Mathematical analysis of volcanoes

Summary: Mathematical models and data analysis can help geologists better understand the activity of volcanoes and the fluid dynamics of their eruptions.

Volcanoes are openings of channels connecting the molten interior of a planet with its surface. Active volcanoes emit magma, ash, and gasses, and inactive volcanoes are reminders of past eruptions, consisting of solidified lava and ash. The science of studying volcanoes is known as “volcanology.” Many scientists and philosophers throughout history, including mathematicians Johannes Kepler and René Descartes, theorized about their nature and formation. Mathematics continues to play a role in modern volcanology through both the coursework and degrees that are required and in the mathematical research prevalent in the exploration of various volcanic phenomena. Computer-based numerical simulations and digital imagery, often from satellite observation, combined with mathematical and statistical methods, such as neural networks and data mining, are increasingly used to model, describe, and visualize the complex mathematical representations of volcanic processes. Predicting eruptions is also a challenge, which is necessary not only for safety and response at the time of the eruption but also for larger issues such as global climate change. Benjamin Santer of Lawrence Livermore National Laboratory, who specializes in mathematical and statistical analyses of climate data, has used volcanoes as one variable in explaining climate change. Scientists at the Yellowstone Volcano Observatory also collect data to monitor and mathematically study the enormous Yellowstone caldera, sometimes known as the Yellowstone supervolcano.

Measuring Volcanoes

The most destructive volcanic effect comes from pyroclastic flow, which is a mixture of solid to semi-solid fragments of rock, ash, and hot gases that flows down the sides of the volcano. It is a type of gravity current, similar to an avalanche, that can be modeled with theories and equations from fluid dynamics. A useful metric for comparing eruptions is the volume of volcanic ejecta. For example, the 1980 eruption of Mount St. Helens produced about 1.3 cubic kilometers of ash, but the ancient eruption of the Toba volcano on Sumatra around 75,000 years ago produced more than two thousand times more ash. It is possible to measure the fragmentation of the airborne volcanic matter, called “tephra,” even for ancient eruptions. Fragmentation is associated with the strength of the volcanic explosion. The dispersion of tephra over an area has been found to be related to the height of the eruption column. Finding and analyzing dispersion allows estimation of heights for ancient eruptions and an additional way to measure heights for modern eruptions. Volcanologists have created the Volcanic Explosivity Index (VEI), which takes into account the volume of ash and the height and duration of the eruption. There are nine types of volcanoes according to VEI, scaled 0–8. For example, the low-strength, low-height Type 0 is called “Hawaiian,” and the high-strength, low-fragmentation Type 6 through Type 8 are called “Plinian eruptions,” named for Roman historian Pliny the Younger, who described in detail the first century eruption of Mount Vesuvius that destroyed Pompeii. Plinian eruptions can have global environmental effects. Similar to the Richter scale, VEI is logarithmic: each level type is about 10 times greater in magnitude than the previous level.

Geometry of Volcanoes

Shapes of volcanoes depend on their explosivity, viscosity of magma, the composition of the surrounding crust, and other geological factors. The familiar, iconic cone shape such as Mount Fuji defines a “stratovolcano,” so named because of its many layers (or “strata”) of ash and hardened lava. Eruptions of these volcanoes have high explosivity and low-viscosity lava, making lava and tephra deposit near the opening in layers of diminishing thickness, thus forming the cone.

In contrast, broad, very fluid lava fields produce shield volcanoes that resemble a rather flat warrior shield. Lava domes, as the name suggests, are proportionally higher than shield volcanoes and more rounded than cone volcanoes, resembling semispheres. Lava domes are formed by high viscosity lava combined with low explosivity, where lava either accumulates under the crust and pushes it up, or flows over the crust and solidifies in the dome shape.

Eruption Forecast

Because volcanic eruptions depend on many variables, eruption forecasting relates to such areas of science and mathematics as chaos theory and systems science. Overall, prediction means collecting multi-variate data in volcano observatories and matching variable patterns to those that occurred before eruptions of similar types of volcanoes in the past. For example, the pattern of earthquakes becoming stronger and shallower with time, called “earthquake swarm,” can be used to forecast the eruption time. Mathematical models of volcanoes are based on equations from thermodynamics, fluid dynamics, and solid mechanics. The systems science principles of prediction describe qualitative trends in variables. For example, the principle of coinciding change says that unrelated, co-evolving trends in several parameters are more significant than changes in any one parameter.

Bibliography

Marti, Joan, and Gerald Ernst. Volcanoes and the Environment. Cambridge, England: Cambridge University Press, 2005.

Zeilinga de Boer, Jelle, and Donalt Sanders. Volcanoes in Human History. Princeton, NJ: Princeton University Press, 2002.