Modulational instability of nonuniformly damped, broad-banded waves: applications to waves in sea-ice

This paper sets out to explore the modulational (or Benjamin-Feir) instability of a monochromatic wave propagating in the presence of damping such as that induced by sea-ice on the ocean surface. The fundamental wave motion is modelled using the spatial Zakharov equation, to which either uniform or non-uniform (frequency dependent) damping is added. By means of mode truncation the spatial analogue of the classical Benjamin-Feir instability can be studied analytically using dynamical systems techniques. The formulation readily yields the free surface envelope, giving insight into the physical implications of damping on the modulational instability. The evolution of an initially unstable mode is also studied numerically by integrating the damped, spatial Zakharov equation, in order to complement the analytical theory. This sheds light on the effects of damping on spectral broadening arising from this instability

Numerical and experimental study on the steady cone-jet mode of electro-centrifugal spinning

This study focuses on a numerical investigation of an initial stable jet through the air-sealed electro-centrifugal spinning process, which is known as a viable method for the mass production of nanofibers. A liquid jet undergoing electric and centrifugal forces, as well as other forces, first travels in a stable trajectory and then goes through an unstable curled path to the collector. In numerical modeling, hydrodynamic equations have been solved using the perturbation method—and the boundary integral method has been implemented to efficiently solve the electric potential equation. Hydrodynamic equations have been coupled with the electric field using stress boundary conditions at the fluid-fluid interface. Perturbation equations were discretized by a second order finite difference method, and the Newton method was implemented to solve the discretized non-linear system. Also, the boundary element method was utilized to solve electrostatic equations. In the theoretical study, the fluid was described as a leaky dielectric with charges only on the surface of the jet traveling in dielectric air. The effect of the electric field induced around the nozzle tip on the jet instability and trajectory deviation was also experimentally studied through plate-plate geometry as well as point-plate geometry. It was numerically found that the centrifugal force prevails on electric force by increasing the rotational speed. Therefore, the alteration of the applied voltage does not significantly affect the jet thinning profile or the jet trajectory

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’

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

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

Laboratory Experiments on Internal Solitary Waves in Ice-Covered Waters

Internal solitary waves (ISWs) propagating in a stably-stratified two-layer fluid in which the upper boundary condition changes from open water to ice are studied for cases of grease, level and nilas ice. The ISW-induced current at the surface is capable of trans-porting the ice in the horizontal direction. In the level ice case, the transport speed of, relatively long ice floes, non-dimensionalised by the wave speed is linearly dependent on the length of the ice floe non-dimensionalised by the wave length. Measures of turbulent kinetic energy dissipation under the ice are comparable to those at the wave density interface. Moreover, in cases where the ice floe protrudes into the pycnocline, interaction with the ice edge can cause the ISW to break or even be destroyed by the process. The results suggest that interaction between ISWs and sea ice may be an important mechanism for dissipation of ISW energy in the Arctic Ocean

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