Peridynamic Discrete Element Method Effectively Captures Changing Sea Ice Dynamics

By Lina Sorg

Sea ice dynamics are an important component of the Earth System Model. They impact planetary albedo and are frequently coupled with ocean and atmospheric systems. Researchers typically model sea ice at the Arctic basin scale. “When we think about sea ice modeling tools that try to model sea ice on this regional scale, spatial scale, or over these kinds of time scales, we most often use Eulerian-based continuum models,” Devin T. O'Connor of the U.S. Army’s Cold Regions Research and Engineering Laboratory said. These types of models work great for continuous time scales when the ice behaves like a viscous fluid. 

However, sea ice regularly presents as a highly discontinuous material that is comprised of many different-sized ice floes (see Figure 1). Continuous assumptions do not always apply to these types of real-world scenarios, such as fissures, cracks, or ridges in the ice. Creating these realistic dynamics is more challenging. During a minisymposium presentation at the 2021 SIAM Annual Meeting, which took place virtually this week, O'Connor used the discrete element method (DEM) to better understand how deformations and stress influence crack formation within sea ice. His DEM examines ice dynamics by focusing on smaller spatial or time scales. Because the DEM is a Lagrangian formulation, it knows the position of sea ice particles at any given time. Specifically, it models the sea ice as a collection of rigid particles, which works nicely in a discrete algorithm framework and captures the fine-scale aspects. Such aspects include pressure regime, divergence fragmentation, and rotation — the kind of dynamics that continuum methods fail to capture.

Because it is difficult to model the way in which sea ice breaks apart using a DEM framework, O’Connor sought a different technique to model deformation and stress within a larger discrete particle. This search inspired the discrete element peridynamic hybrid model — a particle-based method that uses a peridynamic formulation to model the deformation and cracking inside ice floes. It retains the Lagrangian perspective, meaning that users can still track individual particle positions to determine where cracks first appear. The procedure is simple to validate and works well in the highly discontinuous framework that represents sea ice. “The other nice thing about peridynamics is that because it’s a particle method, it also fits nicely into a DEM framework,” O’Connor said. He thus implemented a linear elastic peridynamic solid into his code but continued to use the DEM to assess behavior between floes.

Next, O’Connor incorporated a first-order approach to handle failure—in the form of broken bonds—within his peridynamic model framework. This first-order approach involves a bond stretch failure model, which calculates a critical strength and allows him to visualize damage. “These bonds between particles stretch and reach some critical strain, and then we break them,” O’Connor said. “That’s how cracks develop in the method. Any time you see about 50 percent damage to the bonds, you can assume that a crack has passed through the material.”

In addition, O’Connor included a pressure-ridging-based DEM contact law between particles, which models two floes that come together and create a pressure ridge. Such a scenario is common in sea ice. “This is really important to add to a sea ice modeling software because it is a huge source of energy dissipation and can really influence the overall dynamics of the system,” O’Connor said.

Several examples highlighted the model in action. Every scenario imagined a rectangular block of ice that is 140 kilometers (km) long, 70 km wide, and three meters thick to reflect the regional scale. The block is comprised of roughly 4,000 polygonal particles, each of which is one km in size. O’Connor also noted that he employed an atmospheric and ocean drag function in the code for each situation. The first scenario involved a pre-crack on the left side of the ice block (see Figure 2) that causes the ice to pull apart at a single millimeter per second. O’Connor measured the stress macroscopically, which yielded a maximum tensile stress that corresponds to expectations of sea ice at this particular scale. 

The next simulation reproduced a divergent wind field with the pre-crack in the center. O’Connor utilized three different case studies to explore this divergent wind, which can cause strong cracking patterns in the systems. The three cases—a pure shear case with divergent wind, a divergent field, and a flipped version of the second field—resulted in three different types of cracking: a shear-driven type of deformation that begins to curl upward, a compressive-like fracture and shear-driven failure that develop from the crack’s edges, and tensile-like crack behavior with a bit of shear. 

O’Connor’s third scenario examined a wind gyre field that circles the crack and continues out in space to encompass the particle’s entire domain. He also incorporated a small amount of damage—in the form of pre-cracks—at the simulation’s onset by randomly snipping the bonds of 50 percent of the particles. The initial damage field thus appears quite scattered because small cracks are present throughout the domain, which is more realistic because actual sea ice is not usually of uniform thickness and strength. The gyre causes the developing cracks to curve out from the edges.

O’Connor ended his presentation with an exploration of ice arching behavior in the Nares Strait, a region between Greenland and Canada. To create a similar manifestation in his own code, he considered an idealized channel with a larger, 120-km area that constricts down to about 40 km and then opens up again. A downward wind travels straight through the channel, going from zero to 26 meters per second over a 24-hour period before holding constant for another 12 hours. O’Connor witnessed shear-driven behavior where the ice starts to break along the coastline/edge of the idealized channel. “Then we do actually begin to see arching through the channel, which is a good thing to see,” he said. “Numerous arches start to form through the bottom of the channel.” He also noted some floe formation at the exit (see Figure 3).

In conclusion, O’Connor’s DEM-peridynamic framework effectively captures the deformations and cracks of sea ice dynamics more effectively than traditional continuum methods. These techniques will be particularly valuable as the Arctic continues to undergo significant changes in the coming years due to climate change, which reduces ice thickness and makes floes more susceptible to fractures.

Lina Sorg is the managing editor of SIAM News.