Fragmentation of the sea ice cover by ocean waves is an important mechanism impacting ice evolution. Fractured ice is more sensitive to melt, leading to a local reduction in ice concentration, facilitating wave propagation. A positive feedback loop, accelerating sea ice retreat, is then introduced. Despite recent efforts to incorporate this process and the resulting floe size distribution (FSD) into the sea ice components of global climate models (GCMs), the physics governing ice breakup under wave action remains poorly understood and its parametrisation highly simplified. We propose a two-dimensional numerical model of wave-induced sea ice breakup to estimate the FSD resulting from repeated fracture events. This model, based on linear water wave theory and visco-elastic sea ice rheology, solves for the scattering of an incoming time-harmonic wave by the ice cover and derives the corresponding strain field. Fracture occurs when the strain exceeds an empirical threshold. The geometry is then updated for the next iteration of the breakup procedure. The resulting FSD is analysed for both monochromatic and polychromatic forcings. For the latter results, FSDs obtained for discrete frequencies are combined following a prescribed wave spectrum. We find that under realistic wave forcing, lognormal FSDs emerge consistently in a large variety of model configurations. Care is taken to evaluate the statistical significance of this finding. This result contrasts with the power law FSD behaviour often assumed by modellers. We discuss the properties of these modelled distributions with respect to the ice rheological properties and the forcing waves. The projected output can be used to improve empirical parametrisations used to couple sea ice and ocean wave GCM components.

Sea ice is a distinctive feature of both polar oceans and has a profound influence on our climate.
It blankets a significant fraction of the Earth, is hard to reach, and offers particularly harsh fieldwork conditions.
Consequently, numerical modelling is a valuable tool not only for forecasting ice extent evolution, but also to gain insights, at a global scale, into the physical processes shaping this evolution.
Hindcast results straying away from observations

The marginal ice zone (hereafter MIZ), a belt of loosely to densely packed ice floes, serves as a buffer between the ice-free open ocean and the pack ice; it is a region notably affected by waves

Remote sensing observations of floe sizes (e.g.

For the last decades, wave–ice interaction has been an active field of research, and numerous theoretical models, gradually benefiting from advances in computing power, have been developed.
Classically, the goal was to understand the attenuation imposed on the waves by the ice cover

The reciprocal response of the ice to the waves is unsatisfactorily understood, as direct observations of wave-induced floe breakup are scarce and localised in both time and space.
Numerical assessment of the feedbacks between breakup and wave propagation were pioneered by

^{®} (WW3).
Wave attenuation is determined by a mean floe size, computed assuming a power law FSD.
Their work was extended by coupling WW3 to the sea ice models LIM3

Recent numerical experiments have been conducted to investigate the wave effect on the FSD without a priori assumptions on the distribution shape.

In this study, we model the wave-induced breakup process with the aim of quantifying the resulting FSD.
Breakup happens on timescales shorter than other processes affecting the FSD, such as thermodynamics

This paper unwinds as follows.
In Sect.

We consider surface gravity waves propagating in a two-dimensional fluid domain of constant, finite depth

We place an array of

The system is forced by a monochromatic plane wave of angular frequency

Geometry of the model at rest. Wave forcing would alter the fluid boundaries around

We consider the seabed to be impervious, hence not allowing for normal flow, so that on

The small amplitude forcing allows us to use linear surface wave theory in

We also neglect the surge motion of the floes, meaning that

We finally add the free-edge conditions

The boundary conditions given in Eqs. (

The solution to the boundary value problem described in Sect.

In any sub-domain

The wavenumbers {

Likewise, the wavenumbers

Since

For any wave mode

Finally, the functions

We obtain the solution to the multiple scattering problem of the incident wave by the ice floes by imposing continuity of pressure and normal velocity across the vertical boundaries between adjacent ice-free and ice-covered sub-domains, i.e. at each floe edge.
These conditions are enforced by matching

Considering the scattering by the left edge of floe

By symmetry, the scattering by the right edge of floe

The reflection and transmission matrices depend only on the quantities present in the dispersion relations, Eqs. (

We assess convergence of the numerical procedure by investigating energy conservation after scattering for floes of zero viscosity.
Our analysis proved

Wave fields radiated by adjacent floes are coupled, which is clearly shown by Eqs. (

An array of

As the last floe here is

The vector of unknown coefficients is

Building

To parametrise the breakup, we build upon the commonly used strain-based approach (e.g.

When using the plane stress approximation and our symmetry assumption, the strain

Floe

We situate the breakup point at

The values of the parameters kept fixed across all simulations are given in Table

The model is initialised with a set of physical parameters as input and a single, semi-infinite floe.

The scattered wave field is determined and the strain field evaluated for every floe in the domain. All the floes for which the conditions for breakup are met are split, and the domain is updated.

The second step is repeated until none of the floes break or a prescribed number of iterations reached.

Outline of the numerical experiment.

For a given iteration, all floes are scrutinised for breakup first, then their positions are updated.
As we neglect floe motion, to prevent them from overlapping, they are assigned order-preserving random locations, determined through a process described in Appendix

We note that in the arrangement considered here, the transmission and reflection matrices (Eqs.

The final result extracted from the simulation is the set of newly formed floe lengths, excluding the semi-infinite floe on the right of the domain, considered to be a steady-state FSD.

Fixed model parameters and their values.

We first investigate the FSD our model generates under monochromatic forcing with prescribed wave period

Figure

Impact of varying

We obtain unimodal, right-skewed histograms (Fig.

Increasing

Figure

Various metrics at the end of a simulation.

The final number of floes (number of floes reached when the forcing wave field no longer breaks any floe during a simulation) depends sharply on

The minimum floe size, shown in Fig.

The breakup width, here defined as the cumulated length of ice broken off the semi-infinite floe and shown in Fig.

Our model, as described in Sects.

To estimate the effect of a developed sea on the FSD

This FSD model assumes that different frequency components of the wave spectrum affect the FSD independently from each other. This is a strong assumption, the validity of which is discussed in Sect.

We evaluate Eq. (

For each of the 200 frequencies, we draw an FSD

We consider a reference configuration where

The resulting FSD, shown in Fig.

The associated density function

We obtain a point estimate

We outline the goodness of fit with a quantile–quantile plot shown in Fig.

We do not expect the FSD to have the same shape all across an ice pack or even across the MIZ. To illustrate this effect, we analyse the evolution of the distribution when considering subsets of the domain.

Our experiment design generates distributions in parallel, for various periods in the spectrum.
Therefore, there is no unique definition of the breakup width.
To circumvent this issue, we use a sliding window whose bounds are relative to the local breakup width for each period used in the discretisation of the spectrum.
We estimate the density

The distributions remain remarkably lognormal, with the distribution parameters following regular trends for most positions of the sliding window (Fig.

We expand the analysis conducted in Sect.

We observe a remarkably good fit over most of the parametric space explored, with a few notable limitations.
The smallest waves (significant wave heights between

As can be seen in Fig.

The distribution parameters do not have a clear physical significance by themselves.
Beyond the estimation of summary statistics, their main interest is the generation of floe sizes samples without the numerical cost of running the physical model.
We illustrate such forecasts, and the associated errors, in Fig.

Figure

Figure

The emergence of a lognormal FSD from repeated wave-induced breakup is the key outcome of this paper.
It contrasts with the power laws often assumed in modelling studies

The lognormal model does come with limitations.
Field observations show the extensive spatial variability in the FSD.
For instance,

Our polychromatic forcing simulations (Sect.

The Pierson–Moskowitz spectrum chosen in our polychromatic simulations was selected for its simplicity.
To make sure our results are not qualitatively sensitive to the choice of spectrum, we conducted additional simulations of FSD generation for a range of different spectra, including symmetric ones.
The simulations are described, and results are discussed in Appendix

Additionally to the wave height, the ice thickness, and the strain threshold, we analysed the effect of varying the ice viscosity

A framework to model the evolution of the ice thickness distribution (ITD) has been introduced by

Parametric distributions may never be flawless descriptions of the quantities they model. In this study, we show the relevance of the lognormal distribution when considering wave-induced breakup. We describe the evolution of the distribution shape for a range of ice properties, under various wave forcings. These results aim at being a step towards the parametrisation of wave action in FSD-evolving models.

This section details the process of redistributing the floes after a breakup event. The length of the gap between two floes does not matter inasmuch as the fluid is inviscid, and no wave energy is lost to it. The two main constraints are preserving the order of the floes and ensuring they do not overlap.

Floes are positioned from left to right and localised by their left edge, whose location is drawn from a uniform distribution.
The leftmost floe is placed in an interval of fixed width, symmetric around its location at the previous iteration.
For subsequent floes, the left bound corresponds to the right edge of the last positioned floe (on their left).
The right bound corresponds to the previous right bound, augmented by the length of the last positioned floe.
Locations are drawn between these two bounds; an illustration is given in Fig.

We ran simulations with alternative values to ensure these values do not have any impact on our results.
We used the case presented in Sect.

Distribution sensitivity to various parameters of random positioning. All quantities, except the sample size, are expressed in metres. The statistics are averages computed over 50 realisations. The support is truncated to the 0.5th and 99.5th percentiles.

Illustration of the floe repositioning method. The top row shows current floes with identified breakup location marked by vertical bars and the resulting lengths. Successive rows show the iterative positioning as described in the text. Below each row, a segment shows the interval from which a location will be randomly drawn for the next floe; the cross marks that location.

Complementary CDFs of floe lengths, averaged over 50 model realisations, with varied positioning parameters and model parameters mentioned in the text.

We ran comparisons using alternative weighting functions, represented in Fig.

The lognormal density function has, qualitatively, a shape similar to the Pierson–Moskowitz spectrum expressed as a function of frequency.
However, we obtain similar, skewed unimodal densities with a range of symmetrical weighting functions, as displayed in Fig.

The one case that stands out, with a lot of secondary peaks, is n_md.
It corresponds to a function giving more weights to high-frequency waves (5 to

Details of several weighting methods.

Representation of functions used as alternative weights.
They are all normalised on the positive real half-line; because of the finite range of frequencies supported by our model, truncations imply that some of them integrate to less than unity.
Description of the legend entries can be found in Table

Densities obtained when combining monochromatic model runs with different weighting functions, as represented in Fig.

Histograms and average lognormal fits, as described in Sect.

Histograms and average lognormal fits, as described in Sect.

Histograms and average lognormal fits, as described in Sect.

The Kolmogorov–Smirnov statistic

Instead of rejecting the lognormal hypothesis at an arbitrarily chosen confidence level, we report

We report the results in Figs.

Distribution of

Same as Fig.

Same as Fig.

The model code, software tools developed for analysis, and the resulting preprocessed output presented in this paper are publicly available at

NGAM and FM designed the numerical experiments. NGAM developed the model code, ran the simulations, and conducted the analysis. NGAM prepared the manuscript with significant input from and under the supervision of FM.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank Christopher Horvat for proposing the method of combining FSDs obtained at individual frequencies in order to form the FSD resulting from the breakup by a wave spectrum (Eq. 23).

This research has been supported by the University of Otago (doctoral scholarship), the Marsden Fund (grant nos. 18-UOO-216 and 20-UOO-173), and Antarctica New Zealand (Antarctic Science Platform Project 4).

This paper was edited by Yevgeny Aksenov and reviewed by three anonymous referees.