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Wave scattering by multiple rows of circular ice floes

Published online by Cambridge University Press:  09 October 2009

L. G. BENNETTS*
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand
V. A. SQUIRE
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand
*
Email address for correspondence: lbennetts@maths.otago.ac.nz

Abstract

A three-dimensional model of ocean-wave scattering in the marginal ice zone is constructed using linear theory under time-harmonic conditions. Individual floes are represented by circular elastic plates and are permitted to have a physically realistic draught. These floes are arranged into a finite number of parallel rows, and each row possesses an infinite number of identical floes that are evenly spaced. The floe properties may differ between rows, and the spacing between the rows is arbitrary.

The vertical dependence of the solution is expanded in a finite number of modes, and through the use of a variational principle, a finite set of two-dimensional equations is generated from which the full-linear solution may be retrieved to any desired accuracy. By dictating the periodicity in each row to be identical, the scattering properties of the individual rows are combined using transfer matrices that take account of interactions between both propagating and evanescent waves.

Numerical results are presented that investigate the differences between using the three-dimensional model and using a two-dimensional model in which the rows are replaced with strips of ice. Furthermore, Bragg resonance is identified when the rows are identical and equi-spaced, and its reduction when the inhomogeneities, that are accommodated by the model, are introduced is shown.

Type
Papers
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Bennetts, L. G., Biggs, N. R. T. & Porter, D. 2007 A multi-mode approximation to wave scattering by ice sheets of varying thickness. J. Fluid Mech. 579, 413443.CrossRefGoogle Scholar
Bennetts, L. G., Biggs, N. R. T. & Porter, D. 2009 a The interaction of flexural-gravity waves with periodic geometries. Wave Mot. 46, 5773.CrossRefGoogle Scholar
Bennetts, L. G., Biggs, N. R. T. & Porter, D. 2009 b Wave scattering by an axisymmetric ice floe of varying thickness. IMA J. Appl. Math. 74, 273295.CrossRefGoogle Scholar
Bennetts, L. G. & Squire, V. A. 2009 Linear wave forcing of an array of axisymmetric ice floes. IMA J. Appl. Math. In press.CrossRefGoogle Scholar
Cavalieri, D. J., Parkinson, C. L. & Vinnikov, K. Y. 2003 30-year satellite record reveals contrasting Arctic and Antarctic decadal sea ice variability. Geophy. Res. Lett. 30 (18). DOI: 10.1029/2003GL018031.CrossRefGoogle Scholar
Chamberlain, P. G. & Porter, D. 1995 Decomposition methods for wave scattering by topography with application to ripple beds. Wave Mot. 22, 201214.CrossRefGoogle Scholar
Chou, T. 1998 Band structure of surface flexural-gravity waves along periodic interfaces. J. Fluid Mech. 369, 333350.CrossRefGoogle Scholar
Dixon, T. W. & Squire, V. A. 2001 Energy transport in the marginal ice zone. J. Geophys. Res. 106, 1991719927.CrossRefGoogle Scholar
Kohout, A. L. & Meylan, M. H. 2008 An elastic plate model for wave attenuation and ice floe breaking in the marginal ice zone. J. Geophys. Res. 113, C09016. DOI: 10.1029/2007JC004434.CrossRefGoogle Scholar
Kohout, A. L., Meylan, M. H., Sakai, S., Hanai, K., Leman, P. & Brossard, D. 2007 Linear water wave propagation through multiple floating elastic plates of variable properties. J. Fluids. Struct. 23 (4), 649663.CrossRefGoogle Scholar
Linton, C. 1998 The Green's function for the two-dimensional Helmholtz equation in periodic domains. J. Engng Math. 33, 377402.CrossRefGoogle Scholar
Linton, C. M. & Thompson, I. 2007 Resonant effects in scattering by periodic arrays. Wave Mot. 44 (3), 165175.CrossRefGoogle Scholar
McPhedran, R. C., Botten, L. C., Asatryan, A. A., Nicorovici, N., Robinson, P. & Sterke, C. M. D. 1999 Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders. Phys. Rev. E 60 (6), 76147617.CrossRefGoogle ScholarPubMed
Meylan, M. H. & Masson, D. 2006 A linear Boltzmann equation to model wave scattering in the marginal ice zone. Ocean Model. 11, 417427.CrossRefGoogle Scholar
Peter, M. A. & Meylan, M. H. 2004 Infinite depth interaction theory for arbitrary floating bodies applied to wave forcing of ice floes. J. Fluid Mech. 500, 145167.CrossRefGoogle Scholar
Peter, M. A. & Meylan, M. H. 2007 Water-wave scattering by a semi-infinite periodic array of arbitrary bodies. J. Fluid Mech. 575, 473494.CrossRefGoogle Scholar
Porter, R. & Porter, D. 2001 Interaction of water waves with three-dimensional periodic topography. J. Fluid Mech. 434, 301335.CrossRefGoogle Scholar
Rothrock, D. A., Yu, Y. & Maykut, G. A. 1999 Thinning of the Arctic sea-ice cover. Geophys. Res. Lett. 26 (23), 34693472.CrossRefGoogle Scholar
Serreze, M. C., Holland, M. M. & Stroeve, J. 2007 Perspectives on the Arctic's shrinking sea-ice cover. Science 315 (5818), 15331536. DOI: 10.1126/science.1139426.CrossRefGoogle ScholarPubMed
Squire, V. A. 2007 Of ocean waves and sea-ice revisited. Cold Reg. Sci. Technol. 49, 110133.CrossRefGoogle Scholar
Squire, V. A., Dugan, J. P., Wadhams, P., Rottier, P. J. & Liu, A. K. 1995 Of ocean waves and sea ice. Annu. Rev. Fluid Mech. 27, 115168.CrossRefGoogle Scholar
Wadhams, P., Squire, V. A., Ewing, J. A. & Pascal, R. W. 1986 The effect of the marginal ice zone on the directional wave spectrum of the ocean. J. Phys. Oceanogr. 16 (2), 358376.2.0.CO;2>CrossRefGoogle Scholar
Wadhams, P., Squire, V. A., Goodman, D. J., Cowan, A. M. & Moore, S. C. 1987 The attenuation of ocean waves in the marginal ice zone. J. Geophys. Res. 93 (C6), 67996818.CrossRefGoogle Scholar