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Computations of fully nonlinear hydroelastic solitary waves on deep water

Published online by Cambridge University Press:  17 October 2012

Philippe Guyenne
Affiliation:
Department of Mathematical Sciences, University of Delaware, DE 19716, USA
Emilian I. Pǎrǎu*
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
*
Email address for correspondence: e.parau@uea.ac.uk

Abstract

This paper is concerned with the two-dimensional problem of nonlinear gravity waves travelling at the interface between a thin ice sheet and an ideal fluid of infinite depth. The ice-sheet model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff’s hypothesis, which yields a conservative and nonlinear expression for the bending force. A Hamiltonian formulation for this hydroelastic problem is proposed in terms of quantities evaluated at the fluid–ice interface. For small-amplitude waves, a nonlinear Schrödinger equation is derived and its analysis shows that no solitary wavepackets exist in this case. For larger amplitudes, both forced and free steady waves are computed by direct numerical simulations using a boundary-integral method. In the unforced case, solitary waves of depression as well as of elevation are found, including overhanging waves with a bubble-shaped profile for wave speeds $c$ much lower than the minimum phase speed ${c}_{\mathit{min}} $. It is also shown that the energy of depression solitary waves has a minimum at a wave speed ${c}_{m} $ slightly less than ${c}_{\mathit{min}} $, which suggests that such waves are stable for $c\lt {c}_{m} $ and unstable for $c\gt {c}_{m} $. This observation is verified by time-dependent computations using a high-order spectral method. These computations also indicate that solitary waves of elevation are likely to be unstable.

Type
Papers
Copyright
©2012 Cambridge University Press

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