Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T02:45:46.624Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave scattering by ice floes and polynyas of arbitrary shape

Published online by Cambridge University Press:  28 September 2010

L. G. BENNETTS  and
T. D. WILLIAMS
L. G. BENNETTS*
Affiliation:
Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand
T. D. WILLIAMS
Affiliation:
Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Email address for correspondence: lbennetts@maths.otago.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

An efficient solution method is presented for linear and time-harmonic water-wave scattering by two classes of a three-dimensional hydroelastic system. In both cases, the fluid domain is of infinite horizontal extent and finite depth. The fluid surface is either open, except in a finite region where it is covered by a thin-elastic plate, which represents an ice floe, or fully covered by a plate, except in a finite region where it is open, which represents an ice polynya. The approach outlined herein permits the boundary between the ice-covered and free-surface fluid regions to be described by an arbitrary smooth curve. To solve the governing equations of the full three-dimensional linear problem, they are first projected onto the horizontal plane by using an approximation theory that combines an expansion of the vertical motion of the fluid in a finite set of judiciously chosen modes with a variational principle. This generates a system of two-dimensional partial differential equations that are converted into a set of one-dimensional integro-differential equations using matrices of Green's functions, which are solved numerically through an application of the Galerkin technique. A numerical results section justifies the consideration of an arbitrarily shaped boundary by comparing the response of differently shaped floes and polynyas over a range of relevant wavenumbers. Comparisons are made in terms of the magnitude and direction of the far-field scattering response, and also the maximum average curvature of the floe and the maximum wave elevation within the polynya.

JFM classification

Geophysical and Geological Flows: Sea ice Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 662 , 10 November 2010 , pp. 5 - 35
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. & Stegun, I. 1964 Handbook of Mathematical Functions. Dover.Google Scholar
Bennetts, L. G. 2007 Wave scattering by ice sheets of varying thickness. PhD thesis, University of Reading, Berkshire, UK.Google Scholar
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, 413–443.CrossRefGoogle Scholar
Bennetts, L. G., Biggs, N. R. T. & Porter, D. 2009 Wave scattering by an axisymmetric ice floe of varying thickness. IMA J. Appl. Math. 74, 273–295.Google Scholar
Bennetts, L. G., Peter, M. A., Squire, V. A. & Meylan, M. H. 2010 A three-dimensional model of wave attenuation in the marginal ice zone. J. Geophys. Res., doi:10.1029/2009JC005982.CrossRefGoogle Scholar
Bennetts, L. G. & Squire, V. A. 2009 Wave scattering by multiple rows of circular ice floes. J. Fluid Mech. 639, 213–238.CrossRefGoogle Scholar
Chung, H. & Linton, C. M. 2005 Reflection and transmission of waves across a gap between two semi-infinite elastic plates on water. Q. J. Mech. Appl. Maths 58, 1–15.CrossRefGoogle Scholar
Evans, D. V. & Peter, M. A. 2009 Asymptotic reflection of linear water waves by submerged horizontal porous plates. J. Engng Maths, doi:10.1007/s10665-009-9355-2.Google Scholar
Evans, D. V. & Porter, R. 2003 Wave scattering by narrow cracks in ice sheets floating on water of finite depth. J. Fluid Mech. 484, 143–165.Google Scholar
Fox, C. & Squire, V. A. 1994. On the oblique reflexion and transmission of ocean waves at shore fast sea ice. Phil. Trans. R. Soc. Lond. A 347, 185–218.Google 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.Google Scholar
Langhorne, P. J., Squire, V. A. & Haskell, T. G. 2001 Lifetime estimation for a fast ice sheet subjected to ocean swell. Ann. Glaciol. 33, 333–338.Google Scholar
Linton, C. & Chung, H. 2003 Reflection and transmission at the ocean/sea-ice boundary. Wave Motion 38, 43–52.Google Scholar
Manam, S. R., Bhattarchajee, J. & Sahoo, T. 2006 Expansion formulae in wave structure interaction problems. Proc. R. Soc. Lond. A 462, 263–287.Google Scholar
Meylan, M. H. 2002 Wave response of an ice floe of arbitrary geometry. J. Geophys. Res. 107 (C1), doi:10.1029/2000JC000713.Google Scholar
Meylan, M. H. & Squire, V. A. 1996 Response of a circular ice floe to ocean waves. J. Geophys. Res. 101, 8869–8884.Google Scholar
Norris, A. N. & Vemula, C. 1995 Scattering of flexural waves on thin plates. J. Sound Vib. 181 (1), 115–125.CrossRefGoogle Scholar
Ohkusu, M. & Namba, Y. 2004 Hydroelastic analysis of a large floating structure. J. Fluids Struct. 19, 543–555.Google Scholar
Peter, M. A. & Meylan, M. H. 2009 Water-wave scattering by vast field of bodies. SIAM J. Appl. Maths 70 (5), 1567–1586.CrossRefGoogle Scholar
Peter, M. A., Meylan, M. H. & Chung, H. 2004 Wave scattering by a circular elastic plate in water of finite depth: a closed form solution. Intl J. Offshore Polar Engng 14 (2), 81–85.Google Scholar
Porter, R. & Evans, D. V. 2007 Diffraction of flexural waves by finite straight cracks in an elastic sheet over water. J. Fluids Struct. 23 (2), 309–327.Google Scholar
Porter, R. & Porter, D. 2000 Water wave scattering by a step of arbitrary profile. J. Fluid Mech. 411, 131–164.Google Scholar
Porter, D. & Porter, R. 2004 Approximations to wave scattering by an ice sheet of variable thickness over undulating bed topography. J. Fluid Mech. 509, 145–179.Google Scholar
Porter, D. & Staziker, D. J. 1995 Extensions of the mild-slope equation. J. Fluid Mech. 509, 145–179.Google Scholar
Squire, V. A. 2007 Of ocean waves and sea-ice revisited. Cold Reg. Sci. Technol. 49, 110–133.Google Scholar
Squire, V. A. 2008 Synergies between VLFS hydroelasticity and sea ice research. Intl J. Offshore Polar Engng 18, 1–13.Google Scholar
Squire, V. A. & Dixon, T. W. 2000 An analytic model for wave propagation across a crack in an ice sheet. Intl J. Offshore Polar Engng 10 (3), 173–176.Google 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, 115–168.Google Scholar
Squire, V. A., Vaughan, G. L. & Bennetts, L. G. 2009 Ocean surface wave evolvement in the Arctic Basin. Geophys. Res. Lett. 36 (L22502).CrossRefGoogle Scholar
Sturova, I. V. 2001 The diffraction of surface waves by an elastic platform floating on shallow water. J. Appl. Math. Mech. 65, 109–117.CrossRefGoogle Scholar
Timoshenko, S. & Woinowsky-Krieger, S. 1959 Theory of Plates and Shells, 2nd edn. McGraw-Hill.Google Scholar
Vaughan, G. L., Bennetts, L. G. & Squire, V. A. 2009 The decay of flexural-gravity waves in long sea-ice transects. Proc. R. Soc. Lond. A 465, 2785–2812.Google Scholar
de Veubeke, B. F. 1979 Cours Delasticite. Springer.Google Scholar
Wang, C. D. & Meylan, M. H. 2004 A higher-order-coupled boundary element and finite element method for the wave forcing of a floating elastic plate. J. Fluids Struct. 19, 557–572.Google Scholar
Watanabe, E., Utsunomiya, T. & Wang, C. 2004 Hydroelastic analysis of pontoon-type VLFS: a literature survey. Engng Struct. 26 (2), 245–256.Google Scholar
Williams, T. D. 2006 Reflections on ice: scattering of flexural-gravity waves by irregularities in Arctic and Antarctic ice sheets. PhD thesis, Otago University, Dunedin, New Zealand.Google Scholar
Williams, T. D. & Porter, R. 2009 The effect of submergence on the scattering by the interface between two floating plates. J. Fluids. Struct. 25 (5), 777–793.Google Scholar
Williams, T. D. & Squire, V. A. 2004 Oblique scattering of plane flexural-gravity waves by heterogeneities in sea-ice. Proc. R. Soc. Lond. A 460, 3469–3497.Google Scholar
Williams, T. D. & Squire, V. A. 2010 On the estimation of sea-ice thickness using wave observations. Dyn. Atmos. Oceans 49 (2–3), 215–233.CrossRefGoogle Scholar