The Arctic sea ice cover has changed drastically over the last decades. Associated with these changes is a shift in dynamical regime seen by an increase of extreme fracturing events and an acceleration of sea ice drift. The highly non-linear dynamical response of sea ice to external forcing makes modelling these changes and the future evolution of Arctic sea ice a challenge for current models. It is, however, increasingly important that this challenge be better met, both because of the important role of sea ice in the climate system and because of the steady increase of industrial operations in the Arctic. In this paper we present a new dynamical/thermodynamical sea ice model called neXtSIM that is designed to address this challenge. neXtSIM is a continuous and fully Lagrangian model, whose momentum equation is discretised with the finite-element method. In this model, sea ice physics are driven by the combination of two core components: a model for sea ice dynamics built on a mechanical framework using an elasto-brittle rheology, and a model for sea ice thermodynamics providing damage healing for the mechanical framework. The evaluation of the model performance for the Arctic is presented for the period September 2007 to October 2008 and shows that observed multi-scale statistical properties of sea ice drift and deformation are well captured as well as the seasonal cycles of ice volume, area, and extent. These results show that neXtSIM is an appropriate tool for simulating sea ice over a wide range of spatial and temporal scales.

Sea ice dynamics are very complex and share many characteristics with earth
crust dynamics, such as dynamical triggering and clustering of deformation
events or earth/ice quakes. Both sea ice and the earth's crust are
geophysical solids that can be viewed from the mechanical point of view as
two-dimensional plates due to their very small geometrical aspect ratio
(

There has recently been renewed interest in further development of various aspects of dynamical sea ice models. This includes
research on developing different solvers for the standard VP/EVP
rheology

One of the reasons to redesign or replace the VP and EVP rheologies is that
classical models give a poor representation of ice drift and deformation
statistics and scaling, compared with satellite observations

The main goal of the neXtSIM development is to reproduce the mechanical
behaviour and state of the Arctic sea ice cover on seasonal timescales (over
1

This paper presents the first comprehensive version of the neXtSIM model, a
fully Lagrangian dynamical/thermodynamical sea ice model. A generic
presentation of the model is made in Sect.

List of variables used in neXtSIM.

The sea ice variables used in neXtSIM are the following:

The evolution equations for

The evolution of

The evolution of sea ice velocity derives from the vertically integrated sea ice momentum equation:

The evolution of the internal stress is computed as in

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

The inertial mass

The failure envelope is defined as in

The effect of the concentration on the mechanical response of sea ice is
parameterised here by a decreasing exponential function of the concentration:

The air stress

The oceanic stress

The basal stress

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

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

The change in

The source/sink term for snow thickness,

Thickness changes in level ice and snow are calculated using the zero-layer
model of

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:

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

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.

Most sea ice models use an Eulerian approach for the advection. However, we
believe that a purely Lagrangian approach as in

In the purely Lagrangian approach, the vertices of the element (i.e. the
nodes of the grid) move with the sea ice velocity

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

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

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

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

Model domain projected on a polar stereographic plane, with open boundaries in green. The region delimited by the dashed blue line and the cyan area is used to compute the drift and deformation statistics. The dashed line in magenta shows the area for which the mean ice thickness and ice volume time series are calculated.

The oceanic forcing variables are sea surface height, velocity at
30

The atmospheric forcing consists of the 10

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 (

Parameters used in the model with their values for the simulations performed for this study.

In neXtSIM, as in most classical sea ice models, the air and water drags
depend on four parameters:

Number of occurrence of free drift events identified between 31 October 2007 and 28 April 2008 and selected for the optimisation of the air drag parameter.

By performing a scale analysis, it can be shown that the sea ice momentum equation (Eq.

Scatter plots for the two components of the simulated and observed drift selected from the air drag optimisation procedure (left and middle panels). Cumulative distribution of the velocity errors (right panel).

The calibration method uses a full winter of sea ice drift data (here from
the GlobICE data set,

Scatter plots for the two components of all the simulated and observed drift vectors between 31 October 2007 and 28 April 2008 (left and middle panels). Cumulative distribution of all the velocity errors (right panel).

Sensitivity of the velocity statistics to the healing timescale. The
left panel shows the correlation between the simulated and observed ice drift,
the central panel shows the RMSE and the right panel the velocity mean and median
velocity errors. The dots in green correspond to the reference run (28

By optimising the air drag parameters for these selected free drift vectors,
we find an optimal value of

The quality of the simulated sea ice drift is evaluated by comparing simulated
velocity vectors to the ones provided by the RGPS and GlobICE data sets between
31 October 2007 and 28 April 2008. The high spatial and temporal resolution
of these data sets (about 3

The sensitivity of the correlation, RMSE, and velocity errors to the healing
timescale parameter is presented on
Fig.

Simulated and observed ice drift averaged over the period between 31 October 2007 and 28 April 2008 (left and middle panels). The number of observations is shown in the right panel. The mean fields are built on a regular grid with a resolution of 21 km and are computed by averaging the components of the simulated and observed drift vectors used for the scatter plot. Note that the colour scale for the number of observations is capped at 30 to show that some regions are poorly covered by data.

To identify potential systematic errors, we also look at the mean sea ice
drift by averaging modelled and observed drift over the whole season on a
mesh grid of 21 by 21

Difference between the observed and simulated mean ice drift
shown on Fig.

The mean drift speed, taken over the central Arctic, correlates closely with
the mean wind speed taken over the same area. This is to be expected, since
the wind is the main driver of ice drift. We do, however, expect to see a
significant difference between the ice response to wind in summer and in
winter, due mainly to changes in ice concentration and thickness. In order to
assess this effect, Fig.

Ratio of drift speed over wind speed for the reference simulation forced with ASR-Interim (cyan) and a simulation forced with ERA-Interim (blue). As a reference the same ratio is shown for the IABP buoys drift speed climatology over the ERA-Interim wind speed climatology (green).

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.

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

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

Example of deformation fields simulated by neXtSIM. The divergence rate, shear rate, and vorticity are computed from the Lagrangian displacement simulated between 20 and 21 February 2008. One could note that a large divergence rate coincides with a large shear rate and that landfast ice is present on the Siberian coast and east of Kara Sea.

Mean value of the shear rate distributions corresponding to 13 periods
of 9

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

Scaling analysis of sea ice deformation performed for the period
January–May. The left panel shows the mean shear rate (i.e., the first-order
moment of the distribution) computed for spatial scales ranging from

Temporal scaling analysis of sea ice deformation performed for the period
January–May. Left panel shows the mean deformation rate (i.e., the 1

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

Evolution of the mean shear rate simulated by the model (left panel) and
of the corresponding spatial scaling exponent (right panel). The circles correspond to
the values computed for each of the 125 periods of 3

Modelled seasonal cycle in volume (left panel), extent (centre panel), and
area (right panel). The volume is calculated within the area covered by the ICESat
observations (

The simulated mean value and spatial scaling exponent of the 3-day
deformation evolves during the year of simulation (see
Fig.

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

Modelled ice concentration at the observed extent minimum (19 September 2008). Overlaid are lines for the modelled and observed (AMSR-E) 15 % concentration limit in white and cyan, respectively.

Ice thickness for the initial conditions (left panel), the model state at midwinter (centre panel), and the model state at observed extent minimum (right panel). Note the increasing heterogeneity in the sea ice thickness field emerging from the new physics included in the model.

Figure

Figure

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

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

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.

In addition to concentration, Fig.

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

In this paper we presented the first comprehensive version of the neXtSIM
model, a fully Lagrangian dynamical/thermodynamical model for sea ice. The
model is built around the dynamical core previously described in

In order to be able to run simulations for seasonal timescales we have
developed and implemented the following numerical and physical components
into the model:

local dynamic remeshing accompanied with an efficient and conservative remapping scheme;

a thermodynamics model coupled to a slab ocean;

a healing parameterisation which simulates the restoration of mechanical strength due to refreezing of leads.

In order to evaluate the performance of neXtSIM we used a full Arctic set-up
and ran the model for 13 months, starting on 15 September 2008, and using
realistic atmospheric forcing. The main evaluation results are as follows:

the model reproduces the local motion of sea ice that is in free drift well;

the model also reproduces the drift of the pack ice well, at local (

the model captures the observed spatial multi-fractal scaling of sea ice deformation over 3 orders of magnitude, from

the model captures the observed intermittency of sea ice deformation over 2 orders of magnitude, from 1 to

the model produces seasonal cycles of sea ice volume, area, and extent that are all in good agreement with observations.

In conclusion, for scales smaller than a year, neXtSIM performs very well with respect to several important metrics related to sea ice dynamics and thermodynamics. We believe that in its current stage of development, neXtSIM may already be a useful tool for both the scientific and engineering communities. For longer timescales and to study the interactions between sea ice and the ocean, ecosystems, or the atmosphere, more developments are needed, especially on the coupling with other components and the use of a more advanced sea ice thermodynamics model.

The atmospheric reanalysis data used in this article are available at

We would like to acknowledge T. Williams and P. Griewank for interesting discussions and their contribution to the overall model development. This work was funded by the Research Council of Norway via the SIMech project (no. 231179/F20, 2014-2016), the Oil and Gas Producers (OGP), and TOTAL E&P. This work is the follow-up of research supervised by Jérôme Weiss, David Marsan, and Bernard Barnier at Grenoble and Thierry Fichefet and Vincent Legat at Louvain-la-Neuve. We would like to thank them for their support. Edited by: G. H. Gudmundsson