Numerical study of interfacial solitary waves propagating under an elastic sheet

Abstract

Steady solitary and generalized solitary waves of a two-fluid problem where the upper layer is under a flexible elastic sheet are considered as a model for internal waves under an ice-covered ocean. The fluid consists of two layers of constant densities, separated by an interface. The elastic sheet resists bending forces and is mathematically described by a fully nonlinear thin shell model. Fully localized solitary waves are computed via a boundary integral method. Progression along the various branches of solutions shows that barotropic (i.e. surface modes) wave-packet solitary wave branches end with the free surface approaching the interface. On the other hand, the limiting configurations of long baroclinic (i.e. internal) solitary waves are characterized by an infinite broadening in the horizontal direction. Baroclinic wave-packet modes also exist for a large range of amplitudes and generalized solitary waves are computed in a case of a long internal mode in resonance with surface modes. In contrast to the pure gravity case (i.e without an elastic cover), these generalized solitary waves exhibit new Wilton-ripple-like periodic trains in the far field.

Well-posedness of two-dimensional hydroelastic waves with mass

“…Similarly, the mollified Birkhoff-Rott integral, W ε , is defined the same way as W , using (28), but in terms of the new quantities γ ε , z ε d :…”

“…Most prior works in the literature for the hydroelastic problem have been concerned with the existence or computation of traveling waves, including by Toland in the massless case [27] and by Toland and Baldi and Toland in the case which accounts for the mass [9], [26]. Other papers on hydroelastic traveling waves include [16], [22], [28], [29].…”

Submarine Mass Movements and their Consequences

“…Another notable case, the Notre-Dame-de-la-Salette in 1908, is unique because water and ice displaced by the landslide in quick clays destroyed a portion of the village and caused 33 deaths (Ells 1908). There is little literature on the effect of ice cover on tsunami wave propagation, particularly for shallow depths (Jørstad 1968;Murty and Polavarapu 1979;Vanneste et al 2010;Wang et al 2015). 61.1).…”

Periodic traveling interfacial hydroelastic waves with or without mass

“…Toland and Baldi and Toland have proved existence of periodic traveling hydroelastic water waves with and without mass including studying secondary bifurcations [25], [26], [10], [11]. A number of authors have also computed traveling hydroelastic water waves, finding results in 2D and 3D, computations of periodic and solitary waves, comparison with weakly nonlinear models, and comparison across different modelling assumptions for the bending force [13], [14], [17], [18], [19], [28], [29]. While we believe these computations of hydroelastic water waves are the most relevant such studies to the present work, this is not an exhaustive list, and the interested reader is encouraged to consult these papers for further references.…”