The energy of water waves propagating through sea ice is attenuated due to non-dissipative (scattering) and dissipative processes. The nature of those processes and their contribution to attenuation depends on wave characteristics and ice properties and is usually difficult (or impossible) to determine from limited observations available. Therefore, many aspects of relevant dissipation mechanisms remain poorly understood. In this work, a discrete-element model (DEM) is used to study one of those mechanisms: dissipation due to ice–water drag. The model consists of two coupled parts, a DEM simulating the surge motion and collisions of ice floes driven by waves and a wave module solving the wave energy transport equation with source terms computed based on phase-averaged DEM results. The wave energy attenuation is analysed analytically for a limiting case of a compact, horizontally confined ice cover. It is shown that the usage of a quadratic drag law leads to non-exponential attenuation of wave amplitude

As ocean waves propagate through sea ice, they undergo attenuation due to both non-dissipative and dissipative processes. Whereas attenuation due to non-dissipative scattering has been extensively studied and can be regarded as well understood

Depending on wave forcing and sea ice properties, the relative importance of individual dissipative processes varies; similarly, the relative contribution of scattering and dissipation to the overall wave attenuation is strongly dependent on wave and ice conditions. In general, attenuation due to scattering at floes' edges tends to dominate at relatively low ice concentration and, crucially, when the floe sizes are comparable with wavelengths

Considering the multitude of processes contributing to wave energy attenuation, it is not surprising that the observed attenuation rates span a few orders of magnitude

Among the most crucial characteristics of wave energy attenuation in sea ice are the functional dependence of wave amplitude

In most studies, exponential attenuation is assumed, and no alternative forms of

Another problem with some models predicting exponential attenuation is related to the second of the two attenuation characteristics mentioned above – they produce

In this work – described below and in the companion paper

The present study is to a large extent motivated by the results of

After formulating the assumptions and equations of the sea ice and wave model in the next section, we begin our study with a theoretical analysis of energy dissipation induced by ice–water drag in a special, limiting case of waves propagating through horizontally confined ice (i.e. with zero horizontal velocity). We show that the attenuation equation in this case can be solved analytically and that this model configuration leads to non-exponential attenuation of the form of Eq. (

As already mentioned in the Introduction, the model used here consists of two coupled parts: a sea ice module and a wave module. The sea ice part is based on the DEM by

We consider linear, unidirectional, progressive waves with period

The angular frequency and the wavenumber are related by the following dispersion relation

The choice of dispersion relation (Eq.

It must be noted that in the case of small-amplitude, irrotational water waves propagating under multiple elastic, non-colliding plates floating on the surface, the velocity potential – and thus the velocity components – can be expressed, for each plate, as a sum of transmitted and reflected waves, each in turn consisting of travelling, damped travelling, and evanescent modes

As already mentioned, the model is one-dimensional, i.e. the ice floes are placed along the

As in

As marked explicitly in Eqs. (

For an individual ice floe with bottom surface area

We use a very unsophisticated approximation of overwash effects, the development of which was motivated by the observation that strong overwash occurred in laboratory tests analysed in Part 2. The algorithm described here should be treated as a framework for future parameterizations rather than as an ultimate solution.

Following

In the model described in Sect.

The model is run over

Over the next

New wave amplitude

If overwash effects are taken into account,

In the present model version, when computing

It is worth noting that the number of iterations necessary for convergence increases with the distance over which attenuation is computed – as each location is affected by the situation in the up-wave direction, the convergence criterion is reached very fast close to the ice edge, and the required number of iterations increases with increasing

Before proceeding to an analysis of full DEM simulations with collisions, it is useful to consider a limiting case with ice concentration

Notably, Eq. (

In the case of the mass loading model,

If

Group velocity

This very different behaviour of

We set up the DEM for conditions corresponding to those from LS-WICE series 3000 (see Part 2). The ice sheet is 42 m long, and three floe lengths

As described in Part 2, the ice in LS-WICE was constrained horizontally by a floating boom and a sloping beach. In DEM, an analogous effect is obtained by adding a linear spring force

The time step

As already mentioned, the analysis presented in the remaining parts of this paper concentrates on the role of ice–water drag; i.e. it is limited to results obtained without overwash,

We begin exploring the model behaviour with an analysis of the influence of the restitution coefficient

Computed relative wave amplitude

Another aspect of the results immediately seen in Fig.

Results of simulations with

Time series of the modulus of the wave energy dissipation term

It is also worth noting that the existence of the collisional zone at the ice edge, producing strong attenuation, is directly related to the fact that the ice edge position is fixed in space – by the boom in the laboratory and by the additional spring force in the model. Without that force, the floes drift gradually in the up-wave direction (again, towards lower granular pressure) until the ice concentration drops enough so that collisions become sporadic. We return to this issue in the discussion section.

As can be expected from the analysis in Sect.

Computed relative wave amplitude

The fact that the frequency and character of collisions play a crucial role in shaping floe dynamics and wave energy dissipation in the region close to the ice edge means that the ice concentration, and thus the floe–floe distances, should have a visible influence on attenuation. This is indeed the case (Fig.

Finally, it is worth stressing that the modelled wave attenuation in both regions is strongly dependent on the incident wave amplitude

Computed relative wave amplitude

As noted recently by

The DEM simulations predict a very distinctive pattern of wave attenuation resulting from combined effects of ice–water drag and collisions between ice floes. The results suggest that intense collisions between ice floes can be expected to occur only within a narrow zone close to the ice edge, which is also a zone of lowered ice concentration and of very strong attenuation – provided that the floes are not able to drift in the up-wave direction. In natural conditions, forces keeping the ice edge in place may include compressive stress caused by wave reflection from the ice edge as well as wind and/or average currents with sufficient velocity so that the forces exerted by them on the ice compensate for those related to increased granular pressure. It is interesting to note that the elevated granular pressure can be sustained only by a constant energy input from the waves; otherwise, inelastic floe–floe collisions would lead not to increased, but to decreased, collision rates. This makes the situation very different from the wind-forced sea ice studied by

The fact that the floes tend to accumulate in the inner zone, forming a semi-continuous ice cover with ice concentration close to 100 % and limited horizontal ice motion, means that – if dissipation due to ice–water drag is significant – the expected attenuation rates within that zone should be close to those computed analytically in Sect.

It is also worth noting that the influence of ice–water drag on wave energy attenuation depends very strongly on the drag law used. If, for example, a linear drag law

A very important limitation of the model used here is the fact that it takes into account only the transmitted propagating component

The code of the DESIgn model is freely available at

All authors contributed to the planning of the research and to the discussion of the results. AH developed the numerical model, performed the simulations, and wrote the text.

The authors declare that they have no conflict of interest.

We are very grateful to the two anonymous reviewers for very insightful and constructive comments on the draft of this paper.

The development of the numerical model used in this work has been financed by the Polish National Science Centre research grant no. 2015/19/B/ST10/01568 (“Discrete-element sea ice modeling – development of theoretical and numerical methods”). Sukun Cheng and Hayley H. Shen are supported in part by ONR grant no. N00014-17-1-2862.

This paper was edited by Lars Kaleschke and reviewed by two anonymous referees.