Sea ice is a granular material, that is, a material that is composed of ice floes of different sizes and shapes . In most large-scale models, sea ice is treated as a viscous–plastic continuum. It deforms plastically when the internal stress becomes critical in compression, shear, or tension; it deforms as a very viscous (creep) flow when the internal stress is relatively small e.g.,. The corresponding highly nonlinear sea ice momentum equations can be solved with modern numerical solvers to reproduce, in a qualitative way, observed linear patterns of sea ice deformation within reasonable computing time . These linear kinematic features (LKFs) are places of large shear and divergence . Leads that open along LKFs are responsible for an emergent anisotropy of such models, affecting the subsequent dynamics, mass balance, and the heat and matter exchanges between the ocean, ice, and atmosphere. It is therefore important to investigate whether sea ice fracture is represented accurately in continuum sea ice models.

The sea ice dynamics are complicated because of sharp spatial changes in material properties associated with discontinuities (e.g., along sea ice leads or ridges) and heterogeneity (spatially varying ice thickness and concentration). The sea ice momentum equations are difficult to solve numerically because of the nonlinear sea ice rheology. Since the first sea ice dynamics model, the elastic–plastic sea ice model based on data collected during the Arctic Ice Dynamics Joint Experiment AIDJEX;, several approaches to modeling sea ice have been developed. Sea ice has been modeled as an incompressible fluid , a viscous–plastic (VP) material , an elastic–viscous–plastic (EVP) material , a granular material , an elastic anisotropic plastic (EAP) medium , an elastic–decohesive medium , an elasto–brittle (EB) material , and a Maxwell(viscous)–elastic–brittle (MEB) material . The actual number of approaches to sea ice modeling in the community, however, is much smaller. For example, 30 out of 33 global climate models in CMIP5 use some form of the standard VP rheology .

In spite of its success, the standard VP rheology is not undisputed. critically reviewed the assumptions behind current modeling practice since the original model of , namely the zero tensile strength (ice is a highly fractured material) and isotropy assumptions of the sea ice cover and the rheological model. Originally, assumed sea ice to have cracks in all directions, justifying isotropic ice properties and isotropic rheologies. The use of continuum models such as the standard VP model for high-resolution simulations (grid spacings of 1–10 km) is also debated since the grid size approaches a typical floe size and clearly violates the continuum assumption. For instance, recent high-resolution simulations using the VP model used spatial resolution of approximately 500 m for a regional domain and 1 km for a pan-Arctic domain . It can be argued that if the mode of deformation of a single floe is similar to that of an aggregate of floes, a given rheology developed for a continuum can still be applicable at spatial resolutions of the order of the floe size Appendix C, but the validity of a given flow rule across scales is not clear. At any scale, the assumption of viscous creep for small deformations is not physical, and an elastic model would be appropriate for low stress states. The long viscous timescale, compared to the synoptic timescale of LKFs, of order 30 years , however, allows viscous deformation to be viewed as a small numerical regularization with few implications for the dissipation of mechanical energy from the wind or ocean current , and the ice model can be considered an ideal plastic material. included anisotropy explicitly in a VP model and show that it improved the representation of ice thickness and ice drift compared to an EVP model. Alternative VP rheologies were never widely used in the community. These include a Coulombic yield curve with a normal flow rule , a parabolic lens and a tear drop , a diamond-shaped yield curve with normal flow rules , a Mohr–Coulomb yield curve with a double-sliding deformation law , or a curved diamond .

Previously, fracture lines (LKFs) in the pack ice were explained by brittle fracture . Similar fracture patterns were also observed, from the centimeter scale in the lab to hundreds of kilometers in satellite observations . The scale invariance of the fracture processes at the floe scale has not been shown. This may come from a lack of observations at both high spatial and temporal resolution. Based on satellite observations (e.g., RADARSAT Geophysical Processor System, RPGS, or Advanced Very-High-Resolution Radiometer, AVHRR) and in situ internal ice stress measurements (e.g., from the Surface Heat Budget of the Arctic Ocean, SHEBA, experiment), proposed to model winter sea ice as a material that undergoes brittle failure with subsequent inelastic deformation by sliding along LKFs. This idea was formalized with an additional parameterization to simulate damage associated with brittle fracture in an elasto–brittle (EB) and Maxwell–elasto–brittle (MEB) model . We note that subsequent deformation in this model is considered to be elastic deformation (EB) or visco–elastic deformation (MEB) instead of plastic. That is, in the EB and MEB approaches, the material does not weaken when fracture occurs, but rather the Young's modulus is reduced, leading to larger elastic deformation for the same stress. From the scaling behavior of simulated sea ice deformation fields of EVP models (with 12 km grid spacing), it was found that the heterogeneity and the intermittency of deformation in the VP model are not consistent with Radarsat Geophysical Processor System (RGPS) data . In contrast, VP models were shown to be indeed capable of simulating the probability density functions (PDFs) of sea ice deformations and some of the scaling characteristics over the whole Arctic in agreement with the same observations, either with sufficient resolution or with tuned shear and compressive strength parameters .

High-resolution sea ice models simulate LKF patterns in pack ice, where they appear as lines of high deformation . Previously fractured ice will be weaker and will affect future sea ice deformation fields. The weakening associated with shear deformation results from divergence and a reduction in ice concentration along the LKFs. This mechanism introduces an anisotropy in high-resolution simulations that is similar to observations with comparable spatial resolution. Lead characteristics, including intersection angles between LKFs, were studied a number of times . These studies show that VP models produce LKFs with various confinements, scales, resolutions, and forcings. From observations with different instruments (Landsat, Seasat/SAR, areal photographs, AVHRR), typical fracture angles between intersecting LKFs of (15±15)∘ emerge at scales from 1 to 100 km (; ). present an LKF tracking algorithm and show that fracture angles (half of the intersection angles) between LKFs in RGPS data follow a broad distribution that peaks around 20∘, in line with previous assessments e.g.,. also show that the distribution of fracture angles in a VP simulation with 2 km grid spacing is biased, with a high modal value of 45∘ and with too few small intersection angles between 15 and 25∘. The observed bias motivates the present investigation of the dependence of fracture angles in different VP rheologies and model settings, that is, scale, resolution, boundary conditions, model geometry, and variability in initial ice thickness field.

The simulation of fractures in sea ice models has been studied in idealized model geometries before. investigated the effect of embedded flaws – that favor certain angles of fractures – in idealized experiments using a Coulombic yield curve. showed that LKFs can be simulated with an isotropic VP model using an idealized model geometry. The shape of the elliptical yield curve (ratio of shear to compressive strength) in the standard VP model determines if ice arches can form in an idealized channel experiment . investigated the yield curve's mathematical characteristics and derived angles between the principal stress directions and characteristics directions that depend on the tangent to the yield curve. These results show that stress states exist in plastic materials where no LKFs form and were later used to build a yield curve . To build an anisotropic rheology, used a discrete element model (DEM) in an idealized model domain and showed clear diamond-shaped fracture patterns. Idealized experiments are also used to investigate new rheologies, for example, the Maxwell–elastic–brittle (MEB) rheology or the material-point method (MPM) , or to study the theoretical framework explaining the fracture angles e.g.,with a Mohr–Coulomb yield criterion in an MEB model. Recently, compared simulated fractures by the EVP and EAP models using an idealized model geometry and wind forcing and showed that the anisotropic model creates sharper deformation features. To the best of our knowledge, the dependency of the fracture angles in sea ice on the shape of the yield curve using high-resolution models has not yet been investigated. This is another motivation of this study.

In this paper, we simulate the creation of a pair of conjugate faults in an ice floe with two different VP rheologies in an idealized experiment at a spatial resolution of 25 m. We explore the influence of various parameters of the rheologies and the model geometry (scale, resolution, confinement, boundary conditions, and heterogeneous initial conditions). The remainder of this paper is structured as follows. Section  presents the experimental setup: the VP framework (Sect. ), the definition of the yield curve (Sect. ), and the description of the idealized experiment (Sect. ). Section  presents the results: first the control simulation is presented (Sect. ), then we explore the sensitivity of the setup in Sect.  to scale, resolution and longer run time (Sect. ), modified boundary conditions and lateral confinement (Sect. ), and to heterogeneity in initial conditions (Sect. ). Finally, we consider the effects of two different yield curves with different flow rules in Sect. : the elliptical (Sect. ) and the Coulombic yield curve (Sect. ). Discussion and conclusions follow in Sects.  and .

We use simple uni-axial loading experiments to investigate the creation of pair of conjugate faults and their intersection angle. After presenting the results of simulations with the default parameters (Sect. ), we explore the effects of experimental choices: confining pressure, choice of boundary conditions (i.e., von Neumann versus Dirichlet), domain size, and spatial resolution and inhomogeneities (i.e., localized weakness) in the initial thickness and concentration field (Sect. ). Finally, we study the behavior of two viscous–plastic rheologies with different yield curves and compare these dependencies to what we can infer from smaller- and larger-scale measurements from laboratory experiment and RGPS observations (Sect. ).

Our idealized experiments using the VP rheologies resolve fracture lines as described by and akin to observations . The fracturing of the ice floe creates smaller floes in a manner that appears realistic, for example, compared to Landsat-7 images Fig. 2. At the high resolution of 25 m the original interpretation of the continuity assumption, namely that each grid cell should represent a distribution of floes , is no longer valid, but we show that the fracture angle is independent of resolution and scale as expected. Instead, the emerging discontinuities and the polygonal diamond shape of the fracture lines that appear as floes spanning many grid cells are a consequence of the mathematical characteristics of the VP model . Diamond-shaped floes are observed in the Arctic Ocean and also modeled using a discrete element model (DEM) in an idealized experiment . The elastic anisotropic plastic (EAP) rheology assumes predominately diamond-shaped floes in sea ice . A sea ice model with EAP creates sharper fractures than a model with the elastic–viscous–plastic EVP; rheology . The authors concluded that the anisotropic model may improve the fracturing process for sea ice, especially by creating areas of oriented weaknesses, and particularly at coarse resolution where the fracture is not resolved by the grid spacing. In the experiments presented here, the VP rheologies lead to sharp and anisotropic fracture lines without any additional assumptions.

We explored some experimental choices to separate their effects from those of the rheology parameters. The fracture angles do not depend on the spatial resolution and domain size as expected in our idealized numerical experiment setup (Sect. , Fig. ). The maximum viscosities in the VP model are very high, and consequently, the VP model can be considered an ideal plastic material (i.e., a model with an elastic component that has an infinite elastic wave speed). For this reason, fracture in a VP model occurs almost instantaneously. Observed timescales of fracture are on the order of 10 s for 60 m floe diameters Fig. 6b, and from typical elastic wave speeds of 200–2000 ms-1, large cracks of order 1000 km can form in minutes to hours .

In our setup, the no-slip boundary condition has little effect on the fracture pattern, but our results suggest that in basin-wide simulations the choice of boundary conditions affects the fracture depending on the geometry and stress direction. The no-slip condition appears to be unphysical. It acts to concentrate the stress on the corners of the floe and forces the fracture to occur at this location. This should motivate a more thorough investigation of the boundary conditions for LKFs that form between one shoreline and another. Similar results were obtained from analytical solutions in idealized geometry for the Mohr–Coulomb yield curve with a double sliding deformation law .

The confining pressure (i.e., thin ice imposed on the side of the domain) changes the distribution of stress within the domain. This results in different deformation patterns (shear and divergence) and different fracture angles because the yield curve is convex and uses a normal flow rule. From this we can conclude that by surrounding our floe with open water, we get the most acute angles from the rheology in this uni-axial compression setup. This is not consistent with the behavior of typical granular material for which an angle of fracture is independent of the confining pressure . Details of a heterogeneous ice cover also affect the fracture pattern. LKFs link the weaknesses in the ice cover, but the pattern still depends on the preferred fracture angles implied by the model rheology. In summary, we are confident that our choice of parameters allows us to isolate the effects of the rheology and the yield curve on the fracturing process.

In granular material, large shear resistance is linked to contact normals between floes that oppose the shear motion and lead to dilatation . In our experiments, increasing shear strength in the standard VP model (reducing the ellipse aspect ratio e) does not decrease but increases the fracture angle. This is in contrast to the behavior of granular material where larger shear strength leads to lower fracture angles – think of a moist sand castle versus a dry sand castle. In addition, high shear strength in the VP model with the elliptical yield curve leads to convergence along the fracture plane, whereas observations (e.g., RADARSAT-derived deformation fields) show a range of positive and negative divergence along LKFs – in accordance with laboratory tests of granular material that show a variable internal angle of friction that depends on the distribution of the contact normals between individual floes . Inspection of the stress states in the 2-D stress plane suggests that the intersection of the yield curve with the σ2 axis has an important role in the fracture process. This intersection point appears to determine the fracture angle. In fact, the angle is determined from the intersection of the Mohr's circle of stress with the yield curve to give a theoretical relationship between the fracture angle and the ellipse ratio e. With our experiments, we were able to confirm this relationship empirically.

Arctic-wide simulations improve metrics of sea ice concentration, thickness, and velocity by decreasing the value of e of the standard elliptical yield curve, that is, by adding shear and bi-axial tensile and compressive strength . The representation of sea ice arches improves with smaller e , as do LKF statistics . Our results, however, show that this makes the fracture angles larger, which is in stark contrast to what we expect to be necessary to improve the creation of LKFs in sea ice models.

The fracture angle and the sea ice opening and ridging depending on the deformation states are consistent with the theory of the yield curve analysis developed in and the Mohr's circle framework that we present in Appendix . Interestingly, a change of ice maximum compressive strength P⋆ with a constant e has no influence on the LKF creation, although P⋆ is usually thought of as the principal parameter of sea ice models in climate simulations e.g.,. The effects of bi-axial tensile strength T⋆ on fracture processes require further investigation, especially given the fact that the assumption of zero tensile strength is being challenged . The ice strength parameter C⋆ (the parameter governing the change of ice strength depending on ice concentration; Eq. ) was not studied here, although it appears to be an important tuning parameter, and it also helps to improve basin-wide simulations . The simulations presented in this study are not realistic and cannot be compared directly to observations of ice floe fracture. For instance, our idealized ice floe is homogeneous, while sea ice is known to feature some weaknesses like thermal cracks or melt ponds.

With the Coulombic yield curve, the simulated fracture angle can be smaller than for the elliptical yield curve. For μ=0.7 (ϕ=44∘) theory predicts θMC=22.8∘ (Appendix ). The simulated fracture angle with μ=0.7 of θ=23.5∘ is close to the ≃20∘ described in . developed a different Mohr–Coulomb theory linking the internal angle of friction and the fracture angle. This complex theory takes into account the fractal (or self-similar) nature of sea ice. It gives different results but is inadequate for a single ice floe simulated as presented here. Based on the results of , used observed fracture patterns to design a curved diamond yield curve. But this yield curve also contains a non-differentiable point, which will be problematic for numerical reasons. The Coulombic yield curve used here uses a normal flow, and consequently divergence will always be present along shear lines. In situ measurements, however, show that the deformations follow a non-normal flow rule , and large-scale observations show both divergence and convergence (ridges) along LKFs . There are alternative flow rules still to be explored, for example, a double-sliding law with or without dilatation included .

Motivated by the observation that the intersection angles in a 2 km Arctic-wide simulation of sea ice are generally larger than in the RGPS dataset , the fracturing of ice under compression was studied with two VP rheologies in a highly idealized geometry and with very small grid spacing of 25 m. The main conclusions are given in the following.

In our experimental configuration with uni-axial compression, fracture angles below 30∘ are not possible in a VP model with an elliptical yield curve. Observations suggest much lower values. We find an empirical relationship between the fracture angle and the ellipse ratio e of the elliptical yield curve that can be fully explained by the convexity of the yield curve (Appendix ). In contrast to expectations, increasing the maximum shear strength in the sea ice model increases the fracture angle. Along a fracture line, there can be both divergence and convergence depending on the shear strength of the ice, linked to the flow rule. The simulated ice opens and creates leads with an ellipse ratio e>1 (shear strength is smaller than compressive strength) and ridges for e<1 (shear strength is larger than compressive strength).

With a modified Coulombic yield curve, the fracture angle can be decreased to values expected from observations, but the non-differentiable corner points of this yield curve lead to numerical (convergence) issues and, for some values of the coefficient of internal friction μ, to fracture patterns that are difficult to interpret. At these corner points, two different slopes meet and give two non-unique solutions for fracture angles and deformation directions. We recommend avoiding non-differentiable yield curves (with a normal flow rule) in viscous–plastic sea ice models.

More generally, the model produces diamond-shaped fracture patterns. Later the ice floe disintegrates and several smaller floes develop. The fracturing process in the ice floe in our configuration is independent of the experiment resolution and scale but sensitive to boundary conditions (no-slip or free-slip). The fracture angle in the VP model is also sensitive to the confining pressure. This is not consistent with the notion of sea ice as a granular material. Unsurprisingly, the yield curve plays an important role in fracturing sea ice in a numerical model as it governs the deformation of the ice as a function of the applied stress.

The idealized experiment of a uni-dimensional compression is useful to explore the effects of the yield curve because all other parameters are controlled. Historically, the discrimination between the different yield curves was not possible because of the scarcity of sea ice drift data. Model comparisons to recent sea ice deformation datasets, such as from RADARSAT, imply that we would need to increase the shear strength with the ellipse in the standard VP rheology to match observations . We find that this increases the fracture angles, creating a dilemma. Therefore, the high-resolution idealized experiment presented in this work provides a framework to investigate and discriminate different rheologies – a yield curve and a flow rule.

If Arctic-wide sea ice simulations with a resolution of 25 m are not feasible today because of computational cost, we can still imagine small experiments being useful for process modeling on small scales when local and high-resolution observations (e.g., wind, ice velocities) are available. For example, such process modeling studies could be used to constrain the rheology with data from the upcoming MOSAiC campaign that will provide a full year of sea ice observations in pack ice. Such simulations would also need to take into account the effects of heterogeneous ice cover and wind patterns, with potentially convergent and divergent wind forcing. Most climate models use the standard VP rheology or one of its variants (e.g., EVP). Results presented here, however, imply that a more physical yield curve with a (possibly non-associative) flow rule is required. Such a yield curve would have to be continuous in all representations, differentiable without corners, have some cohesion, and be consistent with available observations of fracture angles in convergent and divergent flow.