The concept of strength envelopes, developed in the 1970s, allowed quantitative predictions of the strength of the lithosphere based on experimentally determined constitutive equations. Initial strength envelopes used an empirical relation for frictional sliding to describe deformation along brittle faults in the upper portion of the lithosphere and power law creep equations to estimate the plastic flow strength of rocks in the deeper part of the lithosphere. In the intervening decades, substantial progress has been made both in understanding the physical mechanisms involved in lithospheric deformation and in refining constitutive equations that describe these processes. The importance of a regime of semibrittle behavior is now recognized. Based on data from rocks without added pore fluids, the transition from brittle deformation to semibrittle flow can be estimated as the point at which the brittle fracture strength equals the peak stress to cause sliding. The transition from semibrittle deformation to plastic flow can be approximated as the stress at which the pressure exceeds the plastic flow strength. Current estimates of these stresses are on the order of a few hundred megapascals for relatively dry rocks. Knowledge of the stability of sliding along faults and of the onset of localization during brittle fracture has improved considerably. If the depth to the bottom of the seismogenic zone is determined by the transition to the stable frictional sliding regime, then that depth will be considerably more shallow than the depth of the transition to the plastic flow regime. Major questions concerning the strength of rocks remain. In particular, the effect of water on strength is critical to accurate predictions. Constitutive equations which include the effects of water fugacity and pore fluid pressure as well as temperature and strain rate are needed for both the brittle sliding and semibrittle flow regimes. Although the constitutive equations for dislocation creep and diffusional creep in single-phase aggregates are more robust, few data exist for plastic deformation in two-phase aggregates. Despite the fact that localization is ubiquitous in rocks deforming both in brittle and plastic regimes, only a limited amount of accurate experimental data are available to constrain predictions of this behavior. Accordingly, flow strengths now predicted from laboratory data probably overestimate the actual rock strength, perhaps by a significant amount. Still, the predictions are robust enough that uncertainties in geometry, mineralogy, loading conditions and thermodynamic state are probably the limiting factors in our understanding. Thus, experimentally determined rheologies can be applied to understand a broad range of topical problems in regional and global tectonics both on the Earth and on other planetary bodies.

Strength of the lithosphere: Constraints imposed by laboratory experiments

Around the globe, and in a variety of settings including active and inactive mountain belts, increases in sedimentation rates as well as in grain sizes of sediments were recorded at approximately 2-4 Myr ago, implying increased erosion rates. A change in climate represents the only process that is globally synchronous and can potentially account for the widespread increase in erosion and sedimentation, but no single process-like a lowering of sea levels or expanded glaciation-can explain increases in sedimentation in all environments, encompassing continental margins and interiors, and tropical as well as higher latitudes. We suggest that climate affected erosion mainly by the transition from a period of climate stability, in which landscapes had attained equilibrium configurations, to a time of frequent and abrupt changes in temperature, precipitation and vegetation, which prevented fluvial and glacial systems from establishing equilibrium states.

Increased sedimentation rates and grain sizes 2-4 Myr ago due to the influence of climate change on erosion rates.

An overview of glacial hazards in the Himalayas

Based on observations of the 1982–1983 surge of Variegated Glacier, Alaska, a model of the surge mechanism is developed in terms of a transition from the normal tunnel configuration of the basal water conduit system to a linked cavity configuration that tends to restrict the flow of water, resulting in increased basal water pressures that cause rapid basal sliding. The linked cavity system consists of basal cavities formed by ice-bedrock separation (cavitation), ∼1 m high and ∼10 m in horizontal dimensions, widely scattered over the glacier bed, and hydraulically linked by narrow connections where separation is minimal (separation gap ≲ 0.1 m). The narrow connections, called orifices, control the water flow through the conduit system; by throttling the flow through the large cavities, the orifices keep the water flux transmitted by the basal water system at normal levels even though the total cavity cross-sectional area (∼200 m^2) is much larger than that of a tunnel system (∼10 m^2). A physical model of the linked cavity system is formulated in terms of the dimensions of the “typical” cavity and orifice and the numbers of these across the glacier width. The model concentrates on the detailed configuration of the typical orifice and its response to basal water pressure and basal sliding, which determines the water flux carried by the system under given conditions. Configurations are worked out for two idealized orifice types, step orifices that form in the lee of downglacier-facing bedrock steps, and wave orifices that form on the lee slopes of quasisinusoidal bedrock waves and are similar to transverse “N channels.” The orifice configurations are obtained from the results of solutions of the basal-sliding-with-separation problem for an ice mass constituting of linear half-space of linear rheology, with nonlinearity introduced by making the viscosity stress-dependent on an intuitive basis. Modification of the orifice shapes by melting of the ice roof due to viscous heat dissipation in the flow of water through the orifices is treated in detail under the assumption of local heat transfer, which guarantees that the heating effects are not underestimated. This treatment brings to light a melting-stability parameter Ξ that provides a measure of the influence of viscous heating on orifice cavitation, similar but distinct for step and wave orifices. Orifice shapes and the amounts of roof meltback are determined by Ξ. When Ξ ≳ 1, so that the system is “viscous-heating-dominated,” the orifices are unstable against rapid growth in response to a modest increase in water pressure or in orifice size over their steady state values. This growth instability is somewhat similar to the jokulhlaup-type instability of tunnels, which are likewise heating-dominated. When Ξ ≲ 1, the orifices are stable against perturbations of modest to even large size. Stabilization is promoted by high sliding velocity ν, expressed in terms of a ν^(−½) and ν^(−1) dependence of Ξ for step and wave cavities. The relationships between basal water pressure and water flux transmitted by linked cavity models of step and wave orifice type are calculated for an empirical relation between water pressure and sliding velocity and for a particular, reasonable choice of system parameters. In all cases the flux is an increasing function of the water pressure, in contrast to the inverse flux-versus-pressure relation for tunnels. In consequence, a linked cavity system can exist stably as a system of many interconnected conduits distributed across the glacier bed, in contrast to a tunnel system, which must condense to one or at most a few main tunnels. The linked cavity model gives basal water pressures much higher than the tunnel model at water fluxes ≳1 m^(3/s) if the bed roughness features that generate the orifices have step heights or wave amplitudes less than about 0.1 m. The calculated basal water pressure of the particular linked cavity models evaluated is about 2 to 5 bars below ice overburden pressure for water fluxes in the range from about 2 to 20 m^(3/s), which matches reasonably the observed conditions in Variegated Glacier in surge; in contrast, the calculated water pressure for a single-tunnel model is about 14 to 17 bars below overburden over the same flux range. The contrast in water pressures for the two types of basal conduit system furnishes the basis for a surge mechanism involving transition from a tunnel system at low pressure to a linked cavity system at high pressure. The parameter Ξ is about 0.2 for the linked cavity models evaluated, meaning that they are stable but that a modest change in system parameters could produce instability. Unstable orifice growth results in the generation of tunnel segments, which may connect up in a cooperative fashion, leading to conversion of the linked cavity system to a tunnel system, with large decrease in water pressure and sliding velocity. This is what probably happens in surge termination. Glaciers for which Ξ ≲ 1 can go into surge, while those for which Ξ ≳ 1 cannot. Because Ξ varies as α^(3/2) (where α is surface slope), low values of Ξ are more probable for glaciers of low slope, and because slope correlates inversely with glacier length in general, the model predicts a direct correlation between glacier length and probability of surging; such a correlation is observed (Clarke et al., 1986). Because Ξ varies inversely with the basal shear stress τ, the increase of τ that takes place in the reservoir area in the buildup between surges causes a decrease in Ξ there, which, by reducing Ξ below the critical value ∼1, can allow surge initiation and the start of a new surge cycle. Transition to a linked cavity system without tunnels should occur spontaneously at low enough water flux, in agreement with observed surge initiation in winter.

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Glacier surge mechanism based on linked cavity configuration of the basal water conduit system

Implications of fault-valve behaviour for rupture nucleation and recurrence

Understanding water movement through a glacier is fundamental to several critical issues in glaci- ology, including glacier dynamics, glacier-induced floods, and the prediction of runoff from glacierized drainage basins. To this end we have synthesized a conceptual model of water movement through a temper- ate glacier from the surface to the outlet stream. Pro- cesses that regulate the rate and distribution of water input at the glacier surface and that regulate water movement from the surface to the bed play important but commonly neglected roles in glacier hydrology. Where a glacier is covered by a layer of porous, perme- able firn (the accumulation zone), the flux of water to the glacier interior varies slowly because the firn tempo- rarily stores water and thereby smooths out variations in the supply rate. In the firn-free ablation zone, in con- trast, the flux of water into the glacier depends directly on the rate of surface melt or rainfall and therefore varies greatly in time. Water moves from the surface to the bed through an upward branching arborescent net- work consisting of both steeply inclined conduits, formed by the enlargement of intergranular veins, and gently inclined conduits, spawned by water flow along the bottoms of near-surface fractures (crevasses). Engla- cial drainage conduits deliver water to the glacier bed at a limited number of points, probably a long distance downglacier of where water enters the glacier. Englacial conduits supplied from the accumulation zone are quasi steady state features that convey the slowly varying water flux delivered via the firn. Their size adjusts so that they are usually full of water and flow is pressurized. In contrast, water flow in englacial conduits supplied from the ablation area is pressurized only near times of peak daily flow or during rainstorms; flow is otherwise in an open-channel configuration. The subglacial drainage system typically consists of several elements that are distinct both morphologically and hydrologically. An up- glacier branching, arborescent network of channels in- cised into the basal ice conveys water rapidly. Much of the water flux to the bed probably enters directly into the arborescent channel network, which covers only a small fraction of the glacier bed. More extensive spatially is a nonarborescent network, which commonly includes cav- ities (gaps between the glacier sole and bed), channels incised into the bed, and a layer of permeable sediment. The nonarborescent network conveys water slowly and is usually poorly connected to the arborescent system. The arborescent channel network largely collapses during winter but reforms in the spring as the first flush of meltwater to the bed destabilizes the cavities within the nonarborescent network. The volume of water stored by a glacier varies diurnally and seasonally. Small, temper- ate alpine glaciers seem to attain a maximum seasonal water storage of ;200 mm of water averaged over the area of the glacier bed, with daily fluctuations of as much as 20 -30 mm. The likely storage capacity of subglacial cavities is insufficient to account for estimated stored water volumes, so most water storage may actually occur englacially. Stored water may also be released abruptly and catastrophically in the form of outburst floods.

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Water flow through temperate glaciers

We develop theoretically a description of a possible subglacial drainage mechanism for glaciers and ice sheets moving over saturated, deformable till. The model is based on the plausible assumption that flow of water in a thin film at the ice-till interface is unstable to the formation of a channelized drainage system, and is restricted to the case in which meltwater cannot escape through the till to an underlying aquifer. In describing the physics of such channelized drainage, we have generalized and extended Rothlisberger’s model of channels cut into basal ice to include “canals” cut into the till, paying particular attention to the role of sediment properties and the mechanics of sediment transport. We show that sediment-floored Rothlisberger (R) channels can exist for high effective pressures, and wide, shallow, ice-roofed canals cut into the till for low effective pressures. Canals should form a distributed, non-arborescent system, unlike R channels. For steep slopes typical of alpine glaciers, both drainage systems can exist, but with the water pressure lower in the R channels than in the canals; the canal drainage should therefore be unstable in the presence of channels. For small slopes typical of ice sheets, only canals can exist and we therefore predict that, if channelized meltwater flow occurs under ice sheets moving over deformable till, it takes the form of shallow, distributed canals at low effective pressure, similar to that measured at Ice Stream B in West Antarctica. Geologic evidence derived from land forms and deposits left by the Pleistocene ice sheets in North America and Europe is also consistent with predictions of the model.

/pdf/channelized-subglacial-drainage-over-a-deformable-bed-2ms4csq5ga.pdf

Channelized subglacial drainage over a deformable bed

We present a mathematical model for the breakdown of porous rock by the growth of ice within cracks. The model is founded upon well-established principles of fracture mechanics and recent advances in soil physics, iJong with the assumption that progressive crack growth results from water migrating to ice bodies in cracks, much as water migrates to ice lenses in freezing soil. Our model predicts crack-growth rates compatible with empirical data. Calculations for a granite and a marble indicate that sustained freezing is most effective in producing crack growth when temperatures range from —4 °C to -15 °C. At higher temperatures, thermodynamic limitations prevent ice pressure from building up sufficiently to produce significant crack growth; at lower temperatures, the migration of water necessary for sustaining crack growth is strongly inhibited. In hydraulically "open" systems, in which pore-water pressure remains near atmospheric pressure during the freezing process, crackgrowth rates during continuous cooling will generally be greatest at low rates of cooling, less than -0.1-0.5 °C/h. At higher rates of cooling, the influx of water to growing cracks is significantly inhibited. The model delineates clearly the role of material parameters (elastic moduli, fracture-mechanical properties, grain size and shape, and crack size), environmental conditions (temperature, temperature gradient, water pressure), and time in frost damage to rocks. Our calculations, along with recent experimental work on water migration in freezing rocks (Fukuda, 1983), lead us to question the widely accepted importance of two phenomena—freezing of water in sealed cracks and freeze-thaw cycling—in the fracture of rock exposed to natural freezing conditions.

A theoretical model of the fracture of rock during freezing

Floods from failures of natural and constructed dams constitute a widespread hazard to people and property. Expeditious means of assessing flood hazards are necessary, particularly in the case of natural dams, which may form suddenly and unexpectedly. We revise statistical relations (derived from data for past constructed and natural dam failures) between peak discharge (Qp) and water volume released (V0) or drop in lake level (d) but assert that such relations, even when cast into a dimensionless form, are of limited utility because they fail to portray the effect of breach-formation rate. We then analyze a simple, physically based model of dam-breach formation to show that the hydrograph at the breach depends primarily on a dimensionless parameter η=kV0/gl/2d7/2, where k is the mean erosion rate of the breach and g is acceleration due to gravity. The functional relationship between Qp and η takes asymptotically distinct forms depending on whether η ≪ 1 (relatively slow breach formation or small lake volume) or η ≫ 1 (relatively fast breach formation or large lake volume). Theoretical predictions agree well with data from dam failures for which k, and thus η, can be estimated. The theory thus provides a rapid means of predicting the plausible range of values of peak discharge at the breach in an earthen dam as long as the impounded water volume and the water depth at the dam face can be estimated.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/97WR01616

Methods for predicting peak discharge of floods caused by failure of natural and constructed earthen dams

Glaciological theory predicts that the subglacial drainage network at the base of gently sloping ice sheets resting on deforming sediment should consist of many wide, shallow, probably braided "canals" distributed along the ice-sediment interface, rather than an arborescent network of relatively few large tunnels, as would develop over a rigid substrate. A corollary prediction examined here is that eskers, which form in large subglacial tunnels, should be rare where subglacial bed deformation occurred, but they may be relatively common where the bed was rigid. Bed deformation would be most likely where subglacial till was relatively continuous, fine-grained, and of low permeability—that is, in regions where till is derived primarily from underlying sedimentary bedrock—but unlikely where discontinuous, coarse-grained, high-permeability till was derived from underlying crystalline bedrock. The observed distribution of eskers in areas covered by the Laurentide and Eurasian (British, Scandinavian, and Barents Sea) ice sheets during the last glaciation shows that most eskers occur over crystalline bedrock overlain by discontinuous, high-permeability till, but are rare or absent over sedimentary bedrock overlain by fine-grained, low-permeability till, thus matching reasonably well our prediction. Glaciological theory and geologic evidence indicate that esker systems on a subcontinental scale are time-transgressive. Sedimentological evidence for a "canal" drainage system appears to be present in fine-grained tills where eskers are largely absent.

Joseph S. Walder

Papers

Water flow through temperate glaciers

Channelized subglacial drainage over a deformable bed

A theoretical model of the fracture of rock during freezing

Methods for predicting peak discharge of floods caused by failure of natural and constructed earthen dams

Subglacial drainage, eskers, and deforming beds beneath the Laurentide and Eurasian ice sheets