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1 - Mathematical Models and the Shallow-Ice Approximation
1.1. Introduction
The interaction between large masses of snow and ice with the earth's climate is of
fundamental importance. The strong coupling between the cryosphere, which includes
the ice and snow fields, sea ice and permafrost, with the other components of the climatic
system cannot be underestimated. Its influence on the global climate acts at long time
scales and is a dominant factor in the variation of the planetary albedo (see, for instance,
[POI).
If glaciation can lower considerably the sea level (possible 100 m or more) an hypothetical climatic warming would contribute significantly to world-wide sea levels, as a
consequence of the melting of glaciers and ice sheets, the "Earth thermometers". For
instance, only a 1% change in the Antartic ice volume would rise the global sea-level
about 73 cm, while the total melting of Greenland and Antartic ice would increase it by
approximately 80m. This fact has motivated several recent studies (see [Huy] and [V],
for instance).
Within the interdisciplinary science of Glaciology, with its various specialized branches
in Geography, Geophysics, Material Science or in Continuum Mechanics, the theoretical
study of the "ice thermo-mechanics" has brought the atention to applied mathematicians
in recent years. In particular, it has been recognized that the mathematical modelling
plays an important and central role in theoretical glaciology (see [Hu] or [MI). The mathematical treatment of ice flow problems and ice sheet-climate behaviour is still in an early
stage, although the intensive use of numerical models and efficient algorithms in this field
of geophysical sciences present new challenges to mathematicians.
In particular, in formulating the mathematical models for the study of the natural
large-scale ice dynamics, which consist primarily in gravity-driven flows of bounded valley
glaciers, grounded ice-sheets and floating ice-shelves, it is necessary to take into account
that the flow domain is not prescribed and is itself part of the solution. On the other
hand, a complete model must also take into account the co-existence of ice and water
regions in zones of melting and refreezing, which may be significant under the climate
warming or in interglacial periods, or simply in the description of the glacier's basal water
system and in the modelling of subglacial lakes due to geothermal heating (see [F2]).
The free surface flow of a glacier or ice sheet and the ice-water interface are typical
examples of free boundary problems, which are considered here, from a mathematical
point of view, in very simple situations in order to illustrate the potential application of
variational methods for three dimensional models in theoretical glaciology.
1.2. Shallow-ice modelling
The mathematical treatment of the thermomechanical coupled flow of the ice driven
by gravity and interacting with the atmosphere at the ice sheet surface and with the
ground at the base is based on the principles of continuum mechanics, hence on equations
of balance (see [Hu], for instance). At the scales of years or at longer time scales, ice
is modelled as an incompressible, non-linearly viscous heat conducting fluid which rate
factor strongly depend on the temperature.
1 - Mathematical Models and the Shallow-Ice Approximation
1.1. Introduction
The interaction between large masses of snow and ice with the earth's climate is of
fundamental importance. The strong coupling between the cryosphere, which includes
the ice and snow fields, sea ice and permafrost, with the other components of the climatic
system cannot be underestimated. Its influence on the global climate acts at long time
scales and is a dominant factor in the variation of the planetary albedo (see, for instance,
[POI).
If glaciation can lower considerably the sea level (possible 100 m or more) an hypothetical climatic warming would contribute significantly to world-wide sea levels, as a
consequence of the melting of glaciers and ice sheets, the "Earth thermometers". For
instance, only a 1% change in the Antartic ice volume would rise the global sea-level
about 73 cm, while the total melting of Greenland and Antartic ice would increase it by
approximately 80m. This fact has motivated several recent studies (see [Huy] and [V],
for instance).
Within the interdisciplinary science of Glaciology, with its various specialized branches
in Geography, Geophysics, Material Science or in Continuum Mechanics, the theoretical
study of the "ice thermo-mechanics" has brought the atention to applied mathematicians
in recent years. In particular, it has been recognized that the mathematical modelling
plays an important and central role in theoretical glaciology (see [Hu] or [MI). The mathematical treatment of ice flow problems and ice sheet-climate behaviour is still in an early
stage, although the intensive use of numerical models and efficient algorithms in this field
of geophysical sciences present new challenges to mathematicians.
In particular, in formulating the mathematical models for the study of the natural
large-scale ice dynamics, which consist primarily in gravity-driven flows of bounded valley
glaciers, grounded ice-sheets and floating ice-shelves, it is necessary to take into account
that the flow domain is not prescribed and is itself part of the solution. On the other
hand, a complete model must also take into account the co-existence of ice and water
regions in zones of melting and refreezing, which may be significant under the climate
warming or in interglacial periods, or simply in the description of the glacier's basal water
system and in the modelling of subglacial lakes due to geothermal heating (see [F2]).
The free surface flow of a glacier or ice sheet and the ice-water interface are typical
examples of free boundary problems, which are considered here, from a mathematical
point of view, in very simple situations in order to illustrate the potential application of
variational methods for three dimensional models in theoretical glaciology.
1.2. Shallow-ice modelling
The mathematical treatment of the thermomechanical coupled flow of the ice driven
by gravity and interacting with the atmosphere at the ice sheet surface and with the
ground at the base is based on the principles of continuum mechanics, hence on equations
of balance (see [Hu], for instance). At the scales of years or at longer time scales, ice
is modelled as an incompressible, non-linearly viscous heat conducting fluid which rate
factor strongly depend on the temperature.