Computations of fully nonlinear hydroelastic solitary waves on deep water

• Published in: Journal of fluid mechanics 2012-12, Vol.713, p.307-329
• Main Authors: Guyenne, Philippe, Pǎrǎu, Emilian I.
• Format: Article
• Language: English
• Subjects: Computation; Deep water; Depression; Earth, ocean, space; Elevation; Exact sciences and technology; External geophysics; Fluid mechanics; Fundamental areas of phenomenology (including applications); Gravity waves; Ice; Mathematical models; Nonlinearity; Physics; Physics of the oceans; Schroedinger equation; Sea ice; Solid mechanics; Solitary waves; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2012.458

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.