Forced and Unforced Flexural-gravity Solitary Waves

Flexural-gravity waves beneath an ice sheet are investigated. Forced waves generated by a moving load as well as freely propagating solitary waves are considered for the nonlinear problem as proposed by Plotnikov and Toland [2011]. In the unforced case, a Hamiltonian reformulation of the governing equations is presented in three dimensions. A weakly nonlinear analysis is performed to derive a cubic nonlinear Schrödinger equation near the minimum phase velocity in two dimensions. Both steady and time-dependent fully nonlinear computations are presented in the two-dimensional case, and the influence of finite depth is also discussed

Numerical study of solitary wave attenuation in a fragmented ice sheet

A numerical model for direct phase-resolved simulation of nonlinear ocean waves propagating through fragmented sea ice is proposed. In view are applications to wave propagation and attenuation across the marginal ice zone. This model solves the full equations for nonlinear potential flow coupled with a nonlinear thin-plate formulation for the ice cover. A key new contribution is to modeling fragmented sea ice, which is accomplished by allowing the coefficient of flexural rigidity to vary spatially so that distributions of ice floes can be directly specified in the physical domain. Two-dimensional simulations are performed to examine the attenuation of solitary waves by scattering through an irregular array of ice floes. Two different measures based on the wave profile are used to quantify its attenuation over time for various floe configurations. Slow (near linear) or fast (exponential-like) decay is observed depending on such parameters as incident wave height, ice concentration and ice fragmentation

Wave-breaking and generic singularities of nonlinear hyperbolic equations

Wave-breaking is studied analytically first and the results are compared with accurate numerical simulations of 3D wave-breaking. We focus on the time dependence of various quantities becoming singular at the onset of breaking. The power laws derived from general arguments and the singular behavior of solutions of nonlinear hyperbolic differential equations are in excellent agreement with the numerical results. This shows the power of the analysis by methods using generic concepts of nonlinear science

Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon

This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the behavior of the flow and in which situations the dead-water effect appears.Comment: To appear in Nonlinearit

Dynamics of fully nonlinear capillary–gravity solitary waves under normal electric fields

Two-dimensional capillary–gravity waves travelling under the effect of a vertical electric field are considered. The fluid is assumed to be a dielectric of infinite depth. It is bounded above by another fluid which is hydrodynamically passive and perfectly conducting. The problem is solved numerically by time-dependent conformal mapping methods. Fully nonlinear waves are presented, and their stability and dynamics are studied

Computations of fully nonlinear hydroelastic solitary waves on deep water

AbstractThis 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.</jats:p