Science News
Week of Feb. 2, 2002; Vol. 161, No. 5
Return to Gallery
It's a Rough World
Fractals help model vexing problems in earth science
Sid Perkins
In the realm of mathematics, perfection abounds. Lines stretch
straight to infinity, planes are flawlessly flat, and spheres are
impeccably round. The real world, however, is almost always
irregular—the jagged spear of a lightning bolt, the rough face of a
broken rock, and the ragged profile of a mountain range are just a
few examples. Although people have always been surrounded by texture,
until recently they could not describe it in anything other than
qualitative terms such as smooth or rugged or lumpy. Long after
scientists developed ways to measure physical properties such as
temperature, weight, and time, the techniques needed to quantify
roughness eluded them.
Then came fractals.
Before these mathematical tools were developed during the 1960s,
scientists usually represented the physical world using only three
dimensions. Lines occupy one dimension; planes, two; and cubes,
three. But fractals freed scientists from the tyranny of integer
dimensions and enabled them to describe objects using fractional
dimensions.
Not only do such representations of objects show roughness and
irregularity, they do so across the entire range of scale. A close-up
look at a jagged nonfractal line would reveal that it’s made up of
small, straight segments. But for a fractal line—say, a
1.3-dimensional line—the view would be the same from afar as from
close-up. Each segment would have the same degree of roughness no
matter the scale at which it’s viewed. Likewise, a microscopic look
at a rock with a 2.7-dimensional surface would display the same
texture as the telescopic view of a cliff face with the same
fractional dimension.
Far from being merely mathematical abstractions, fractals pop up
throughout the natural and the humanmade world. They appear in the
swirling patterns on a flowing liquid’s surface (SN: 1/23/93, p. 53)
and the patchy mosaics of urban sprawl (SN: 1/6/96, p. 8).
Scientists have used fractals to analyze all sorts of irregular
objects or phenomena, from swings in the stock market to the
frequency of natural disasters, says Donald L. Turcotte, a
geophysicist at Cornell University. One of the first uses of fractals
in geosciences was a simple one, describing the roughness of
Britain’s coastline. Today, researchers are applying fractals to
far-ranging topics—the probability of wildfire, as well as the spread
of toxic fluids through rocks and soil.
Natural disasters
Earthquakes, wildfires, and other large-scale phenomena are the focus
of many fractal analyses. The relationship between the frequency and
the magnitude of earthquakes is fractal, Turcotte notes. When
researchers plot these two quantities on a logarithmic graph, the
data points fall on a line. In such a correspondence between the two
quantities, called a power-law relationship, the slope of that line
is an exponential power that can be employed as a fractal dimension.
Fractal equations allow scientists to more easily develop computer
models to simulate how often earthquakes occur.
Scientists first noted the power-law relationship between the
frequency and magnitude of earthquakes in the 1950s—well before
fractals were invented, says Bruce D. Malamud, a mathematician at
King’s College in London. The relation holds for earthquakes of
almost all sizes.
More recently, Malamud and his colleagues showed that a power-law
relationship exists between the frequency and size of wildfires. The
researchers discovered that the fractal relationship held true for
four different sets of wildfire data, even though the sets were
compiled for areas of the world radically different in terrain and
vegetation. Malamud says that the consistent relationship between the
size and the frequency of wildfires—even for time periods as short as
2 years—gives hazard-management personnel an opportunity to use
limited amounts of data to predict the likelihood of wildfires of
various sizes in an area.
Scientists are now subdividing the four wildfire data sets to see if
intervention—either extinguishing fires before they die out naturally
or setting preventative fires to remove dried brush—changes the
relationship between the size and frequency of blazes. The new
analyses could reveal whether certain firefighting policies affect
the location and intensity of wildfires in unexpected ways, Malamud
says.
Fractal fractures
Many phenomena related to the fracture of rocks and minerals, from
tiny bits of volcanic ash to the large blocks of Earth’s crust that
scrape past each other during earthquakes, can be described or
modeled using fractals. North of Los Angeles, the two sides of the
San Gabriel Fault sandwich a 6-meter-thick layer of pulverized
material. This crushed rock, called gouge, can be found between the
surfaces of many faults, says Charles G. Sammis, a geophysicist at
the University of Southern California in Los Angeles.
The power-law relationship between the number and the size of the
particles, from large rocks to fine mineral grit, has a fractal
dimension of 1.6, Sammis notes. In other words, chunks of rock with a
cross-sectional area of 1.6 units are about one-tenth as common as
those with a cross-sectional area of 1 unit. This fractal
relationship holds across a wide range of particle sizes, he adds.
Gouge forms in the early life of a fault, says Sammis. Stress
transfers from one side of the fault to the other through bridges of
large rocks. When the stress along the fault is high enough to break
the gouge particles, it fractures only those that are bearing the
load. As the load shifts to new bridges, those rocks fracture, and
the debris is gradually ground into finer and finer grains, all the
while maintaining a size distribution with a fractal dimension of
1.6.
Computer models that simulate particles of rock trapped in a moving
fault, as well as laboratory tests with pellets of crushed granite,
also generate a distribution of particle sizes with a fractal
dimension of 1.6. This specific value is interesting, Sammis
explains, because it seems to mark a transition point for the
behavior of faults. For values more or less than 1.6, the gouge does
not transfer stress across the fault because bridges don’t form. In
those cases, the two faces of the fault slide past one another
without triggering an earthquake. When the value equals 1.6, however,
the rock bridges lock the fault and stress builds up until a temblor
occurs.
Many other characteristics of fractured rock are fractals, notes
Christopher C. Barton, a geophysicist with the U.S. Geological Survey
in St. Petersburg, Fla. For example, the relation between the number
and the lengths of fractures in many types of rock follow a fractal,
power-law relationship on scales that range from long fissures in
Earth’s crust down to microscopic cracks in small stones.
Flowing through the rock
Scientists have developed computer models that use fractals to
simulate the movement of fluids through permeable rock, such as
limestone. Rina Schumer, a hydrologist at the Desert Research
Institute in Reno, Nev., and her colleagues model the spread of fluid
through fractured and porous rock. A rock’s porosity—the percentage
of the material’s volume that’s occupied by space between solid
grains—can easily be expressed as a fractal dimension.
By taking just a few measurements of rock and soil characteristics
near the site of a toxic spill such as a fuel leak, for example,
researchers can use fractals to estimate the directions and speeds at
which the fluid will travel through the surrounding earth. This
information can be useful to emergency-response personnel, and it may
eventually help government officials develop plans to protect
aquifers from spills.
Models that use fractals to simulate the presence of fractures in
porous rock may provide better answers than those that assume the
rock is uncracked. For example, Department of Energy scientists
modeling the potential seepage of fluids from Nevada’s Yucca Mountain
Project—an underground repository being designed to hold radioactive
waste from nuclear plants (SN: 1/19/02, p. 39:
http://www.sciencenews.org/20020119/fob7.asp)—estimate that if a leak
were to occur, it would take about 1,000 years for the first traces
of tainted fluid to reach 5 kilometers from the site. However, that
analysis assumes that waste simply seeps through unfractured rock,
says David T. Purvance, a hydrogeologist at hydrOhm Environmental
Geophysics in Reno.
A fractal analysis that incorporates the effect of fractures in the
rock predicts a much faster travel of liquid radioactive waste. A
model that uses a previously measured distribution of crack sizes in
the region’s so-called fractured welded tuff—a cracked, porous rock
made up of volcanic ash particles that have been fused together under
extreme heat and high pressure—estimates that spilled waste could
travel a distance of 5 km in only 200 years.
Volcanic dimensions
Although the roughness of the glassy particles spewed out by volcanic
activity doesn’t keep the same fractal dimension across all scales,
measuring that parameter of the ash bits nevertheless may help
scientists analyze the pattern of a volcano’s past eruptions. Anton
H. Maria and Stephen N. Carey, both volcanologists at the University
of Rhode Island in Narragansett, studied ash ejected from five
different eruptions, including some deep undersea.
The researchers found that the combination of fractal dimensions of
the ash particles measured at two different scales enabled them to
discriminate among the eruptions. The surface roughness of the ash on
the larger scale was linked to the abundance of large bubbles in the
molten rock that had erupted from the volcano. The explosive power of
the eruption shattered the rock along these bubbles, leaving small
particles with ragged edges. At smaller scales, the ash particles’
roughness was linked to the abundance of small bubbles trapped within
the material as it cooled. These two factors explained more than 90
percent of the differences among the ash particles from the five
eruptions, says Maria.
The size of the bubbles in the volcanic ash particles seems to be
influenced by several factors, including the type and amount of gases
that accompanied the eruption and whether the eruption was explosive
or relatively calm. Ash particles from eruptions on land, where the
ash cloud spewed directly into the air, typically have different
fractal dimensions from those produced by underwater volcanic
activity.
These distinctions are particularly important because they may help
scientists track a volcano’s long-term history in areas where there
are few if any written records, says Maria. By analyzing particles
from different ash layers in a long sequence of eruptions,
researchers might be able to tell whether all of a volcano’s
eruptions have been explosive or its strong eruptions were
interspersed with calmer belchings of ash.
Return to Gallery
References and Sources
References:
Barton, C.C. 2001. Scaling of rock fractures: An overview (Abstract
NG32A-05). American Geophysical Union 2001 Fall Meeting. Dec. 10-14.
San Francisco.
Maria, A.H., and S.N. Carey. 2001. Fractal spectrum technique for
quantitative analysis of volcanic particle shapes (Abstract
V42D-1044). American Geophysical Union 2001 Fall Meeting. Dec. 10-14.
San Francisco.
Sammis, C.G. 2001. Fractal fragmentation, friction, and
faulting(Abstract NG32A-07). American Geophysical Union 2001 Fall
Meeting. Dec. 10-14. San Francisco.
Schumer, R., D.A. Benson, and M.M. Meerschaert. 2001. Modeling
multidimensional contaminant plume growth with fractal scaling
(Abstract NG31B-0369). American Geophysical Union 2001 Fall Meeting.
Dec. 10-14. San Francisco.
Turcotte, D.L. 2001. The legacy of Benoit Mandelbrot in geophysics
(Abstract NG32A-02). American Geophysical Union 2001 Fall Meeting.
Dec. 10-14. San Francisco.
Ventosa, I.P., and B.D. Malamud. 2001. Cellular automata models: A
useful modelling tool to define environmental policy for forest fires
(Abstract NG51B-0464). American Geophysical Union 2001 Fall Meeting.
Dec. 10-14. San Francisco.
Further Readings:
Burroughs, S.M., and S.F. Tebbens. 2001. Upper-truncated power laws
and limits to scale invariance in natural systems (Abstract
NG31B-0376). American Geophysical Union 2001 Fall Meeting. Dec.
10-14. San Francisco.
Malamud, B.D., G. Morein, and D.L. Turcotte. 1998. Forest fires: An
example of self-organized critical behavior. Science 281(Sept.
18):1840-1842. Abstract available at
http://www.sciencemag.org/cgi/content/abstract/281/5384/1840.
Mandelbrot, B.B. 2001. Fractals as the measure of roughness in the
earth sciences (Abstract NG12B-01). American Geophysical Union 2001
Fall Meeting. Dec. 10-14. San Francisco.
______. 2001. Problems of geophysics that inspired fractal geometry.
American Geophysical Union 2001 Fall Meeting. Dec. 10-14. San
Francisco.
Peterson, I. 1996. The shapes of cities. Science News 149(Jan. 6):8-9.
______. 1993. From surface scum to fractal swirls. Science News
143(Jan. 23):53.
______. 1983. Fractals for modeling ecosystems. Science News 123(June
11):381.
Purvance, D.T. 2001. Travel times of nonlocal dispersion and their
geoelectric approximation in Nevada's fractured welded tuffs. Water
Resources Research 37(December):2915.
http://www.sciencenews.org/20020202/bob10.asp
From Science News, Vol. 161, No. 5, Feb. 2, 2002, p. 75.
Copyright (c) 2002 Science Service. All rights reserved.