Researchers in various areas develop theoretical frameworks within which they can interpret their experimental data. These theoretical frameworks, or models, incorporate as complete a description of the physical processes connecting different aspects of the experimental data as is possible. Comparison of the modeling results with the experimental data allows the researchers to verify their intuitions regarding the underlying physical processes that control the system being studied. A good model can be used in a predictive fashion to describe the behavior of a system subject to a different environment or at a different time in its evolution. Within the constraints of the theoretical framework, researchers can extend their vision beyond the measurable surface of the phenomenon, allowing them to literally ``see'' inside the system which they are experimentally observing.
Physics-based modeling of this type is intrinsically dependent on the modern computer to make meaningful sense of the wealth of experimental data available for many fields of study. Traditionally modelers have used restrictive assumptions and simplified domains to allow for the analytic solution of the equations describing their physical system. With the advent of modern computers numerical methods which yield an approximate solution can be applied to equations which are not amenable to analytic evaluation, thereby allowing the researchers to solve the exact equations without restrictive assumptions or simplified domains.
In this paper a numerical procedure for solving differential equations arising from consideration of conserved quantities is described. This numerical technique, called the finite-element method, is a robust alternative to other traditional numerical techniques such as the finite-difference method. The technique is applied to a specific area of research, the modeling of ice sheets. Differential equations describing the behavior of ice sheets arise from consideration of conservation of mass, conservation of momentum, and conservation of energy, none of which can be solved analytically for realistic domains.
Modeling ice sheets is intrinsically tied to the understanding of past and future climatic change. As we try to understand the climatic system of the Earth, we develop more and more complex models of the Earth's atmosphere, hydrosphere, cryosphere, and even biosphere to understand the interactions that control climate. Many of these models behave well in recreating the present climate, but to give these models a predictive capability we must test them against different climates of the past. A time period with a reasonably well-understood and yet very different climate was the most recent ice age, climaxing approximately 18,000 years ago. During this time period major ice sheets thousands of meters thick covered much of North America and Eurasia, presenting to the atmospheric circulation a completely different terrain, both in elevation and in albedo, than presently exists on the planet. At the same time a wealth of datable climate data exists from the oceans and the unglaciated land surfaces. The recentness of the event on the geological time scale makes the determination of the past climate far more accurate than for paleo-climates in the more distant past.
Ice sheet modeling provides critical input to the atmospheric models in the form of the ice surface elevation during the ice ages. Geological mapping can determine the outline of the ice sheet from glacial indicators such as moraines, but only through model calculations can we arrive at a reasonable estimate for the shape and elevations of these vast mountain ranges of ice.
Modeling physical phenomena involves the generation and solution of differential equations. The accuracy and efficacy of these models depends ultimately on the veracity and completeness of the underlying assumptions used to generate these equations. Both analytic and numerical solutions can be used, although both have their limitations. Often in an attempt to obtain an analytic solution, limited or simplified assumptions are made about the underlying phenomena. Numerical solutions allow a more complete and exhaustive treatment of a physical problem, although we are left with the less satisfying condition of being dependent on the computer to obtain the solution, and of having no analytic solution which one can examine. With the analytic solution the physics is often obvious in the form of the equations, whereas in the numerical arena we must examine ``results'' of various simulations and model-runs with differing boundary and initial conditions in order to get a feeling for the connection between various components of the phenomenon.