Michael H. Meylan

The response of ice floes to ocean waves

A precise linear mathematical theory is reported to model the response of a solitary ice floe in ocean waves, allowing the floe to bend with the passing wave. Both infinite and finite water depths are considered, and the model is also extended to include a pair of separated floes of different length, the case of n-floes then being a natural and straightforward development. For a single ice floe, perfect transmission is achieved whenever the wavelength beneath the ice couples perfectly to the length of the ice floe. Then the strains induced in the bending ice floe reach a maximum, and because multiple-cycle tuning can occur in floes which are long compared to the wavelength, strain response amplitude operators (RAOs) are complicated. By considering the case of infinite stiffness, heave and roll RAOs are also found for typical ocean wave periods, and these agree well with two-dimensional rigid-body models. Finally, ice floes of different diameters but constant 1-m thickness are subjected to spectral forcing, and the strain spectral density for each is found. Spectral density envelopes increase gradually with floe diameter, achieving a maximum value for a floe of about 102m. Thereafter strains never decrease below those for the 80-m curve owing to multiple-cycle tuning. This may explain the presence of zones within an ice field where floe size never exceeds some prescribed value. Results obtained from the complete theory for two adjacent floes do not differ significantly from those found by applying the single-floe model serially except when the separation is very small.

Storm-induced sea-ice breakup and the implications for ice extent

The propagation of large, storm-generated waves through sea ice has so far not been measured, limiting our understanding of how ocean waves break sea ice. Without improved knowledge of ice breakup, we are unable to understand recent changes, or predict future changes, in Arctic and Antarctic sea ice. Here we show that storm-generated ocean waves propagating through Antarctic sea ice are able to transport enough energy to break sea ice hundreds of kilometres from the ice edge. Our results, which are based on concurrent observations at multiple locations, establish that large waves break sea ice much farther from the ice edge than would be predicted by the commonly assumed exponential decay. We observed the wave height decay to be almost linear for large waves—those with a significant wave height greater than three metres—and to be exponential only for small waves. This implies a more prominent role for large ocean waves in sea-ice breakup and retreat than previously thought. We examine the wider relevance of this by comparing observed Antarctic sea-ice edge positions with changes in modelled significant wave heights for the Southern Ocean between 1997 and 2009, and find that the retreat and expansion of the sea-ice edge correlate with mean significant wave height increases and decreases, respectively. This includes capturing the spatial variability in sea-ice trends found in the Ross and Amundsen–Bellingshausen seas. Climate models fail to capture recent changes in sea ice in both polar regions. Our results suggest that the incorporation of explicit or parameterized interactions between ocean waves and sea ice may resolve this problem.

Response of a circular ice floe to ocean waves

A new model is presented to reproduce the behavior of a solitary, circular, flexible ice floe brought into motion by the action of long-crested sea waves. The intended application of the work is ultimately a fully three-dimensional analogue of a marginal ice zone (MIZ) through which ocean waves propagate, allowing the attenuation and directional advance to be forecast and validated against observations. (Existing theory does not treat directional changes correctly.) To enable a check to be made on the model, two independent methods are developed: an expansion in the eigenfunctions of a thin circular plate, and the more general method of eigenfunctions used to construct a Greens function for the floe. Displacement and three-dimensional scattering patterns in the water surrounding the floe are given for several floe geometries. The model is also used to investigate the strain field generated in the floe, its surge response, and the energy initiated in the water encircling it. Finally, with the aim of understanding how floes herd together to form cohesive structures in the MIZ, the force induced on floes of various thicknesses and diameters is plotted.

In situ measurements and analysis of ocean waves in the Antarctic marginal ice zone

In situ measurements of ocean surface wave spectra evolution in the Antarctic marginal ice zone are described. Analysis of the measurements shows significant wave heights and peak periods do not vary appreciably in approximately the first 80km of the ice-covered ocean. Beyond this region, significant wave heights attenuate and peak periods increase. It is shown that attenuation rates are insensitive to amplitudes for long-period waves but increase with increasing amplitude above some critical amplitude for short-period waves. Attenuation rates of the spectral components of the wavefield are calculated. It is shown that attenuation rates decrease with increasing wave period. Further, for long-period waves the decrease is shown to be proportional to the inverse of the period squared. This relationship can be used to efficiently implement wave attenuation through the marginal ice zone in ocean-scale wave models.

Toward realism in modeling ocean wave behavior in marginal ice zones

The model of Meylan and Squire [1996], which treats solitary ice floes as floating, flexible circular disks, is incorporated into the equation of transport for the propagation of waves through a scattering medium, assumed to represent open ice pack in a marginal ice zone. The time-independent form of the equation is then solved for homogeneous ice conditions allowing for dissipation due to scattering, together with extra absorption from interactions between floes, losses in the water column, and losses arising from the inelastic character of the sea ice including local brash. The spatial evolution of wave spectra as they progress through the pack is investigated with the aim of explaining the field data of Wadhams et al. [1986]. Specifically, the change toward directional isotropy experienced by waves as they travel into the ice interior is of interest. In accord with observations, directional spread is found to widen with penetration until eventually becoming isotropic, the process being sensitive to wave period. The effect of absorption on the solution is investigated.

The linear wave response of a floating thin plate on water of variable depth

We present a solution for the linear wave forcing of a floating two-dimensional thin plate on water of variable depth. The solution method is based on reducing the problem to a finite domain, which contains both the region of variable water depth and the floating thin plate. In this finite region, the outward normal derivative of the potential on the boundary is expressed as a function of the potential. This is accomplished by using integral operators for the radiating boundaries and the boundary under the plate. Laplaces equation in the finite domain is solved using the boundary element method and the integral equations are solved by numerical integration. The same discretisation is used for the boundary element method and to integrate the integral equations. The results show that there is a significant region where the solution for a plate with a variable depth differs from the simpler solutions for either variable depth but no plate or a plate with constant depth. Furthermore, the presence of the plate increases the frequency of influence of the variable depth.

Infinite-depth interaction theory for arbitrary floating bodies applied to wave forcing of ice floes

We extend the finite-depth interaction theory of Kagemoto & Yue (1986) to water of infinite depth and bodies of arbitrary geometry. The sum over the discrete roots of the dispersion equation in the finite-depth theory becomes an integral in the infinite-depth theory. This means that the infinite dimensional diffraction transfer matrix in the finite-depth theory must be replaced by an integral operator. In the numerical solution of the equations, this integral operator is approximated by a sum and a linear system of equations is obtained. We also show how the calculations of the diffraction transfer matrix for bodies of arbitrary geometry developed by Goo & Yoshida (1990) can be extended to infinite depth, and how the diffraction transfer matrix for rotated bodies can be calculated easily. This interaction theory is applied to the wave forcing of multiple ice floes and a method to solve the full diffraction problem in this case is presented. Convergence studies comparing the interaction method with the full diffraction calculations and the finite- and infinite-depth interaction methods are carried out.

Spectral solution of time-dependent shallow water hydroelasticity

The spectral theory of a thin plate floating on shallow water is derived and used to solve the time-dependent motion. This theory is based on an energy inner product in which the evolution operator becomes unitary. Two solution methods are presented. In the first, the solution is expanded in the eigenfunctions of a self-adjoint operator, which are the incoming wave solutions for a single frequency. In the second, the scattering theory of Lax-Phillips is used. The Lax-Phillips scattering solution is suitable for calculating only the free motion of the plate. However, it determines the modes of vibration of the plate-water system. These modes, which both oscillate and decay, are found by a complex search algorithm based contour integration. As well as an application to modelling floating runways, the spectral theory for a floating thin plate on shallow water is a solvable model for more complicated hydroelastic systems.

A variational equation for the wave forcing of floating thin plates

Abstract A variational equation is presented for floating thin plates subject to wave forcing. This equation is derived from the thin plate equation of motion by including the wave forcing using the free-surface Green function. This variational equation combines the standard method for solving the motion of a thin plate (a variational equation) with the standard method for solving the wave forcing of a floating body (the free-surface Green function method). Solutions of the variational equation are presented for some simple thin plate geometries using polynomial basis functions. The variational equation is extended to the case of plates of variable properties and to multiple plates and further solutions are presented.

Relating wave attenuation to pancake ice thickness, using field measurements and model results

Wave attenuation coefficients (α, m A1) were calculated from in situ data transmitted by custom wave buoys deployed into the advancing pancake ice region of the Weddell Sea. Data cover a 12 day period as the buoy array was first compressed and then dilated under the influence of a passing low-pressure system. Attenuation was found to vary over more than 2 orders of magnitude and to be far higher than that observed in broken-floe marginal ice zones. A clear linear relation between α and ice thickness was demonstrated, using ice thickness from a novel dynamic/thermodynamic model. A simple expression for α in terms of wave period and ice thickness was derived, for application in research and operational models. The variation of α was further investigated with a two-layer viscous model, and a linear relation was found between eddy viscosity in the sub-ice boundary layer and ice thickness.