This paper presents theoretical foundations, numerical implementation and examples of application of the two-dimensional Discrete-Element bonded-particle Sea Ice model – DESIgn. In the model, sea ice is represented as an assemblage of objects of two types: disk-shaped “grains” and semi-elastic bonds connecting them. Grains move on the sea surface under the influence of forces from the atmosphere and the ocean, as well as interactions with surrounding grains through direct contact (Hertzian contact mechanics) and/or through bonds. The model has an experimental option of taking into account quasi-three-dimensional effects related to the space- and time-varying curvature of the sea surface, thus enabling simulation of ice breaking due to stresses resulting from bending moments associated with surface waves. Examples of the model's application to simple sea ice deformation and breaking problems are presented, with an analysis of the influence of the basic model parameters (“microscopic” properties of grains and bonds) on the large-scale response of the modeled material. The model is written as a toolbox suitable for usage with the open-source numerical library LIGGGHTS. The code, together with full technical documentation and example input files, is freely available with this paper and on the Internet.

Sea ice cover in polar and subpolar seas is a complex assemblage of ice blocks of various sizes, thicknesses, ages, structures and properties resulting from their genesis, typically consisting of multiple cycles of partial melting, (re)freezing and mechanical deformation resulting from the action of external agents (wind, waves, solar radiation, etc.) and from interactions with surrounding ice. In favorable conditions, the ice blocks may join (freeze) to form larger blocks (ice floes), behaving like semi-rigid bodies, so that the deformation of ice is localized, limited to narrow shear and compression zones. This type of ice cover is characteristic of the compact, central Arctic ice pack. Close to the ice edge, extensive breaking primarily caused by ocean waves produces ice behaving as a polydisperse granular material composed of individual, relatively small floes of various diameters. In all cases, many important aspects of sea ice dynamics are directly related to its discrete, discontinuous nature. Consequently, although some of the large-scale effects of those processes can be parametrized in continuum sea ice models, mechanisms underlying them can be investigated and understood only by means of models that properly take into account the fundamental physics.

Probably the most appreciable way in which the granular nature of sea ice
influences other processes, including its own behavior, is through the floe
size (mean and the floe-size distribution, FSD). Examples include mechanical
weakening of ice after storms resulting from its fragmentation

The Discrete-Element, bonded-particle Sea Ice model – DESIgn – presented here has been developed as a tool for studying the processes mentioned above at the floe level, with hope that it will help to deepen our understanding of ice dynamics at different scales, and possibly to develop parametrizations of relevant processes for continuum models.

This paper presents the theoretical background, underlying assumptions and
equations of the model, its numerical implementation and examples of
applications to sea ice problems. The model is an extension of the earlier
versions described in

The model is two-dimensional (2-D), but it enables one to take into account some wave-related effects, i.e., stresses resulting from flexural moments acting on sea ice when surface waves are present. It can be applied to a wide range of sea ice types, although it is worth stressing here that the word “granular” in the present context describes macroscopic, large-scale ice properties, i.e., the fact that it is composed of individual ice floes. It does not refer in any way to smaller-scale material structures at the level of (groups of) ice crystals. Also, in view of specific assumptions underlying the model (e.g., the above-mentioned two-dimensionality), it is not suitable for early stages of sea ice formation, like frazil and grease ice; pancake ice can be regarded as a rough lower limit of the model validity in terms of floe size. On the other hand, although the model was designed for sea ice, it can be applied to other 2-D materials composed of disk-shaped grains.

The paper is structured as follows. The next section contains a short review
of previous attempts to account for the granular nature of sea ice in
numerical models of its dynamics. Section

From the point of view of processes analyzed here, one of the most important
properties of sea ice is the FSD. Since the seminal paper of

A number of parametrizations have been developed to improve the performance
of continuum sea ice models in situations when granular effects have a
significant influence on the large-scale sea ice behavior. One group of
parametrizations is related to the so-called collisional rheology, describing
stress in fragmented sea ice due to inelastic collisions between floes,
relevant especially in the MIZ. In the existing collisional rheology models,
stress is calculated from the average collision frequency and momentum
transferred during collisions, assuming uniform distribution of the floes on
the sea surface and either constant

Following the well-established theory describing the evolution of the
ice-thickness distribution in continuum sea ice models, analogous equations
for the floe-size distribution

Although continuum models remain a standard tool for simulating sea ice
dynamics, especially at large scales, a number of discrete-element models
have been developed in recent decades in which sea ice is represented as an
assemblage of interacting objects (“particles”); although the models share
the same underlying idea, they differ in terms of the shape and properties of
their building blocks, details of the contact mechanics formulations,
parametrization of physical processes not explicitly accounted for in the
model (e.g., ridging), as well as numerical algorithms used to solve the
model equations. Some models combine an Eulerian, grid-based approach typical
for continuum models with a Lagrangian, particle-based approach used in DEMs
– examples include particle-in-cell (PIC) models

In a series of papers by Hopkins and coworkers

Finally, the works by

The sea ice model proposed in this work describes the motion and interactions
of two types, or classes, of objects. Following the nomenclature used in
similar models of rocks and soils, we will refer to those two types of
objects as “grains” or “particles”, and “bonds”. (In the documentation
of the numerical libraries on which the numerical model is based, the terms
“atoms” and “bonds” are used independently of the type of those objects;
see Sect.

In the context of sea ice, we define the particles as disk-shaped sea ice
blocks moving within a 2-D space representing the sea surface. The bonds form
when the neighboring disks freeze together. In other words, bonds represent
new, usually thinner ice filling cracks, leads and other open spaces between
thicker ice blocks. Throughout this paper, sets of bonded particles will be
called floes – similarly as in

Two approaches to the usage of grains and bonds in DESIgn: without
bonds, when ice floes are identical to grains and the FSD is
prescribed

There are two essentially independent mechanisms of interactions between neighboring particles. The first, based on repulsive and frictional forces between particles, requires that they are in direct contact with each other. The second requires that the particles are connected with an elastic bond. Crucially, whereas forces are transmitted in both cases, bonds are also able to transmit momentum. Another substantial difference between the two interaction types results from the fact that bonds have a certain tensile strength. Consequently, if they are present in a material, it attains a tensile strength at a macroscopic level as well.

From the point of view of sea ice modeling, the bonded-particle approach has
a number of important advantages and provides an opportunity to overcome
several drawbacks of the existing models. In particular, the model is
suitable for arbitrary ice concentration

In typical soil and rock applications of bonded-particle models, the
simulations are initiated with fully bonded material, and the bonds are
allowed to break but not to recover during a simulation

Let us consider an ensemble of

The horizontal space

Let

Analogously, let

Finally, the net external force acting on disk

Let us define a unit vector pointing upward,

Each of the normal and tangential forces

The details of the formulation of the contact forces depend on the selected
contact model and on the geometry of the interacting particles, as described
in Sect. S1 of the Supplement, which also contains the derivation of the full
set of equations used in the present sea ice model. Both alternative
formulations available are based on the Hertzian contact mechanics.
Additional details of the Hertzian model in a general context can be found,
e.g., in

Notably, in the quasi-3-D version of the model, the contact forces are calculated in the same way as in the 2-D version, i.e., without taking into account the tilt between grains – which amounts to the assumption that the orientations of the axes of symmetry of neighboring grains do not deviate significantly from each other.

In the following, the bonds are identified by the pair of indices of grains
that they connect. Each bond, cuboid in shape, is characterized by the
following set of properties: thickness

Geometry of two grains,

The forces and torques acting on the grains connected with a bond result from
the (finite) relative displacement and rotation of those grains; they can be
decomposed into axial, tangential, bending and twisting components

The components of the force due to the relative displacement are calculated
from a linear elastic material law, in which the force is proportional to the
displacement, given by

Analogously, twisting and bending moments due to changes in the relative
orientation of grains

In total,

According to the classical beam theory, the shear stress acting on the bond
can be calculated as

In terms of the formulation of forces acting on the grains, the model is very flexible and enables one to specify any combination of forces that may be space- and time-varying and depend on the properties of the individual grains (e.g., their mass or size). To make the configuration of the model more convenient, formulae describing the forces most relevant to the motion of sea ice on the sea surface have been implemented in the code and the corresponding forces can be activated easily by means of simple commands described in the User's Guide. These forces include the Coriolis force and the skin and form drag due to the wind and surface current.

Forces and torques acting on an elastic bond connecting grains

The Coriolis force acting on the

In most real-world situations, the dominating surface forces acting on sea
ice floes are the atmospheric and oceanic skin drag,

In the MIZ, as well as in regions with low ice concentrations, sea ice is
affected by surface waves (wind waves and, most importantly, swell). Flexural
stresses related to the curvature of the sea surface are one of – or
presumably

The wave-related effects available in the present version of the model are
under development and should be treated as a starting point for more advanced
models. At present, two wave-related processes have been implemented: forces
due to the oscillating surface current, and a net moment of buoyancy forces
due to the time-varying sea surface slope and curvature. These two mechanisms
tend to be relevant in different conditions. The alternating convergence and
divergence associated with oscillatory motion of the sea surface, and the
resulting tensile and compressive stress, influence pancake ice formation and
dynamics

Obviously, there are a number of other wave-related effects that are very
important but not included in the present model. Most crucially, the wave
properties have to be prescribed – they may be spatially and temporarily
variable, but are unaffected by the ice, which means that, for example, wave
scattering and reflection at the floes' edges cannot be taken into account.
Similarly, although the tilt of the grains is calculated, this is not the
case for their vertical movement. Whereas some of these additional aspects of
wave–ice interactions can be relatively easily implemented in the type of
model described here, others, like for example rafting and ridging during
compression, would require either a fully 3-D computation

Let us suppose that the sea surface elevation

If the oscillating current is to be taken into account in the model,

As mentioned above, the presence of waves and the associated space- and
time-varying slope of the sea surface induces torque and rotation around the
horizontal axes of the ice grains. In the following, an assumption is made
that the

For unbonded grains, their angular momentum resulting from
Eq. (

A sketch of a circular grain on a sloping sea surface, illustrating the variables involved in calculation of the wave-induced torque (see text for details).

The numerical model is based on two libraries designed for effective
simulation of large systems of objects interacting through a variety of
short- or long-range forces: LAMMPS

Details regarding the numerical aspects of the model can be found in the User's Guide (available in the “doc” folder in the attached material) and in the documentation of LAMMPS/LIGGGHTS. DESIgn uses the standard methods of the solution of the governing equations implemented in LIGGGHTS. Therefore, only the most important facts regarding the numerical aspects of the model are given here.

The momentum equations are integrated in time using the energy-conserving
velocity Verlet solver suitable for finite-size grains and taking into
account not only the position and velocity of the center of mass of the
particles, but also their angular velocity. Due to numerical stability and
so-called energy drift issues, the computations require very small time
steps. For sea ice simulations, in which grains typically have diameters of

All simulations described in

Physical and numerical model parameters used in the reference
simulations in Sect.

In the first set of simulations, a rectangular sample of compact sea ice
(densely packed grains with uniform size distribution, fully connected with
their neighbors) is subject to a prescribed uniaxial tensile, uniaxial
compressive, or shear strain. The strain rate is obtained by setting to zero
the velocity of the grains located at the lower boundary of the domain, and
moving the grains located at the upper boundary with a specified velocity
until terminal failure

Example damage patterns obtained in simulations of an initially
compact sample under uniaxial tensile

In all cases, the initial increase in strain results in a fast, approximately
linear increase in stress. The rate of the stress accumulation in the
material depends on its properties. In particular, it increases with
increasing mean bond thickness

Amplitude of the maximum normal

As in Fig.

Temporal evolution of the fraction of broken bonds and the rate of
bond breaking

The role of

In all simulations initialized with

A very important aspect of the model, mentioned in the introduction, is that
both types of interactions – those due to bonds and those due to a direct
contact between grains – act mutually and contribute to the overall
properties of the modeled sea ice. In the simulations discussed above, the
majority of grains were in direct contact with their neighbors. Even though,
macroscopically, the stress due to pairwise interactions tends to be more
than an order of magnitude lower than the stress transmitted through bonds
(e.g., it never exceeded 2–3 % in the reference model run), it in many
ways influences the sea ice behavior. In particular, pairwise interactions
play a crucial role in the development of the fracture zones, as, obviously,
after bond breaking they are the only type of grain–grain interactions
present. Thus, friction between grains influences sliding along deformation
zones, as well as their width. Figures

In the second group of simulations, compact, undamaged sea ice is subject to
flexural stresses generated by regular deep-water surface waves with a
prescribed, spatially uniform amplitude

This 1-D setting is very close to a 2-D one in which a unidirectional wave
propagates through a regular matrix of identical grains, each bonded to its
four neighbors. Because in this case the stresses acting on bonds oriented
parallel to the wave crests, related to the Poisson effect, are close to
zero, the model produces long parallel stripes with a breaking pattern
exactly the same as that obtained with a 1-D model version. Therefore, in
Sect.

As in Fig.

Instantaneous normal stress (color scale; in Pa) acting on
individual grains in simulation under uniaxial compressive strain shortly
after terminal failure of the material (

Instantaneous shear stress (color scale; in Pa) acting on individual
grains due to bond interactions

Let us consider a set of

The wave forcing is calculated directly from Eq. (

In the reference simulation summarized in Table

Results of 1-D simulations of sea ice response to waves (without
breaking): tilt of the grains

As in Fig.

Physical and numerical model parameters used in the reference simulations in Sect.

Understandably, if only one end of the floe is allowed to move freely, the
above-mentioned influence of the exact ratio of the floe size to the
wavelength is much less pronounced (Fig.

For large floes, the amplitude of their flexural motion is largest in the
vicinity of their free ends: in the cases tested, it has a roughly
exponential profile within the distance of 10–20 wavelengths from the free
floe's ends (Figs.

Wave-induced breaking of an ice floe with initial size

Two aspects of these results are worth stressing. First, this breaking
pattern, progressing from the ice edge towards inner regions, has been
obtained with a spatially constant wave amplitude. In a more realistic
configuration, with wave amplitude decreasing with the distance from the ice
edge due to attenuation, this effect would be even stronger. Second, with a
constant wave amplitude, progressive breaking is possible only if the
wave-induced stress acting on the ice far from its edge remains smaller than
its strength. In real-world situations, such “tuning” of the wave steepness
to the ice strength is presumably rare in stationary settings, but can be
expected to occur frequently during periods of wave amplitude increasing in
time, e.g., when the onshore wind strengthens or when swell from distant
locations arrives at the ice edge with gradually increasing amplitude. Thus,
breaking similar to that shown in Fig.

As already mentioned in the introduction to this section, a 2-D model initialized with a regular, square matrix of identical grains produces floes in the form of long stripes parallel to the wave crests. However, in order for the model to be applicable to more general conditions, e.g., with waves coming from different directions, it is desirable that the model produces realistic results (in terms of both floe sizes and shapes) when it is initialized with randomly distributed grains of different sizes, and thus contains bonds with a range of spatial orientations not aligned with the wave direction.

Fragments of the model domain at the end of the reference
simulation, with

As can be expected, most aspects of the behavior of the 2-D model are fully
analogous to those of the 1-D model. In particular, the dependence of the
amplitude of the flexural and rigid motions of the floes on their size,
described in the previous section, is very similar in 1-D and 2-D. As
previously, the model has been run for a reference run
(Table

Rank-order statistics of floe sizes (

Physical and numerical model parameters used in the reference simulations in Sect.

It was not possible to obtain a wide distribution of floe sizes in the 1-D
model, which supports the notion that this feature is directly linked to
irregular floe shapes in the 2-D model. Also, it must be remembered that
these features tend to disappear if some irregularity is introduced to the
model (e.g., if the forcing is specified as a superposition of more than one
elementary wave with different directions and phases), so that it may seem
irrelevant in more realistic model settings than those analyzed here, but
nevertheless it signals an undesired feature of the model. The tendency to
produce non-convex floes decreases with increasing shear stiffness and
decreasing shear strength of bonds (relative to their normal stiffness and
compressive strength, respectively), but it remains to be investigated
whether adjusting these parameters accordingly does not produce any other
negative effects. In any case, the general conclusion is that the
orientations of bonds in the model do determine the resulting geometry of the
floes. In particular, although the floes tend to be elongated in the
direction of the wave crests (Fig.

The modeling results presented in this paper have been limited to very simple
configurations, with the goal of testing the basic model features and
analyzing the influence of the model parameters on its behavior. Further work
is necessary to verify the model – its underlying assumptions, numerical
algorithms, etc. – in a wider range of configurations. Importantly, the
modular structures of LIGGGHTS in general and of the sea ice toolbox in
particular make it relatively easy to modify and/or replace some parts should
better or alternative solutions become available. In the same way, the model
may be extended with new features, including time variability of the
properties of grains and bonds (e.g., changes in their thickness due to
thermodynamic processes), other contact models or bond breaking criteria, or
a model of wave–sea ice interactions in which the wave characteristics are
not prescribed but modified based on the properties of the ice cover. As
noted earlier in Sect.

The fact that DESIgn is a toolbox of LIGGGHTS offers a possibility to
relatively easily combine this model with a wide range of functionalities
offered by LIGGGHTS. In particular, the mesh-free SPH method available in the
open-source LIGGGHTS version could be used to simulate sea ice in a manner
similar to

Among the challenges related to the usage of DESIgn (and other, similar
models) in realistic sea ice problems, validation with observational data
undoubtedly belongs to the most urgent ones. Although substantial progress
has been made in recent years in terms of availability of high-resolution
remote-sensing data, the temporal and spatial resolution of those data is
still too low to capture some rather subtle effects described in

The code of DESIgn, together with full technical documentation and example
input files, is freely available with this paper and at the internet page

All comments, questions, suggestions and critiques regarding the functioning of the DESIgn model can be directed to the author of this paper.

The development of the sea ice model would not be possible without LAMMPS and
LIGGGHTS. In particular, the bond-related code of DESIgn is based on the
experimental version of the bond package available through the LIGGGHTS home
page. Part of the simulations presented in this paper have been conducted at
the Academic Computer Center in Gdansk (TASK,