one hand, and transient, time-dependent processes on the other hand. This separation is performed under the assumption that steady-state stresses are larger than the corresponding transient counterparts, suggesting a linearization of the transient equations with regard to the stresses. The steady-state equations are then analysed for the special case of an infinitely long, nearly parallel-sided slab. With the assumption that bottom undulations are small as compared to the glacier thickness it is shown that the original non-linear boundary-value problem can be decomposed into an infinite hierarchy of boundary-value problems defined on the simpler domain of the exactly parallel-sided slab, all of which are linear except for the lowest order one. Since its solution is readily available, the determination of the velocities and stresses due to bedrock protuberances is basically a linear problem, even though the constitutive response may be non-linear.Assuming harmonic bedrock undulations we show for a Navier–Stokes fluid that the transfer of the bedrock undulations to the surface strongly depends on the mean inclination of the slab, but, more importantly, does now show a maximum when plotted as a function of wavelength λ. This result is contradictory to the corresponding results of Budd (1970[a]) and implies serious drawbacks to his calculations of longitudinal stresses and strain-rates in his subsequent article (Budd, 1970[b]). Yet, it is not true that for maximal transfer of bottom protuberances to the surface a distinct wavelength would not exist. The calculations of Budd must rather be extended to include non-linear constitutive behaviour, variations of temperature with depth, and sliding at the bed. It then turns out that under certain circumstances maximal transfer of bottom undulations to the surface in a distinct wavelength domain (3 < λ < 5) may indeed exist. Sliding at the bed and vertical temperature variation thereby play a decisive role.Equally important is the stress distribution at the base, in particular the influence of the longitudinal strain effects on the latter. Rheological non-linearities, vertical temperature variations, and the sliding law at the bed play an important role and are investigated in detail.For non-linear constitutive behaviour and spatially dependent temperature-variation solutions must be sought numerically. The finite-difference scheme used suggests a generalization of Glen’s flow law so as to account for a nearly linear behaviour at low strain-rates.We conclude with a perspective of possible extensions of the general theory to various other time-dependent and time-independent problems.

Study of Deformations within Ice Sheets due to Bottom Undulations by Means of Radio Echo-Sounding

Abstract In order to test the theory of glacier flow over bedrock undulations presented by S. J. Johnsen, K. Rasmussen, and N.Reeh in the accompanying abstract, data from the E.G.I.G. and the Dye-3 flow lines on the Greenland ice sheet have been analysed. The data comprise surface profiles measured by conventional techniques, and ice thicknesses and depths of internal isochronic layers obtained by the Technical University of Denmark by means of radio echo-soundings.

Study of Deformations within Ice Sheets due to Bottom Undulations by Means of Radio Echo-Sounding

AbstractIn order to test the theory of glacier flow over bedrock undulations presented by S. J. Johnsen, K. Rasmussen, and N.Reeh in the accompanying abstract, data from the E.G.I.G. and the Dye-3 flow lines on the Greenland ice sheet have been analysed. The data comprise surface profiles measured by conventional techniques, and ice thicknesses and depths of internal isochronic layers obtained by the Technical University of Denmark by means of radio echo-soundings.