Effect of small floating disks on the propagation of gravity waves

Abstract

A dispersion relation for gravity waves in water covered by disk-like impurities embedded in a viscous matrix is derived. The macroscopic equations are obtained by ensemble-averaging the fluid equations at the disk scale in the asymptotic limit of long waves and low disk surface fraction. Various regimes are identified depending on the disk radii and the thickness and viscosity of the top layer. Semi-quantitative analysis in the close-packing regime suggests dramatic modification of the dynamics, with orders of magnitude increase in wave damping and wave dispersion. A simplified model working in this regime is proposed. Possible applications to wave propagation in ice-covered ocean is discussed and comparison with field data is provided.

On the Ocean Wave Attenuation Rate in Grease‐Pancake Ice, a Comparison of Viscous Layer Propagation Models With Field Data

“…We notice the following facts: For small and fixed finite values of the other dimensionless parameters (including ν 2 / ν 1 ), we find q CP > q TLV > q Keller , meaning that, to obtain a given damping, a smaller value of the effective viscosity is required by the CP model, than it is by the TLV model and the Keller model. The result, already noted in De Santi and Olla (), will be confirmed in the coming analysis. For ν 2 ≈ ν 1 , the prediction by the TLV model coincides with the result by Lamb () for wave propagation in a homogeneous viscous fluid. From equations and , the Keller's model is retrieved for .…”

“…A comparison with field data, of the numerically obtained values of the damping, will be carried out in the coming sections. Analytical approximate expressions for q ( ω ) can nevertheless be obtained by exploiting the smallness, compared to the wavelength, of two key length scales of the problem: the ice thickness h and the thickness of the viscous boundary layer in the wave field (De Santi & Olla, ), …”

“…For ≠ 0 the tangential stress is proportional to (minus) the horizontal compression rate and does not depend on the pancakes radii R. The normal stress modification can be determined as an average effect from the flow perturbation by individual pancakes and thus depends on their size. However, it can be proved that for ≠ 0 the role of R is tiny (De Santi & Olla, 2017). Since the pancakes are assumed much smaller than a typical wavelength, no wave scattering contribution to the dynamics is considered (for a general discussion of wave scattering by ice floes, see, e.g., Squire (2007)).…”

“…Different models of wave propagation in GPI-covered ocean have been proposed (De Carolis & Desiderio, 2002;De Santi & Olla, 2017;Keller, 1998;Lamb, 1932;Weber, 1987;Wang & Shen, 2010a). All these models represent the ice-water system as a two-layer fluid with different density and viscosity.…”

“… Abstract The ability of viscous layer models to describe the attenuation of waves propagating in grease‐pancake ice covered ocean is investigated. In particular, the Keller's model (Keller, ; https://doi.org/10.1029/97JC02966), the two‐layer viscous model (De Carolis & Desiderio, ; https://doi.org/10.1016/S0375-9601(02)01503-7) and the close‐packing model (De Santi & Olla, ; arXiv:1512.05631) are extensively validated by using wave attenuation data collected during two different field campaigns (Weddell Sea, Antarctica, April 2000; western Arctic Ocean, autumn 2015). We use these data to inspect the performance of the three models by minimizing the differences between the measured and model wave attenuation; the retrieved ice thickness is then compared with measured data.…”