NeXtSIM uses INI-style config files, which are read in using the |boost| |program_options| library. The neXtSIM config file is sorted into 18 sections: |simul|, |debugging|, |numerics|, |setup|, |mesh|, |moorings|, |drifters|, |restart|, |output|, |ideal_simul|, |dynamics|, |thermo|, |nesting|, |forecast|, |coupler|, |wave_coupling|, |statevector|, and |age|. Importantly, |simul| controls basic parameters of each simulation, such as start date and time step. The |setup| block sets the initial and boundary conditions, and |dynamics| and |thermo| control parameters and options for the model's dynamics and thermodynamics, respectively.

Initial conditions are set depending on whether the user requests restarting from a previous model run or starting from a given set of initial conditions. The restart files are a set of custom binary files that describe both the model mesh and all required fields. Restart files are not guaranteed to work correctly if moved from one computer to another, although in practice, they will as long as the system's CPUs use the same endianness.

NeXtSIM supports a substantial number of different initial conditions when not starting from a restart file, through the |setup.ice-type| option. This includes reading in sea-ice thickness and concentration from the TOPAZ4, GLORYS, and PIOMAS reanalysis, as well as fields from AMSR2, AMSR2, IceSat, and CryoSat2 in various combinations.

The main program loop consists of checking and updating the computational mesh, running the thermodynamics and dynamics routines, reading forcing fields, and outputting diagnostic and restart outputs. Mesh handling, thermodynamics, and dynamics are discussed in more detail in Sects. , , and . Here, it is important to note that remeshing impacts both reading of forcing files and writing to regularly spaced netCDF files.

While neXtSIM uses a Lagrangian moving mesh, the model assumes that the mesh remains static between remeshings for most input and output operations. In practice, this means that the functions that interpolate the input and output data between the mesh and the source or destination grids use interpolation weights that are recalculated only when the mesh is updated (see Sect. ). This also holds for routines interfacing with the OASIS coupler. In addition to gridded outputs (Sect. ), neXtSIM also provides outputs which are snapshots of the computational grid and preserve the instantaneous positions of the grid nodes. These files are written in the same customised format as the restart files. Finally, restart files are written at the end of the simulation or at given intervals, as requested by the user.

The finite element discretisation is based on a triangular grid, with ice velocities defined at the vertices and tracers treated as constant within the elements consider e.g.for details beyond what is given below. To solve the momentum equation in this framework, we define a set of test functions to obtain the weak formulation of the equation. Using a continuous Galerkin discretisation, we select the test functions, Nje, as linear functions associated with vertex j of element e of the triangular mesh. This function equals one at the vertex j and zero at all other vertices. The ice velocity field in element e is thus 2u=∑e(j)ujNje, where the sum is over all the vertices of e.

The gradient of the velocity field is related to the gradient of the test functions, referred to here as the shape coefficients. These are calculated by defining the test functions on the three nodes of a reference element in the coordinates {ξ,υ}, as 3N1(ξ,υ)=1-ξ-υ4N2(ξ,υ)=ξ5N3(ξ,υ)=υ, and then projecting this onto e using the transformation 6x(ξ,υ)=N1ex1e+N2ex2e+N3ex3ey(ξ,υ)=N1ey1e+N2ey2e+N3ey3e, where the coordinates xi and yi are those of the three vertices of e. The shape coefficients are then computed as 7∂Nje∂x∂Nje∂y=1det(J)y2-y3y3-y1y1-y2x2-x3x3-x1x1-x3, where det(J) is the determinant of the Jacobian of the transformation Eq. (), computed as 8det(J)=x2y3+x3y1+x1y2-x2y1-x3y2-x1y3, and is equal to twice the element area. The coordinates xi and yi are taken to start at j and go around e in a counter-clockwise fashion.

The strain-rate equations can now be calculated using the shape coefficients. The strain-rate equations are 9ε˙11=∂u∂x10ε˙22=∂v∂y11ε˙12=12∂u∂y+∂v∂x, where ε˙ij are the components of the strain-rate tensor ε˙, the dot symbolising the time derivative, and u and v are the zonal and meridional components of u. The strain-rate equations are projected onto the test functions to give element-wise strain rates: 12ε˙11e=∑j(e)uj∂Nje∂x13ε˙22e=∑j(e)vj∂Nje∂y14ε˙12e=∑j(e)uj∂Nje∂y+vj∂Nje∂x. In neXtSIM, we solve the equations on an {x,y} plane (usually using a polar stereographic projection), so we do not need to account for metric terms that arise due to the curvature of the earth.

To solve the momentum equation, we follow and use nodal quadrature in all terms that do not involve spatial derivatives. This allows us to solve the BBM, EVP, and mEVP models efficiently using explicit time stepping. Multiplying the momentum equation by the test functions, integrating over the domain, and integrating the rheology term by parts gives the following at vertex j: 15∫mNj∂u∂tdS=∫NjhnσdΓ-∫∇NjhσdS+∫NjAτa+τw+τb+fk×udS-∫mg∇HoNjdS, where S is the domain, Γ the boundary of the domain and n the normal vector to that boundary.

The numerical integration of Eq. () proceeds as follows: The left-hand side can be written as 16∫mNj∂u∂tdS=∑kMjkmk∂uk∂t, where the summation is taken over all vertices, k. The lump mass matrix, M is diagonal and its non-zero entries are 17Mii=13∑e(i)Ae, where the sum is taken over all elements containing the vertex i. The nodal mass mj is calculated as the area-weighted mean of all the neighbouring element thickness values. The left-hand side then becomes simply 18∫mNj∂u∂tdS=mj∂uj∂tMjj. Similarly, the third term on the right-hand side becomes 19∫NjAτa+τw+τb+fk×ud.=AτaS+τwj+τbj+fk×ujMjj. The first term on the right-hand side is the boundary contribution, which we set to zero under the assumption of a no-slip boundary condition.

The second and fourth terms on the right-hand side – the internal stress term and sea-surface tilt term, respectively – are more involved than the terms already discussed and require summation over the elements e(j) surrounding j. The x-component of the internal stress term becomes 20-∫∇NjhσdSx=-13∑e(j)Aeheσ11e∂Nj∂x+σ12e∂Nj∂y and the y-component becomes 21-∫∇NjhσdSy=-13∑e(j)Aeheσ12e∂Nj∂x+σ11e∂Nj∂y, with σij the components of the stress tensor σ. The sea-surface tilt term requires a double summation, where the x-component becomes 22-∫mg∇HoNjdSx=gmj13∑e(j)Ae∑k(e)Ho,k∂Nk∂x and the y-component 23-∫mg∇HoNjdSy=gmj13∑e(j)Ae∑k(e)Ho,k∂Nk∂y.

The time-splitting method we use in neXtSIM is based on that proposed by for the EVP model, where they solve the column physics and advection equations on the main model time step Δt, but apply a sub-time stepping to solve the momentum equation at a shorter time step, Δtm. In neXtSIM, we use this time-stepping scheme for the BBM as in and the EVP model, whereas the mEVP requires a small modification to the time-stepping and internal stress calculation (see Sect. ).

The time stepping of the momentum equation follows the semi-implicit solution derived by . Following the discretisation discussion above, this then takes the form 24α2+β2ujn+1=αuj+βvj+ΔtmmjMjjα-∫∇NjhjσdSx+τx+β-∫∇NjhjσdSy+τy and (correcting for a sign error in on the last β term) 25α2+β2ujn+1=αuj-βvj+ΔtmmjMjjα-∫∇NjhjσdSy+τy-β-∫∇NjhjσdSx+τx where all values on the right-hand side are taken at time n to produce u and v at time n+1 on the left-hand side. The integrals are calculated following Eqs. () and (). Additionally we have 26c′=ρwCw|uw-u|27α=1+Δtmmjc′cos⁡θw+Cb28β=Δtmf+c′sin⁡θwmj and 29τx=τa,x+c′uwcos⁡θw-vwsin⁡θw-∫mg∇HoNjdSx30τy=τa,y+c′vwcos⁡θw-uwsin⁡θw-∫mg∇HoNjdSy, where the integrals are calculated following Eqs. () to (). Here, ρw is the ocean water density, Cw the ice-ocean drag coefficient, uw the ocean velocity (and uw and vw its components), θw the ocean turning angle, and τa,x and τa,y are the x and y components of the ice-ocean stress 31τ=ρaCm|u|u, with ρa the atmospheric density and Cm the atmospheric drag coefficient for momentum exchange e.g..

The Brittle Bingham-Maxwell rheology (BBM) is described in detail in ; here we briefly discuss it for completeness, focusing on aspects relevant to neXtSIM. The BBM rheology is a damaging Bingham-Maxwell constitutive model. Using damage mechanics for sea-ice modelling was introduced with the elasto-brittle (EB) model by , but damage mechanics are widely used in other communities, e.g. in rock and crustal mechanics e.g.).

The EB model simulates the ice as a damaging elastic sheet, where each grid element can be considered a spring in a mechanics sense. In the EB model, the stress is calculated in every grid cell, and if the stress is outside the Mohr–Coulomb yield criterion, the elasticity is reduced in that grid cell, and the sea ice experiences a reversible deformation. The Mohr–Coulomb criterion is 32τ=σNμ+C, where μ is a constant friction coefficient, C, is the material cohesion, and τ and σN are the shear and normal stresses: 33τ=12σ11-σ222+4τ12234σN=12σ11+σ22. Elasticity, E, in the model is thus 35E=Y(1-d), where Y is the ice's Young modulus, and d with 0≤d<1, is a scalar damage variable.

At the start of the simulation, d=0 everywhere, but d is then increased to ensure that all stresses in the ice are always within the yield criterion. A local increase in d represents damaging of the ice, which is modelled as a reduction in elasticity. The reduction in elasticity in an element means that this element deforms more easily than before, and so the distribution of stresses in the neighbouring elements must change. This causes a stress redistribution and a cascade of damage increases, emulating the multiplicative cascade that suggested was the reason for the spatial scaling of sea-ice deformation they observed in satellite remote sensing data.

The MEB model introduced a viscous element, or a dashpot, in series with the elastic one. This is intended to simulate the viscous dissipation of internal stresses and the larger, irreversible deformations (as opposed to the elastic counterpart) that occur along faults once the material is highly damaged. The viscosity of the dashpot is very high when the ice is undamaged, but should decrease faster than the spring's elasticity when damage increases. This is formulated as: 36η=η0(1-d)α, where η is the dashpot's viscosity, α is a material-dependent constant, and η0 is the viscosity of undamaged ice. This should tend towards infinity, but is set to 10⁷ s⁻¹ for practical purposes.

The Bingham–Maxwell model consists of a friction block and a dashpot in parallel, connected with a spring in serial. A schematic of the constitutive model is shown in panel a of Fig. . The spring's elasticity and the dashpot's viscosity evolve as a function of the damage variable, as in the MEB rheology. The friction element is introduced to emulate the resistance to ridging, so that, for small compressive stresses, the ice remains fully elastic regardless of the level of damage. This leads to the model having three regimes in stress space: a visco-elastic converging one for high compressive normal stress, a purely elastic one for compressive normal stress less than the threshold Pmax, and a visco-elastic diverging one for divergent normal stress. These three regimes are shown schematically in of Fig. b.

The constitutive equation for BBM is 37σ˙=EK:ε˙-σλ1+P̃+λd˙1-d, where σ and ε˙ are the stress and strain rate tensors, respectively, λ=η/E=λ0(1-d)α-1 is the viscous relaxation time, with λ0=η0/Y the undamaged viscous relaxation time. We assume plane stress conditions, so the stiffness tensor operation K:ε˙ is 38K:ε˙11K:ε˙22K:ε˙12=11-ν21ν0ν10001-νε˙11ε˙22ε˙12 where ν is Poisson's ratio. The resistance to ridging is encoded in P̃ as 39P̃=PmaxσNforσN<-Pmax-1for-Pmax<σN<00forσN>0 where 40Pmax=Phh0fe-C(1-A), with P a constant ice strength parameter, h0=1 m, A is ice concentration and C is a constant regulating the decrease of Pmax as the concentration decreases. The factor f∈[1,2] was set to f=3/2 by , who also briefly explored the impact of varying f. The three cases for P̃ correspond to the cases I, II, and III in Fig. b, corresponding to visco-elastic (ridging), pure elastic, and diverging regimes. Equation () for Pmax determines the compressive stress required to move from the elastic to the visco-elastic regime, i.e. when the ice can ridge.

Equation () is solved together with the momentum equation using Eqs. () and (). The time stepping of Eq. () is done in two steps. First, we calculate an intermediate stress, σ′ through an Euler forwards iteration of Eq. (), assuming d˙=0, wich gives 41σ′-σnΔt=EK:ε˙-σ′λ1+P̃, or, after rearranging 42σ′=λΔtmEK:ε˙+σnλ+Δtm1+P̃, where σn is the stress at the start of the time step.

If σ′ is outside the yield criterion, an updated damage value is calculated as 43dn+1=dn+1-dcrit1-dn, and a consistent stress correction is applied 44σn+1=σ′dcrit. Here, dcrit is the factor needed to relax the stresses back onto the yield criterion, which is calculated as 45dcrit=-N/σNifσN<-N,c/τ+μσNifτ>c-μσN, where c and μ are the cohesion and internal friction coefficient values for the Mohr–Coulomb criterion, respectively. A capping with a large value of N is needed to prevent numerical instabilities see, where N is chosen sufficiently large to avoid affecting the solution. In neXtSIM, the default value is four orders of magnitude larger than the cohesion. Equation () follows the damaging scheme of rather than that of , as did. The consequences of which scheme is chosen are discussed in Sect. . If σ′ is inside the yield criterion, i.e. dcrit<0 or dcrit>1, damage remains unchanged and σn+1 is set to σ′.

We note that the finite-element discretisation used in neXtSIM results in the co-location of all the stress tensor components. This co-location is advantageous for the BBM implementation, because all stress tensor components are needed to calculate σN and τ of Eq. (). Models using a staggered grid, where different components of the stress tensor lie at different locations (e.g. the B- or C-grid) must apply extra effort to achieve the co-location of the stress components needed to get an accurate estimate of dcrit e.g..

The elasto-viscous-plastic EVP and the modified elasto-viscous-plastic mEVP rheologies are numerically efficient and easily parallelisable approaches to solving the equations of the original visco-plastic rheology VP. In the VP model, the internal stress tensor is diagnosed from the strain rates as 46σN=ζε˙I-P/247τ=ηε˙II where 48ε˙I=12ε˙11+ε˙2249ε˙II=12ε˙11-ε˙222+4ε˙122 are the first and second strain rate invariants and 50ζ=P2Δ+Δmin51η=ζ/e252Δ=ε˙I2+ε˙II2/e253P=P*he-C(1-A) with e and P* constants controlling the ellipse eccentricity and the ice strength, and Δmin a constant regulating the maximum viscosity in the model.

The VP equations must be solved implicitly using an iterative scheme, such as a fixed-point iteration or a Picard solver. To avoid using an implicit solver, introduced an elastic term, which allows the equations to be solved explicitly through sub-iterations. The VP rheology is recovered as the steady state limit of the EVP rheology of or in the limit of an infinitely large elasticity. For the EVP, the momentum equation is solved using the explicit time-stepping scheme detailed above in Eqs. () and ().

In neXtSIM, we follow in solving the EVP constituent equations by defining σ1=σ11+σ22 and σ2=σ11-σ22 and solving for those as 54σ1n+1=σ1n+2ζε11+ε22-Δ-σ1n/α155σ2n+1=σ2n+2ζε11-ε22/e2-σ2n/α256σ12n+1=σ12n+2ζε12/e2-σ12n/α2, where α1=2T/Δtm and α2=α1/e2, with T=Δt/3 as the elastic time scale.

From a technical standpoint, the mEVP consists of slight modifications of the time stepping of both the momentum and constituent equations of EVP. The constituent equations of mEVP are the same as for EVP (namely Eqs. –), except we set α1=α2=α′, where α′ is a constant tuning parameter introduced (as just α) by . In neXtSIM, we take advantage of this similarity to reduce code duplication.

We also exploit similarities between the EVP/BBM momentum solver and the mEVP solver to reduce code duplication. For mEVP, the momentum equation time stepping takes the form 57β′mΔtmun+1-un=mΔtmu0-un+1+∇⋅(σh)+Aτa+τw+τb+mfn×u-mg∇η, where β′ is the numerical tuning parameter introduced (as just β) by , u0 is the ice velocity at the start of the sub iterations, and un and un+1 are the velocities at the previous and current sub-iteration step, respectively. Using the approach of , this can be rewritten as modified versions of Eqs. () and (): 58α2+β2ujn+1=αuj+βvj+αuj0-ujnb+βvj0-vjnb+ΔtmbmjMjjα-∫∇NjhjσdSx+τx+β-∫∇NjhjσdSy+τy and 59α2+β2vjn+1=αvj-βuj+αvj0-vjnb-βuj0-ujnb+ΔtmbmjMjjα-∫∇NjhjσdSy+τy-β-∫∇NjhjσdSx+τx, where b=β′+1 and α and β become 60α=1+Δtmbmjc′cos⁡θw+Cb61β=Δtmbf+c′sin⁡θwmj. Importantly, the BBM and EVP equations are recovered by setting b=1 and u0=un.

In neXtSIM, switching between the BBM and mEVP rheologies is done by calling separate functions to calculate the stress tensor, σ. The momentum equation is then solved using Eqs. () and (), setting β′=-1 and u0=un when using BBM or EVP.

In general, a sea–ice model's grid cell must be divided into at least two categories: ice and open water e.g.. Many models also use categories of different ice thicknesses, with five thickness categories being a common choice e.g.. Contrary to pure thickness categories, neXtSIM includes a multi-category model based on state of development, considering three categories: consolidated ice, young ice and open water see Appendix A2 ofwho refer to the categories as “thick” and “thin” and not “consolidated” and “young”.

In this model, young ice is described by its concentration, Ay, volume per unit area, Hy, and snow volume per unit area, Hs,y. Consolidated ice is described by the concentration, A, volume per unit area, H, and snow volume per unit area, Hs. We assume that the young ice has no mechanical strength and simply follows the motion of the surrounding consolidated ice. Note the total ice concentration and volume per unit area are A+Ay and H+Hy, and the total snow volume per unit area is Hs+Hs,y.

Young ice thickness is considered to be uniformly distributed with thickness hy=Hy/Ay required to be between the parameters hmin and hmax, which have the default values hmin=5 cm and hmax=27.5 cm. These parameter values affect the ice thickness in a similar way to h0 parameter . The evolution equations for A, H, Hs, At, Ht and Hs,t have the following form: 62DϕDt=-ϕ∇⋅u+Ψϕ+Sϕ, where DϕDt is the material derivative that is defined for any scalar as 63DϕDt=∂ϕ∂t+(u⋅∇)ϕ. Here ∇⋅u is the divergence of the horizontal velocity, Ψϕ a sink/source term due to ridging, and Sϕ a thermodynamical sink/source term. Volume conservation is imposed by setting ΨH=-ΨHy and ΨHs=-ΨHs,y and an additional constraint is that Ay+A≤1.

The evolution of A, H, Ay and Hy is computed following three main steps:

Advection: the scheme solves the equation:64DϕDt=-ϕ∇⋅u,for each conserved scalar quantity (A, H, Ay, Hy, etc.), using the advection Lagrangian scheme (see Sect. ). After this step, the concentration may exceed 1.

Vertical ice growth of the ice is calculated by solving the heat diffusion equation in one dimension. We have implemented two classical approaches: the zero-layer approach of , and the two-layer approach of . The zero-layer approach assumes no heat capacity in the ice and snow, and is generally considered insufficient for longer simulations e.g.. It is, however, sufficient for very thin ice, and so we use this for the young-ice class. We use the two-layer approach for the consolidated ice class.

The two-layer approach of discretises the heat-diffusion equation using two internal temperature points in the ice and assumes zero heat capacity in the snow. It also accounts for the change in ice's internal enthalpy due to the presence of brine. Although substantially simpler than the more complete schemes e.g., this approach can capture the seasonal cycle and the main characteristics of thermodynamic melt and growth. The zero-layer approach of assumes that the ice has no internal temperature and no heat capacity. This straightforward approach is insufficient for general use , but suffices to reproduce the melt and growth of ice in the young ice class, as this is generally thinner than about 30 cm. Therefore, neXtSIM uses the zero-layer approach for ice in the young-ice class, as it is both simpler and faster in execution. It uses the two-layer approach for the consolidated ice.

For both two- and zero-layer models, it is assumed that the ice temperature at the base is at the freezing point of the seawater below. The surface temperature is calculated by balancing the heat flux through the ice with the fluxes from the surface into the atmosphere as derived from 80Q+∂Q∂TsΔTs=kikskshi+kihsTs+ΔTs-Ti. Here, Q is the sum of the latent, sensible, short-wave, and long-wave fluxes at the surface, Ts is the surface temperature at the previous time step, ΔTs is the change in surface temperature, ks and ki are the snow and ice heat conductivities, respectively, and hs is the snow thickness. Ti is the ice temperature at the upper-temperature point, T1 in the case of the two-layer model, or the temperature of the ice base in the case of the zero-layer model. For the two-layer model, hi is the thickness of ice above the T1 temperature point, while for the zero-layer model, hi is the total ice thickness. Once we have calculated the surface fluxes, we calculate ice and snow thickness changes and internal ice temperatures for the two-layer model, following for the zero-layer model and for the two-layer model.

The reanalysis prescribes the incoming short-wave and long-wave fluxes, while the model calculates the albedo, short-wave penetration, and outgoing long-wave radiation. The albedo is calculated using a simplified meltpond scheme (see Sect. ), and the outgoing long-wave radiation and its derivative are simply the black-body radiation and its derivative.

The turbulent fluxes are calculated using bulk formulas that relate the near-surface atmospheric and ice-surface states to surface fluxes, since neXtSIM has so far always been forced by atmospheric model results. The bulk formulas for sensible and latent heat fluxes are taken from : 81Qs=Chρacp|ua|Ts-θ82Ql=ChρaLs|ua|qs-qa, where Ch is a drag coefficient for turbulent heat exchange, ρa atmospheric density, cp the specific heat capacity of air, Ls the latent heat of sublimation, |ua| is the wind speed, θ the potential temperature at the reference height, qs the specific humidity at the surface, and qa the specific humidity at the reference height. The specific humidity may be provided by reanalysis or calculated from the mixing ratio or the dew-point temperature.

The drag coefficient for turbulent heat exchange, Ch, as well as the one for momentum exchange, Cm, do not account for surface roughness; they only consider atmospheric stability. The drag coefficients at the current time step then become 83Chn+1=k2ln⁡zz0-Ψh(ζ)284Cmn+1=k2ln⁡zz0-Ψm(ζ)2, where k is the von Kármán constant, z is the reference height, z0 the surface roughness length, Ψh and Ψm the stability functions for heat and momentum, and ζ=z/L the stability parameter, where L is the Obukov length. We use the stability functions from and for both momentum and heat, and compute separate drag coefficients and stability functions for momentum and heat, using different reference heights (10 and 2 m by default, respectively). The inverse of the Obukov length is 851L=-kgw′θv′‾su*3θ‾v, where g is gravitational acceleration, (w′θv′‾)s is the surface virtual potential temperature flux, u* the friction velocity, and θ‾v is the mean virtual temperature. Here, the inverse Obukov length is calculated by approximating some key parameters. The surface virtual potential temperature flux is approximated as 86w′θv′‾s=w′θ′‾(1+ar)+aθw′r′‾, where the potential temperature flux is approximated as 87w′θ′‾=Chn|u|Ts-θ, the mixing ratio flux as 88w′r′‾=C^hn|u|qs-qa1-qs1-qa, where a=0.6078, r=qa/(1-qa) is the mixing ratio, θ‾v=θ(1+aqa) is the virtual potential temperature, and θ=T+Γdzh is the potential temperature, with Γd being the adiabatic lapse rate. The frictional velocity is approximated as 89u*=Cmn|u|. The inverse Obukov length is limited to 1/L∈[-1,1] m⁻¹.

As advecting additional prognostic variables is very cheap, we set the drag coefficients as prognostic variables and can calculate Chn+1 and Cmn+1 from Chn and Cmn. In models where advection is more expensive, a common approach is to solve for drag iteratively, starting from the neutral drag and recalculating it about 5 times to obtain a more accurate estimate of the stability-dependent drag. The atmospheric stability changes slowly enough that using the drag coefficient from the previous time step is sufficient to calculate the Obukov length and the drag coefficient for the current time step.

The neXtSIM model includes two albedo parameterisations that account for the evolution of melt ponds. The first one is the same as the “CCSM3” albedo scheme . The CCSM3 scheme is a shortwave radiation scheme that includes a melt pond parameterisation, representing the melt pond effect by reducing the albedo value when the surface temperature of sea ice increases. The second one is a recent addition and explicitly represents melt ponds.

The neXtSIM melt-pond scheme largely follows the work of and . It relies on the following set of equations: 90Vpnd=hpndfpnd91hpnd=minaempfpnd,0.9hi92ρwdVpnddt=(1-r)ρidhtopdt+ρsdhsdt+Frain where Vpnd is the melt pond volume, hpnd the pond depth, fpnd the pond fraction, ρi and ρs the ice and snow densities, aemp is the slope from a linear fit between the pond fraction and the pond depth from SHEBA observations , and r is the fraction of surface meltwater that runs off the ice floes and is not collected by the ponds.

Like in , the melt pond volume is a virtual reservoir that does not impact the freshwater and salt fluxes with the ocean. Like and , we first estimate the change in Vpnd from the accumulated water using Eq. (), then estimate the fraction of ponded ice fpnd by combining Eqs. () and (). We then check that hpnd is not higher than 0.9 hi to obey Eq. (); if this is the case, we reduce hpnd by reducing Vpnd while keeping fpnd untouched. This can be interpreted as the vertical draining of ponds that are almost as deep as the ice is thick.

For the sake of simplicity, we use a constant albedo αp for the melt ponds. To avoid having a significant effect from very shallow ponds, we introduce a low limit threshold to the pond depth set to 0.05 m and reduce fpnd if needed.

In contrast to and , we consider r, the fraction of meltwater that runs off the ice, to be constant instead of a function of sea ice concentration. This change is motivated by new insights from MOSAIC data . suggest that, after an initial peak at ≃40 %, the fraction of surface freshwater that is stored in ponds (i.e., 1-r) reduces to ≃10 %. They acknowledge that their data may be biased low due to high lateral draining at the location of their measurements, close to the edge of a floe. However, they find relatively similar results applying the same analysis to data from SHEBA, even though the fraction bounces back slowly to 20 % by the end of the summer. From these observations, a reasonable range for r should be between ≃60 % and ≃90 %, while it is ≃20 % in compact ice using the formulation in and .

Since neXtSIM lacks an explicit representation of lateral and vertical draining, obtaining albedo and melt pond fraction values consistent with observations requires using a large value of r to avoid accumulating too much water in ponds by the end of the summer. Therefore, we recommend using a relatively high value for r (e.g., r≃0.9).

We also allow the aemp value to be changed. In and , aemp is a constant set to 0.8 from , but the MOSAIC data presented in (their Figs. 9 and 10) would suggest a higher slope value (aemp≃1) than the one obtained doing linear fit on SHEBA data only. Low ponded-ice fractions associated with deep ponds increase the value of aemp and are also associated with draining, hence strongly depending on the location. Again, as draining is missing in our model, it is possible to increase the value of aemp so that deeper ponds are not associated with large ponded-ice fractions.

In freezing conditions, a lid forms over the pond, which reduces the amount of meltwater in the pond: 93dVliddt=QiaρiLf94dVpnddt=-dVliddtρiρw, where Vlid is the change in lid sea ice volume, Qia is the heat lost by the ice to the atmosphere, and Lf is the latent heat of fusion. If the meltwater under the lid keeps freezing: 95dVliddt=minQia-Qic,0+QicρiLffpnd, with Qic the heat lost by the meltwater in the pond: 96Qic=Tpnd-TikiVpnd/fpnd with Tpnd the temperature of meltwater in the pond that we assume to be equal to the freezing temperature of water with the same salinity as sea ice (constant in neXtSIM). If the ice lid thickness Vlid/fpnd reaches 1 m, the melt pond and its lid are removed (Vpnd=Vlid=0).

In the end, the total albedo atotal is equal to: 97atotal=A0aow+acons_ice+ayoung_ice, where aow is the albedo of open water, acons_ice the total albedo of the consolidated ice-covered area and ayoung_ice the total albedo of the young ice-covered area. These terms are computed as follows: 98acons_ice=A-fs-fpndai+fsas+fpndasurf_pnd99ayoung_ice=Ay-fs,yai+fs,yas, with ai, as the albedos of bare ice and snow, respectively, and fs and fs,y the fractions of consolidated and young sea ice covered by snow, which are estimated as: 100fs=hshs+0.02101fs,y=hs,yhs,y+0.02, where hs and hs,y are the snow thickness on consolidated and young ice, respectively. In Eq. (), asurf_pnd is the albedo at the surface of the pond (including the lid), which we compute as: 102asurf_pnd=ai+apnd-aie-κ*Vlid/fpnd, with κ a constant related to the extinction coefficient of sea ice. Taking κ=4 and ai=0.76 gives a dependency of the lid albedo to its thickness, similar to the logarithm law suggested by .

NeXtSIM tracks the volume ratio of ridged ice throughout the simulation. Convergence in the Lagrangian framework is represented by a shape change of the model element, while mass in the element is conserved. When this happens, the mean thickness in the element increases, which can be ascribed to ridging. This is easily converted to a volume fraction, which can then be tracked. Ridges form only when the element is fully ice-covered; otherwise, convergence reduces the open-water fraction.

The following is based on the model's prognostic variables for the consolidated ice category: H, the mean ice thickness over the element (or volume per unit area) and A∈[0,1], the fractional ice concentration in the grid cell, as well as the diagnostic variable R, the volumetric ridge ratio. The mean thickness of level ice is thus (1-R)H/A, the volume of level ice is (1-R)H, and so on. The superscripts n and n+1 denote the current and following time steps, respectively. Even though ridging only occurs when the concentration is 100 %, the following equations also hold when the concentration is less than that.

Tracking the volume ratio in the Lagrangian reference frame requires no assumptions on ridge characteristics such as shape, keel depth, sail height, or frequency, while some assumptions about thermodynamic melt and growth of ridges must be made. Taking into account the ridging of young ice (when using the young-ice class), four cases must be considered, for which we make the following assumptions:

Ridging of consolidated ice: assume the conservation of mean thickness of level ice. This assumption can be written as1031-RnHnAn=1-Rn+1Hn+1An+1,which can be rewritten as104Rn+1=1-1-RnAn+1HnAnHn+1,where F=Hn+1/Hn. The ratio Hn/Hn+1 can also be related to the Lagrangian change in element area, S, since volume conservation105SnHn=Sn+1Hn+1can be written as106SnSn+1=Hn+1Hn=F.

introduced the tracers Amy and Hmy, which are, respectively, the concentration and volume of multi-year ice (MYI) – defined as ice that survived the summer melt (for a full description of these tracers, see ). They are affected by the following processes:

Freezing contributes to an increase in FYI volume and concentration only.

An average sea ice age estimate was not included in the study of . We estimate the sea ice age by a surface average age (αA) and a volume average age (αV). The first is intended to relate to the age deduced from satellite observations, while the second is a more physical measure of the total age of the ice. The volume-averaged age tracer follows the same ideas as, e.g. , while we are not aware of an implementation of an area-averaged ice age tracer in the community.

The two tracers evolve according to 110aαAn+1=wAαAn+Δt+1-wAΔt,110bwA=min1,AnAn+1,110cαVn+1=wVαVn+Δt+1-wVΔt,110dwV=min1,HnHn+1. The definitions of wV and wA ensure that in melting conditions (H and A are decreasing), the age increases by Δt, while in freezing conditions, new ice is given the age Δt and the weights wV and wA are the fractions of ice determined by volume or area (respectively) that were already present at the previous time step.

NeXtSIM uses a triangular mesh with moving nodes and a remeshing scheme, as previously documented in , and . In this section, we provide an overview of the Lagrangian advection used in neXtSIM and detail the conservative remapping scheme added since .

The neXtSIM mesh is an unstructured triangular mesh on an {x,y}-plane, usually a polar-stereographic plane, although other projections are technically possible. The initial model mesh can be constructed using tools such as Gmsh or any mesh generator that can produce a mesh in the Gmsh format. The initial mesh sets the mesh resolution, and subsequent adapted meshes retain that same resolution. This is achieved by creating a spatially varying field of mean edge length for the initial mesh, and then requiring that the lengths of the vertices of later adapted meshes must fall within 20 % of the mean edge length of the initial mesh.

Move the nodes of the mesh

In step 1, we move the mesh nodes by the displacement Δx=Δtu, where Δt is the model time step and u is the ice velocity, which is trivial. Nodes on land boundaries are fixed with u=0. For most neXtSIM setups, one or more sections of the domain edge are open, allowing ice to flow in or out of the domain. In- and out-flux through open boundaries is ensured by keeping the open-boundary nodes fixed and not updating these prognostic variables on elements that border an open boundary. This means that ice will flow out as if there was no resistance and flow in as if the ice state outside the boundary was the same as that inside it.

In step 2, we consider the changes in area of the elements, S (using the same notation as in Sect. ). To conserve the ice and snow volume and the ice area in each element, these prognostic variables must be multiplied by the fraction F=Sn/Sn+1, where n and n+1 denote the state before and after the nodes are moved. This follows directly from volume and area conservation 111SnHn=Sn+1Hn+1,112SnAn=Sn+1An+1, where H denotes snow or ice volume and A∈[1,0] the concentration fraction. The ice concentration, ice thickness, and snow thickness are the only prognostic variables affected by changes in element area due to Lagrangian advection. This is because those are defined per unit area, so when the area changes, their value must change. The value of all other advected variables (such as temperatures and stresses) remains unchanged. Velocities are defined on the nodes and are, therefore, not affected by the deformation of the element.

The mesh adaptation in step 3(a) is performed using a port of the bi-dimensional anisotropic mesh generator, BAMG . The main advantage of BAMG is that it can adapt the mesh only where it is too deformed, leaving other nodes in their original position. The mesher in BAMG returns C++ objects with complete element and node connectivity, making it possible to quickly address nodes and elements connected to any given node or element of the mesh. The mesh adaptation is described in more detail in .

In previous versions of neXtSIM i.e., step 3(b) was completed using an algorithm which identified areas where the old and new meshes differed (“cavities”) and then interpolated the fields from the old mesh to the cavities of the new mesh in a conservative manner. This approach was very efficient and highly optimised, but ultimately very complex and occasionally fragile. When developing the coupling routines to couple neXtSIM with an ocean model (see Sect. ), we created a more robust conservative remapping approach, described in the following subsection. We could use this approach to map between the old and adapted meshes with only minimal code changes, and the relative simplicity of the scheme and its robustness, in addition to substantial code reuse, more than made up for the fact that the new scheme is less efficient than the old one.

The conservative remapping approach is fully conservative, but remeshing introduces some local diffusion because a set of deformed elements is replaced with less deformed ones. A quantitative assessment of this diffusion is non-trivial and outside the scope of this paper. We note, however, that by construction, any translation or rotation of features such as leads and ridges is non-diffusive. This is a major advantage for neXtSIM, as the BBM rheology produces particularly well-localised leads.

Step 3(c) is necessary because, although the model is parallelised using MPI , the remeshing process is not. Therefore, all mesh information and model fields must be gathered onto a single MPI processor to adapt the mesh: remapping from the old to the new mesh and partitioning the new mesh, which is then scattered to the other MPI processes. The mesh is partitioned using the METIS partitioner.

Performing mesh adaptation and field remapping on a single MPI process creates a substantial bottleneck for large problems, even though the Lagrangian advection itself is highly efficient. This was already identified by , but their solution, an arbitrary Lagrangian-Eulerian (ALE) approach, was technically challenging and was abandoned. As a result, neXtSIM is, in practice, limited to meshes with O(105) elements running on O(100) MPI processors. For larger problems, the model scales poorly with the number of MPI processors, so execution time remains constant regardless of the number of processors. For regional Arctic setups, this limits the model use to resolutions coarser than about 5 km.

NeXtSIM uses a conservative remapping scheme, in which values from a donor mesh are mapped to a receiver mesh via weighted averaging, with the area of overlap between elements in the two grids as weights. The weights are computed once and then applied across many fields at a very low computational cost. The weight calculation algorithm below relies on two features of BAMG: the complete connectivity information of the mesh and a BAMG utility, which allows one to find the element covering any point in the domain. The algorithm is simple (see Algorithm 1) and relies on recursion, two checks for element nodes, and one check for element-element intersections. The algorithm returns two lists: a list of elements of the donor mesh overlapping a given element of the receiver mesh, L, and a list of the areas of overlaps, used as weights, W. In practice, a single check routine is called for the receiver element to build L and W.

The first step of the algorithm is to find an element in the receiver mesh, Er∈Ωr, that overlaps with the barycentre of a given element in the donor mesh, B(Ed)∈Ωd. This is done using an interpolation utility in BAMG, which uses the underlying quadtree to find the element in a given mesh that overlaps a set of points in the domain. This is by far the most time-consuming part of the algorithm. Once all overlapping elements have been found, the algorithm iterates over all donor elements, Ed, performs three simple checks, and then recursively checks neighbour elements, as needed.