Glacier and ice sheet volume

Glacier area can be well mapped from satellite imagery and aerial photographs, but global ice volume is more difficult to estimate. Mapping of ice thickness requires surface or airborne radar surveys. Electromagnetic wave frequencies of 5-50 MHz are typically used for glacier depth sounding, as these relatively low frequencies limit signal attenuation and allow penetration of radar pulses through several hundred meters of glacier ice. Resolution at these frequencies is of the same order as the wavelength, 6-60 m for the range 5-50 MHz. The ice-bed interface provides a strong electrical contrast, so these systems have good success in mapping subglacial topography and ice thickness. Other geophysical techniques (e.g., seismic and gravity surveys) have also been applied to ice-thickness mapping.

Detailed (1-5 km) survey grids have been flown over the Greenland and Antarctic ice sheets, providing the basis for ice-volume estimates in table 6.1. Directly translated, the ice volumes in the Greenland and Antarctic ice sheets are equivalent to 7.4 and 63.8 msl (meters of global eustatic sea level equivalent), respectively. Eustatic sea level is defined as the mean height of the world's oceans relative to the fixed surface of the geoid, assuming global ocean area is fixed.1 In reality, portions of the ice sheets are grounded below sea level, particularly in West Antarctica, and the ice shelves are floating; if this ice were to melt, it would not contribute to global sea level rise. The actual global sea level rise associated with complete, instantaneous loss of all of the ice in Greenland and Antarctica would give a sea level rise of about 64 m: 7.2 msl from Greenland and 56.6 msl from Antarctica.

This is approximate, as ocean basin area changes as a function of water volume. In addition, both the seafloor and the continents experience elastic and long-term viscous (isostatic) responses to changes in water and ice loading. The subglacial bedrock in much of Greenland and Antarctica has been depressed by more than 1000 m. Where the bed lies below sea level in Antarctica, removal of the ice sheet would cause initial flooding of the area—essentially an expansion of the Southern ocean.

Over centuries to millennia, isostatic relaxation would decrease the volume of the subglacial cavities and contribute to additional sea level rise.

Instantaneous removal of the part of the West Antarctic ice sheet (WAIS) that is susceptible to marine icesheet instabilities would give a global eustatic sea level rise of 3.2 m. Elastic rebound of the deglaciated WAIS bed would contribute an additional 0.06 msl, and the long-term (e.g. 10-kyr) isostatic adjustment gives a further 0.4 msl. Large portions of Antarctica and Greenland are also grounded below sea level: 41% of the Antarctic ice sheet, including most of West Antarctica, and 22% of the Greenland ice sheet. Deglaciation of either ice mass would expose extensive marine basins. In East Antarctica and Greenland, most of these would isostatically rebound to elevations above mean sea level over a tim-escale of centuries to millennia. There are important regional exceptions to this, where deep marine channels incise into the interior of each ice sheet.

Outside of Greenland and Antarctica, ice thickness measurements have been made on several dozen individual valley glaciers and ice caps. The Icelandic ice caps, in particular, are exceptionally well mapped, but this is unusual; glacier volume is poorly known in most other regions. Several approaches have been applied to estimation of ice thickness or volume based on other aspects of glacier geometry. Three common methods include (i) estimation of ice thickness as a function of distance from the ice margin, based on ice rheology and including assumptions about the subglacial topography (e.g., a flat or uniformly sloping bed), (ii) volume-area scaling, and (iii) estimates of local ice thickness from surface slope.

Of these methods, volume-area scaling is the most simple and popular method for estimation of glacier volume. Ice-volume data from approximately 100 glaciers around the world give the empirical relationship

where V(106 m3) is the volume and A(km2) the surface area of the glacier. The parameters c and g require regional calibration, and it is also common to adopt different values of the scaling parameters for different glacier sizes. Empirical data for the worldwide distribution of glaciers give the exponent g = 1.36. A power-law scaling relationship between glacier area and volume has also been theoretically derived, based on the rheologically determined surface profiles and aspect ratios of steady-state valley glaciers and ice caps. This gives g = 1.375 for valley glaciers, and the classical parabolic geometry of ice caps corresponds with g = 1.25.

Volume-area scaling is believed to be generally applicable to a large ensemble of glaciers. Individual glaciers can deviate by more than 50% from the aggregate relationship as a result of complex or deeply eroded bed topography, unusual ice flow regimes (e.g., extensive glacier sliding), or as a result of being far out of equilibrium. The relationship is not intended to be used on individual glaciers.

Estimates in table 6.1 for the volume of ice locked up in the world's mountain glaciers and ice caps are primarily based on volume-area scaling. Differences stem from contrasting assumptions about the scaling-law coefficients, glacier size distributions, the distinction in form between glaciers and ice caps, and uncertainty in estimates of glacier area. Excluding the peripheral icefields in Antarctica and Greenland, the estimates span a range from 51 X 103 to 164 X 103 km3, which translates to a eustatic sea level equivalent of 13-41 cm. There is an additional, poorly constrained area and volume of ice in the peripheral ice caps in Greenland and Antarctica; including the Antarctic Peninsula, this may amount to an additional 40-60 cm of global sea level equivalent. Combining the minimum and maximum estimates for different regions gives an estimated 56-97 cm of total eustatic sea level equivalent in global ice masses that are dynamically independent of the Greenland and Antarctic ice sheets.

As an alternative to volume-area scaling, first-order approximations of ice thickness can be made from the surface slope, based on the empirical observation that glacier ice has a gravitational driving (shear) stress, td, of about 100 kPa at the base. This is a result of ice rheol-ogy once again; ice deforms through nonlinear viscous flow in a way that is predictable, with an effective viscosity that allows glacial ice to support about this amount of shear stress. Glaciers thicken until they reach values close to this, and further thickening or steepening leads to increased ice flow, self-regulating to give td. 100 kPa. Taking advantage of this relationship, it is possible to estimate local ice thickness, H, through the equation

Pi gds where pi is the ice density, g is gravity, and Vs is the surface slope. The relation is local by definition, and it breaks down when surface slopes become small (or zero), as occurs in the accumulation area of large icefields and ice sheets. Average glacier slope is sometimes applied in this case.

This relation provides rough estimates of ice thickness, but there are large local and regional exceptions to the "100 kPa rule of thumb." Steep, deep parts of Greenland's Jakobshavn Isbrae ice stream, for instance, have shear stresses of 200-300 kPa. In contrast, the low-sloping Siple Coast ice streams in West Antarctica operate at shear stresses of 20-40 kPa. In this situation, basal flow prevails over internal deformation of the ice, so (6.2) is not valid. Like volume-area scaling, then, inferences from surface slope only provide a rough approximation of ice thickness.

The fact that we only know the volume of the world's mountain glaciers to a factor of two is humbling with respect to our knowledge of the planet, but it also points to an obvious target for advances in Earth system observation.

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