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Wave diffraction by multiple arbitrary shaped cracks in an infinitely extended ice sheet of finite water depth

Published online by Cambridge University Press:  21 April 2020

Zhi Fu Li ,
Guo Xiong Wu Open the ORCID record for Guo Xiong Wu [Opens in a new window]  and
Kang Ren Open the ORCID record for Kang Ren [Opens in a new window]
Zhi Fu Li
Affiliation:
School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang212003, China
Guo Xiong Wu*
Affiliation:
Department of Mechanical Engineering, University College London, Torrington Place, LondonWC1E 7JE, UK
Kang Ren
Affiliation:
Department of Mechanical Engineering, University College London, Torrington Place, LondonWC1E 7JE, UK
*
† Email address for correspondence: [email protected]
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Abstract

Flexural-gravity wave interactions with multiple cracks in an ice sheet of infinite extent are considered, based on the linearized velocity potential theory for fluid flow and thin elastic plate model for an ice sheet. Both the shape and location of the cracks can be arbitrary, while an individual crack can be either open or closed. Free edge conditions are imposed at the crack. For open cracks, zero corner force conditions are further applied at the crack tips. The solution procedure starts from series expansion in the vertical direction based on separation of variables, which decomposes the three-dimensional problem into an infinite number of coupled two-dimensional problems in the horizontal plane. For each two-dimensional problem, an integral equation is derived along the cracks, with the jumps of displacement and slope of the ice sheet as unknowns in the integrand. By extending the crack in the vertical direction into the fluid domain, an artificial vertical surface is formed, on which an orthogonal inner product is adopted for the vertical modes. Through this, the edge conditions at the cracks are satisfied, together with continuous conditions of pressure and velocity on the vertical surface. The integral differential equations are solved numerically through the boundary element method together with the finite difference scheme for the derivatives along the crack. Extensive results are provided and analysed for cracks with various shapes and locations, including the jumps of displacement and slope, diffraction wave coefficient, and the scattered cross-section.

JFM classification

Geophysical and Geological Flows: Ice sheets Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 893 , 25 June 2020 , A14
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover Press.Google Scholar
Balmforth, N. J. & Craster, R. V. 1999 Ocean waves and ice sheets. J. Fluid Mech. 395, 89–124.CrossRefGoogle Scholar
Barrett, M. D. & Squire, V. A. 1996 Ice-coupled wave propagation across an abrupt change in ice rigidity, density, or thickness. J. Geophys. Res. 101 (C9), 20825–20832.CrossRefGoogle Scholar
Bennetts, L. G. & Squire, V. A. 2012 On the calculation of an attenuation coefficient for transects of ice-covered ocean. Proc. R. Soc. Lond. A 468 (2137), 136–162.CrossRefGoogle Scholar
Bennetts, L. G. & Williams, T. D. 2010 Wave scattering by ice floes and polynyas of arbitrary shape. J. Fluid Mech. 662, 5–35.CrossRefGoogle Scholar
Burden, R. L. & Faires, J. D. 2010 Numerical Analysis. Brooks Cole.Google Scholar
Chung, H. & Fox, C. 2002 Calculation of wave-ice interaction using the Wiener–Hopf technique. New Zealand J. Math. 31 (1), 1–18.Google 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), 1–15.CrossRefGoogle Scholar
Eliasson, K., Ulfarsson, G. F., Valsson, T. & Gardarsson, S. M. 2017 Identification of development areas in a warming arctic with respect to natural resources, transportation, protected areas, and geography. Futures 85, 14–29.CrossRefGoogle Scholar
Evans, D. V. & Davies, T. V.1968 Wave–ice interaction. Report. Davidson Laboratory, Stevens Institute of Technology.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.CrossRefGoogle Scholar
Fox, C. & Squire, V. A. 1990 Reflection and transmission characteristics at the edge of shore fast sea ice. J. Geophys. Res. 95 (C7), 11629–11639.CrossRefGoogle 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 (1682), 185–218.Google Scholar
Herman, A., Evers, K. U. & Reimer, N. 2018 Floe-size distributions in laboratory ice broken by waves. Cryosphere 12 (2), 685–699.CrossRefGoogle Scholar
Kohout, A. L., Williams, M. J., Toyota, T., Lieser, J. & Hutchings, J. 2016 In situ observations of wave-induced sea ice breakup. Deep-Sea Res. II 131, 22–27.Google Scholar
Li, Z. F., Wu, G. X. & Ji, C. Y. 2018a Interaction of wave with a body submerged below an ice sheet with multiple arbitrarily spaced cracks. Phys. Fluids 30 (5), 057107.CrossRefGoogle Scholar
Li, Z. F., Wu, G. X. & Ji, C. Y. 2018b Wave radiation and diffraction by a circular cylinder submerged below an ice sheet with a crack. J. Fluid Mech. 845, 682–712.CrossRefGoogle Scholar
Li, Z. F., Wu, G. X. & Shi, Y. Y. 2018c Wave diffraction by a circular crack in an ice sheet floating on water of finite depth. Phys. Fluids 30 (11), 117103.CrossRefGoogle Scholar
Linton, C. M. & Chung, H. 2003 Reflection and transmission at the ocean/sea-ice boundary. Wave Motion 38 (1), 43–52.CrossRefGoogle Scholar
Linton, C. M. & Mciver, P. 2001 Handbook of Mathematical Techniques for Wave/Structure Interactions. CRC Press.CrossRefGoogle Scholar
Martin, P. A. & Rizzo, F. J. 1989 On boundary integral equations for crack problems. Proc. R. Soc. Lond. A 421 (1861), 341–355.Google Scholar
Meylan, M. H. 2002 Wave response of an ice floe of arbitrary geometry. J. Geophys. Res. 107 (C1), 5.CrossRefGoogle Scholar
Meylan, M. H. & Squire, V. A. 1994 The response of ice floes to ocean waves. J. Geophys. Res. 99 (C1), 891–900.CrossRefGoogle Scholar
Meylan, M. H. & Squire, V. A. 1996 Response of a circular ice floe to ocean waves. J. Geophys. Res. 101 (C4), 8869–8884.CrossRefGoogle Scholar
Montiel, F., Squire, V. A. & Bennetts, L. G. 2016 Attenuation and directional spreading of ocean wave spectra in the marginal ice zone. J. Fluid Mech. 790, 492–522.CrossRefGoogle Scholar
Norris, A. N. & Wang, Z. 1994 Bending wave diffraction from strips and cracks on thin plates. Q. J. Mech. Appl. Maths 47 (4), 607–627.CrossRefGoogle Scholar
Peter, M. A., Meylan, M. H. & Chung, H. A. 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. 2006 Scattering of flexural waves by multiple narrow cracks in ice sheets floating on water. Wave Motion 43 (5), 425–443.CrossRefGoogle 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.CrossRefGoogle Scholar
Robin, G. Q. 1963 Wave propagation through fields of pack ice. Phil. Trans. R. Soc. Lond. A 255 (255), 313–339.Google Scholar
Sahoo, T., Yip, T. L. & Chwang, A. T. 2001 Scattering of surface waves by a semi-infinite floating elastic plate. Phys. Fluids 13 (11), 3215–3222.CrossRefGoogle Scholar
Smith, J. A., Meylan, M. H. & Mcphedran, R. C. 2011 Scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems. J. Sound Vib. 330 (16), 4029–4046.Google Scholar
Squire, V. A. 2007 Of ocean waves and sea-ice revisited. Cold Reg. Sci. Technol. 49 (2), 110–133.CrossRefGoogle Scholar
Squire, V. A. 2011 Past, present and impendent hydroelastic challenges in the polar and subpolar seas. Phil. Trans. R. Soc. Lond. A 369 (1947), 2813–2831.CrossRefGoogle ScholarPubMed
Squire, V. A. 2020 Ocean wave interactions with sea ice: a reappraisal. Annu. Rev. Fluid Mech. 52 (1), 37–60.CrossRefGoogle 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. & Dixon, T. W. 2001a How a region of cracked sea ice affects ice-coupled wave propagation. Ann. Glaciol. 33, 327–332.CrossRefGoogle Scholar
Squire, V. A. & Dixon, T. W. 2001b On modelling an iceberg embedded in shore-fast sea ice. J. Engng Maths 40 (3), 211–226.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 (1), 115–168.CrossRefGoogle Scholar
Squire, V. A., Robinson, W. H., Langhorne, P. J. & Haskell, T. G. 1988 Vehicles and aircraft on floating ice. Nature 333 (6169), 159–161.CrossRefGoogle Scholar
Timoshenko, S. P. & Woinowsky, K. S. 1959 Theory of Plates and Shells. McGraw-Hill.Google Scholar
Tkacheva, L. A. 2001 Hydroelastic behavior of a floating plate in waves. J. Appl. Mech. Tech. Phys. 42 (6), 991–996.CrossRefGoogle Scholar
Tkacheva, L. A. 2004 The diffraction of surface waves by a floating elastic plate at oblique incidence. Z. Angew. Math. Mech. 68 (3), 425–436.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 (4), 557–572.CrossRefGoogle Scholar
Williams, T. D. & Squire, V. A. 2006 Scattering of flexural-gravity waves at the boundaries between three floating sheets with applications. J. Fluid Mech. 569, 113–140.CrossRefGoogle Scholar
Wu, G. X. & Eatock Taylor, R. 1995 Time stepping solutions of the two-dimensional nonlinear wave radiation problem. Ocean Engng 22 (8), 785–798.CrossRefGoogle Scholar