In this paper, we evaluate the neXtSIM sea ice model with respect to the observed scaling invariance properties of sea ice deformation in the spatial and temporal domains. Using an Arctic setup with realistic initial conditions, state-of-the-art atmospheric reanalysis forcing and geostrophic currents retrieved from satellite data, we show that the model is able to reproduce the observed properties of this scaling in both the spatial and temporal domains over a wide range of scales, as well as their multi-fractality. The variability of these properties during the winter season is also captured by the model. We also show that the simulated scaling exhibits a space–time coupling, a suggested property of brittle deformation at geophysical scales. The ability to reproduce the multi-fractality of this scaling is crucial in the context of downscaling model simulation outputs to infer sea ice variables at the sub-grid scale and also has implications for modeling the statistical properties of deformation-related quantities, such as lead fractions and heat and salt fluxes.

The fact that sea ice deformation maps look similar at different scales, with highly localized deformation features intersecting with a wide range of intersection angles

In the spatial domain, deformation is observed as being concentrated along quasi one-dimensional, so-called linear kinematic features (LKFs) organized in “web-like arrays”

A quantitative indication of scale invariance in sea ice deformation is given by the shape of the distribution of deformation rate invariants (i.e., shear and divergence) and the total deformation rates, which we refer to here as

Localization in the time and space domains is revealed by scaling analysis of the deformation rate invariants. In such analyses, deformation rates are estimated at different spatial and temporal scales, by such methods as “coarse-graining” (see Sect.

The fact that sea ice deformation rates are characterized by a heavy-tailed statistical distribution, i.e., dominated by extreme events, also indicates that the mean (moment of order 1) is not a sufficient quantity to describe the distribution of deformation rates at a given timescale or space scale. Higher moments of the distribution of deformation rates, such as the variance (order 2) and skewness (order 3), should indeed be explored to better describe the distribution and the associated process of sea ice deformation and be considered in temporal and scaling analyses, as proposed in this study.

While the value of the scaling exponents

Spatial scaling analysis of sea ice deformation retrieved from radar or buoy drift data show a clear multi-fractal scaling expressed by a power-law scaling of the first, second and third moments, ranging from the resolution of the data up to hundreds of kilometers

These properties of sea ice deformation imply that observations of these quantities available at large scales can be statistically related, i.e., downscaled, to the same quantities at smaller, unresolved scales. In the case of model simulations, downscaling of outputs could be particularly valuable to infer quantities at the sub-grid and/or sub-time-step scale. In this context, the capability of these properties to reproduce in mono-fractal versus multi-fractal manner becomes very important. Indeed, if one was to estimate the distribution of a variable at the sub-grid scale based on a model that would not reproduce the observed multi-fractality but only a mono-fractality, the downscaled distribution of this variable would greatly underestimate extreme values.

The multi-fractal behavior of sea ice has been the subject of a large number of interesting studies and is hypothesized to be of significant importance for sea ice rheology

The observed self-similarity and multi-fractality in the deformation and related characteristics of sea ice actually poses great challenges to the development of sea ice models, in particular in the continuum framework. On the one hand, the momentum and evolution equations for sea ice properties are based on mean variables. On the other hand, however, the observed multi-fractality in sea ice deformation implies that there is not a clear separation of scales between the strain rate due to mesoscale (50–100 m) heterogeneities in the ice (leads, ridges, etc.) and the strain rates at the 10 to 100 km scale. Consequently, no scale appears appropriate for homogenizing sea ice motion and thereby define a mean velocity or deformation rate for model resolutions ranging from 50 to 100 km.

In the absence of a characteristic space scale and timescale for the sea ice deformation, perhaps the best a continuum framework for sea ice modeling can do is to correctly reproduce the statistics of deformation from the smallest scales resolved (the nominal scale) to the largest scale, i.e., from the resolution of the grid in space and the model time step in time to the size of the Arctic basin and the timescale of a season. This is one of the motivation in developing neXtSIM, the numerical model used in the current study.
Such localization at the nominal scale is the most faithful representation of the discontinuous nature of sea ice possible in a continuum model. Knowing the importance of essentially discontinuous features, such as leads, for atmosphere–ocean interactions modulated by sea ice

This paper is the last step in validating neXtSIM against sea ice deformation statistics. While previous works have shown that the model reproduces the observed scaling of sea ice deformation

Section

neXtSIM is a finite elements sea ice model that uses a moving Lagrangian mesh.
Its original dynamical component was based on the elasto-brittle (EB) mechanical framework of

However, the elasto-brittle model does not, by definition, include a physical mechanism for irreversible deformations, as it is based on a strictly linear-elastic constitutive law. It therefore cannot represent the transition between the small, elastic deformations associated with the fracturing of the ice cover and the permanent, potentially large, post-fracturing deformation that dissipates internal stresses. It is therefore not suited to represent the dynamical behavior of a fractured ice cover over long (

The recent Maxwell elasto-brittle (MEB) rheology addresses this limitation of the EB framework by including a mechanism for the relaxation of internal stresses that depend on the degree of fracturing of the sea ice cover

All of the relevant equations of the current version of neXtSIM are presented in the appendices (Sect.

The initial mesh is generated in preprocessing over a pan-Arctic region by using the mesh generator presented in

The atmospheric forcing consists of the 10 m wind velocity, 2 m air temperature, mixing ratio, mean sea level pressure, total precipitation amount and snow fraction, and incoming shortwave and longwave radiation. All of these quantities are provided as 3-hourly means and on a 30 km spatial resolution grid from the atmospheric state of the Arctic System Reanalysis

The ice-ocean surface stress is computed from monthly ocean surface geostrophic currents derived, following

Our reference simulation starts on 15 November 2006. The level of damage in the ice cover (see Appendix

We use the Lagrangian displacement data produced by the RADARSAT Geophysical Processor System (RGPS) as described in

Scaling analyses of sea ice deformation can be performed using two approaches: a so-called coarse-graining method as in, e.g.,

Drifters in the model are seeded at the location of the RGPS grid points as of 3 December 2016. The RGPS grid for this initialization is undeformed and the points are regularly spaced by 10 km. The positions of the simulated drifters are updated at each model time step until the end of the simulation or until the ice concentration drops to zero (through melting or opening of a lead). Both the RGPS and simulated trajectories are filtered for the presence of coasts, with a proximity threshold of 100 km. Only the trajectories spanning the same time periods in both the simulation and RGPS dataset are considered in the calculation of the deformation and their statistics. This selection led to discarding only about 1 % of the total trajectory dataset and does not affect the results of the analyses presented in this paper. However, we apply this selection in order to make our comparison between model and observations as consistent and clean as possible.

Triplets of drifting points are defined as the result of Delaunay triangulation of the initial positions of the tracked RGPS points, which ensures that the associated polygons are independent, i.e., nonoverlapping. The exact same triplets of points are considered in the model for the analysis, meaning that we follow the exact same set of triplets of trajectories (or triangles) in the model and in the observations. The polygons after initiation are defined by the positions of their three nodes at any given time. We stress the fact that the simulated trajectories are not re-initialized every 3 d to match the RGPS positions; only the initial positions are identical between the RGPS and the model trajectories.

To perform a spatial scaling analysis of sea ice deformation, one needs to consider triplets of points with different spacing, i.e., different sizes of polygons. In order to obtain sets of polygons of different surface areas, we perform successive Delaunay triangulation through the clouds of points defined by the initial positions of the RGPS points, using increasingly subsampled clouds of these points. Each set of contiguous polygons obtained using this process is associated to a spatial scale,

To perform a temporal scaling analysis of sea ice deformation, one also need to consider the positions of triplets of drifters separated by different times

For each available polygon, the total deformation rate is calculated as follows:

The distribution of total deformation rates is constructed from each given coupled space scale–timescale (

Below we discuss some issues that are inherent to the data and our method and their impact in terms of the robustness of the statistics calculated here.

With time, the triangular elements can become too distorted, in which case their length scale,

The RGPS trajectories are not sampled at regular time intervals, contrary to the model, due to the irregular interval between the two satellite orbits. The mean sampling is of about 3 d, and 90 % of trajectories are sampled with a frequency between 2.5 and 3 d. Because sea ice deformation depends on the timescale (see results of Sect.

The 3 d RGPS sampling additionally places a lower bound on the timescales one can explore when comparing the simulated and observed deformation rates. In the present analysis, we therefore restrict ourselves to timescales equal to or greater than 3 d.

We find that the relative number of available polygons is what has the largest impact on the deformation statistics. Some facts therefore need to be kept in mind when performing a scaling analysis over a finite period of time. In the time domain in particular this means that sea ice deformation is better sampled, i.e., more triplets are available, for the early than for the late part of the period and for short timescales

Figure

Divergence, shear and total sea ice deformation rates per day (top to bottom), as simulated by the model

Probability density functions of the absolute divergence, shear and total deformation rates shown in the maps of Fig.

Figure

Spatial scaling analysis of the observed (black) and simulated (blue) total deformation rate calculated over a timescale of 3 d from the motion of the same triplets in the model and the RGPS dataset.

Using successive and contiguous snapshots throughout the winter, a time series of the value of the spatial scaling exponent

Time series of spatial scaling exponents for the mean total deformation (i.e.,

We further characterize the properties of the spatial scaling for both the model and observations by exploring its dependence on the temporal scale,

We also note a decrease in the multi-fractal character of the spatial scaling (i.e., the curvature of

Same as Fig.

Same as Fig.

Curvature of the structure function as a function of the timescale

The results of the temporal scaling analysis for

Temporal scaling analysis of the total deformation rate derived from the motion of the same triplets with initial surface area of

We note, however, that the third moments of the distributions are slightly underestimated by the model at all timescales. This means that the proportion of extreme deformation events compared to lower ones is too small or that their values are too low in the simulation. This may come from the inaccuracy of the relatively coarse (30 km) atmospheric reanalysis we use to force our model and that is known to poorly resolve the most extreme low pressure systems, a common shortcoming of all the available global or regional atmosphere reanalysis to date. Another explanation could be the fact that we have not tuned the MEB rheology parameters to reproduce the proportion of extreme deformation events versus the less extreme events. In this rheology, the coupling between the damage and the mechanical behavior of sea ice is nonlinear and it is therefore expected that varying parameter values can change the proportion of the simulated extreme events, i.e., the skewness of the distribution of deformation rates.

As in the spatial domain, the temporal scaling is found to be multi-fractal for the model and observations, and the match is remarkably good. The curvature values (i.e., the coefficient

We also investigate the dependence of the temporal scaling on the spatial scale of observation,

Same as Fig.

Same as Fig.

Curvature of the structure function as a function of the space scale

Our statistical analyses have shown that the neXtSIM model correctly reproduces the distribution of sea ice deformation rates, its scaling properties in both the space and time domains and its multi-fractal behavior. In particular, it is the first time that multi-fractality in the time domain is shown to be reproduced in a sea ice model.

The MEB rheology was developed with the aim of improving the representation of the physics of sea ice continuum models by including the ingredients hypothesized by

We show here that the spatial scaling of sea ice deformation simulated in a realistic setup by neXtSIM holds down to the nominal resolution of the mesh, a result that is in agreement with previous analyses of the MEB model in idealized simulations

We also show that this spatial localization and the multi-fractal character of the simulated mean sea ice deformation is resolution-independent in the model (see Fig.

Spatial scaling analysis of the simulated deformation derived from the motion of triplets over a timescale of

A model that allows for reproducing sea ice deformation and its scaling properties down to its nominal resolution does not preclude the need for appropriate sub-grid-scale parameterizations. On the contrary, we believe that physically sound parameterizations are indeed required and that the knowledge of the distribution of deformation rates at the sub-grid scale made possible by neXtSIM could be highly valuable in terms of informing these parameterizations. An appropriate sub-grid-scale parametrization should link the deformation simulated at the scale of the grid cell with the scale at which deformation really occurs within the ice cover, which is the size of individual leads and ridges.

Moreover, we argue that, as sea ice deformation is strongly tied to other model variables, such as drift, lead fraction and thickness distribution, a proper simulation of these variables is a necessary prerequisite to using models for investigating various coupled ocean–ice–atmosphere processes, and their impact on their immediate vicinity and on the polar climate system. For example, the accuracy of neXtSIM in reproducing the observed statistical properties of sea ice deformation as demonstrated in this paper is thought to go hand-in-hand with its capability of representing the observed properties of lead fraction. This is the subject of a parallel study that is in preparation.

In this study we have compared the deformation rates simulated by neXtSIM to those derived from RGPS observations on the basis of their distributions and have shown how these distributions scale in time and space. The conclusions of our analysis are as follows.

The neXtSIM model reproduces the first, second and third moments of the statistical distribution of observed sea ice deformation rates and how it scales in space and time well. In particular, this is the first time the observed scaling invariance in the temporal domain (i.e., intermittency) of sea ice deformation is shown to be reproduced by a model on a realistic pan-Arctic setup over such a large range of scales.

Sea ice deformation rates calculated over a temporal scale of 3 d scale in space from the scale of the model (mesh resolution) and observations up to about 700 km in a multi-fractal manner.

Sea ice deformation rates calculated over a spatial scale of 7.5 km scale in time over the range of 3 d to 3 months in a multi-fractal manner.

A space–time coupling in the scaling properties of sea ice deformation is shown to be reproduced by the model. This suggests that neXtSIM could be the proper tool for studying the physical meaning and origin of this coupling, in the context of brittle deformation of geophysical solids.

The simulated mean sea ice deformation rates and their associated scaling invariance characteristics are resolution-independent, i.e., when running the neXtSIM model at resolutions of 7.5, 15 or 30 km. The most extreme deformation events may be missed, however, if running at coarser resolutions, i.e., the second- and third-order moments may be underestimated compared to the high-resolution run.

As the mono-fractal versus multi-fractal character of the scaling of deformation rates is the discriminating factor for the heterogeneity and intermittency of the deformation, we suggest that a multi-fractal scaling analysis could be considered a meaningful validation step before further analyzing sea ice model outputs that could be influenced by sea ice dynamics.

The good agreement between the model and observations motivates the use of neXtSIM as a tool to further investigate physical processes that are highly sensitive to sea ice deformation.

The datasets used for this study (forcing data, ocean topography, sea ice initial conditions) are all publicly available at the URL addresses mentioned in the main text. Outputs of the simulations analyzed in this study are not publicly accessible but are available upon request. The actual code of the numerical model neXtSIM, despite being hosted on Github (

This section presents the dynamical and thermodynamical components of neXtSIM. The wave-in-ice module implemented by

List of variables used in neXtSIM.

Parameters used in the model with their values for the simulation at 7.5 km resolution used for this study.

The evolution equation for sea ice velocity comes from vertically integrating the horizontal sea ice momentum equation as follows:

Following

As in

The evolution of the damage is controlled by the location of the predicted stress state relative to the failure envelope, which is defined in terms of the principal stress components

Here, the envelope combines a Mohr–Coulomb failure criterion and maximum tensile and compressive stress criteria. These are given by

When one of the damage criteria is met,

Healing is included here to represent the counteracting effect of refreezing of water within leads on the level of damage of the ice cover. It is implemented via a constant term in the damage evolution equation:

neXtSIM includes a multi-category model inspired from

Note that the total ice concentration and volume per unit area are

Thin ice thickness is considered to be uniformly distributed with thickness

The evolution of

Compute the new open-water concentration as follows:

Compute the new thin ice concentration as follows:

Compute the transfer of thin ice if

Here, we have transferred ice and snow volume from thin to thick ice in a conservative manner,
but we will not try to conserve ice area;

Compute the new thick ice concentration as follows:

Apply more ridging if

This work is the result of a long-term team effort at the Nansen Centre in Norway carried out by, in alphabetical order, SB, VD, EO, PR, AS and TW to develop the new sea ice model neXtSIM. PR led the research and performed the analyses presented in this paper; VD and PR led the writing with contributions from EO, SB, TW and AK. AS implemented the parallel C

The authors declare that they have no conflict of interest.

This paper was prepared thanks to the financial support of the Research Council of Norway (RCN) through the FRASIL project (grant no. 263044). However, the development of the neXtSIM model has been supported since 2013 through several projects from the RCN, in particular the SIMECH (grant no. 231179) and NEXTWIM (grant no. 244001) projects. TOTAL E&P are also thanked for their continuous support over the period 2013–2017 through the KARA project. And finally, we thank Jerome Weiss and David Marsan for fruitful discussions.

This paper was edited by Christian Haas and reviewed by two anonymous referees.

This research has been supported by the Research Council of Norway (grant nos. 263044, 231179 and 244001).