Recent high-resolution pan-Arctic sea ice simulations show
fracture patterns (linear kinematic features or LKFs) that are typical of
granular materials but with wider fracture angles than those observed in
high-resolution satellite images. Motivated by this, ice fracture is
investigated in a simple uni-axial loading test using two different
viscous–plastic (VP) rheologies: one with an elliptical yield curve and a
normal flow rule and one with a Coulombic yield curve and a normal flow rule
that applies only to the elliptical cap. With the standard VP rheology, it is
not possible to simulate fracture angles smaller than 30∘. Further,
the standard VP model is not consistent with the behavior of granular
material such as sea ice because (1) the fracture angle increases with ice
shear strength; (2) the divergence along the fracture lines (or LKFs) is
uniquely defined by the shear strength of the material with divergence for
high shear strength and convergent with low shear strength; (3) the angle of
fracture depends on the confining pressure with more convergence as the
confining pressure increases. This behavior of the VP model is connected to
the convexity of the yield curve together with use of a normal flow rule. In
the Coulombic model, the angle of fracture is smaller (θ=23∘)
and grossly consistent with observations. The solution, however, is unstable
when the compressive stress is too large because of non-differentiable
corners between the straight limbs of the Coulombic yield curve and the
elliptical cap. The results suggest that, although at first sight the large-scale patterns of LKFs simulated with a VP sea ice model appear to be
realistic, the elliptical yield curve with a normal flow rule is not
consistent with the notion of sea ice as a pressure-sensitive and dilatant
granular material.
Introduction
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 .
Experimental setupViscous–plastic model
We use the Massachusetts Institute of Technology general circulation model
MITgcm; with its sea ice package that
allows for the use of different rheologies .
All thermodynamic processes have been turned off for our experiments. The
initial sea ice conditions, mean (grid cell averaged) thickness h and
fractional sea ice cover A, are advected by ice drift velocities with a
third-order flux limiter advection scheme .
Ice drift is computed from the sea ice momentum equations
ρh∂u∂t=-ρhfk×u+τair+τocean-ρh∇ϕ(0)+∇⋅σ,
where ρ is the ice density, h is the grid cell averaged sea ice
thickness, u is the velocity field, f is the Coriolis parameter,
k is the vertical unit vector, τair is the
surface air stress, τocean is the ocean drag,
∇ϕ(0) is the gradient of sea surface height, and
σ is the vertically integrated internal ice stress
tensor. The form of σ defines the rheology. In the case
of the standard VP model described in , the
components of σ are defined as
σij=2ηijε˙ij+ζ-ηε˙kkδij-P2δij,
where δij is the Kronecker delta, and summation over equal indices
is implied. η and ζ are the shear and bulk viscosities,
ϵ˙ij is the strain rate tensor defined as
ϵ˙ij=12∂ui∂xj+∂uj∂xi,
and P is the maximum compressive stress defined as a function of the ice
strength parameter P⋆, mean sea ice thickness h, and the sea ice
concentration A:
P=P⋆he-C⋆(1-A),
where C⋆ is a free parameter.
The stress tensor σ is often expressed in terms of
principal stresses σ1 and σ2 or stress invariants
σI and σII. The principal stresses
σ1 and σ2 are the principal components or eigenvalues of the
stress tensor on a sea ice element. Eigenvalues always exist because the
stress tensor is by definition symmetric. The principal stresses σ1
and σ2 can be expressed as a function of σij as
5σ1=12σ11+σ22+(σ11-σ22)2+4σ122,6σ2=12σ11+σ22-(σ11-σ22)2+4σ122.
This change of coordinates can then be represented as a rotation of the
coordinates by ψ (Fig. ). This angle is
tan(2ψ)=2σ12σ11-σ22.
Any linear combination of the principal stresses consists of stress invariants. One
common set of stress invariants is the mean normal stress (σI)
and the maximal shear stress (σII). They can be written as
8σI=12(σ1+σ2)=12(σ11+σ22),9σII=12(σ1-σ2)=12(σ11-σ22)2+4σ122.
Yield curve
The VP rheology was originally developed to simulate ice motion on a basin
scale (e.g., Arctic Ocean, Southern Ocean) . In
this model, stochastic elastic deformation is parameterized as highly viscous
(creep) flow . Ice is set in motion by surface air
and basal ocean stresses moderated by internal ice stress. When the internal
sea ice stress reaches a critical value in compression, tension, or shear, sea
ice fails and relatively large plastic deformation takes place. Internal ice
stress below these thresholds leads to highly viscous (creep) flow that
parameterizes the bulk effect of many small reversible elastic deformation
events. The timescale of viscous deformation is so high
(≃30 years) that viscous deformation can be seen as
regularization for better numerical convergence in the case of small
deformation. Plastic deformations are relatively large and irreversible.
Viscous deformations are negligibly small; in contrast to elastic
deformation, they are also irreversible. The yield criterion is expressed as
a 2-D envelope either in principal stress space or stress invariant space with
a normal flow rule. The constitutive equations (Eq. ) are derived
assuming that the principal axes of stress coincide with the principal axes
of strain. The stress state on the yield curve together with the normal flow
rule therefore determines the relative importance of divergence (positive or
negative) and shear strain rate at a point. The magnitude of the deformation
is such that the stress state remains on the yield cure during plastic
deformation.
In this study, we use two different yield curves: an elliptical yield curve
and a Coulombic yield curve
. The elliptical yield curve is used in
conjunction with a normal flow rule, while the Coulombic yield curve uses a
normal flow rule on the elliptical cap and a flow rule normal to the
truncated ellipse for the same first principal stress
Appendix A. For the elliptical yield curve
(Fig. , black line), η and ζ are given by
10ζ=P2Δ,11η=ζe2,
with the abbreviation
Δ=ϵ˙I2+1e2ϵ˙II2.
In this abbreviation, the strain rate invariants are the divergence
ϵ˙I=ϵ˙11+ϵ˙22, and the maximum
shear strain rate ϵ˙II=(ϵ˙22-ϵ˙11)2+4ϵ˙122. e=a/b is the
ellipse aspect ratio with the semi-major half-axes a and b (shown in blue
in Fig. ). The ellipse aspect ratio e defines the shear
strength S⋆=P⋆/2e of the material as a fraction of its
compressive strength . For the Coulombic yield
curve (Fig. , red curve), the shear viscosity η is capped on
the two straight limbs:
ηMC=minη,1ϵ˙IIμP2-ζ⋅ϵ˙kk-c,
where μ is the slope of the Mohr–Coulomb limbs (Fig. ), and c
is the cohesion value (the value of σII for σI=0) defined
relative to the tensile strength by c=μ⋅T⋆, where T⋆ is
defined as a fraction of P⋆.
Elliptical yield curve (black) with ellipse aspect ratio e=a/b=2. Coulombic yield curve (red) and elliptical capping with internal angle of
friction (μ). Both e and μ are measures of the shear strength of
the material. The normal flow rule applies only to the elliptical part of the
yield curves. For the two straight limbs of the Coulombic yield curve, the
flow is normal to the truncated ellipse (dashed-dotted line) with the same first
stress invariant. Note that the axes σ1, σ2 and
σI, σII do not have the same scale.
The theoretical angle of fracture θ can be calculated from the Mohr's
circle of stress and yield curve written in the local (reference) coordinate
system . Details are described in Appendix . For a Mohr–Coulomb yield
criterion, θ follows immediately from the internal angle of friction
or the material shear strength. An instructive analogue is the slope of a
pile of sand on a table. Moist sand has a higher shear strength, and hence the
slope angle can be steeper (i.e., the angle θ is smaller).
Idealized experiment
An idealized compressive test is used to investigate the modes of sea ice
fracture (Fig. ). This experiment is standard in
engineering . The numerical
configuration is inspired by and similar
to the one shown in . All experiments presented
below use the same setup unless specified otherwise. The values of
parameters and constants are presented in Table .
Model domain with a solid wall on the southern (red) boundary
(Dirichlet boundary conditions with u=0) and prescribed southward
velocities on the northern orange boundary (u=0, v=av⋅t+vi; Eq. ) and open boundaries to the east and the
west (green) with von Neumann boundary conditions. θ is the measured
fracture angle with the blue line representing an LKF.
Model parameters of the reference simulation.
SymbolDefinitionValueUnitρDensity of ice910kgm-3P⋆Ice strength27.5kNm-1CStrength reduction parameter20ΔminMaximum viscosity10-10s-1Δx, ΔyGrid spacing25mCwWater drag coefficient5.21×10-3Nx, NySize of the domain400×1000Lx, LySize of experiment10×25kmlx, lyIce floe's size8×25kmAInitial ice concentration100%hInitial ice thickness1.0mNlinNo. linear iteration1500NnlinNo. nonlinear iteration1500ϵerrMax. error in LSR10-11ms-1dtTime step0.1seEllipse ratio (a/b)2.0viInitial velocity0ms-1avAcceleration5⋅10-4ms-2
The model domain is a rectangle of size 10km×25km,
except for the experiments presented in Sect.
and . An ice floe of
size 8km×25km, surrounded by 1 km of open
water on the eastern and western sides, is compressed with a linearly (in
time) increasing strain rate from the north against a solid southern
boundary. The eastern and western strips of open water avoid interesting
dynamics being confounded by the choice of lateral boundary conditions along
the open boundaries. We use a no-slip condition for the southern boundary,
constraining lateral ice motion. Note that the results presented below are
not sensitive to the choice of boundary condition on the eastern and western
boundaries. Because the simulation time and the ice velocities are small, the
Coriolis force in the momentum equations are neglected. Ocean and sea ice are
initially at rest. The only term left in the momentum equation
(Eq. ) that is relevant for our experiment is the stress
divergence term, ∇⋅σ. The ice floe has a uniform
concentration of 100 % and a thickness of 1 m. The spatial resolution
of the model is 25 m. The angle of fracture is measured with the
angle measuring tool of the GNU Image Manipulation Program (GIMP,
https://www.gimp.org/, Version 2.8.22, last access: 4 April 2019). All angles measured in this study have an error
of approximately 1∘. The finite size of the grid spacing
widens the deformation line, and the fracture spreads over several pixels
because of the obliquity of the fracture. Automatic algorithms for measuring
LKF intersection angles are described in and
.
We solve the nonlinear sea ice momentum equations with a Picard or fixed
point iteration with 1500 nonlinear or outer-loop iterations. Within each
nonlinear iteration, the nonlinear coefficients (drag coefficients and
viscosities) are updated and a linearized system of equations is solved with
a line successive (over-)relaxation (LSR) . The
linear iteration is stopped when the maximum norm of the updates is less than
ϵLSR=10-11ms-1, but we also limit the
number iterations to 1500. Typically, 1500 nonlinear iterations are required
to reach a state close enough to the converged solution. Note that this
criterion is much stricter than that proposed by
– this is so because of slow convergence due
to the highly nonlinear rheology term and the high spatial resolution.
On the open eastern and western boundaries, we use von Neumann boundary
conditions for velocity, thickness and concentration, and ice can escape the
domain without any restrictions:
∂u∂xE,W=∂v∂xE,W=∂A∂xE,W=∂h∂xE,W=0,
where E and W denote the eastern and western boundaries, respectively.
Strain is applied to the ice at the northern boundary by prescribing a
velocity that increases linearly with time:
vN(t)=av⋅t+vi;uN=0;∂A∂yN=∂h∂yN=0,
where av is the prescribed acceleration, and N denotes the northern boundary.
Results
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. ).
Uni-axial compressive test – default parameters
With default parameters (Table ), a diamond-shaped
fracture appears in the shear strain rate and divergence fields after a few
seconds of integration (Fig. ). After one time step (or
0.1 s), the stress states already lie on the yield curve, and the
fracture is readily seen in the deformation fields (divergence and shear). We
iterate for a total of 20 s in order for the signal to be apparent in
the thickness and concentration fields. We do this to more clearly show the
link between position of the stress states on the yield curve and the
resulting deformation defined by the normal flow rule in the standard VP
rheology of . The shear deformation
(ϵ˙II) shows where the ice slides in friction and deforms
plastically. From Fig. , the simulated intersection angle is
θ=(34±1)∘.
(a) First and (b) second strain invariants,
(c) ice thickness anomaly (Δh=h-1), and
(d) stress states in normalized stress invariant space along with
the elliptical yield curve after 5 s of integration. The first and second
strain invariants represent the divergence and maximum shear strain rate,
respectively. The modeled angle of fracture is θ=(34±1)∘.
After a few time steps, the ice thickness decreases, particularly along the
LKFs (Fig. c) where divergence is maximal. Note that the loading
axis in our simple 1-D experiment is also the second principal axis, and
consequently the stress states are migrating along the σ2 axis as the
strain rate at the northern boundary increases. Fracture occurs after plastic
failure when the stress state reaches the yield curve and the ice starts to
move in divergence. This occurs in the half of the ellipse closer to the
origin (for e>1) where the normal to the flow rule points in the
direction of positive divergence (or first strain rate invariant) (see
Fig. ). This explains the simulated divergent flow field
and lower ice thickness particularly along LKFs.
Schematic of stress states and failure in principal stress space.
Black arrows show how stresses move from zero at the beginning of loading
towards the yield curve until failure. Red points show the stress states at
failure – the intersection point between the second principal axis 2 (in
red) and the elliptical yield curve – for different ellipse ratios e=2,
1, 0.7. The red arrows show the direction of deformation with a normal flow
rule. The blue points and arrows show the case when the ice floe is confined
and the loading will lead to extra stress in the direction of σ1.
Sensitivity experiments
In this section, we test the sensitivity of the standard VP model simulation
(Sect. ) to the choice of resolution,
scale, and run time (Sect. ), boundary conditions
and confinement pressure (Sect. ),
and heterogeneity in the initial sea ice mass field
(Sect. ).
Maximum shear strain rate (second strain invariant) after 10 s of integration for the default domain size and Δx=100m(a) and 500m(b) and for the
default Δx and a doubled domain size of 20km×50km(c). Note that for the case of the double
domain (c), the southward velocity at the northern boundary was also
doubled to keep the deformation rate constant and that this simulation is
limited to 2 s for numerical efficiency.
Domain size, spatial resolution, and length of integration
The angle of intersection between a pair of conjugate faults does not change
with domain size and spatial resolution (Fig. ). This is
expected because non-dimensionalizing the divergence of the internal ice
stress term (the only term that remains in this simple uni-axial test
experiment) by setting u′=u/U, x′=x/L gives the same equations in
non-dimensional form, irrespective of the initial ice thickness or spatial
resolution. Consequently, the control and sensitivity experiments are scale-independent, and the behavior of the standard VP model can be readily
compared with results from RGPS, AVHRR, or laboratory experiments.
Continuing the integration to 2700 s (45 min), compared to 20 s
in the reference simulation, leads to the creation of smaller diamond-shaped
ice floes due to secondary and tertiary fracture lines
(Fig. ). The openings are visible in the thickness and
concentration fields with thinner, less concentrated ice in the lead. In this
longer experiment, the sea ice also ridges, for instance, at the center of the
domain, where the apex of the diamonds fails in compression. There is also
some thicker ice at the northern boundary induced by the specified strain
rate at the northern boundary. The fracture pattern and presence of secondary
and tertiary fracture lines are in line with results from laboratory
experiments and with AVHRR and RGPS
observations.
Sea ice thickness (a), concentration (b), maximum
shear strain rate (c), and divergence (d) after 45 min of
integration (2700 s) in a uni-axial loading test. To make these longer simulations possible, both nonlinear and linear iterations
are limited to 150 per time step. Results show the development of secondary
fracture lines in all fields after the first fracture line has
formed.
In the following, we always show results after 5 s of integration
because our main focus is on the initial fracture of the ice, that is, the
instant when the ice breaks for the first time under compression.
Maximum shear strain rate after 5 s of integration in a reduced
size domain (8km×10km) with free-slip (a)
and no-slip (b) boundary conditions. Note that the no-slip boundary
condition forces the fracture to occur at the corner of the domain, leading
to a larger angle of θ=39∘ versus 34±1∘ in the
control experiment. This suggests that the choice of boundary conditions in
current sea ice models needs to be revisited.
Boundary conditions and geometry
The dynamics responsible for the ice fracture and location of the fracture
(presented above) take place far away from the eastern and western boundaries
and therefore do not depend on the choice of the corresponding boundary
conditions. We now investigate the sensitivity of the results to the choice
of boundary condition at the southern boundary. To this end, we force the
fracture line to intersect the southern boundary by reducing the domain size
to 10km×10km with an ice floe of
8km×10km in the interior. In this case, the fracture
develops from corner to corner, and the angle is solely determined by the
geometry of the ice floe, that is, θ=arctan(lx/ly)
(Fig. b). With a free-slip boundary condition at the
southern boundary, the fracture angle is similar to the one from the control
simulation (Fig. a). That is, the no-slip condition
concentrates the stress to the corner of the ice floe touching the boundary
and predetermines the fracture location. A free-slip boundary condition is
therefore considered more physical in such idealized experiments where
fractures lines can extend from one boundary to another. This result can have
implications for simulation of LKFs in the Arctic that would extend from one
boundary to another, for instance in the Beaufort Sea.
No-slip or free-slip boundary conditions have little impact on the fracture
angle in the larger domain used in the control run simulation because the
LKFs always only touch one boundary and end in open water (results not
shown). With the free-slip boundary conditions, the stresses and strains are
only different south of the diamond fracture pattern because ice can move
along the southern boundary, and the second fracture cannot form.
We now explore the effect of confining pressure on the eastern and western
boundaries on the angle of fracture when using a (convex) elliptical yield
curve with a normal flow rule. To do so, we replace the open boundaries to
the east and the west with solid walls and the open water gaps with ice of
thicknesses hc. Note that the ice strength is linearly related to the ice
thickness (Eq. ). Therefore the normal stress at the edge of
the floe is completely defined by the thickness of the surrounding ice.
With an increasing lateral confinement pressure (i.e., an increasing ice
thickness hc next to the main floe), all stress states are moved to higher
compressive stresses (blue curve in Fig. ), and the
fracture angle increases (Fig. ). In this case, the stress
states are again migrating in a direction parallel to the σ2 axis but
with a non-zero σ1 value. The stress states of the ice along the
fracture are therefore located in a region of higher compressive stresses on
the yield curve where the divergence is reduced or even changes sign. With
increasing confinement, the stress states of the ice floe move to more
negative values of σ1 along a line of constant σ2 (blue
line in Fig. ) with deformation moving towards more
convergent states. Between hc=0.2 and hc=0.3, the regime changes
from lead opening to ridging, as the fracture angle increases to values above
45∘. This is inconsistent with the behavior of a granular
material where the angle of fracture is independent of confining pressure in
uni-axial loading laboratory experiment.
Maximum shear strain rates (left) and stress state in stress
invariant space (right) after 5 s of integration for different confinement
pressure: hc=0.05m(a) and hc=0.3m(b). Note how stress states with divergent strain
rates (a) migrate left towards convergent strain
rates (b).
Effects of the heterogeneity
So far, all initial conditions have been homogeneous in thickness and
concentration within the ice floe. In practice, sea ice (in a numerical
model but also in reality) is not homogeneous. A local weakness in the
initial ice field is likely the starting point of a crack within the ice
field e.g.,her Fig. 5c. Local
failures raise the stress level in adjacent grid cells, and a crack can
propagate. Note that the crack propagation in an “ideal” plastic model such
as the VP model is instantaneous, and this propagation is not seen between
time steps. As a consequence, lines of failure will likely develop between
local weaknesses. The location of weaknesses in the ice field together with
the ice rheology (yield curve and flow rule) both determine the fracture
angles . The influence of
previous leads on subsequent lead creation have been studied with a
discrete element model and has been used to
constrain new anisotropic rheologies that include the effects of embedded
anisotropic leads .
Sea ice thickness with two ice-free areas (a) and maximum
shear strain rates for two different ellipse aspect ratios (b, c)
after 5 s of integration. The position of the ice weaknesses determines the
location and angle of the fracture lines, and also the rheology parameter e
has an entirely different effect. The main fractures lines are at angles of
25 and 34∘ for e=2.0 and 57.6∘ for e=1.0.
To illustrate this behavior, we start new simulations from an initial ice
field with two areas of zero ice thickness and zero ice concentration, hence
weaker ice (Fig. a). After 5s these simulations
yield fracture patterns that are dramatically different from those of the
control run simulation (Sect. ): the
fracture lines now start and terminate at the locations of the weak ice
areas. Still, changing the shear strength of the ice (by changing e)
changes the fracture pattern (Fig. b and c). With e=1, the
angles are much wider than with e=2, which is consistent with the general
dependence of fracture angles on e (see Sect. ). Our simulations cannot lead to conclusive
statements about the relative importance of heterogeneity of initial
conditions and yield curve parameters for the fracture pattern, but we can
state that both affect the simulations in a way that requires treating them
separately to avoid confounding effects. Details are deferred to a dedicated
study.
Effects of the yield curve on the fracture angleElliptical yield curve
Keeping P⋆=27.5kNm-1 at its default value, the maximal
shear strength S⋆=P⋆/2e is varied by changing the ellipse
ratio e. Scaling the absolute values of P⋆ and S⋆ while
keeping e constant does not change the fracturing pattern as the tangent to
the ellipse stays the same (not shown). Changing the ellipse aspect ratio e
has a large effect on the fracture angle. The fracture angle decreases
monotonically as the shear strength of the material (or e) decreases, from
61∘ for e=0.7 to 32∘ for e=2.6. This is clearly
inconsistent with the behavior of a granular material; in the sand castle
analogue this would correspond to a dry sand castle with steeper walls than a
moist sand castle. From the simple schematic of Fig. , it
becomes clear that with increasing e, the intersection of the σ2
axis with the yield curve gradually migrates from the left side of the
ellipse to the right, where the normal to the yield curve points increasingly
towards convergent motion. We present a theoretical explanation for the
sensitivity of the fracture angle to the shear strength of the material (e,
for the ellipse) in Appendix by
rewriting the elliptical yield curve in local coordinates in the fracture
plane (σ,τ) instead of principal or stress invariant coordinates.
The fracture angle is then determined from the slope of the tangent to the
yield curve in local coordinates, and this angle follows from the Mohr's
circle see, for instance,.
suggest a smaller ellipse aspect ratio (e.g.,
e=0.7) to obtain a closer match with RADARSAT-derived distribution of
deformation rates in pan-Arctic simulations at 10 km resolution. From
Figs. and , the corresponding fracture angle
is θ=(61±1)∘, that is, much larger than that is derived
from RADARSAT images. e also changes the distribution of the stress states
on the yield curve. As the stress state migrates along the principal stress
σ2 until it reaches the yield curve in our uni-axial compressive
test, the stress states are in the second half of the ellipse for e<1 and
the resulting deformation is in convergence (or ridging). The ice thickness
increases due to ridging along the shear lines (Fig. ). In a
longer simulation with e=0.7 (not shown) the ice does not open but only
ridges, with thicker ice building up within the ice floe. This is in strong
contrast to the results with e=2.0 presented in
Sect. , where the initial floe breaks up and
separate floes form.
Fracture angles as a function of ellipse aspect ratio e with
constant P⋆ (red, bottom scale;
Sect. ). The theoretical relationship
θth,ell=12arccos121-1e2 (dashed black curve;
Eq. in the Appendix) fits the modeled angles almost
perfectly with R2=0.9995 and VAR=0.089. The simulated
fracture angles for the Coulombic yield curve as a function of the slope of
the Mohr–Coulomb limbs (blue, top scale; Sect. )
fit the theoretical relationship
θth,c=12arccosμ only for μ≤0.7 (black line; Eq. in the Appendix). The errors bars
mean that there was more than one unique fracture line: for a small μ,
the ice breaks easily along the lateral edges of the floe. For μ>0.7
(ϕ=44∘), the ambiguity appears because the stress states are
both on the linear limbs and on the elliptical cap. For μ≥0.9 (blue
line), the fracture angle is the same as for the ellipse for e=1.4.
Maximum shear strain (a), ice thickness
anomaly (b), divergence (c), and stress state in stress
invariant space (d) after 5 s of integration for a smaller ellipse
aspect ration (e=0.7 compared to e=2 in the reference run in
Sect. ). Compared to the control run on
Fig. , the angle of fracture is larger (θ=(61±1)∘), the stress states are in the second half of the ellipse (with
strain rates pointing into the convergent direction), and there is convergence
along the fracture lines (b) in agreement with the schematic in
Fig. .
Coulombic yield curve
In this section, we replace the elliptical yield curve with a Coulombic yield
curve . This yield curve consists of a
Mohr–Coulomb failure envelope – two straight limbs in principal or stress
invariant space with a slope μ – capped by an elliptical yield curve
for high compressive stresses. Note that the flow rule applies only to the
elliptical cap in this yield curve. For the two straight limbs, the yield
curve is normal to the truncated ellipse with the first stress invariant
σI. For a Mohr–Coulomb yield curve, the fracture angle depends
directly on the slope of the Mohr–Coulomb limb of the yield curve.
Appendix provides a theoretical
explanation of how the angle of fracture depends on the internal angle of
friction.
Maximum shear strain (top) and stress state in stress invariant
space (bottom) for different internal angles of friction. (a)μ=0.7 or ϕ=44∘, (b)μ=0.85 or ϕ=58∘ and
(c)μ=0.95 or ϕ=72∘ after 5 s of
integration. The angles of fracture are θ=23, (28±2) and
41∘. Figure illustrates how θ depends
on μ for a Coulombic yield curve.
The slope of the Mohr–Coulomb limbs of the Coulombic yield curve μ is
varied between 0.3 and 1.0 (corresponding to an internal angle of friction
ϕ=arcsin(μ) of 17.5 to 90∘) to study how the fracture
angle depends on the shear strength of the material. In all experiments with
the Coulombic yield curve, we use a tensile strength of 5% of
P⋆ and an ellipse ratio e=1.4, following
. The tensile strength is introduced mainly for
numerical reasons. With zero tensile strength, the state of stress in a
simple uni-axial compressive test with no confinement pressure is tangential
to the yield curve at the origin (failure in tension) and on the two straight
limbs (failure in shear) simultaneously, resulting in a numerical
instability. With tensile stress (or confinement pressure) included, the
state of stress reaches the yield only on the two limbs of the yield curve
(see Fig. a).
For the Coulombic yield curve, there are two distinct regimes of failure.
When the σ2 axis intersects the yield curve on the two straight
limbs, which happens for our configuration for angles of friction ϕ<45∘ (Fig. a, left hand side for μ=0.7 or
ϕ=44∘), the angle of fracture θ=π/4-ϕ/2 as per
standard theory (Appendix ). When the
σ2 axis intersects the yield curve on the elliptical cap, which
happens for ϕ>45∘ (Fig. c, for μ=0.95 or
ϕ=72∘), we observed a discontinuity in the fracture angle
associated with the non-differentiable corner in the yield curve. Note that
this corner cannot be removed (by changing the P⋆ and e of the
elliptical cap) as the two straight Mohr–Coulomb limbs are defined as a
truncation of the ellipse. For ϕ≈45∘ in our
configuration, the numerical solver has difficulties reaching convergence
because of the non-differentiable corner in the yield curve between the
elliptical cap and the two straight limbs (Fig. b
for μ=0.8 or ϕ=53∘). Finally for very small angles ϕ, a
large number of fractures, as opposed to single well defined fracture lines,
appear because of the weakness of the material in shear. This behavior is not
something that is typically observed in a uni-axial compressive test of a
granular material which generally have higher shear resistance. Note that the
value of ϕ that is characteristic of the individual regimes depends on
the amount of tensile strength.
Discussion
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
.
Conclusions
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.
Code and data availability
No datasets were used in this article. All simulation
data have been obtained with the MITgcm (http://mitgcm.org, last
access: 4 April 2019). Model configuration and code modifications are
described in detail in the paper. Additionally they are available on GitHub
(https://github.com/dringeis/MITgcm/tree/obcs_seaice_cont+mc, last
access: 4 April 2019).
Fracture angle
Below, we derive a relationship between the fracture angle and the internal
angle of friction for a Mohr–Coulomb yield criterion for completeness. We
consider an arbitrary piece of a 2-D medium (Fig. a) that is
subject to stresses in physical stress space σij (i=1,2).
Computing the change of coordinates as described in Eq. (), we
can consider the principal stresses (σ1,σ2) applied on the
medium (Fig. b). From the force balance, the normal stress
σ and the shear stress τ on a plane at an angle θ from the
principal stress axis can be written as (see Fig. b and
)
A1σdA=σ2sin(θ)sin(θ)dA+σ1cos(θ)cos(θ)dA,A2τdA=-σ2cos(θ)sin(θ)dA+σ1cos(θ)sin(θ)dA,
where dA is the area of the friction plane on which the stresses are
applied (in 2-D it is just a line). The second trigonometric term comes from
the fact that this surface is tilted compared to the direction of stresses
σ1 and σ2. Using THE angle sum and difference identities of
trigonometry, we can write the stresses σ and τ in terms of the
principal stresses σ1 and σ2 as
A3σ=12(σ1+σ2)+12(σ1-σ2)cos(2θ),A4τ=12(σ1-σ2)sin(2θ).
In terms of the stress invariants σI and σII, this gives
A5σ=σI+σIIcos(2θ),A6τ=σIIsin(2θ).
The Mohr–Coulomb failure criterion can be written in the fracture plane
stress space (see Fig. ) as
τ=-tan(ϕ)σ+c,
where ϕ is the internal angle of friction, and c the cohesion when no
stresses are applied . Substituting
Eqs. () and () in Eq. (), we get
σIIsin(2θ)=-tan(ϕ)σI-tan(ϕ)σIIcos(2θ)+c,
and after multiplying both sides by cos(ϕ),
σIIsin(2θ)cos(ϕ)+cos(2θ)sin(ϕ)=A9-σIsin(ϕ)+ccos(ϕ).
By geometrical construction (see Fig. ), the MC criterion is
satisfied when see alsoSect. 20.4σII=-σIsin(ϕ)+ccos(ϕ),
so that Eq. () becomes
sin(2θ)cos(ϕ)+cos(2θ)sin(ϕ)=sin(2θ+ϕ)=1,
from which we get the classical result of material deformation physics:
2θ+ϕ=π2⇒θ=π4-ϕ2.
Stress state in physical stress space (a) and in an
arbitrary coordinate system oriented at an angle θ with respect to the
principal stress axes (b). The principal stresses are the
eigenvalues of the stress tensor in an arbitrary coordinate system, and the
angle ψ is derived from the rotation matrix composed of the two
eigenvectors. Note that in the study above there is no shear stress
(σ12=0, so principal axes and physical axes are aligned (ψ=0).
Mohr's circle of stress (black) with the Mohr–Coulomb yield criterion
(red) of the angle of internal friction ϕ (red) and cohesion c in
(σ,τ) space. From Eq. (), the deformation is
created with an angle θ that can be represented in Mohr's circle
(blue).
Fracture angle and yield curve
A yield curve can be defined in the local stress (σij), principal
stress (σ1,2), or stress invariant (σI,II) spaces. The
latter gives the center and radius of the Mohr's circle of stress defining
all equivalent stress states (σ,τ) for all angles with respect to
a reference coordinate system. This allows the translation of the elliptical
yield curve from the standard principal or stress invariant space to a local
stress coordinate system (σij). In this sense, we can plot the
yield curve in (σ,τ) space as the envelope of all Mohr's circles
for each point on the elliptical yield curve defined in stress invariant
coordinates (see Fig. for an illustration with the
elliptical yield curve). In the following, we refer to this envelope of all
Mohr's circles as the reconstructed yield curve. The tangent to this curve
can be expressed as (Fig. )
sin(ϕ)=tan(γ)=μ=∂σII∂σI.
We can then express the fracture angle for stress states on the yield curve
envelope by placing Eq. () in Eq. ():
θ(σI)=π4-12arcsin∂σII∂σI(σI)B2=12arccos∂σII∂σI(σI).
This is the same relation presented and
used previously but is obtained within the
(σ,τ) stress space.
Elliptical yield curve
From the previous equations, some implications about the elliptical yield
curve immediately follow. As shown in Fig. , in a
uni-directional compressive setup the slope of a tangent to the yield curve
changes with the ellipse ratio. The convexity of the ellipse implies that the
ratio τσ=tan(ϕ) of shear strength τ to
compressive strength σ becomes smaller with smaller e. If we compute
the slope of the tangent to the elliptical yield curve at the intersection
point between the yield curve and the σ2 axis, we get
∂σII∂σIσ1=0=121-1e2.
Inserting this relationship into Eq. () gives the angle of
fracture for the uni-axial compressive experiment with an ellipse ratio e:
θth,ell(e)=12arccos121-1e2.
Illustration of the Mohr's circle applied to the elliptical yield
curve (black ellipse) in σ,τ space, some examples of Mohr's
circles (blue), and the reconstructed yield curve (red) in the fracture plane
space. The orange Mohr's circle illustrates the case in which no fracture lines
exists, for |μ|>1.
Mohr's circle of stress with an arbitrary yield curve (black line)
in the fracture plane reference. tan(γ)=μ is the tangent to the
yield curve, and ϕ is the internal angle of friction as described in
Appendix . We note that
sin(ϕ)=tan(γ)=μ (for |μ|≤1). For a slightly different
Mohr's circle (grey), the blue and red tangents meet in the same point on
the σ axis.
Note that a yield curve in (σI,II) space with a tangent slope above
unity does not have a Mohr's circle that can be tangent to the yield curve in
(σ,τ) space (orange circle on in Fig. ). This
implies that no angle of fracture can be derived for these stress states.
This is the case for the elliptical yield curve for low and high compressive
stresses. It is still unclear what happens in the VP model for stress states
on the yield curve that have a tangent with a slope higher than unity
see also. Note also that for some
(σI,σII) states, the ice will actually fail in tension, as
the reconstructed yield curve with a few points in the first and fourth
quadrant.
The shear and bulk viscosities are symmetrical about the center of the
ellipse. This implies that they are equal for divergence and convergence.
Clearly this is not physical since, for shear deformations where ice floes
continue to interact with one another (termed the quasi-static flow regime
, divergent flow counterintuitively should have
more ice–ice interactions and higher viscosities than convergent flow –
because divergent flow is the result of a higher number of contact normals
opposing the shear. When the divergence is large and floes no longer
interact, the shear and bulk viscosities are still symmetrical about the
center of the ellipse. While this is nonphysical, it does lead to more
numerical stability because the extra viscosity or dissipation of energy
regularizes the problem. We also note that a yield curve with a tangent that
has a slope smaller than 1 (in absolute value) in the first and fourth
quadrant (positive first principal stress) is unphysical because it would
lead to a diamond-shaped pair of ice fracture, even in a uni-axial tensile
test, which is inconsistent with laboratory experiments
. We conclude that adding tensile
strength to the elliptical yield curve may not be physical. The behavior of
the elliptical yield curve in uni-axial tensile tests will be explored
elsewhere.
Coulombic yield curve
Applying Mohr's circle to the Coulombic yield curve explains why the
non-differentiable corners in the yield curve lead to numerical problems
(Fig. ). The tangent does not vary smoothly, and the
reconstructed yield curve in the failure plane (σ,τ) becomes
discontinuous (Fig. , red line). As shown in
Sect. , when the stress states fall on only one of
the two parts (ellipse or limb) the conjugate faults form as expected. Using
Eq. (), with μ as the slope of the Mohr–Coulomb limbs of
the Coulombic yield curve, the fracture angle is given by
θth,c(μ)=12arccos(μ),
which is identical to Eq. ().
The Coulombic yield curve with an internal angle of friction of 1 (μ=1)
and no cohesion (c=0) also called the truncated ellipse method,
TEM;Appendix only has one possible solution with an
angle of fracture equal to 0∘ (i.e., conjugate pairs of fracture are not
possible). Zero cohesion implies that the ice will deform, even for nearly no
stress. This yield curve also appears unphysical to us.
Mohr's circle applied to the Coulombic yield curve (in black) in
σ,τ space, the Mohr's circle for the cusps between the elliptical
cap and the Mohr–Coulomb linear limbs (blue circle), and the yield curve in
(σ,τ) space (red). We can see the effect of combining two
regimes; for the same Mohr's circle, two different angles coexist (red
circles) and are apart from each other.
Author contributions
DR designed the experiments, ran the simulations, and interpreted the results with the help of ML and LBT.
NH contributed to the discussion on LKFs in simulations and observations. DR
prepared the paper with contributions from all
co-authors.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
We would like to thank Jennifer Hutchings and Harry Heorton for their helpful
reviews on this paper, as well as Amélie Bouchat and Mathieu Plante for
useful discussion during this work. This project was supported by the
Deutsche Forschungsgemeinschaft (DFG) through the International Research
Training Group “Processes and impacts of climate change in the North
Atlantic Ocean and the Canadian Arctic” (IRTG 1904 ArcTrain). The authors
would like to thank the Isaac Newton Institute for Mathematical Sciences for
support and hospitality during the program “Mathematics of sea ice
phenomena” when work on this paper was undertaken. This work was supported by EPSRC
grant numbers EP/K032208/1 and EP/R014604/1. This work is a contribution to
the Canadian Sea Ice and Snow Evolution (CanSISE) network funded by the
Natural Sciences and Engineering Research Council (NSERC) of Canada, the
Marine Environmental Observation Prediction and Response (MEOPAR) Network, and
the NSERC Discovery Grants Program awarded to L. Bruno Tremblay.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.
Review statement
This paper was edited by Daniel Feltham and reviewed
by Harry Heorton and Jennifer Hutchings.
References
Aksenov, Y. and Hibler, W. D.: Failure Propagation Effects in an
Anisotropic Sea Ice Dynamics Model, in: IUTAM Symposium on
Scaling Laws in Ice Mechanics and Ice Dynamics, edited by:
Dempsey, J. P. and Shen, H. H., Solid Mechanics and Its Applications,
363–372, UTAM Symposium, Fairbanks, Alaska, USA, 13–16 June 2000, Kluwer Academic Publishers, 2001.Babić, M., Shen, H. H., and Shen, H. T.: The stress tensor in granular
shear
flows of uniform, deformable disks at high solids concentrations, J.
Fluid Mech., 219, 81–118, 10.1017/S0022112090002877,
1990.Balendran, B. and Nemat-Nasser, S.: Double sliding model for cyclic
deformation of granular materials, including dilatancy effects, J. Mech.
Phys. Solids, 41, 573–612, 10.1016/0022-5096(93)90049-L, 1993.Bouchat, A. and Tremblay, B.: Energy dissipation in viscous-plastic sea-ice
models, J. Geophys. Res.-Oceans, 119, 976–994,
10.1002/2013JC009436,
2014.Bouchat, A. and Tremblay, B.: Using sea-ice deformation fields to constrain
the mechanical strength parameters of geophysical sea ice, J. Geophys.
Res.-Oceans, 122, 5802–5825, 10.1002/2017JC013020, 2017.Bröhan, D. and Kaleschke, L.: A Nine-Year Climatology of Arctic
Sea
Ice Lead Orientation and Frequency from AMSR-E, Remote Sensing,
6, 1451–1475, 10.3390/rs6021451, 2014.Coon, M., Kwok, R., Levy, G., Pruis, M., Schreyer, H., and Sulsky, D.: Arctic
Ice Dynamics Joint Experiment (AIDJEX) assumptions revisited and
found inadequate, J. Geophys. Res.-Oceans, 112, C11S90,
10.1029/2005JC003393,
2007.
Coon, M. D., Maykut, A., G., Pritchard, R. S., Rothrock, D. A., and
Thorndike,
A. S.: Modeling The Pack Ice as an Elastic-Plastic Material,
AIDJEX Bulletin, 24, 1–106, 1974.Cox, G. F. N. and Richter-Menge, J. A.: Tensile Strength of Multi-Year
Pressure Ridge Sea Ice Samples, J. Energ. Resour.-ASME, 107,
375–380, 10.1115/1.3231204, 1985.Dansereau, V., Weiss, J., Saramito, P., and Lattes, P.: A Maxwell
elasto-brittle rheology for sea ice modelling, The Cryosphere, 10,
1339–1359, 10.5194/tc-10-1339-2016, 2016.Dansereau, V., Démery, V., Berthier, E., Weiss, J., and Ponson, L.: Fault
orientation in damage failure under compression, arXiv:1712.08530 [cond-mat,
physics:physics], available at: http://arxiv.org/abs/1712.08530 (last access: 4 April 2019), 2017.Dempsey, J. P., Xie, Y., Adamson, R. M., and Farmer, D. M.: Fracture of a
ridged multi-year Arctic sea ice floe, Cold Reg. Sci. Technol.,
76–77, 63–68, 10.1016/j.coldregions.2011.09.012,
2012.Dethloff, K., Rex, M., and Shupe, M.: Multidisciplinary drifting
Observatory for the Study of Arctic Climate (MOSAiC), EGU General
Assembly Conference Abstracts, 18, available at:
https://ui.adsabs.harvard.edu/#abs/2016EGUGA..18.3064D/abstract (last
access: 4 April 2019), 2016.Dumont, D., Gratton, Y., and Arbetter, T. E.: Modeling the Dynamics of the
North Water Polynya Ice Bridge, J. Phys. Oceanogr.,
39, 1448–1461, 10.1175/2008JPO3965.1,
2009.
Erlingsson, B.: Two-dimensional deformation patterns in sea ice, J.
Glaciol., 34, 301–308, 1988.Feltham, D. L.: Sea Ice Rheology, Annu. Rev. Fluid Mech., 40,
91–112, 10.1146/annurev.fluid.40.111406.102151,
2008.Girard, L., Weiss, J., Molines, J. M., Barnier, B., and Bouillon, S.:
Evaluation of high-resolution sea ice models on the basis of statistical and
scaling properties of Arctic sea ice drift and deformation, J.
Geophys. Res.-Oceans, 114, C08015, 10.1029/2008JC005182,
2009.Girard, L., Bouillon, S., Weiss, J., Amitrano, D., Fichefet, T., and Legat,
V.:
A new modeling framework for sea-ice mechanics based on elasto-brittle
rheology, Ann. Glaciol., 52, 123–132,
10.3189/172756411795931499, 2011.Heorton, H. D. B. S., Feltham, D. L., and Tsamados, M.: Stress and
deformation
characteristics of sea ice in a high-resolution, anisotropic sea ice model,
Philos. T. R. Soc. A, 376, 20170349, 10.1098/rsta.2017.0349,
2018.Herman, A.: Discrete-Element bonded-particle Sea Ice model DESIgn, version
1.3a – model description and implementation, Geosci. Model Dev., 9,
1219–1241, 10.5194/gmd-9-1219-2016, 2016.Hibler, W. D.: A viscous sea ice law as a stochastic average of plasticity,
J. Geophys. Res., 82, 3932–3938, 10.1029/JC082i027p03932, 1977.Hibler, W. D.: A Dynamic Thermodynamic Sea Ice Model, J. Phys.
Oceanogr., 9, 815–846,
10.1175/1520-0485(1979)009<0815:ADTSIM>2.0.CO;2, 1979.Hibler, W. D. and Schulson, E. M.: On modeling sea-ice fracture and flow in
numerical investigations of climate, Ann. Glaciol., 25, 26–32,
10.3189/S0260305500190019,
1997.Hibler, W. D. and Schulson, E. M.: On modeling the anisotropic failure and
flow of flawed sea ice, J. Geophys. Res.-Oceans, 105, 17105–17120,
10.1029/2000JC900045, 2000.Hibler, W. D., Hutchings, J. K., and Ip, C. F.: sea-ice arching and multiple
flow States of Arctic pack ice, Ann. Glaciol., 44, 339–344,
10.3189/172756406781811448,
2006.Hundsdorfer, W., Koren, B., vanLoon, M., and Verwer, J. G.: A Positive
Finite-Difference Advection Scheme, J. Comput. Phys.,
117, 35–46, 10.1006/jcph.1995.1042,
1995.Hunke, E. C.: Viscous–Plastic Sea Ice Dynamics with the EVP
Model: Linearization Issues, J. Comput. Phys., 170,
18–38, 10.1006/jcph.2001.6710,
2001.Hunke, E. C. and Dukowicz, J. K.: An Elastic–Viscous–Plastic Model
for Sea Ice Dynamics, J. Phys. Oceanogr., 27, 1849–1867,
10.1175/1520-0485(1997)027<1849:AEVPMF>2.0.CO;2,
1997.Hutchings, J. K., Jasak, H., and Laxon, S. W.: A strength implicit correction
scheme for the viscous-plastic sea ice model, Ocean Model., 7, 111–133,
10.1016/S1463-5003(03)00040-4,
2004.Hutchings, J. K., Heil, P., and Hibler, W. D.: Modeling Linear Kinematic
Features in Sea Ice, Mon. Weather Rev., 133, 3481–3497,
10.1175/MWR3045.1, 2005.Hutter, K. and Rajagopal, K. R.: On flows of granular materials, Continuum
Mech. Therm., 6, 81–139, 10.1007/BF01140894, 1994.Hutter, N., Martin, L., and Dimitris, M.: Scaling Properties of Arctic
Sea Ice Deformation in a High-Resolution Viscous-Plastic
Sea Ice Model and in Satellite Observations, J. Geophys.
Res.-Oceans, 123, 672–687, 10.1002/2017JC013119,
2018.Hutter, N., Zampieri, L., and Losch, M.: Leads and ridges in Arctic sea ice
from RGPS data and a new tracking algorithm, The Cryosphere, 13, 627–645,
10.5194/tc-13-627-2019, 2019.
Ip, C. F., Hibler, W. D., and Flato, G. M.: On the effect of rheology on
seasonal sea-ice simulations, Ann. Glaciol., 15, 17–25, 1991.Kwok, R.: Deformation of the Arctic Ocean Sea Ice Cover between
November 1996 and April 1997: A Qualitative Survey, in: IUTAM
Symposium on Scaling Laws in Ice Mechanics and Ice Dynamics,
edited by: Dempsey, J. P. and Shen, H. H., no. 94 in Solid Mechanics and
Its Applications, Springer Netherlands, 315–322,
10.1007/978-94-015-9735-7_26,
2001.Lemieux, J.-F. and Tremblay, B.: Numerical convergence of viscous-plastic sea
ice models, J. Geophys. Res.-Oceans, 114, C05009, 10.1029/2008JC005017,
2009.Lemieux, J.-F., Tremblay, B., Sedláček, J., Tupper, P., Thomas, S.,
Huard,
D., and Auclair, J.-P.: Improving the numerical convergence of
viscous-plastic sea ice models with the Jacobian-free Newton–Krylov
method, J. Comput. Phys., 229, 2840–2852,
10.1016/j.jcp.2009.12.011,
2010.Lindsay, R. W. and Rothrock, D. A.: Arctic sea ice leads from advanced very
high resolution radiometer images, J. Geophys. Res.-Oceans,
100, 4533–4544, 10.1029/94JC02393,
1995.Linow, S. and Dierking, W.: Object-Based Detection of Linear
Kinematic Features in Sea Ice, Remote Sensing, 9, 493,
10.3390/rs9050493, 2017.Losch, M., Menemenlis, D., Campin, J.-M., Heimbach, P., and Hill, C.: On the
formulation of sea-ice models. Part 1: Effects of different solver
implementations and parameterizations, Ocean Model., 33, 129–144,
10.1016/j.ocemod.2009.12.008,
2010.Marko, J. R. and Thomson, R. E.: Rectilinear leads and internal motions in
the ice pack of the western Arctic Ocean, J. Geophys. Res., 82, 979–987,
10.1029/JC082i006p00979, 1977.Marsan, D., Weiss, J., Larose, E., and Métaxian, J.-P.: Sea-ice thickness
measurement based on the dispersion of ice swell, J. Acoust. Soc. Am., 131,
80–91, 10.1121/1.3662051, 2012.Marshall, J., Adcroft, A., Hill, C., Perelman, L., and Heisey, C.: A
finite-volume, incompressible Navier Stokes model for studies of the
ocean on parallel computers, J. Geophys. Res.-Oceans, 102,
5753–5766, 10.1029/96JC02775,
1997.Menge, J. A. R. and Jones, K. F.: The tensile strength of first-year sea ice,
J. Glaciol., 39, 609–618, 10.3189/S0022143000016506,
1993.Miller, P. A., Laxon, S. W., and Feltham, D. L.: Improving the spatial
distribution of modeled Arctic sea ice thickness, Geophys. Res.
Lett., 32, L18503, 10.1029/2005GL023622,
2005.
Overland, J. E., McNutt, S. L., Salo, S., Groves, J., and Li, S.: Arctic sea
ice as a granular plastic, J. Geophys. Res., 103,
21845–21868,
1998.
Popov, E. P.: Mechanics of Materials, 2nd edn., Prentice Hall, Englewood
Cliffs, N.J., USA, 1976.Pritchard, R. S.: An Elastic-Plastic Constitutive Law for Sea
Ice, J. Appl. Mech., 42, 379–384, 10.1115/1.3423585, 1975.Pritchard, R. S.: Mathematical characteristics of sea ice dynamics models,
J. Geophys. Res.-Oceans, 93, 15609–15618,
10.1029/JC093iC12p15609,
1988.Rampal, P., Bouillon, S., Ólason, E., and Morlighem, M.: neXtSIM: a new
Lagrangian sea ice model, The Cryosphere, 10, 1055–1073,
10.5194/tc-10-1055-2016, 2016.Rothrock, D. A.: The steady drift of an incompressible Arctic ice cover,
J. Geophys. Res., 80, 387–397, 10.1029/JC080i003p00387,
1975.Rothrock, D. A. and Thorndike, A. S.: Measuring the sea ice floe size
distribution, J. Geophys. Res.-Oceans, 89, 6477–6486,
10.1029/JC089iC04p06477,
1984.Schmidt, G. A., Kelley, M., Nazarenko, L., Ruedy, R., Russell, G. L.,
Aleinov,
I., Bauer, M., Bauer, S. E., Bhat, M. K., Bleck, R., Canuto, V., Chen, Y.-H.,
Cheng, Y., Clune, T. L., Genio, A. D., Fainchtein, R. d., Faluvegi, G.,
Hansen, J. E., Healy, R. J., Kiang, N. Y., Koch, D., Lacis, A. A., LeGrande,
A. N., Lerner, J., Lo, K. K., Matthews, E. E., Menon, S., Miller, R. L.,
Oinas, V., Oloso, A. O., Perlwitz, J. P., Puma, M. J., Putman, W. M., Rind,
D., Romanou, A., Sato, M., Shindell, D. T., Sun, S., Syed, R. A., Tausnev,
N., Tsigaridis, K., Unger, N., Voulgarakis, A., Yao, M.-S., and Zhang, J.:
Configuration and assessment of the GISS ModelE2 contributions to the
CMIP5 archive, J. Adv. Model. Earth Syst., 6, 141–184,
10.1002/2013MS000265,
2014.Schreyer, H. L., Sulsky, D. L., Munday, L. B., Coon, M. D., and Kwok, R.:
Elastic-decohesive constitutive model for sea ice, J. Geophys.
Res.-Oceans, 111, C11S26, 10.1029/2005JC003334,
2006.Schulson, E. M.: Compressive shear faults within arctic sea ice: Fracture
on scales large and small, J. Geophys. Res.-Oceans, 109, C07016,
10.1029/2003JC002108, 2004.Sirven, J. and Tremblay, B.: Analytical Study of an Isotropic
Viscoplastic Sea Ice Model in Idealized Configurations, J. Phys.
Oceanogr., 45, 331–354, 10.1175/JPO-D-13-0109.1, 2014.Spreen, G., Kwok, R., Menemenlis, D., and Nguyen, A. T.: Sea-ice deformation
in a coupled ocean–sea-ice model and in satellite remote sensing data, The
Cryosphere, 11, 1553–1573, 10.5194/tc-11-1553-2017, 2017.Stern, H. L., Rothrock, D. A., and Kwok, R.: Open water production in
Arctic
sea ice: Satellite measurements and model parameterizations, J.
Geophys. Res.-Oceans, 100, 20601–20612, 10.1029/95JC02306,
1995.Stroeve, J., Barrett, A., Serreze, M., and Schweiger, A.: Using records from
submarine, aircraft and satellites to evaluate climate model simulations of
Arctic sea ice thickness, The Cryosphere, 8, 1839–1854,
10.5194/tc-8-1839-2014, 2014.Sulsky, D., Schreyer, H., Peterson, K., Kwok, R., and Coon, M.: Using the
material-point method to model sea ice dynamics, J. Geophys.
Res.-Oceans, 112, C02S90, 10.1029/2005JC003329,
2007.Tremblay, L.-B. and Mysak, L. A.: Modeling Sea Ice as a Granular
Material, Including the Dilatancy Effect, J. Phys.
Oceanogr., 27, 2342–2360,
10.1175/1520-0485(1997)027<2342:MSIAAG>2.0.CO;2,
1997.Tsamados, M., Feltham, D. L., and Wilchinsky, A. V.: Impact of a new
anisotropic rheology on simulations of Arctic sea ice, J.
Geophys. Res.-Oceans, 118, 91–107, 10.1029/2012JC007990,
2013.Ungermann, M., Tremblay, L. B., Martin, T., and Losch, M.: Impact of the
Ice Strength Formulation on the Performance of a Sea Ice
Thickness Distribution Model in the Arctic, J. Geophys. Res.-Oceans,
122, 2090–2107, 10.1002/2016JC012128, 2017.
Verruijt, A.: An Introduction to Soil Mechanics, Theory and
Applications of Transport in Porous Media, Springer International
Publishing, available at: http://www.springer.com/gp/book/9783319611846
(last access: 4 April 2019), 2018.Walter, B. A. and Overland, J. E.: The response of lead patterns in the
Beaufort Sea to storm-scale wind forcing, Ann. Glaciol., 17,
219–226, 10.3189/S0260305500012878,
1993.Wang, K.: Observing the yield curve of compacted pack ice, J.
Geophys. Res.-Oceans, 112, C05015, 10.1029/2006JC003610,
2007.Wang, K., Leppäranta, M., and Kõuts, T.: A study of sea ice dynamic
events in a small bay, Cold Reg. Sci. Technol., 45, 83–94,
10.1016/j.coldregions.2006.02.002, 2006.Wang, Q., Danilov, S., Jung, T., Kaleschke, L., and Wernecke, A.: Sea ice
leads in the Arctic Ocean: Model assessment, interannual variability
and trends, Geophys. Res. Lett., 43, 7019–7027, 10.1002/2016GL068696,
2016.Weiss, J., Schulson, E. M., and Stern, H. L.: Sea ice rheology from in-situ,
satellite and laboratory observations: Fracture and friction, Earth
Planet. Sc. Lett., 255, 1–8, 10.1016/j.epsl.2006.11.033,
2007.Wilchinsky, A. V. and Feltham, D. L.: Anisotropic model for granulated sea
ice
dynamics, J. Mech. Phys. Solids, 54, 1147–1185,
10.1016/j.jmps.2005.12.006,
2006.Wilchinsky, A. V. and Feltham, D. L.: Modeling Coulombic failure of sea ice
with leads, J. Geophys. Res.-Oceans, 116, C08040, 10.1029/2011JC007071,
2011.Wilchinsky, A. V. and Feltham, D. L.: Rheology of Discrete Failure
Regimes of Anisotropic Sea Ice, J. Phys. Oceanogr., 42,
1065–1082, 10.1175/JPO-D-11-0178.1,
2012.Wilchinsky, A. V., Feltham, D. L., and Hopkins, M. A.: Effect of shear
rupture on aggregate scale formation in sea ice, J. Geophys. Res.-Oceans,
115, C10002, 10.1029/2009JC006043, 2010.Wilchinsky, A. V., Feltham, D. L., and Hopkins, M. A.: Modelling the
reorientation of sea-ice faults as the wind changes direction, Ann.
Glaciol., 52, 83–90, 10.3189/172756411795931831,
2011.Zhang, J. and Hibler, W. D.: On an efficient numerical method for modeling
sea ice dynamics, J. Geophys. Res.-Oceans, 102, 8691–8702,
10.1029/96JC03744, 1997.Zhang, J. and Rothrock, D. A.: Effect of sea ice rheology in numerical
investigations of climate, J. Geophys. Res.-Oceans, 110, C08014,
10.1029/2004JC002599, 2005.