Discrete Element Simulations of Sea Ice Dynamics in Nares Strait

By Brendan West, Devin O’Connor, Matthew Parno, Max Krackow, and Chris Polashenski

In recent years, Arctic sea ice has undergone drastic changes due to the impacts of climate change. Numerical models allow researchers to (i) study how the ice changes in response to different environmental factors and (ii) predict future sea ice states. To date, many large-scale (i.e., arctic-scale) sea ice models use continuum approaches. These approaches treat the ice as a continuous material that is discretized with an Eulerian mesh and capture the material behavior with constitutive models, such as the viscous-plastic or elastic-viscous-plastic rheologies [4, 5]. Research has shown that these methods accurately capture large-scale ice motion at scales on the order of hundreds of kilometers. However, sea ice is highly discontinuous at much smaller scales and in regions like the marginal ice zone (MIZ); in these scenarios, interactions between discrete pieces of ice—known as floes—drive the dynamics.

Particle-based methods—including the discrete element method (DEM)—provide an alternative means of modeling sea ice. The DEM treats the ice as a collection of rigid bodies or “particles” that are driven by external wind or ocean drag loads and the forces generated during impacts with neighboring particles. It uses linear and angular momentum to define a system of differential equations that describes each particle’s motion according to the different body loads and contact forces that act upon them. Its discrete nature makes the DEM well suited to capture discontinuities in the ice pack that are common near the ice edge and in the MIZ. In addition, several modeling efforts have extended the classic DEM approach to utilize inter-particle bonds that couple the motion of neighboring particles and let the method capture continuum-like behavior. This development allows stresses and strains to transmit across large regions of the modeled ice. Importantly, the bonds can fail and enable the propagation of fractures—known as leads—through the ice.

Figure 1. Diagram of nonlocal failure model for a single particle and its nearest neighbor particles. We evaluate the Cauchy stress tensor (1) for each particle and use the direction of the largest tensile principal stress to define a fracture surface. We then break all bonds on one side of this surface to ensure that the ice fails in tension. Figure courtesy of Brendan West.

We build upon these recent bonded-DEM efforts and incorporate an inter-particle bond model that captures the response of Euler-Bernoulli beams located between particles [1]. We also include a Mohr-Coulomb bond failure model that computes a nonlocal evaluation of the ice’s stress state around each DEM particle. Mohr-Coulomb is widely used in the sea ice modeling community and can describe sea ice failure that is observed in the field [7]. Our nonlocal stress approach computes the Cauchy stress tensor for each DEM particle, which accounts for all of its bonded nearest neighbor particles within a certain distance:

\[\sigma_i=\frac{1}{\Omega_i}\Bigg(\frac{1}{2}\Sigma_jr_{ij}\otimes f_{ij}+f_{ij}\otimes r_{ij}\Bigg).\tag1\]

Here, \(\Omega_i\) is the volume of particle \(i\), \(r_{ij}\) is the vector from particle \(i\) to neighboring particle \(j\), and \(f_{ij}\) is the force vector between \(i\) and \(j\). If the resultant stress state lies outside the Mohr-Coulomb failure surface, we find the direction that corresponds to the largest tensile principal stress and define a failure surface that is normal to that direction. We then break all of the bonds that fall on one side of this surface (see Figure 1). This method ensures that cracks through the simulated ice form in tension, thus matching field observations of lead formation [6].

Previous studies have demonstrated that a similar nonlocal approach can reproduce accurate crack patterns in DEM simulations of elastic brittle materials and rock mechanics [1, 2]. By utilizing this physics-based failure model in a bonded DEM ice model, we can represent both discrete fractures and emergent aggregate behavior of the ice as it deforms prior to failure.

We use the DEM software library ParticLS [3] and generate particles via centroidal Laguerre diagrams to discretize the ice extent in satellite imagery from the Moderate Resolution Imaging Spectroradiometer (MODIS) into polygons that approximate the shape and extent of ice in nature. We tested our model by simulating sea ice in Nares Strait, which is located between Greenland and Canada’s Ellesmere Island [8]. This region is a popular choice for sea ice models because it exhibits complex ice dynamics that repeat almost every year. During the winter months, large, stable ice arches form in the strait and effectively halt all ice flow throughout the region. These arches fail in the spring, resulting in large amounts of fracture and deformation that are accompanied by a highly dynamic outflow of ice within the strait. 

Animation 1 presents the model results. The left side depicts the Von Mises stress within the ice as it deforms, and the right side illustrates places where inter-particle bonds have failed to indicate the locations of crack formation. These preliminary results suggest that our nonlocal Mohr-Coulomb failure model qualitatively captures many salient features of ice arch failure in Nares Strait. The arch fails into many large floes and a series of concentric cracks form up channel. Significant crushing occurs where the ice pushes into the coastlines, and regions of landfast ice remain in many of the smaller channels. In particular, the sea ice exhibits regions of highly localized damage that correspond to areas with large stresses that relax during intermittent fracture events. Future work will focus on validating results against observations, and incorporating more realistic ocean, wind, and ice thickness data from field measurements.

References
[1] André, D., Iordanoff, I., Charles, J.-L., & Néauport, J. (2012). Discrete element method to simulate continuous material by using the cohesive beam model. Comput. Methods Appl. Mech. Eng., 213-216, 113-125.
[2] André, D., Levraut, B., Tessier-Doyen, N., & Huger, M. (2017). A discrete element thermo-mechanical modelling of diffuse damage induced by thermal expansion mismatch of two-phase materials. Comput. Methods Appl. Mech. Eng., 318, 898-916.
[3] Davis, A.D., West, B.A., Frisch, N.J., O'Connor, D.T., & Parno, M.D. (2021). ParticLS: Object-oriented software for discrete element methods and peridynamics. Computation. Part. Mech., 1-13.
[4] Hibler III, W.D. (1979). A dynamic thermodynamic sea ice model. J. Phys. Oceanogr., 9(4), 815-846.
[5] Hunke, E.C., & Dukowicz, J.K. (1997). An elastic-viscous-plastic model for sea ice dynamics. J. Phys. Oceanogr., 27(9), 1849-1867.
[6] Timco, G.W., & Weeks, W.F. (2010). A review of the engineering properties of sea ice. Cold Reg. Sci. Technol., 60(2), 107-129.
[7] Weiss, J., Schulson, E.M., & Stern, H.L. (2007). Sea ice rheology from in-situ, satellite and laboratory observations: Fracture and friction. Earth Planet. Sci. Lett., 255(1-2), 1-8.
[8] West, B., O'Connor, D., Parno, M., Krackow, M., & Polashenski, C. (2021). Bonded discrete element simulations of sea ice with non-local failure: Applications to Nares Strait. Preprint, arXiv:2105.05143v2.

	Brendan West is a research mechanical engineer at the U.S. Army Corps of Engineers’ Cold Regions Research and Engineering Laboratory (CRREL). He is interested in the development of computational tools and models to study cold climate processes, including sea ice mechanics and snow mechanics.
	Devin O’Connor is a research mechanical engineer at CRREL who specializes in modeling the mechanics of snow and ice and developing discrete element models for sea ice dynamics.
	Matthew Parno is a research assistant professor at Dartmouth College. His research interests are at the intersection of Bayesian statistics, measure transport, and sea ice modeling.
	Max Krackow is a research physical scientist at CRREL. His research interests are in both applied and nonlinear physics.
	Chris Polashenski is a field sea ice geophysicist at CRREL and Dartmouth College. He has expertise in sea ice dynamics and thermodynamics as well as remote and autonomous observing techniques.