Following , we consider sea ice to behave as a 2D granular material described by the 2D momentum equation (neglecting the Coriolis and sea surface tilt terms): 1ρihDuDt=∇⋅σ+τ, where ρi is the ice density, h is the mean ice thickness (ice volume over an area), u=ux^+vy^ is the ice velocity vector, σ is the vertically integrated internal stress tensor acting in the y^ direction on a face with a unit-outward-normal pointing in the x^ direction, τ is the sum of water stress and surface air stress, and DDt=∂∂t+u⋅∇ is the Lagrangian derivative operator. The Coriolis and sea surface tilt terms are neglected from the momentum equation to ease the comparison with the analytical solution and simple 1D problem. Note that using the Lagrangian derivative operator naturally incorporates the advection of momentum in the ice dynamics – a term that is typically neglected for most continuum-based Eulerian sea-ice models. The surface air stress and the water stress can be written using the bulk formulation as 2τ=ρaCa|ua-u|(ua-u)+ρwCw|uw-u|(uw-u),3≈ρaCa|ua|(ua)+ρwCw|uw-u|(uw-u), where ρa and ρw are air and water densities, ua and uw are air and water velocity vectors, Ca and Cw are the air and water drag coefficients, and u is neglected in the formulation of the wind stress since u≪ua. The continuity equations for the mean ice thickness h and the ice concentration A can be written as 4DhDt+h∇⋅u=0,5DADt+A∇⋅u=0, where the thermodynamic source terms are omitted. Note that the thickness and concentration are independent prognostic variables in a two-category model , resulting in a singularity when thickness reaches zero. To avoid this singularity and for a more mathematically correct treatment of the mass equation, introduced a continuous solution where the concentration asymptotes to zero and one. In the following, we ignore melting and consider cases where only convergent motion is present and the use of a two-category model does not have an impact on the simulated results.

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 6σij=2ηϵ˙ij+[(ζ-η)ϵ˙kk-Pr(1-kt)2]δij,7ϵ˙ij=12(∂uj∂xi+∂ui∂xj)=12(∇u+∇u⊺), where ϵ˙ij is the symmetric part of the strain-rate tensor, ζ and η are the non-linear bulk and shear viscosities, Pr is the replacement pressure, kt is the tensile strength factor, and δij is the Kronecker delta. Following we write 8ζ=P(1+kt)2Δ*,9η=ζe2=ζ(2SP(1+kt))2,10Δ*=max(Δ,Δmin),11Δ=[(ϵ˙112+ϵ˙222)(1+e-2)+4e-2ϵ˙122+2ϵ˙11ϵ˙22(1-e-2)]1/2, where P=P*hexp(-C(1-A)) is the ice strength , P* and C are respectively the ice compressive strength and ice concentration parameters, S is the ice shear strength, and e is the ellipse aspect ratio. In the limit where Δ goes to zero, ζ and η tend to infinity. To avoid this situation, the deformation Δ is capped to Δmin=2×10-9s-1. Using the Δ* formulation, the replacement pressure Pr can be written as 12Pr=PΔΔ*, which ensures that the stresses are zero when the strain rates are zero.

To solve the system of equations in the SPH framework, equations involving spatial derivatives (Eqs. , , and ) are reformulated (see Appendix  for more details on the SPH theory) using Eqs. (––) with the particle subscripts p and q (see Fig. ), and a temporal evolution for the ice particle position is defined: 13DxpDt=up,momentum14ρihpDupDt=ρp∑q=1Nmq(σqρq2+σpρp2)⋅∇pWpq+τp,momentum15DhpDt+hpρp∑q=1Nmq(uq-up)⋅∇pWpq=0,continuity16DApDt+Apρp∑q=1Nmq(uq-up)⋅∇pWpq=0,continuity17ϵ˙p=12[(∑q=1Nmqρq(uq-up)⊗∇pWpq)+(∑q=1Nmqρq(uq-up)⊗∇pWpq)⊺].constitutive It is important to make the distinction between the intrinsic ice density ρi and the particle densities ρp. For consistency reasons with the standard VP rheology, we consider the following definition of density independent of ice concentration in contrast to previous work (see Results section for discussion): 18ρp=ρihp. By formulating density as in Eq. (), the continuity Eq. () has the same form as the more commonly used continuity density equation : 19DρpDt=-ρp∇⋅up=∑q=1Nmq(up-uq)⋅∇pWpq, except for the fact that the divergence of the velocity field is scaled by the ice material density ρi DρpDt=ρiDhpDt. Overall, a particle can be seen as an unresolved collection of floes scattered within the support domain A that can converge, ridge over one another, break and drift apart. Note that since the particle density ρp definition is independent of Ap, the concentration can be interpreted as a quantity that measures the compactness of the sea ice at the particle location. It describes the probability of ice floes carried by a particle, which is a point in space, to come in “contact” with ice floes of another particle and get repulsed according to the ice strength.

Following , we use a second-order predictor–corrector scheme to evolve in time the SPH ice system of equations (see Algorithm  below). This integration scheme takes a given function f (here f can be x, u, A and h) and uses a predictor step to calculate its value fn+1/2 at time t=(n+12)Δt (where Δt is the time step) followed by a correction step to calculate the solution fn+1 at time t=(n+1)Δt from fn+1/2: 20fpn+1/2=fpn+Δt2DfpnDt+O(Δt2),21fp correctedn+1/2=fpn+Δt2Dfpn+1/2Dt,22fpn+1=2fp correctedn+1/2-fpn+O(Δt3). In the above equations, O(Δt2) and O(Δt3) represent higher-order terms, which are ignored in the proposed scheme. Following , a simple 1D model taking into account only the viscous term – the most restrictive condition – leads to the following stability criterion: 23Δt≤ρihlmin2ηmax=e2ρilmin2ΔminP*(1+kt), where lmin is the minimum smoothing length across all the particles. The stability criterion imposes a strict limitation on the time step (∼10-4 to 10-2 s for particles with a radius of 1 to 10 km); this cannot be avoided using a pseudo-time step because particles in an SPH framework are irregularly placed and move within the domain at each time step. This makes the parallelization of the particle interactions algorithm mandatory for any practical applications. On the positive side, the explicit time stepping also eliminates possible convergence issues of the numerical solver. A pseudo-code for the proposed algorithm is shown below (Algorithm ).

Following , we use the bucket search algorithm parallelized using shared memory multiprocessing (OpenMP) to find all the neighbours of each particle in favour of the tested tree algorithm , which involve pointers and complex memory structure that are not easy to manipulate in OpenMP. The proposed OpenMP parallelization is rudimentary, and one time step in a domain with 40 000 particles takes ≈0.1 s. For this reason, the model requires more computational resources for the effective resolution when compared with a continuum approach. This could be greatly improved by taking advantage of CPU clusters or GPUs .

After the neighbour search, the interactions between pairs of particles are computed using the Wendland C6 kernel – Wendland kernels have the best stability properties for wavelengths smaller than the smoothing kernel – which is written as 24W(|rp-rq|,lp)=WC6(R)=αd(1-R)8(32R3+25R2+8R+1),0≤R<1,0,R≥1,25∂W(|rp-rq|,lp)∂|rq-rp|=∂WC6(R)∂|rq-rp|=αd-22R(16R2+7R+1)(1-R)7κlp,0≤R<1,0,R≥1, where αd is a normalization factor depending on the dimension of the problem. Note that R (=κ|rp-rq|/lp) is the normalized distance between particles in the referential rp-rq. Consequently, we always integrate from 0 to lp (the smoothing length) independently of the kernel instead of 0 to κlp as shown by . The constant αd becomes 78κ27πl2 in 2D, with a factor of κ2 difference from the usual definition. Note that the scaling factor κ has a value of 1 for the Wendland C6 kernel. The choice of kernel was validated using stability tests with six different kernels including the original Gaussian kernel ; a quartic-spline Gaussian approximation ; a quintic-spline Gaussian approximation ; a quadratic kernel ; and the Wendland C2, C4 and C6 kernels .

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 lp is too small, there may not be enough particles in the support domain violating the kernel moment requirements. If the smoothing length lp is too large, all the local properties of particles would be smoothed out over a large number of neighbours and the computation time would increase with the number of interactions. In 2D, the optimal number of neighbours interacting with any particle p should be about 20 to balance the precision and the computational cost . We therefore implement a variable smoothing length that evolves in time and space to maintain this approximate number of neighbours. To this end, we keep the mass of particles constant in time and evaluate the smoothing length from the particle density. Note that keeping the mass of a particle constant has the advantage of ensuring mass conservation. This assumption is justified in our case since we are only interested in sea-ice dynamics, and ridging changes the area covered by ice floes but not their mass. However, fixing the ice mass is only valid when neglecting the thermodynamics and needs to be modified for synoptic-scale simulations.

The initial mass of a particle is defined from the ice area it represents within its support domain (ΔAp in Fig. ). To avoid creating porosity in the medium, we divide the space in equal square area (=Lp2) that covers the whole domain. Since we want approximately 20 neighbours for every particle, we introduce α (=3 in all simulations), a parameter that stands for the approximate number of particles desired in any direction within the support domain. The parameter α can also be interpreted as the proportionality constant between the particle spacing Lp and the smoothing length lp. Note that to increase the accuracy of the particle approximation, α can be increased by any desired factor (see Fig. ). The mass carried by a particle is therefore written as 26mp=ΔApρih0p=Lp2ρih0p, where h0p is the initial mean thickness of the particle. The smoothing length is then updated at each time step diagnostically from 27lp=αLp=αmpρp.

The smoothing length lp is capped to 10 times its initial value when the particle density tends to zero. This capping prevents conservation of mass for a density lower than 1 % of its initial value (see Eq. ). We justify this capping because such small densities do not affect the ice dynamics.

We implemented the boundary treatment of because of its simplicity, versatility and low computational cost. The boundaries are set up by placing stationary particles with fixed smoothing length lb and a mass mb equal to the average ice particle mass mp. The boundary smoothing length lb is chosen in a way that only one layer of ice particles initially interact with the boundary (this makes lb resolution-dependent). The boundary particles are (equally) spaced apart by a factor of 1/4 of their smoothing length (lb/4). In this manner, all ice particles p within a support domain lb will interact with approximately four boundary particles (denoted by the subscript b) at a time, resulting in a net normal repulsive force FNp, 28FNp=∑b=1Nbμ(rp-rb)|rp-rb|2Wpb2mbmp+mb, which is added to their momentum equation. In Eq. (), μ is a constant with units of kgm4s-2 used to adjust the repulsion strength and is also simulation-dependent because it needs to counterbalance the particle acceleration and prevent them from escaping the domain. This free parameter is not suited for complex pan-arctic simulations but is sufficient in our idealized experiments. For all the simulations, a free-slip boundary condition, i.e., no tangential friction force between boundary particle and ice particle, is applied.

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 (A=1). In this case, the 1D SPH sea-ice dynamic equations (Eqs. –) form a system of three equations and three unknowns (x, u and h): 29DxpDt=up,30DupDt=Γ∑q=1Nmqρi2(1hq+1hp)xpq|xpq|∂W∂xpq+τp,31DhpDt=-1ρi∑q=1Nmq(uq-up)xpq|xpq|∂W∂xpq, where xpq is a short form for xp-xq and 32Γ=P*2[±(e-2+1)1/2-1]. In the above, we made use of the following 1D normal stress for convergent plastic motion seefor 1D normal stress derivation: 33σ=σxx=P*2[±(e-2+1)1/2-1]h=Γh. Linearizing around a mean state (u‾=0 and h‾=h0), considering small perturbations (δx, δu and δh) and ignoring the second-order term, we obtain 34DδxpDt=δup,35DδupDt=Γρi∑q=1NΔAqx‾pq|x‾pq|(-1h0(δhq+δhp)∂W∂x‾pq+2(δxp-δxq)∂2W∂x‾pq2),36DδhpDt=-h0∑q=1NΔAqx‾pq|x‾pq|(δuq-δup)∂W∂x‾pq, where ΔAq=mqρih0=mqρq (Eq. ) and where we have used the binomial expansion 1h=1h0+δh≈1h0(1-δhh0). Following , we do a perturbation analysis on the system of Eqs. ()–() and assume a wave solution of the form δf=f^exp(i(kx‾-ωt)), where i is the imaginary number; k is the wavenumber; ω is the angular velocity; and f is a dummy variable standing for u, x and h. Substituting δf in Eqs. ()–(), the resulting set of equations in the reference frame following the ice motion reduces to 37x^=iωu^,38u^=iΓωρi∑q=1NAqx‾pq|x‾pq|([-h^h0(1+exp(-ikx‾pq))]∂W∂x‾pq+2x^(1-exp(-ikx‾pq))∂2Wpq∂x‾pq2),39h^=-ih0u^ω∑q=1NAqx‾pq|x‾pq|(exp(-ikx‾pq)-1). Note that since the ice is initially at rest, the Lagrangian and the Eulerian frameworks are equivalent. For a large enough wavelength (so that the perturbation can be resolved across multiple particles with high accuracy, i.e., λ≥lp and N→∞), the summations can be approximated by integrals over the space; i.e., ∑q=1NAqx‾pq|x‾pq| becomes ∫-∞∞dx‾pq. Taking advantage of the kernel properties – i.e., all moments higher than 0 vanish – we can write Eqs. ()–() as 40u^=-iΓωρi∫-∞∞(h^h0∂W∂x‾pq+2x^∂2Wpq∂x‾pq2)exp(-ikx‾pq)dx‾pq=Γωρi(h^h0k+i2k2x^)W̃,41h^=-ih0u^ω∫-∞∞exp(-ikx‾pq)∂W∂x‾pqdx‾pq=h0u^kωW̃, where the integrals have been converted to Fourier transform using F(∂W∂x‾pq)=∫-∞∞(∂W∂x‾pq)exp(-ikx‾pq)dx‾pq=ikF(W)=ikW̃. Finally, Eqs. (), () and () represent a system of three equations for three unknowns (x^, u^, h^) that we solve by substitution. This leads to the following form for the phase speed of the plastic wave (ωk): 42cSPH=ωk=±W̃-Γρi(2W̃-1). For wavelengths much larger than the smoothing length (λ∝1k≫lp), the Fourier transform of the kernel tends to 1 (W̃≈1) and the SPH formulation reduces to the viscous–plastic theory without SPH approximations see for instance, i.e., 43cVP=±-Γρi, with a plastic wave propagation speed cVP≈5.7 ms-1 for typical sea-ice parameters (see Table ). Consequently, a major difference of SPH with the FDM framework is that the plastic wave speed is dispersive with a phase velocity cSPH that is dependent on the wavelength and the smoothing length. In general, only the plastic waves with a wavelength between approximately 1 and 11 times the smoothing length will have their travelling speed modified by more than 1 %. More specifically, in the limit where the wavelength λ approaches the smoothing length lp, the plastic wave speed increases in the SPH framework to a maximum value of ≈6.7 ms-1 (see Fig. a). Note that for wavelengths smaller than the smoothing length, the summations in Eqs. ()–() cannot be written as integrals, but the particles still respond partially to the perturbations. This sometimes leads to the tensile and the zero-energy mode instabilities . As mentioned above, showed that Wendland kernels can diminish the tensile instability and the pairing of particles. A deeper analysis of unresolved waves (λ<lp) in the context of sea-ice SPH dynamic equations is beyond the scope of the current study.

For the more general case when the base state allows for a variable concentration (linearized around a mean state A‾=A0) and considering the classical – denoted by a superscript C – particle density definition (ρpC=ρihpAp) used by , , and , the plastic wave speed becomes 44cA,SPHC=±W̃-Γ*ρi(CA0-3+2W̃), where Γ*=Γexp(-C(1-A0)). We argue that the plastic wave speed cA,SPHC obtained with the classical density definition does not converge (see Fig. b) to the viscous–plastic theory, cA,VP, derived from FDM seefor a derivation, 45cA,VP=±-Γ*ρi(CA0+1), because the ice concentration is taken into account in both the definition of ρpC and implicitly in the definition of the average thickness hp. When we consider the new formulation of particle density independent of concentration as proposed above (Eq. ), the wave speed equation becomes 46cA,SPH=±W̃-Γ*ρi(CA0-1+2W̃), which reduces to the FDM VP theory (Eq. ) when the wavelength is large compared to the smoothing length (see Fig. c). Note that the perturbation analysis presented above is not valid for the classical density definition proposed by , , and since they use a different set of momentum, continuity and constitutive equations to describe sea ice. In a similar manner to the plastic wave speed with a fixed concentration (Eq. ), the wave speed cA,SPH (Eq. ) is dispersive and the wavelengths between 1 and 11 times the smoothing length are those that are mostly affected (more than 1 %). However, in this case, the plastic wave speed is damped for wavelengths close to the smoothing length for a mean concentration state higher than 0.1. Note that while the plastic wave speed is defined for all A, it does not have a physical meaning for A<0.85 since there are negligible ice–ice interactions.

We validate our implementation of the SPH model (with the new definition of particle density ρp) in a 1D ridging experiment for which we can validate against the simulated field from a viscous–plastic sea-ice model based on the FDM – the 1D version of McGill-SIM model used in the SIREx studies – and against the analytical solution seefor a derivation: 47-dσdx=ρaCa|ua|ua⇒dhdx=2ρaCa|ua|uaP*(e-2+1+1), i.e., a linear profile in thickness with a slope proportional to the square of the wind velocity and inversely proportional to the ice strength. We consider a rectangular domain of 1000 by 2000 km including the boundary (the ice field is 1900 km to ensure that no particles escape on the open side) with 37 240 particles; an initial homogeneous smoothing length lp of 21.429 km (spacing lp/α=7.14 km); and a smaller – to limit boundary effect – boundary particle smoothing length lb of 4 km (spacing lb/4=1.0 km) to represent the wall (see Fig. ). Particles are initialized with an average thickness h=1 m and a concentration A=1. They are forced against the wall by a constant unidirectional wind of 5 ms-1. Note that the water stress is removed in the simulation for a faster convergence to the steady state, which enables higher resolution – a water current of 0 ms-1 would slow down the ice and the ridge formation since it is driven by the advection speed. The Coriolis force should normally also have to be considered with this domain size at classical polar latitude – the Rossby number is O(10-2) – but is neglected in this idealized experiment to conserve the symmetry of the solution and compare it to the theoretical 1D equation (Eq. ). In results presented below (Figs. –), the particle properties are averaged over a grid of approximately 10 by 5 km cells for plotting purposes.

Results show that the simulated thickness field converges to the analytical solution (within an error of ≈1%) after ≈5 d with a slope of 1.33×10-3 mkm-1, compared to 1.34×10-3 mkm-1 for the theory – lower-resolution simulations were run for a longer time and also converged to this stable state (results not shown). This is comparable to the precision obtained by the 1D McGill-SIM FDM model which reaches a slope of 1.35×10-3 mkm-1. Artifacts are observed close to the boundary where the repulsive force prevents the particles from reaching the “wall”. Additionally, when a particle comes into contact with the boundary with a certain inertia (due to the 1/r dependence of the boundary force), we observe oscillations in the motion of particles which can propagate far in the domain (e.g., Fig. a, at x≈[50,300] km and t=[30,45] h). The oscillations are damped, and the energy is dissipated by the rheology term with time until an equilibrium is reached. Note that reintroducing the water drag diminishes the oscillation coming from the boundary but does not remove them completely. A more physical boundary treatment is beyond the scope of this study.

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 h=0.5 m and A=0.5. This ensures that both h and A covary in time such that hA remains constant – note that A and h follow the same continuity Eqs. () and () or () and () when omitting the SPH approximations, and therefore they should vary identically in time until A reaches 1 – in the marginal ice zone (MIZ), which we define as the area where the sea-ice concentration ranges between 0.15 and 0.85 and where low ridging by ice collision occurs (see Fig. b). To accomplish this, the domain was extended to 4000 km (3800 km, excluding the boundaries) and the initial particles spacing changed from 7.14 km to 10.0 km for a corresponding initial smoothing length lp of 30.0 km and total number of particles of 38 000. In this configuration, the model converges to a steady-state solution in ≈10 d with a slope of 1.36×10-3 mkm-1, in agreement with theory within an error of ≈1% (see Fig. b). Results at x=300 km, away from boundary effects, show that (as desired) thickness and concentration evolve coherently – d(h/A)dt≈0 – before ice concentration reaches ≈85% (see Fig. a). At that point (t≈22 h), ice–ice interactions emerge and the ridging process starts d(h/A)dt>0. One key difference with the simulation initialized at A=1 is a thickness build-up (above 1 m) at the edge of the ridge in the MIZ. At this location, the continuity equation for sea-ice concentration is capped, while that of the mean ice thickness remains continuous. This results in a local increase in ice thickness to ≈1.1 m. This process is akin to the wave radiation drag in the MIZ . A detailed analysis of simulations in simple convergent ice flow in the MIZ with ice concentration close to 100 % will be considered in future work.

In the ridge building phase, the speed of advance of the ridge front increases until a maximum concentration is reached after ≈70 h (see Fig. c). Subsequently, the ice drift speed reduces, and the rate of advance of the ridge slows down. When the ice thickness gradient is in balance with the surface wind stress (after ≈200 h), d(h/A)dt reaches a steady state. Overall, we can observe three stages in the ridge formation (see Fig. ): first, a rapid compaction stage when ice particles are drifting close to their free-drift speed since the ice strength is weak; second, a transition stage between A≈0.85 and 1.00 when ridging occurs in the MIZ analogous to the wave radiation drag mentioned above; and third, a ridging stage with changes in ice thickness that are about 1 order of magnitude higher than during the transition stage. Note that the amplitude of oscillations between particles within the domain or at the boundaries in ridging experiments diminishes when incorporating the water drag (a damping term). The water drag also increases the time needed to reach steady state because the ice drift speed is slower.

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. ) with an upstream reservoir 5 times the length of the channel (L) to minimize the boundary confinement effect without sacrificing the spatial resolution.

The set of simulations uses a domain with L=60 km. The initial conditions for ice thickness, concentration and velocity are h=1 m, A=1 and u=0 ms-1. The ice is forced with a constant unidirectional wind of -7.5 ms-1 in the y^ direction, and ocean current is fixed to uw=0 ms-1. The corresponding surface stress is ≈0.04 kNm-2, and the total integrated stress at the entry of the channel is slightly smaller than P* (∫05Lτadx=26.325 kNm-1). We use a weaker wind than commonly used in Nares Strait ice arch simulations (≈10 ms-1) to limit the ridging phase prior to the formation of the ice arch. In this experiment, we limit ourselves to ice with no tensile strength (kt=0) and a shear strength of 6.875 kNm-2, i.e., an ellipse aspect ratio of 2.

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 , and the two numerical frameworks compute their quantities with one-to-one interactions. Consequently, we first test whether the SPH approach has the same sensitivity to the relative size of particles with respect to the channel width as in the DEM . Results showed that no stable arch can be formed with the specified forcing for all particle diameter sizes tested (7.5, 5, 3.75 km) (see ice velocity field in Fig. ). Instead, a “continuous” slow flow of ice is present in the channel. The discontinuity at the entry of the channel is visible in the concentration, thickness and velocity fields (Fig. ) and can be interpreted as an intermittent (unstable) ice arch formation. Also, we noted that larger particles are not more prone to ice jam than smaller ones. This is contrary to what is known from granular material theory and to results from that show a transition from stable to no ice arch formation for floe sizes ranging from approximately 1/4 to 1/16 of the strait's width. We explain this difference between SPH and DEM from the continuum description of the ice dynamics equation. In the present model, the constitutive laws prescribe the repulsion of the particles from one another according to the ice strength, which is a function dependent on the ice concentration and mean thickness, not on the particle size. We conclude that to enforce granularity within the SPH framework, the constitutive laws would need to be adapted to account for contact force and particle size which could then reproduce a similar behaviour to that observed in the DEM. However, even though the increase in resolution – or particle size – has no effect on the arch stability, it enables smaller fracture resolutions that are visible at the entrance of the channel (see ϵI and ϵII in Fig. ). In our SPH model, the stress invariants σI and σII shows oscillation patterns in regions where the ice is in the viscous regime (see the tree-like structure in the normal and shear stress fields in Fig. ). From our experiments, the “tree-like” peak stresses appear during the transient phase and at steady state. However, the particles never stop moving even in a steady state because viscous deformations are always present. We hypothesized that stress patterns are associated with over-damped elastic waves associated with small movements (but large internal stresses) of the particle in the viscous regime. Those structures are not symmetric, despite symmetrical initial conditions, because of the domino effect between interacting viscous waves. Note that they are absent from the strain-rate fields since viscous deformations are extremely small. They are also absent in sea-ice model based on a continuum approach , but these tree-like structures are qualitatively similar to the stress structure between floes observed in the DEM e.g.,Fig. 5c. Despite the fact that the model solves the same continuum equations as other FDM models, we believe that stress networks can be observed with the SPH method because particles interact in a pairwise fashion according to their relative distance. This can create less-dense ice areas within the medium which can lead to oscillations in the stress field. It is known that SPH can have spurious behaviour in some cases when the stress is solved at the same location as the particle centre (as done here). This can be avoided using stress particles seefor details. More investigations are required to test whether this behaviour is physical. This is left for future work.

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 kNm-1 (or wind speed of 5.3 ms-1) to a value below the ice compressive strength (P*) to avoid completely ridging the north of the channel and jump immediately to the arch-forming stage. In this case, the results show a clear stable arch (see Fig. ) with a shape that is qualitatively similar to the one presented by , , and . The formation of a stable arch in an SPH model is possible with the standard shear strength (e=2), in contrast to continuum models that require an increase in shear strength e.g., – it is important to keep in mind that the domain configurations were different in each of those studies. This suggests that SPH has a different sensitivity of ice arching to the ellipse aspect ratio e and ice thickness h. With a no-slip boundary condition and the same default yield curve (same P* and ellipse aspect ratio e), preliminary results suggest that no arches form – the pack is undeformed – and instead a higher surface wind stress is required to form an arch. Note that in the SPH simulations, only one arch forms close to the outlet. Presumably, the number of arches would increase and location would change if the model was run at higher resolution, with different boundary conditions or in a non-idealized domain geometry. Overall, this shows that SPH is able to capture large-scale features coming from small-scale interactions. The simulation of a stable ice arch (Fig. ) also shows how SPH can fracture and create discontinuities in the ice field as seen in DEM models. This behaviour is similar to that of the elastic-decohesive sea-ice constitutive model of or the FEM model of . Finally, in the SPH framework, a lead or polynya can be defined by an absence of particles for leads larger than particle size – akin to DEM – or by particles with reduced concentration for sub-particle size leads – akin to FDM.