Wave energy attenuation in fields of colliding ice floes – Part 1: Discrete-element modelling of dissipation due to ice–water drag

• Main Authors: A. Herman, S. Cheng, H. H. Shen
• Format: Article
• Language: English
• Published: Copernicus Publications 2019-11-01
• Series: The Cryosphere
• Online Access: https://www.the-cryosphere.net/13/2887/2019/tc-13-2887-2019.pdf
• ISSN: 1994-0416; 1994-0424

<p>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 <span class="inline-formula"><i>a</i></span> with distance <span class="inline-formula"><i>x</i></span>, of the form <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M3" display="inline" overflow="scroll" dspmath="mathml"><mrow><mi>a</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn mathvariant="normal">1</mn><mo>/</mo><mo>(</mo><mi mathvariant="italic">α</mi><mi>x</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>/</mo><msub><mi>a</mi><mn mathvariant="normal">0</mn></msub><mo>)</mo></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="103pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="2e65d2ca52fdd6059ee99428a28e21b3"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="tc-13-2887-2019-ie00001.svg" width="103pt" height="14pt" src="tc-13-2887-2019-ie00001.png"/></svg:svg></span></span>, with the attenuation rate <span class="inline-formula"><i>α</i></span> linearly proportional to the drag coefficient. The dependence of <span class="inline-formula"><i>α</i></span> on wave frequency <span class="inline-formula"><i>ω</i></span> varies with the dispersion relation used. For the open-water (OW) dispersion relation, <span class="inline-formula"><i>α</i>∼<i>ω</i>⁴</span>. For the mass loading dispersion relation, suitable for ice covers composed of small floes, the increase in <span class="inline-formula"><i>α</i></span> with <span class="inline-formula"><i>ω</i></span> is much faster than in the OW case, leading to very fast elimination of high-frequency components from the wave energy spectrum. For elastic-plate dispersion relation, suitable for large floes or continuous ice, <span class="inline-formula"><i>α</i>∼<i>ω</i>^<i>m</i></span> within the high-frequency tail, with <span class="inline-formula"><i>m</i></span> close to 2.0–2.5; i.e. dissipation is much slower than in the OW case. The coupled DEM–wave model predicts the existence of two zones: a relatively narrow area of very strong attenuation close to the ice edge, with energetic floe collisions and therefore high instantaneous ice–water velocities, and an inner zone where ice floes are in permanent or semi-permanent contact with each other, with attenuation rates close to those analysed theoretically. Dissipation in the collisional zone increases with an increasing restitution coefficient of the ice and with decreasing floe size. In effect, two factors contribute to strong attenuation in fields of small ice floes: lower wave energy propagation speeds and higher relative ice–water velocities due to larger accelerations of floes with smaller mass and more collisions per unit surface area.</p>