Mathematics in Geosciences, September 2001 - June 2002

During the past decade, geoscientists have come to appreciate the often-dominant role played by complex and nonlinear rheologies in the deformation of geologic materials. This is nowhere more evident than in the earth's crust and mantle where effective viscosities can vary over many orders of magnitude and different modes of deformation occur. On the largest scale, plate tectonics occurs because of the poorly understood multi-rheological behavior of crustal and mantle rocks. On a smaller scale, complex geomorphic features, such as those in evidence in the Sierra Nevada, are the outcome of flows which are dramatically influenced by variable viscosity, phase transitions, and other physical and chemical properties. Nonlinear rheology, through a number of microscopic processes and macroscopic interactions such as thermal feedback, results in a rich variety of flow behavior including strain localization and instability leading to faulting. Such behaviors are ultimately responsible for such diverse geologic phenomena as plate tectonics and earthquakes. Pyroclastic flows, associated with volcanic events, occur when these environments contribute to a punctuated or even explosive discharge of molten materials, particularly in the presence of water which not only catalytically alters the rheology but, when superheated, produces steam under very great pressure and very abruptly changes the dynamic equilibrium. Water, in its interaction with other earth materials, also has a profound effect on the landscape. Alone, its erosive effects modify topography on a long geologic time scale. However, if a flow transports granular material, those grains impact and erode irregularities in underlying materials, an effect that is demonstrably unstable. This mechanism, associated with catastrophic flooding, can have an overnight effect on the landscape. The clays that are produced are an important agent in the formation of topography and have an especially complex rheology. Wind and ice can operate in a similar way, completing the triad of aeolian influences. Another invasive process, known as "undercutting," results from the hydrostatic injection of water or steam into cracks in rock; its cooling and freezing, and subsequent expansion, cause existing cracks to grow as well as new cracks to be initiated. Water, even in microscopic quantities, has a dramatic influence on the strength of rocks and could contribute to the migration of seismicity and the triggering of seismic events at great distances from an earthquake. Sea ice is another excellent example of a material with a nonlinear rheology.

Nonlinear rheology, taken in the broadest sense, may be the single most important aspect of the behavior of earth materials. While rheology is not an issue for flows in the earth's liquid outer core, understanding the mechanisms and nonlinear interactions involved in the generation and reversal of the earth's magnetic field by dynamo action in the core remains a great challenge in geophysics. Though progress has been made toward the solution of the coupled nonlinear equations of motion and electrodynamics for the extreme parameter values appropriate to the rapidly rotating and relatively inviscid core of the earth, novel approaches are required to proceed substantially further.

Keywords: mantle, lithosphere, seismic creep, pyroclastic flows, aeolian flows, drainage networks, rheology, plate tectonics, dynamo, earthquakes, erosion, core.

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2001-2002 IMA Thematic Year on Mathematics in the Geosciences