Computer Simulation Reveals Unexpected Links Between Wave Angles And Coastline Shapes

DURHAM, N.C. - By applying the lessons of chaos and complexity research to shorelines, Duke University geologists found in computer simulations that simple interactions between sand and incoming waves can cause a coastline's shape to change in surprising ways.

When waves approach shore from some angles, isolated zones of coastal erosion and buildup are inevitable, their study in the Nov. 15 issue of the journal Nature found. In those cases, bumps in a shoreline can grow and interact with each other, leading to the spontaneous development of coastline shapes. These shapes may eventually grow as large as the capes and broad bays found on the Carolina coastline, the researchers reported.

The surprisingly long-range effects in the novel computer model, which operates at larger scales of distance and time than is the norm, suggest the need for a new type of coastal management planning, suggested principal author A. Brad Murray in an interview.

"What you do in this beach town might affect what happens in another town tens of kilometers away and in future decades," said Murray, an assistant professor of coastal processes and geomorphology at Duke's Nicholas School of the Environment and Earth Sciences.

The study's authors are Murray, graduate student Andrew Ashton and Olivier Arnoult, a former visiting scientist at Duke now at ‰£ole Normale Supeeure in Paris. Their research was funded by the Andrew W. Mellon Foundation.

"Alongshore sediment transport that is driven by waves is generally assumed to smooth a coastline," wrote the investigators in Nature. "But when the angle between the waves and the shoreline is sufficiently large, small perturbations to a straight shoreline will grow."

The article added that "our simulations show growth of coastline perturbations that interact to produce large-scale features that resemble various kinds of natural landforms, including the capes and cuspate forelands observed along the Carolina coast of southeastern North America."

Far from shore, low-angle waves are those less than about 45 degrees, while those creating sand transport instabilities along the shoreline are those greater than 45 degrees, Murray said.

Waves don't actually hit beaches at such higher angles, because they are "refracted" into lower angles when they begin to "feel the bottom" near the surf zone, he added. It is only in deeper water, about one kilometer or more out in an area like the Outer Banks of North Carolina, where waves can move at higher angles. "Sitting on the beach, you never see the waves in deep water," he said.

"The starting point of this exercise was noticing that if waves approach a shore at a fairly high angle, then all sorts of interesting things are going to happen. At such angles, if you start out with a more or less straight shoreline with some random perturbations, those bumps are always going to grow.

"This isn't part of people's normal mindsets about coastal behavior."

Murray and his co-investigators delved further into the link between waves angles and the "self-organization" of beach shapes by running a computer program that operates on larger spatial scales (kilometers) and time scales (decades to 10,000 years) than beach processes are usually modeled.

Their studies also added the insights of non-linear dynamics, the study of chaos and complexity, to traditional shoreline models such as those used by the U.S. Army Corps of Engineers.

"One of the features of our model that other models don't have is we can grow really large features that can interact with other large features far away," said Ashton, who took over and improved Arnoult's initial coding work and has since been running the simulations.

"The way others do the modeling is appropriate for the questions they're asking," Murray emphasized. "The other modelers have all assumed that the shore will only deviate moderately from a straight line. Their models can handle a shoreline that gets a little bit curvy. We're just asking different questions about shorelines that can take on very crazy shapes.

"By using this more innovative style, we've developed an algorithm (a set of mathematical rules) that can handle the shoreline evolving itself into these complex forms."

In their Nature article, the authors described running simulations of waves approaching a hypothetical beach from various angles and directions. "We have found that features grow for all distributions weighted toward high-angle waves," they wrote.

In some cases, "features can overtake each other and merge," they wrote. In other cases, "slightly larger features shelter their neighbors from the highest-angle waves," their article said. "Simulations reveal that the larger features will then grow relative to the smaller features.

When high-angle waves tend to approach the beach "asymmetrically," meaning from one direction more than others, large-scale features variously called "cuspate spits," "sandwaves," "ords" and "humps" can develop over time, they wrote. Examples include the hook-shaped cuspate spits on the shore of the Sea of Azov in the Ukraine.

When the wave distribution was less asymmetric, "cuspate forelands" resembling the shapes of Cape Hatteras and other North Carolina Outer Banks capes evolved over a 10,000 year period in the simulations.

Even though sea levels can rise and fall substantially over that long a time, caused by the melting or buildup of polar ice sheets, the authors suggested that "the Carolina Capes could have continued to form over several sea level high stands."

Ashton's review of wave direction and angles estimates from the Corps of Engineers over the past few decades at a station near Cape Hatteras is also "in agreement with our hypothesis that there are, in fact, more high-angle than low-angle waves along the stretch of the Carolina coast that includes the capes," Murray said.

Murray cautioned that this computer model was not written to describe a specific geographic feature but rather to study general relationships between wave angle and beach shapes. "By no means did we set out to simulate these capes," he said.

On the other hand, "this modeling, combined with the field data that Andrew has put together, suggests that this is a plausible explanation for the fundamental cause of those capes," Murray said. "We can also say with a fair amount of confidence that there will be isolated zones of erosion in storms."

He said he plans to further investigate using these findings to predict the long-range impacts of projects designed to stabilize a beach at a specific site by pumping in extra sand or by building engineered structures such as seawalls.

"That's going to influence the shape of the shoreline in a much greater area than we're used to thinking about," Murray said.

Note to editors: A. Brad Murray can be contacted at (919) 681-5069 or (919) 383-1105, or by e-mail, abmurray@duke.edu.