The viscous–plastic (VP) rheology with an elliptical yield curve and normal flow rule is implemented in a Lagrangian modelling framework using the smoothed particle hydrodynamics (SPH) meshfree method. Results show, from a perturbation analysis of SPH sea-ice dynamic equations, that the classical SPH particle density formulation expressed as a function of sea-ice concentration and mean ice thickness leads to incorrect plastic wave speed. We propose a new formulation for particle density that gives a plastic wave speed in line with theory. In all cases, the plastic wave in the SPH framework is dispersive and depends on the smoothing length (i.e., the spatial resolution) and on the SPH kernel employed in contrast to its finite-difference method (FDM) implementation counterpart. The steady-state solution for the simple 1D ridging experiment is in agreement with the analytical solution within an error of 1 %. SPH is also able to simulate a stable upstream ice arch in an idealized domain representing the Nares Strait in a low-wind regime (5.3

Sea ice is an important component of the Earth’s system to consider for accurate climate projection. Generally, numerical models used for geophysical sea-ice simulations and projections are based on a system of differential equations assuming a continuum. The equations that predict the sea-ice dynamics are a combination of the momentum equations, which describe the drift of sea ice under external and internal forces, and the continuity equations which ensure mass conservation. The external forces (per unit area) generally include surface air stress, water drag, sea surface tilt and the Coriolis effect, and the internal forces are related to the ice stress term. This internal stress term is based on various constitutive relations which can differ between models. The more commonly used constitutive laws are the standard viscous–plastic model

Even though the VP (and EVP) rheologies are commonly used to describe sea-ice dynamics and are able to capture important large-scale deformation features

In recent decades, the spatial resolution of sea-ice models has become comparable to the characteristic length of the ice floes. This makes the continuum assumption of current FDM, FVM and FEM models questionable. Also, Eulerian models are known to have difficulties determining the precise locations of inhomogeneity, free surfaces, deformable boundaries and moving interfaces

Despite the shortcomings of the continuum approaches, FDM, FVM and FEM are still the most commonly used frameworks in the community because they have been developed and tested for a longer period and are well understood, computationally more efficient and easily coupled for large-scale simulations. In an attempt to take advantage of both continuum and discrete formulations, blends between the two approaches have been proposed – e.g., the finite- and discrete-element methods

SPH has been applied successfully for modelling other granular materials such as sand, gravels and soils

In this work, we use the standard VP sea-ice model with an elliptical yield curve and normal flow rule

The paper is organized as follows. In Sect. 2, a description of the sea-ice VP rheology, momentum and continuity equation implementations in the SPH framework is presented. Results of a plastic wave propagation analysis, ridging experiments and ice-arching simulations are presented in Sect. 3. Finally, Sect. 4 discusses the SPH advantages and limitations of the SPH framework, future model development, and the main conclusions from the work.

Following

The constitutive relations for the viscous–plastic ice model with an elliptical yield curve, a normal flow rule and tensile strength can be written as

To solve the system of equations in the SPH framework, equations involving spatial derivatives (Eqs.

Following

Sea-ice SPH.

Domain shape and boundaries, Spatial resolution, Total integration time

initialize particle and boundary according to input

monitor particle interaction statistics

output

Following

After the neighbour search, the interactions between pairs of particles are computed using the Wendland

The smoothing or correlation length is a key element of SPH and has a direct influence on the accuracy of the solution and the efficiency of the computation. For instance, if

The initial mass of a particle is defined from the ice area it represents within its support domain (

Graphical representation of the initial position of the particles and the relevant parameter for the smoothing length evolution: the ice area carried by the particle

The smoothing length

We implemented the boundary treatment of

Physical parameters used in ridging and arch simulations.

Values of the parameters used for the simulations are the same as the ones presented in

We first compare the plastic wave speed for the VP dynamic equations with and without the SPH approximations. To this end, we do a perturbation analysis for a 1D case with a fixed sea-ice concentration (

For the more general case when the base state allows for a variable concentration (linearized around a mean state

SPH plastic wave speed as a function of the normalized wavelength (

Idealized domain of the ridging experiment. The blue circles represent the ice particles, and the black ones are the boundary particles. The grey arrow shows the wind forcing. More particles than shown in this schematic were used during the simulation.

Temporal evolution of simulated sea-ice thickness along the central horizontal line of the domain for

We validate our implementation of the SPH model (with the new definition of particle density

Results show that the simulated thickness field converges to the analytical solution (within an error of

Evolution in time of

We also repeated the ridge experiment with the same forcing and total sea-ice volume but letting the sea-ice concentration evolve with time. Specifically, the initial average thickness and concentration were set to

In the ridge building phase, the speed of advance of the ridge front increases until a maximum concentration is reached after

We next compare the SPH approach with the FDM and DEM sea-ice models in a second well-studied idealized experiment: the ice arch formation. To this end, we run the SPH model in an idealized domain representing the Nares Strait (see Fig.

The set of simulations uses a domain with

We suspect that the SPH and DEM frameworks have a similar behaviour in certain circumstances even though they have different (implicit) rheologies because of their Lagrangian nature. Indeed, the interpretation of the numerical representation of a particle in SPH as a collection of ice floes is also present in the DEM

Idealized domain of the ice arch experiments. The blue circles represent the ice particles, and the black ones are the boundary particles. The grey arrow shows the wind forcing.

Ice concentration, thickness and total velocity (

Strain rate and stress invariants (

Second, we explored the ability of the model to produce stable ice arches. To this end, we reduce the total integrated surface stress at the entry of the channel to 13.146

Thickness field at time

In this paper, we have presented a first implementation of the viscous–plastic rheology with an elliptical yield curve and normal flow rule in the framework of SPH with the long-term goal of simulating synoptic-scale sea-ice dynamics. We have described the basics of the SPH approach and how the sea-ice dynamic equations can be formulated in this framework along with the implementation of key components of the numerical method such as the smoothing length, the kernel, the boundaries and the time integration technique. We proposed a different definition of the particle density and showed that the more commonly used density definition involving the ice concentration

From the simple ridging experiment with fixed sea-ice concentration (

When compared to other numerical frameworks, the SPH model is able to reproduce stable ice arches in an idealized domain of a strait with an ellipse aspect ratio of 2 and a wind forcing of 5.3

Even though we successfully implemented the standard sea-ice viscous–plastic rheology with an elliptical yield curve and a normal flow rule in an SPH framework, the current model does not outperform a classical FDM model. In fact, there are inherent difficulties and instabilities in SPH that do not exist in FDM. It is known that the SPH framework trades consistency – i.e., the ability to correctly represent a differential equation in the limit of an infinite number of points with a null spacing between them – for stability, which gives the SPH a distinct feature of working well for many complicated problems with good efficiency but less accuracy. However, the classical formulation of SPH used and described in the present work does not usually respect zeroth-order consistency because of the unstructured particle position in space

In its current state, the model reproduces very similar behaviour to other FDM continuum models and does not constitute a large improvement. Nevertheless, we believe that SPH enables the possibility to describe sea ice as a continuum at large scale using what is already known from continuum models and to explore some new avenues at small scales, where the continuity approximation is questionable. Indeed, SPH also has interesting properties that could be exploited. For example, SPH can be used with little change for problems involving several fluids, whether liquid, gas or dust fluids

For future work and before exploring new features enabled by the SPH numerical framework, a more physical treatment of the boundary conditions should be investigated to properly simulate the grounding of sea ice near the coast enabling the no-slip conditions. Subsequently, the model could be tested against other benchmark problems in an idealized domain to further understand and compare the effect of the SPH method

The SPH method is at the interface between the finite-element and discrete-element methods. In this framework any function

Graphical representation of the SPH kernel

From the above approximations, we reformulate differential operators relevant to our study in their discrete SPH forms. We write the divergence of a vector field (

Vector operators take different forms in the SPH framework because they only operate on the smoothing kernel

First, the divergence of vector needs to be changed into a form that can be symmetrized. To do so, we use the identity of the divergence of a scalar function times a vector and chose the scalar function to be the density as follow:

Note that in the following demonstration, the Einstein summation convention is used to simplify the calculation and the tensor representation. We start with the divergence of a 2D tensor divided by the density:

To demonstrate Eq. (

Finally using the particle approximation (

Our FORTRAN SPH sea-ice model code is public and can be found at

Output data from the SPH sea-ice model simulations along with a version of the model used and the analyzing programs are available at

OM coded the model, ran all the simulations, analyzed results and led the writing of the manuscript. BT participated in weekly discussions during the course of the work and edited the manuscript. JFL and MI participated in monthly discussions during the course of the work and edited the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

Oreste Marquis is grateful for the support from McGill University, Québec-Océan and Arctrain Canada.

This project was partially funded by grants and contributions from the Natural Science and Engineering Research Council Discovery Program (ONR-N00014-11-1-0977) and the National Research Council (A-0038111) awarded to Bruno Tremblay.

This paper was edited by Jari Haapala and reviewed by three anonymous referees.