Simple mathematical models can give insight into the behavior
of complex geophysical systems.
Seepage channel networks
River networks appear when small tributaries meet to
form creeks, which combine to form larger streams, and so on up
to the scale of major rivers. In typical networks, the water originates
from rainwater or melting ice, and follows the path of least resistance
as it flows over the surface of the ground. This overland flow, and
the erosion it causes, has been a popular subject of study for some time.
Seepage channel networks, though similar, have a different origin.
The water in a seepage channel originates at a spring, where groundwater
seeps through the soil to emerge at the surface. In the process,
the water carries with it a small amount of sediment, causing the spring
to slowly move over the course of many years.
Surprisingly, a set of simple linear growth laws can accurately
predict the movement of springs (the "fingertips") and the large-scale
channel structure in a seepage-driven network. This is remarkable because it
means that the physical environment over regions tens of kilometers in size
can be determined by simple growth laws, rather than unpredictable
For more information, please see "Growth laws for channel networks."
The shape of growing valleys
Valleys can be carved by a range of geophysical processes. Often, the
valley "heads" show an intriguing amphitheatre-shaped structure. With
the mild assumption of curvature-driven growth dynamics (i.e., rate of
erosion proportional to curvature of the valley lip), our theory makes the
remarkable prediction that valley head shapes should have an aspect
ratio of pi. In fact, this prediction seems to be correct for a large set of
valleys varying in size from centimeters to hundreds of kilometers, and
not just on Earth! Mars channel networks also seem to show this
aspect ratio, perhaps hinting at the process through which they formed.
For more information, please see "Geometry of valley growth."
Historical climate oscillations
During the past million years, the earth's climate has regularly oscillated
between cold and hot phases, known as "ice ages" and "interglacials." Ice ages have
recurred on a fairly regular schedule, usually about every 100,000 years.
The most widely accepted explanation for this oscillation proposes that very
long period oscillations in the earth's orbit cause changes in the amount of
heat reaching the earth from the sun. This explanation can also account for
another strong climate signals with a period of 41,000 years.
Unfortunately, there remains a disconnect between our understanding of orbital
mechanics, and our understanding of the global climate on earth. Perhaps
better analytical models of the key elements in earth's climate system will allow
that connection to be made, reinforcing our confidence in our understanding of
What causes hills and valleys? This seemingly basic question has many answers.
The theory of plate tectonics, invoking subduction and uplift, gives the most
common explanation. Glacial action may also have played a role in many world regions.
However, even in regions that have a long history of little tectonic activity,
we find varying topography.
Taking the florida panhandle as an example, we find low-relief hills and troughs at almost
all spatial scales throughout the region. Understanding their origin may have more to do
with stochastic differential equations than traditional tools of gromorphology.