Stability of waves on fluid of infinite depth with constant vorticity

The stability of periodic travelling waves on fluid of infinite depth is examined in the presence of a constant background shear field. The effects of gravity and surface tension are ignored. The base waves are described by an exact solution that was discovered recently by Hur and Wheeler (J. Fluid Mech., vol. 896, 2020). Linear growth rates are calculated using both an asymptotic approach valid for small-amplitude waves and a numerical approach based on a collocation method. Both superharmonic and subharmonic perturbations are considered. Instability is shown to occur for any non-zero amplitude wave

Ship wave patterns on floating ice sheets

This paper aims to explore the response of a floating icesheet to a load moving in a curved path. We investigate the effect of turning on the wave patterns and strain distribution, and explore scenarios where turning increases the wave amplitude and strain in the ice, possibly leading to crack formation, fracturing and eventual ice failure. The mathematical model used here is the linearized system of differential equations introduced in Dinvay et al. (J. Fluid Mech. 876:122–149, 2019). The equations are solved using the Fourier transform in space, and the Laplace transform in time. The model is tested against existing results for comparison, and several cases of load trajectories involving turning and decelerating are tested

Solitary flexural–gravity waves in three dimensions

The focus of this work is on three-dimensional nonlinear flexural–gravity waves, propagating at the interface between a fluid and an ice sheet. The ice sheet is modelled using the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis, presented in (Plotnikov & Toland. 2011 Phil. Trans. R. Soc. A 369, 2942–2956 (doi:10.1098/rsta.2011.0104)). The fluid is assumed inviscid and incompressible, and the flow irrotational. A numerical method based on boundary integral equation techniques is used to compute solitary waves and forced waves to Euler's equations. This article is part of the theme issue ‘Modelling of sea-ice phenomena’

Evolution of wave directional properties in sea ice

Ocean waves and sea ice properties are intimately linked in the marginal ice zone (MIZ), nevertheless a definitive modelling paradigm for the wave attenuation in the MIZ is missing. The evolution of wave directional properties in the MIZ is a proxy for the main attenuation mechanism but paucity of measurements and disagreement between them contributed to current uncertainty. Here we provide an analytical evidence that viscous attenuation tilts the mean wave direction orthogonal to the sea ice edge and the narrows directionality. Departure from this behaviour are attributed to bimodality of the spectrum. We also highlight the need for high quality directional measurements to reduce uncertainty in the definition of the attenuation rate

Finite depth effects on solitary waves in a floating ice sheet

A theoretical and numerical study of two-dimensional nonlinear flexural-gravity waves propagating at the surface of an ideal fluid of finite depth, covered by a thin ice sheet, is presented. 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. From a Hamiltonian reformulation of the governing equations, two weakly nonlinear wave models are derived: a 5th-order Korteweg–de Vries equation in the long-wave regime and a cubic nonlinear Schrödinger equation in the modulational regime. Solitary wave solutions of these models and their stability are analysed. In particular, there is a critical depth below which the nonlinear Schrödinger equation is of focusing type and thus admits stable soliton solutions. These weakly nonlinear results are validated by comparison with direct numerical simulations of the full governing equations. It is observed numerically that small- to large-amplitude solitary waves of depression are stable. Overturning waves of depression are also found for low wave speeds and sufficiently large depth. However, solitary waves of elevation seem to be unstable in all cases

An operator expansion method for computing nonlinear surface waves on a ferrofluid jet

We present a new numerical method to simulate the time evolution of axisym- metric nonlinear waves on the surface of a ferrofluid jet. It is based on the reduction of this problem to a lower-dimensional computation involving surface variables alone. To do so, we describe the associated Dirichlet–Neumann op- erator in terms of a Taylor series expansion where each term can be efficiently computed by a pseudo-spectral scheme using the fast Fourier transform. We show detailed numerical tests on the convergence of this operator and, to illus- trate the performance of our method, we simulate the long-time propagation and pairwise collisions of axisymmetric solitary waves. Both depression and elevation waves are examined by varying the magnetic field. Comparisons with weakly nonlinear predictions are also provided

A dissipative nonlinear Schrodinger model for wave propagation in the marginal ice zone

Sea ice attenuates waves propagating from the open ocean. Here we model the evolution of energetic unidirectional random waves in the marginal ice zone with a nonlinear Schrödinger equation, with a frequency dependent dissipative term consistent with current model paradigms and recent field observations. The preferential dissipation of high fre quency components results in a concurrent downshift of the spectral peak that leads to a less than exponential energy decay, but at a lower rate compared to a corresponding linear model. Attenuation and downshift contrast nonlinearity, and nonlinear wave statistics at the edge tend to Gaussianity farther into the marginal ice zone

The deformation of an elastic cell in a circulatory fluid motion

The deformation of a two-dimensional inextensible elastic cell in an inviscid uniform stream with circulation is investigated. An asymptotic expansion based on a conformal mapping is used to obtain equilibria for low far-field flow speeds, and fully nonlinear solutions are obtained numerically. Expanding upon the results of Blyth and Părău (2013) and Yorkston et al. (2020) for an elastic cell in a uniform stream with zero circulation, it is shown that the nature of the cell deformation in response to circulation depends on whether the transmural pressure exceeds a series of critical values. Below the first of these critical values, the deformed cell is elongated vertically against the stream, and the circulation acts to reduce the deformation of the cell from the circular rest-state, while above this critical pressure the deformed cell elongates horizontally parallel to the flow, with stronger circulation resulting in more severe cell deformation until self-intersection. The solution branches which emerge at the second critical transmural pressure are found to form a closed loop in parameter space, which shrinks in size as the circulation is increased to a critical value at which the solution branch vanishes. We also present a set solution branches distinct from those found by Yorkston et al. (2020), which become dominant for large values of circulation

Flexural-gravity waves generated by different load sizes and configurations on varying ice cover

Three-dimensional nonlinear flexural-gravity waves generated by moving loads on the surface of an ice cover floating upon water of infinite depth are computed using a boundary-integral equation approach. A hybrid preconditioned Newton–Krylov (HPNK) method is used to increase the speed of the numerical computations. The effect of variable ice cover focusing on ridge and channel configuration is investigated. In addition, we consider different distributions of pressure to model the moving loads to determine configurations which lead to smaller deflections, reducing the strain in the ice