Glaciers once covered 30% of Earth’s land area, leaving diverse landforms; glacial geologists study glaciers to understand the formation of these landforms. Glacier ice is a metamorphic rock deforming at temperatures close to the melting point; structural geologists study glaciers to learn more about the origin of similar structures in other rocks. Ice cores from glaciers contain a well-dated record of climatic fluctuations over millennia, so climatologists study glaciers to understand the drivers of Earth’s climate. Failure of glacier dams can cause floods that engineers and town officials seek to prevent. Anthropogenically induced global warming is causing retreat of ice sheets and mountain glaciers; planners and policy makers want to know how to stop this retreat, and how fast it will raise sea level, impacting coastal infrastructure. A quantitative understanding of the physics of glaciers is essential for rigorous analysis of many of these problems. Glaciers occur in spectacular remote areas, unscarred by human activities; these environments appeal to many glaciologists.

Where a glacier bed is at the pressure melting point, ice moves past rigid obstacles by a combination of regelation and plastic flow. Equations can be written to describe the sliding speed due to these two processes when the ice is clean. Clastic debris and impurities decrease the speed. Basal ice commonly contains debris, liquid impurities, and liquid water on crystal boundaries. Thus, it is rheologically different from ice higher in the glacier. Water in cavities in the lee of bumps on the bed is commonly under pressure, and acts as a hydraulic jack, increasing the flow rate. Clasts gripped in basal ice moving over bedrock abrade the bed. Where a layer of till is present between the ice and the bedrock, the till may deform, increasing the glacier flow rate. Till obeys the classic Mohr-Coulomb failure criterion, but once it begins to deform, the deformation rate increases rapidly with stress, as in a perfectly plastic substance. Factors that control the depth of deformation in till are not well understood. Ice may regelate downward into till; this is a possible mechanism of till entrainment. Deformation of subglacial till may lead to the formation of drumlins and flutes.

In this chapter, the equations developed in Chapter 9 are used to solve for the stress and velocity distribution in an idealized “glacier” consisting of a slab of ice of infinite extent on a uniform slope. Solutions are first obtained for ice with a perfectly plastic rheology.Both the shear stress and the surface-normal stress increase linearly with depth. The shear stress equals the plastic yield stress at the bed. Longitudinal normal stresses vary non-linearly, and are compressive in the ablation area and extending in the accumulation area.The surface-normal velocity also varies linearly with depth. The surface-parallel velocity varies non-linearly, with a high gradient near the bed. The stress solutions and the solution for the surface-normal velocity are essentially the same in a non-linear material, except that the shear stress does not reach a limiting value at the bed.However, the surface-parallel velocity, while varying with depth in a similar way, is now dependent on the longitudinal strain rate, and the solution is much more complicated. Interestingly, however, it does not matter whether the longitudinal strain rate is compressive or extending.

Ice cores contain a record of climate change going back close to, and in some cases more than a million years.The record is shorter but more detailed in cores from places with high accumulation rates. The age of ice in cores can be determined by counting sedimentary layers defined by depth hoar or dust, and by analysis of certain chemical species that vary seasonally. The most important of the latter is δ18O. Where records are too low resolution to detect seasonal variations, numerical modeling can be used, although it is less precise. Through study of ice cores we know that concentrations of CO2 in the atmosphere have varied in phase with ice sheet volume over the past 900 ky or so, but it is not clear whether the CO2 led or lagged ice sheet volume.Analysis of a core from the Allen Hills Blue Ice area suggests that the amplitude of oscillations in atmospheric CO2 was smaller prior to 900 ka, when ice sheet volume was varying on a 40 ky time scale. Core records from the Holocene show that atmospheric CO2 and CH4 began to increase when humans began farming; had humans not begun to modify the atmosphere at that time, we would likely be in the middle of an ice age now.

Numerical modeling is a way of solving complex sets of equations that cannot be solved analytically.In finite difference modeling the infinitesimal step (e.g. dx) in the governing differential equation is replaced by a finite step (e.g. "∆x" ), and the variation over this step is assumed to be linear. Integration starts at a boundary and continues stepwise across the domain. In finite element and finite volume models, the domain is discretized into elements of unequal size and equations are written relating parameters on the boundaries of these elements. This results in a system of equations that must be solved simultaneously.Because the deformation rate in a glacier is a function of temperature, and conversely, full solutions require coupling of energy balance and momentum balance equations.Examples are given involving the role of subglacial permafrost in ice sheet behavior, estimation of prior ice sheet behavior from characteristics measured in ice cores and boreholes, and use of non-deterministic models to estimate sea level rise.

Glaciers are classified according to size, shape, and temperature. Temperate glaciers are at the pressure melting point throughout, whereas polar glaciers are below the pressure melting point except, in some places, at the bed. Ice is basically incompressible, which simplifies many analyses. Stresses, strains, and strain rates are second rank tensor quantities, so nine quantities are needed to describe them. Ice flows in response to stresses in excess of hydrostatic, or deviatoric stresses. To a good approximation, the strain rate, “ε”̇, can be described by “ε” ̇= A“σ” ^n in which A is a temperature-dependent rate factor, “σ”isthe stress, and n is a constant, usually taken to be 3.

Water flowing through veins along 3-grain intersections enlarges veins, forming conduits. At depth, the conduits are likely normal to equipotential planes that dip upglacier at ~11-times the slope of the glacier surface.Along the bed, the water flows normal to the intersection of the equipotential planes with the bed. Basal conduits are likely broad and low. Their size reflects a balance between melting by energy dissipated by the flowing water and closure due to pressure in the ice that slightly exceeds that in the water. In a steady state, water pressure in these conduits decreases with increasing discharge, leading to an arborescent drainage network. Water at the bed also moves through a distributed drainage system consisting of cavities linked by small conduits. In this system, water pressure may increase with discharge. Water pressures building up in this system may lead to surges. Conduits on till are more complicated because conduit shape is adjusted to provide a sediment transport rate equal to the rate of sediment supply. Near the glacier margin, sediment deposited in subglacial conduits may form eskers. Tunnel valleys form when subglacial lakes drain catastrophically.

As an initially cubic element of ice is advected through a glacier, it is gradually deformed. By the time it melts out at the glacier margin, it may be hundreds of times longer and fractions of a percent as thick as when it started. The deformation of such an element is described by three parameters, the natural octahedral unit shear, the angle through which a material line that becomes a principal axis in the strained state has rotated, and the angle that the greatest principal axis makes with the x-axis.Once primary features such as snow stratigraphy and crevasse fillings are deformed by this cumulative strain, their origin is difficult or impossible to determine, and the resulting penetrative layering is properly called foliation.Foliation is most highly developed where shear strain rates are highest - along the bed and adjacent to valley walls. Once formed, foliation may become deformed by changes in flow regime. An advance or retreat of a glacier can lead to recumbent folds in basal ice, and surges can result in sweeping folds in foliation defined by medial moraines. Foliation planes in ice sheet margins are not shear planes.

To develop some facility with application of the principles discussed in the preceding three chapters, some examples of classical problems are discussed in this chapter. The first problem involves the collapse of a cylindrical hole in ice. This solution is applied to problems of determining flow law constants from closure of tunnels and boreholes, ofclosure of water conduits in response to melting by energy dissipated in the flowing water. In the second, force balance calculations are used to study the role of seasonal and shorter-term variations in water pressure on glacier motion.Then, the need to consider longitudinal coupling in calculations of deformation rates is explored. In calculating velocity profiles at a point on a glacier, one needs to take into consideration not only the surface slope and ice thickness at that point, but also their variation up- and downglacier from that point.Finally, the theoretical basis for analysis of borehole deformation data to determine flow law constants is presented. Longitudinal strain rates and vertical advection need to be taken into consideration in analyzing borehole tilt data.

In this chapter, we find that snow accumulation and its transformation to ice lead to stratigraphy that persists for thousands of years and can be used to date ice. A perturbation analysis is then used to show that net balances are sensitive to summer temperature in continental areas and to both winter balance and summer temperature in maritime ones. Radiation balance could play a role in either environment. Dynamic thickening or thinning, and terminus advance or retreat are found to be closely linked. A rapid advance accompanied by commensurate thinning does not change mass. However, an advance may result in rapid terminus melting and retreat. Calving and bottom melting are also important mass balance components. On tidewater glaciers calving can lead to retreats unrelated to climate. Calving is a dominant process of mass loss on ice sheets. Bottom melting is important on some valley glaciers and ice shelves. Variations in atmospheric circulation patterns may result in asynchronous mass-balance patterns on glaciers only a few hundred kilometers apart. Finally, estimates of global mass balance, and their contribution to sea level rise are summarized.

In this chapter, a conservation of mass approach is used to develop a broad picture of the flow field in a glacier or ice sheet. Vertical velocities are shown to be downward in the accumulation area and upward in the ablation area, and horizontal velocities to increase with distance from the head of the glacier, reaching a maximum just below the equilibrium line. Conservation of momentum is then used to calculate the variation, with depth of horizontal and vertical velocity. Effects of valley sides and laterally non-uniform basal boundary conditions on the flow field are explored. Next patterns of internal reflectors imaged by radar are shown to reflect variations in effective strain rate and ice fabric. The reflectors can also be used to measure vertical strain rates and sub-ice shelf melt rates, and to document changes in the flow field over millennial time scales. Drifting snow also affects the flow field in polar environments, leading to development of narrow accumulation zones along the margin, and thus to formation of ice-cored moraines somewhat upglacier from the margin. Finally, inhomogeneous bed conditions beneath ice sheets can lead to streaming flow.

Active ice streams move one to two orders of magnitude faster than the ice bounding them, resulting in heavily crevassed shear margins. The high velocity of ice streams is attributed to lubrication by water at the glacier bed. Some shear margins are stabilized by valley walls, but others appear in places with no obvious underlying topographic control. The latter may be unstable against perturbations in the speed of the ice stream or in the influx of ice from outside the margin. Changes in basal hydrology can result in changes in ice stream width and velocity, and may result in shutting down the flow. Ice streams commonly transition into ice shelves in grounding zones, a few kilometers wide, in which coupling of the ice with the bed gradually diminishes. The stress and velocity distribution in ice shelves differs from that in grounded glaciers owing to the lack of any traction on the bed. The mass balance of ice shelves is affected by melting of the shelf base by sea water circulating beneath the shelf. Ice shelves can disintegrate in a matter of days or weeks when atmospheric temperatures are warm enough to result in appreciable melting on the shelf surface.

This chapter leads off with a review of ice crystal structure and the role of dislocations in deformation. Rate-limiting processes are climb at high stresses, grain-boundary slip at intermediate stresses, and diffusion at low stresses. In polycrystalline ice, stress concentrations drive recrystallization by grain growth, polygonization, and nucleation of new grains. The latter and rotation of grains as slip occurs on basal planes leads to preferred orientations of c-axes, and softens the ice. Using a deformation mechanism map in grain size-stress space we show that early experiments spanned the boundary between dislocation creep and creep limited by grain boundary sliding. Much of the deformation in natural ice masses, however, occurs in the latter regime. Next, we introduced Glen’s flow law and related it to these creep mechanisms. Temperature and pressure are incorporated in the flow law by rigorous, physically-based modifications, whereas microfabric and water content must be included empirically. Finally, we introduced linear elastic fracture mechanics and used it to study crevasse depths. Fracturing weakens ice, and this may be included in the flow law with a damage factor.

At any point in a glacier, there are three normal and six shear stresses.Coordinate axes can be chosen so that the shear stresses vanish. The remaining normal stresses are known as the principal stresses. Certain combinations of the stresses do not vary with the orientation of the coordinate axes. These are known as invariants of the stress tensor. The second invariant is one half the sum of squares of all nine stresses in the tensor. This stress is used in the common flow law for ice, so the deformation rate depends on all the stresses acting, not just on those acting in the direction of the deformation.Balancing forces on an element of ice at a point leads to an equation for the conservation of linear momentum. The strain along a line is defined as the change in length per unit length. There are also three normal and six shear strain rates. Again, axes can be chosen so that the shear strain rates disappear.The remaining normal strain rates are called the principal strain rates.In an isotropic material the principal axes of stress and strain rate coincide. Ice is commonly assumed to be isotropic for purposes of theoretical calculations, although this is clearly not true.