The evolution equations for h, hs, and A (here denoted ϕ) have the following generic form: DϕDt=-ϕ∇⋅u+Sϕ, where DϕDt is the material derivative of ϕ, ∇⋅u is the divergence of the horizontal velocity, and Sϕ is a thermodynamical sink/source term. The actual form of Sh, SA, and Shs are defined in Sect. . An additional constraint for the concentration is that A≤1.

The evolution of d is given by DdDt=ΔdΔt+Sd, where Δd is the damage source term (defined below) and Sd is a thermodynamical sink term (defined in Sect. ). The damage is an ice volume tracer and is equal to 0 for newly formed ice. When new ice is formed, the new damage is decreased and calculated as a volume-weighted average over the old and new ice, meaning that the sea ice partly recovers its mechanical strength.

The evolution of sea ice velocity derives from the vertically integrated sea ice momentum equation: mDuDt=∇⋅(σh)-∇P+τa+τw+τb-mfk×u-mg∇η, where m is the inertial mass, P is a pressure term, τa is the surface wind (air) stress, τw is the ocean (water) stress and τb is the basal stress in case of grounded ice. All these terms are defined in Sect. . The other symbols in Eq. () are f, the Coriolis parameter; k, the upward pointing unit vector; g, the gravity acceleration; η, the ocean surface elevation; and σ, the internal stress tensor.

The evolution of the internal stress is computed as in in two steps that correspond to DσDt=σ(n+1)-σ′Δt+σ′-σ(n)Δt, where superscripts n and n+1 correspond to the previous and current time steps, respectively. The first step accounts for the elastic deformation without considering the damaging process and gives a first estimate of the internal stress, σ′, by σ11′-σ11(n)σ22′-σ22(n)σ12′-σ12(n)=E(A,d)1-ν21ν0ν10001-ν2ϵ˙11ϵ˙222ϵ˙12Δt, where E(A,d) is the effective elastic stiffness, ν is Poisson's ratio, and ϵ˙ is the strain rate tensor defined by ϵ˙=12∇u+(∇u)T. The effective elastic stiffness is parameterised as E(A,d)=Yf(A)(1-d), where Y is the sea ice elastic modulus (Young's modulus) and f(A) is a decreasing function of A parameterising the effect of the compactness (defined in Sect. ). The second step accounts for the damaging process. With this estimate of the internal stress, the failure criteria are checked. For the elements where the estimated internal stress σ′ falls outside the failure envelope (defined in Sect. ), the damage factor Ψ is set to the value for which the stress state, σ(n+1)=Ψσ′, is scaled back onto the failure envelope. For the elements for which the estimated internal stress σ′ is inside the failure envelope, Ψ is simply set to 1. The damage source term Δd corresponding to the decrease of σ has been derived in as Δd=(1-Ψ)(1-d(n)).

In this subsection, we detail each term of the sea ice momentum equation (Eq. ) and the underlying parameterisations.

The inertial mass m depends on the assumption made about the motion of the water present in leads: either the water in the leads moves as the ocean below or as the ice around it . We choose the second hypothesis for our model, as we think it is more relevant for high-resolution models and when the rheology allows for sharp transitions of sea ice concentration within the ice cover. Using this approach, the inertial mass m corresponds to the mass of ice and snow plus the mass of the water in the leads as m=ρih+ρshs+ρwhw, where ρw is the reference density of seawater and hw is the volume of water from the base of the ice to the sea surface per unit area. By isostasy, ρwhw(1-A)=ρih+ρshsA, and m is then given by m=ρih+ρshsA.

The failure envelope is defined as in in terms of the principal stress components σ1 and σ2 defined by σ1=-σ11+σ222+σ11-σ2222+σ122σ2=-σ11+σ222-σ11-σ2222+σ122 respectively, following the convention that compressive stresses are positive. The envelope represents a combination of a Mohr–Coulomb criterion, a tensile stress criterion, and a compressive stress criterion, defined by σ1≤qσ2+σc(Mohr–Coulomb criterion),σ1+σ22≥σNmin(tensile stress criterion),σ1+σ22≤σNmax(compressive stress criterion), where q and σc are defined as in by q=μ2+11/2+μ2,σc=2cμ2+11/2-μ, where μ is the internal friction coefficient and c is the cohesion. σNmin<0 and σNmax>0 are the maximal tensile stress and the maximal compressive stress, respectively. The friction coefficient μ for sea ice is chosen equal to 0.7, which is a common value for geo-materials and seems to be scale-independent . The values of the cohesion c, σNmin, and σNmax seem to be inversely proportional to the square root of the spatial scale . Here we use this scaling relationship to define their values at 10 km as equal to c=8kPa, σNmin=-9.52kPa, and σNmax=150kPa from values measured in the field (Weiss et al., 2007) and in lab experiments (Schulson, 2009).

P is a vertically integrated sea ice pressure term that is set to avoid excessive convergence when the ice concentration in a cell is at 100 % and at the same time highly damaged. Without this term, one obtains unrealistically large local thickness of the ice cover, for example north of Greenland and the Canadian Archipelago. This term implies no memory effect, meaning that it cannot be included in the evolution equation of the internal stress. This term is parameterised as P=P∗h2f(A)Φ∇⋅u, where P∗ is the pressure parameter, f(A) is the same function as in Eq. (), and Φ determines if the pressure term is active or not. In our case, Φ is defined as a function of the divergence rate at the previous time step and is computed as Φ=0if ∇⋅u(n)≥01∇⋅u(n)+ϵmin⁡if ∇⋅u(n)<0, where ϵmin⁡ is a parameter set to a small value to regularise the transition when the divergence rate is close to 0. The quadratic dependence on the mean thickness and the value of P∗ used in this study (P∗=12kPa) comes from and corresponds to the redistribution scheme of when it is applied to only one ice thickness category.

The effect of the concentration on the mechanical response of sea ice is parameterised here by a decreasing exponential function of the concentration: f(A)=eα(1-A), where α is the compactness parameter seefor more details.

The air stress τa is computed following the quadratic expression: τa=ρacaua-uua-ucos⁡θa+k×ua-usin⁡θa, where ua is the wind velocity, ρa the air density, θa the atmospheric turning angle, and ca the atmospheric drag coefficient.

The oceanic stress τw is also computed following a quadratic expression, namely τw=ρwcwuw-uuw-ucos⁡θw+k×uw-usin⁡θw, where uw is the ocean velocity, ρw the reference density of seawater, θw the water turning angle, and cw the water drag coefficient.

The basal stress τb is a term added to simulate grounded fast ice. It is parameterised as in by the expression: τb=k2-uu+u0max⁡(0,h-hc)e-Cb1-A, where k2 is the maximum basal stress parameter, u0 is the basal stress velocity parameter, and Cb is the basal stress concentration parameter. The critical thickness from which the parameterisation starts acting is defined as hc=AH+ηk1, where k1 is the critical thickness parameter and H is the ocean depth.

In neXtSIM damaged sea ice recovers its mechanical strength (i.e., decrease of the damage) through time via two processes: formation of new ice in open water and leads and thermodynamical healing. Sea ice melting is supposed to have no direct impact on the damage. New ice formation is naturally treated by updating the value of the local damage as a volume-weighted average over the old and new ice. When sea ice volume in a cell increases due to ice formation, the damage then automatically decreases as new ice is supposed to have a damage equal to zero. For the thermodynamical healing process, more assumptions need to be established. We assume here that the thermodynamical healing process is driven by the local temperature gradient between the bottom of the ice and the snow–ice interfaces and decreases the effective compliance, defined as C=11-d, at a constant rate. This is based on the fact that the cooler the environment, the faster the ice will freeze, so presumably low temperatures result in fast healing and warm temperatures in slower healing, with no healing occurring for temperatures over the freezing point. The damage relaxation term Sd of Eq. () is then computed by SdΔt=1-1C(n+1)-d(n), with C(n+1) given by C(n+1)=max⁡11-d(n)-ΔtTC,1, and where TC is the compliance relaxation time. This relaxation time is assumed here to be inversely proportional to the temperature difference ΔT between the bottom and snow–ice interface, which is given by ΔT=Tb-Ts1+kihsksh, where Tb is the temperature at the ice base, Ts the temperature at the ice or snow surface, ki is the heat conductivity of ice, and ks is the heat conductivity of snow. The compliance relaxation time is then defined as TC=max⁡Td100040KΔT,0, where Td is the parameter controlling the healing rate. We use the constant 1000 and 40 K, which are typical values of effective compliance and temperature difference given by the model during winter, so that Td can be interpreted as the time needed to heal the ice in winter conditions. The sensitivity to this parameter is discussed in Sect. . We also limit TC to be positive so that melting conditions alone (Ts>Tb) cannot damage the ice.

The other components of the thermodynamical model are similar to those in classical sea ice models. There are three thermodynamical source and sink terms corresponding to Sϕ in Eq. (), one for each of ice volume, concentration, and snow volume. The source/sink term for the ice volume stems from the conservation of mass and can be written as ShΔt=AΔh+(1-A)Δhow, where Δh is the change in level ice volume and Δhow is ice formation in open water.

The change in A is calculated by assuming a given thickness for the ice forming over open water (Δhow). We use a constant, h0, for this thickness, giving a source/sink term for A as SAΔt=(1-A)max⁡(Δhow,0)/h0+Amin⁡(ShΔt,0)/2h. Ice formation in the open water portion of the grid cell, how, is calculated such that heat loss from the ocean that would cause supercooling is redirected to ice formation. This is an adaptation of the form suggested by in that he uses prescribed growth rates, but we calculate those depending on the atmosphere and ocean states, as described below.

The source/sink term for snow thickness, Shs, also stems from the conservation of mass and is ShsΔt=AΔhs+hsmin⁡(0,SAΔt)+Amin⁡(0,D-h/A)ρi/ρs. The first term accounts for snowfall and snowmelt, the second term removes snow due to the lateral melt of ice, and the third term converts snow into ice whenever the ice freeboard, D, falls below the water surface due to snow loading. Energy needed to melt snow due to the lateral melt of ice is removed from the ocean as an additional heat flux.

Thickness changes in level ice and snow are calculated using the zero-layer model of , using the same parameter values, unless otherwise stated. This is arguably the simplest usable thermodynamic sea ice model, and it has some well-known deficiencies most notably. It does, however, suffice for short runs with a stand-alone ice model, like the ones discussed in Sect. 3.4. In this model the incoming radiative fluxes are interpolated from the forcing data, applying constant albedos to the incoming short-wave radiation of αi=0.64 for the ice and αs=0.85 for the snow . The turbulent heat fluxes are calculated using bulk formula for the sensible heat flux: Qsh=ρacpCt|ua|(Ta-Ts) and the latent heat flux: Qlh=ρaLvCq|ua|(qa-qs). Here, cp is the atmospheric heat capacity and Lv is the latent heat of sublimation. The temperature difference, Ta-Ts is taken between the ice surface and the atmosphere at 2 m, the same as the difference in specific humidity, qa-qs. We calculate the specific humidity using the formulation of . The drag coefficients Ct and Cq are set to 1.3×10-3 based on . Fluxes between the ice and ocean, Qoi, are calculated assuming the ocean underneath the ice must always be at the freezing point.

In order to produce realistic heat fluxes through the ice, the thermodynamical ice model must be coupled to an ocean model. Here we use a simple slab ocean model that consists of a single ocean layer with one temperature and salinity point per grid cell. The flux of heat at the ocean surface is calculated as a weighted average of the ocean–ice and ocean–atmosphere fluxes: Qw=AQoi+(1-A)Qoa, where Qoa is the ocean–atmosphere flux. The ocean–atmosphere heat flux is calculated using the bulk formulas (Eqs.  and ), with Ct=0.83×10-3 and Cq=1.5×10-3 for the turbulent heat fluxes, while the radiative heat fluxes are read in from the atmospheric forcing, applying a constant albedo of 0.07 to the short-wave flux.

The change in ocean salinity is calculated assuming the total salt content of the ice–ocean system is conserved, resulting in a change in salinity of ΔS=(So-Si)ρiΔh+So(Δhsρs-FfwΔt)Hmlρw-Δhρi-(Δhsρs-FfwΔt) for a change in ice and snow area mean thickness of Δh and Δhs respectively, and where So and Si are the ocean salinity and ice salinity, respectively, ρi and ρs are the ice and snow density, respectively, Ffw is freshwater flux at the surface, Hml the mixed layer depth, and Δt the model time step. We assume a constant ice salinity of Si=5 psu.

When using a slab ocean, simulated ocean temperature and salinity have to be artificially maintained at realistic values because of the missing representation of both vertical and horizontal heat and salt exchanges within the ocean. Here, we use Newtonian nudging to relax the simulated ocean temperature and salinity towards the values of the uppermost ocean layer from a full ocean model (in this case the TOPAZ4 system, see Sect. ). The local mixed layer depth from the full ocean model is also used as the mixed layer depth (Hml) of the slab ocean model.

Most sea ice models use an Eulerian approach for the advection. However, we believe that a purely Lagrangian approach as in may be more suitable to preserve highly localised features (i.e. one cell wide ridged or open water areas) generated by the model. Continuous, purely Lagrangian schemes require unstructured meshes and a procedure for mesh adaptation. Local mesh modifications can be done in parallel and introduce very low numerical dissipation . They also show local conservation .

In the purely Lagrangian approach, the vertices of the element (i.e. the nodes of the grid) move with the sea ice velocity u. The material derivative is then simply equal to the temporal derivative ∂ϕ∂tX relative to the moving mesh so that the quantities are naturally transported with the ice. The sea ice thickness and concentration, for example, are simply updated by h(n+1)=h(n)S(n)S(n+1), and A(n+1)=min⁡A(n)S(n)S(n+1),1, where S(n) and S(n+1) are the surface of the element at time steps n and n+1. The variables defined at the nodes do not need to be updated.

In this approach the model mesh deforms as the ice cover itself deforms. When the mesh becomes too distorted the results of the finite element method are no longer reliable and the mesh must be adjusted, a process referred to as remeshing. In the current implementation the mesh is considered too deformed when the smallest angle of any triangle of the mesh is smaller than 10∘. Using this criterion and the set up we use here, the mesh needs to be adapted on average every model hour.

In order to save computational time the forcing fields are only interpolated onto the model grid after remeshing, or when new forcing fields are required. This means that even though the model grid drifts and deforms in-between remeshings the forcing seen by the nodes and elements of the model does not change. We checked that this method gives virtually identical results as when we interpolate the forcing fields every time step. Indeed as the remeshing criterion is global the error in the position of the forcing field is in practice never larger than a single model element. Given the high resolution of the model grid in our tests, the forcing fields are too smooth and too coarsely resolved for this error to have any substantial effect.

The approach used for the slab-ocean is similar to that used for the forcing in that the fields are only interpolated after remeshing. The slab-ocean model resides on the same mesh as the ice model, but the relative displacement of ice and ocean is ignored between remeshing steps. When the model mesh becomes too deformed and therefore needs to be remeshed, the temperature and salinity are interpolated from the old onto the new mesh using a linear interpolation and ignoring the displacement of the old mesh. This ensures that the temperature and salinity fields do not drift with the ice as the ice-model mesh moves.

The new mesh is created by a version of the BAMG mesh generator by taken from the Ice Sheet System Model . This mesh generator can be instructed to preserve as many of the nodes from the old mesh as possible. The mesh is thus only modified in a limited number of locations, hereafter called cavities, at each remeshing. Doing this allows the model to track large expanses of drifting ice that is deforming very little without any artificial diffusion, since it is only necessary to interpolate values from the new grid to the old one inside the cavities. Outside the cavities the tracer values are not affected by the remeshing. For the variables defined at the nodes (i.e. the sea ice velocities, etc.), a non-conservative linear interpolation is performed for the new nodes.

For the quantities that are defined at the centre of the elements, a conservative remapping scheme is applied to each cavity independently. The cavities are defined as the smallest partitions of the mesh for which the external edges are the same before and after the remeshing. We implemented an algorithm that uses the information provided by BAMG (i.e. node-element connectivity and previous numbering of the preserved nodes) to efficiently detect the cavities and the intersections between the triangles of the old and new meshes. For each intersection, the corresponding integrated quantities are transferred from the old mesh to the new one. The process is fully conservative and generates only limited numerical diffusion.

For the model evaluation we force our model with the ocean state of the TOPAZ4 reanalysis see, and with the atmospheric state of the Arctic System Reanalysis, Interim version (ASR-Interim hereafter) (http://rda.ucar.edu/datasets/ds631.4/, Byrd Polar Research Centre/The Ohio State University (2012). Accessed 1 January 2014.) TOPAZ4 is a coupled ocean–sea ice system combined with a state-of-art ensemble Kalman filter data assimilation scheme of both ocean and sea ice variables, running at an average resolution of 12.5 km in the Arctic. The TOPAZ4 bathymetry is based on the 1 arcminute GEBCO bathymetry and the coastline is derived from the 5 m isobath. The main benefit of using the TOPAZ reanalysis in this context is its accuracy in simulating the location of the ice edge, and therefore to provide realistic forcing for ocean temperature and salinity. The ASR-Interim is a high-resolution atmospheric reanalysis (30 km) known to reproduce the near-surface wind fields particularly well .

In order to simplify the forcing of the slab ocean with TOPAZ4, the domain of our model is defined from TOPAZ4 coastlines and open boundaries. The resulting mesh covers the Arctic and North Atlantic oceans, extending from an open boundary at 43∘ N in the North Atlantic to an open boundary in the Bering Strait (see Fig. ). The resolution of the finite element mesh is about 11 km, where the resolution is defined as the square root of the mean element area. Note that the ocean depth H used for the basal stress parameterization comes from the 1 arcminute ETOPO1 global topography .

The oceanic forcing variables are sea surface height, velocity at 30 m depth, and sea surface temperature and salinity, which are provided as daily means by the TOPAZ4 system. We interpolate these quantities temporally and spatially onto the model mesh at run time using linear and bilinear interpolation methods, respectively. The slab-ocean temperature and salinity are nudged towards TOPAZ4 and we found 30 days to be an appropriate nudging timescale for this set-up. This value allows our model to reproduce the location of the ice edge well without unduly affecting heat fluxes in the central Arctic. The heat flux resulting from the nudging is usually slightly below 0.5 Wm-2 in the central Arctic in midwinter, which is a reasonable value see e.g..

The atmospheric forcing consists of the 10 m wind velocity, the 2 m temperature and mixing ratio, mean sea level pressure, total precipitation and the fraction of that which is snow, and the incoming short-wave and long-wave radiation. All of these quantities are provided as 3-hourly means by the ASR-Interim. Similarly to the oceanic variables, we interpolate them temporally and spatially onto the model grid at run time using a linear and bilinear interpolation method, respectively.

In order to prevent an initial shock to the system when the model is started, the strength of applied wind and ocean currents is increased linearly from zero to full strength over the period of 1 day.

Our reference simulation starts on 15 September 2007 and ends on 9 October 2008. We choose this year because this is the only winter when the GlobICE (http://www.globice.info) and RGPS data sets overlap. The values of the model parameters that are used for this simulation are listed in Table . Damage is initially set to zero where sea ice is present. Initial sea ice concentration and thickness are interpolated from the sea ice of the TOPAZ4 reanalysis. The modelled ice thickness in TOPAZ4 is known to be unrealistically low on average compared to other Arctic ice–ocean coupled systems (http://marine.copernicus.eu/documents/QUID/CMEMS-ARC-QUID-002-003.pdf). We therefore scale it uniformly so that the initial total ice volume is the same as that from the PIOMAS reanalysis data downloaded from ftp://pscftp.apl.washington.edu/zhang/PIOMAS/ on 4 February 2014. Initial snow thickness is from the climatology, but we limit the snow thickness to 20 % of the ice thickness so that we do not get unrealistically thick snow on the relatively thin ice that was present at the start of the simulation.

In neXtSIM, as in most classical sea ice models, the air and water drags depend on four parameters: ca the air drag coefficient, θa the air turning angle, cw the water drag coefficient, and θw the water turning angle. The value of these four parameters has to be calibrated depending on which atmospheric and oceanic forcing are being used. The purpose of this section is to present how we proceed with this calibration for the present study.

One of the main differences of neXtSIM compared to other sea ice models is the rheology, which defines the link between internal stress and deformation. For the internal stress, only a few observations are available and cannot be directly used for a complete evaluation.

However, since we have calibrated the two other dominant terms of the momentum equation (i.e. the oceanic and atmospheric drag terms), then we can use an evaluation of the overall drift and deformation of the ice as an evaluation of the internal tress. A good way to evaluate the new rheology is then to compare the simulated deformation fields to the large amount of data available from satellite products. The data used here are produced from the RGPS sea ice drift data set with the method proposed by .

An interesting specificity of sea ice deformation is its strong localisation in space (see Fig. ) and in time.

This makes a comparison of the geographical location of the observed and simulated deformation features impractical, since small errors in the applied forcing are bound to result in significant changes in the simulated location, compared to the extent of the features. Instead we compare simulated and observed deformation in a statistical sense using, among others, the multi-scale metrics introduced by and .

The comparison with observation is focussed on the period January–April 2008, which has been identified as the period for which deformation is typically lower than during the rest of the year (see the annual cycle presented later in the paper). Figure  shows time series of the observed and simulated mean shear rate from January to April 2008. The deformation rates are computed at a spatial scale of 20 km and for 13 periods of 9 days. As in the previous sea ice drift evaluation, the model data are built to match the observations spatially and temporally. The correlation between the observed and simulated mean shear values is satisfactory, but we note that the model systematically underestimates the mean shear rate during this period.

Spatial scaling properties of sea ice deformation (or the degree of heterogeneity of sea ice deformation) can be studied from the analysis of Lagrangian trajectories, as e.g. in who applied it to the trajectories of the RGPS data set. Here we perform this analysis for both the model and the satellite observation, following the method used in , which is very similar to the one of , and which also gives an estimate of the error on the spatial scaling exponent. The spatial scaling analysis has been first applied to all the 13 snapshots corresponding to the 9-day periods between 1 January and 28 April 2008. Four snapshots had to be discarded because the power-law model fit was not significant in the least squared sense due to the excessive noise in the observed deformation fields. The values of the first-order moment of the shear rate (here denoted ϵ˙s) distribution obtained when gathering the selected snapshots is shown in Fig.  for different spatial scales. The power law 〈ϵ˙s〉∼L-β that fits the data (grey and black lines) corresponds to a scaling exponent β of -0.16 for the observations and -0.11 for the model. The departure from the power-law fit at L<20km and L>500km comes from finite size effects (model resolutions and size of the Arctic basin, respectively). The scaling exponents β(q) for the other moment orders of the shear distribution fit well with a quadratic function of q, whose curvature is equal to 0.13 for the observation and 0.07 for the model. This shows that like observed deformations , the deformation simulated by the model is characterised by a multi-fractal spatial scaling.

Temporal scaling properties of sea ice deformation (or the degree of intermittency of sea ice deformation) can be studied from the dispersion of passive tracers . Note that this approach is inspired by a classical methodology developed originally to study fluid turbulence . Here we perform the temporal scaling analysis for both the model and the observations following the same method as in , using pairs of vertices of the model mesh and pairs of tracking points of the RGPS trajectory data set, respectively. Indeed, in a Lagrangian modelling framework, each vertex of the mesh can be considered as a passive tracer of sea ice, and directly compared with tracking points of the RGPS data set. For each pair of vertices/RGPS points initially separated by a distance L of ∼ 30km on average, a proxy of sea ice deformation D is measured by looking at the relative variation ΔL/L of the distance between the two vertices/RGPS points for different time intervals Δt. The deformation rate D˙ is estimated as D˙=ΔLLΔt. For the model the intervals are 0.25, 0.5, 1, 2, 4, 8, 16, 32, and 64 days, whereas the analysis for the RGPS trajectories only starts from a time interval of 2 days due to the limited temporal resolution of these data. The number of analysed measurements is similar to and decreases from 630 000 for the 0.25 day interval to 2600 for the 64 days interval. Figure  shows the mean value of |D˙| for the different timescales for the model and the observations. A power-law model 〈|D˙|〉∼T-α from T=2 to 64 days fits both the observed and simulated data very well, with the same scaling exponent α=0.3. Previous studies based on buoy data indicate that the scaling should also hold for smaller scales. This is not the case for the model data and cannot be verified from the RGPS data used in this study. The right panel of Fig.  indicates that the model only gives the right scaling exponent for healing timescales equal and larger than 7 days. We note that the order of magnitude of the healing timescale obtained here is consistent with the optimal value obtained in the previous section when analysing the sensitivity of the model with respect to sea ice drift. The low sensitivity to the healing time parameter for values larger than 14 days may indicate that the thermodynamical healing term is not needed and that the healing due to new ice formation is sufficient. However, as this may not be true for all model configurations, we prefer keeping the thermodynamical healing term in the description of the model.

The simulated mean value and spatial scaling exponent of the 3-day deformation evolves during the year of simulation (see Fig. ), with much lower mean deformation between January and April, and a more negative scaling exponent in summer than in winter. We note that this behaviour, as well as the high variability, compares well with the results found by from the analysis of the whole RGPS data set.

We now consider the modelled seasonal cycle in total ice volume and area. This section is intended to demonstrate that the model produces a reasonable seasonal cycle and to explore briefly its sensitivity to key parameters. An in-depth evaluation and tuning of these aspects of the model's behaviour would require several multi-decadal runs, which we consider outside the scope of this paper. For this purpose results from three runs, in addition to the reference run are shown: a run with fixed albedos of αi=0.7 and αs=0.9 (high albedo case), a run with temperature-dependent albedos , and a run forced with the ERA-Interim reanalysis results.

Figure a shows the modelled total ice volume compared to monthly mean outputs from PIOMAS and observations from ICESat data downloaded from http://rkwok.jpl.nasa.gov/icesat/download.html on 6 March 2015. The estimates are calculated for the region delimited by the magenta dashed line of Fig. . The shaded areas around the PIOMAS results show the uncertainty assigned to the PIOMAS results by , and the horizontal and vertical error bars on the ICESat data points are the time span for the observations and the uncertainty assigned to those observations by . Both uncertainty estimates are probably upper bounds according to their authors. It is difficult to assess model performance in terms of total ice volume due to the lack of reliable observations. The uncertainty on the ICESat observations is substantial and the quality of the October–November estimate in particular is suspect. Because of this lack of data we chose to also plot the total volume from the PIOMAS model, but this should also only be considered a reference and not an accurate representation of the state of the ice cover. With these caveats in mind we see that the performance of the reference run is acceptable when it comes to ice volume. The melt rate can also be substantially affected by tuning the albedos, as expected.

Figure b and c show the modelled sea ice extent and area respectively, compared to satellite observations. The area is simply the total area covered by sea ice, while the extent is the total area of grid cells covered with more than 15 % of sea ice. The observations shown are the mean values and extremes for daily observations using the ASI algorithm AMSR-E data obtained from the Integrated Climate Date Center, University of Hamburg, Germany, http://icdc.zmaw.de, OSI-SAF , NASA Team , and bootstrap products. Using these four products gives a good idea of the uncertainty involved in the satellite observations .

In terms of extent, the results of the neXtSIM model are within the limits for the uncertainty estimates for the observations until the start of May. At this point the modelled melt is considerably more rapid than the observed one, leading to a difference of about 1.5 millionkm2 at the beginning of June. From then, however, the observed melt becomes much more rapid and the end result is that the modelled extent at the extent minimum is within the limits of the observations uncertainties. Changing the forcing or albedos does affect the melt substantially, but it does not alter the fact that the model fails to capture the two-phased melt observed, a slow phase from early April to early June and a rapid phase from early June to early September.

In terms of total ice area, the model slightly overestimates the ice area during the freeze-up, but is in good agreement with observations for the rest of the model run. This is, however, not the case when using the ERA-Interim forcing or the temperature-dependent albedos since in those cases the melt is too rapid, resulting in total ice area that is about 1.5 millionkm2 smaller than in the reference run.

There is, therefore, a discrepancy between the modelled extent and area when compared to the observations, in that the modelled extent is too low during melt but the modelled area is correct. This seems to indicate that as the ice concentration is reduced during melt, the ice compacts too easily, resulting in the correct area but too low extent. This issue is currently under investigation.

The spatial distribution of concentration is shown in Fig. . The concentration map shows that the sea ice distribution at minimum extent is well captured. The largest differences between the modelled and observed ice extent occur in the regions where the modelled ice concentration is low and the ice is easily influenced by the wind. have shown the shape of the ice extent minimum to be heavily influenced by the wind, but we have not investigated this in our model.

In addition to concentration, Fig.  shows the spatial distribution of ice thickness at the beginning of the simulation, in midwinter and at the sea ice minimum. These distributions clearly show the high degree of heterogeneity that appears in the model, despite very smooth initial conditions. The midwinter map shows substantial amounts of fracturing and ridge formation in the Beaufort Gyre and the Transpolar Drift Stream in particular. This heterogeneity persists until the end of the melt season, even if the melt does smooth it out somewhat.

Overall, the model performs well in terms of total volume, area, and extent. This behaviour is largely controlled by the atmospheric and oceanic forcing. However, a poorly tuned or conceived ice model is still free to diverge considerably from the observed state, and it is reassuring to see that this is not the case here. The only genuine discrepancy between the model results and observations is that the model does not capture the two phases of melting observed in the extent. The model is sensitive to changes in the surface albedo, which is to be expected and albedos are probably the most widely used tuning parameters for ice and ice–ocean models. The model also shows some sensitivity to the lateral melt formulation, which is limited and was not shown. Sensitivity to the oceanic nudging timescale and various dynamical parameters is negligible within reasonable ranges for these parameters. For longer simulations a more sophisticated thermodynamics model is needed though, such as or .

The atmospheric reanalysis data used in this article are available at http://rda.ucar.edu/datasets/ds631.4/. The ocean reanalysis data used in this article are available at http://marine.copernicus.eu/. The sea ice concentration data sets used in this study are available at http://osisaf.met.no, http://icdc.zmaw.de, and https://nsidc.org/data/search/.