The equations of motion describe a nonlinear viscous fluid with an effective viscosity that is intermediate between that of liquid water and that which characterizes Earth's mantle; ice deforms slowly, but measurably. Where there is basal friction, vertical shear deformation is the dominant type of strain in glaciers and ice sheets. one implication of the nonlinear flow law is that the bulk of the shear deformation is concentrated near the bed. Another consequence is that the surface velocity (or column-integrated ice velocity) associated with shear flow is a highly nonlinear function of ice thickness and gravitational driving stress (surface slope): üd ? H4 Vs3. The thermal structure of glaciers reinforces this and can provide a positive feedback on shear deformation, as strain heating warms and softens the ice near the bed. In some cases (e.g., Ja-kobshavn Isbrae), this creates the development of a thick, strongly deforming temperate layer in the basal ice. Stiff, brittle surface ice that is -40°C can overlie a 1000-m-thick layer of relatively ductile ice at the pressure melting point.
In floating ice, where basal shear stress vanishes, ice deforms through longitudinal spreading, with essentially a free-slip condition at the base because the underlying water offers little frictional resistance. This flow is resisted by longitudinal stresses in the ice and, in most situations, horizontal shear stresses ("side drag") from the walls of a fjord or embayment. These additional stresses are also active in the inland part of ice sheets, where deformation is dominated by vertical shear flow, but are often secondary in this setting. Where ice-bed coupling is weak, as occurs in many ice streams, longitudinal and horizontal shear stresses assume an important role. They are also important in regions of complex ice dynamics, such as near the grounding line (the transition from grounded to floating ice), at ice divides, and in valley glaciers, where velocity and thickness gradients are steep and side drag from valley walls is significant.
Surface velocities of tens of meters per year are typical of the interior regions of the polar ice sheets and in most alpine glaciers. Thick outlet glaciers in steep valleys experience flow rates of 100 m yr-1 or more via internal (vertical shear) deformation. Basal motion is usually at play wherever glaciers and ice sheets have higher flow rates than this, such as Antarctic ice streams and in glacier surges. The main exceptions to this are for floating glaciers and in outlet glaciers that occupy deep fjords. Spreading flow rates in floating glaciers and ice shelves are often hundreds of meters per year. Where grounded glaciers discharge through fjords, as occurs in major outlet glaciers of the Greenland ice sheet, deep channels have been carved by glacial erosion, giving glaciers that are exceptionally thick and steep. This promotes high rates of internal deformation and surface velocities that reach thousands of meters per year.
Most of these fast-flow situations are problematic for glaciological models. The physical controls of basal flow are not well understood or parameterized, and deep, narrow fjord environments are poorly resolved in continental-scale models, which typically operate at spatial resolutions of 10-50 km. This is a pressing challenge for ice sheet models that aim to describe interannual-and decadal-scale variability in outlet glacier dynamics and iceberg discharge (and the associated sea level rise) in Greenland and Antarctica. It also means that some important mechanisms of temporal variability, such as glacier surge cycles, rarely arise naturally in glacier and ice sheet models.
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