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

Proceedings of an ESA-NASA Workshop on a Joint Solid Earth Program

The NASA geodynamics program; spaceborne magnetometry; spaceborne gravity gradiometry (characterizing the data type); terrestrial gravity data and comparisons with satellite data; GRADIO three-axis electrostatic accelerometers; gradiometer accommodation on board a drag-free satellite; gradiometer mission spectral analysis and simulation studies; and an opto-electronic accelerometer system were discussed

A boundary perturbation method to simulate nonlinear deformations of a two-dimensional bubble

Nonlinear deformations of a two-dimensional gas bubble are investigated in the framework of a Hamiltonian formulation involving surface variables alone. The Dirichlet--Neumann operator is introduced to accomplish this dimensional reduction and is expressed via a Taylor series expansion. A recursion formula is derived to determine explicitly each term in this Taylor series up to an arbitrary order of nonlinearity. Both analytical and numerical strategies are proposed to deal with this nonlinear free-boundary problem under forced or freely oscillating conditions. Simplified models are established in various approximate regimes, including a Rayleigh--Plesset equation for the time evolution of a purely circular pulsating bubble, and a second-order Stokes wave solution for weakly nonlinear shape oscillations that rotate steadily on the bubble surface. In addition, a numerical scheme is developed to simulate the full governing equations, by exploiting the efficient and accurate treatment of the Dirichlet--Neumann operator via the fast Fourier transform. Extensive tests are conducted to assess the numerical convergence of this operator as a function of various parameters. The performance of this direct solver is illustrated by applying it to the simulation of cycles of compression-dilatation for a purely circular bubble under uniform forcing, and to the computation of freely evolving shape distortions represented by steadily rotating waves and time-periodic standing waves. The former solutions are validated against predictions by the Rayleigh--Plesset model, while the latter solutions are compared to laboratory measurements in the case of mode-2 standing waves.Comment: 69 pages, 15 figure

Numerical Simulation of Solitary-Wave Scattering and Damping in Fragmented Sea Ice

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. Distributions of ice floes can be directly specified in the physical domain by allowing the coefficient of flexural rigidity to be spatially variable. Dissipation due to ice viscosity is also taken into account by including diffusive terms in the governing equations. Two-dimensional simulations are performed to examine the attenuation of solitary waves by scattering and damping through an irregular array of ice floes. Wave attenuation over time is quantified for various floe configurations

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

Long wave expansions for water waves over random topography

In this paper, we study the motion of the free surface of a body of fluid over a variable bottom, in a long wave asymptotic regime. We assume that the bottom of the fluid region can be described by a stationary random process $\beta(x, \omega)$ whose variations take place on short length scales and which are decorrelated on the length scale of the long waves. This is a question of homogenization theory in the scaling regime for the Boussinesq and KdV equations. The analysis is performed from the point of view of perturbation theory for Hamiltonian PDEs with a small parameter, in the context of which we perform a careful analysis of the distributional convergence of stationary mixing random processes. We show in particular that the problem does not fully homogenize, and that the random effects are as important as dispersive and nonlinear phenomena in the scaling regime that is studied. Our principal result is the derivation of effective equations for surface water waves in the long wave small amplitude regime, and a consistency analysis of these equations, which are not necessarily Hamiltonian PDEs. In this analysis we compute the effects of random modulation of solutions, and give an explicit expression for the scattered component of the solution due to waves interacting with the random bottom. We show that the resulting influence of the random topography is expressed in terms of a canonical process, which is equivalent to a white noise through Donsker's invariance principle, with one free parameter being the variance of the random process $\beta$. This work is a reappraisal of the paper by Rosales & Papanicolaou \cite{RP83} and its extension to general stationary mixing processes

Large-amplitude internal solitary waves in a two-fluid model

We compute solitary wave solutions of a Hamiltonian model for large-amplitude long internal waves in a two-layer stratification. Computations are performed for values of the density and depth ratios close to oceanic conditions, and comparisons are made with solutions of both weakly and fully nonlinear models. It is shown that characteristic features of highly nonlinear solitary waves such as broadening are reproduced well by the present model

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

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

Fully Dispersive Models for Moving Loads on Ice Sheets

The response of a floating elastic plate to the motion of a moving load is studied using a fully dispersive weakly nonlinear system of equations. The system allows for accurate description of waves across the whole spectrum of wavelengths and also incorporates nonlinearity, forcing and damping. The flexural-gravity waves described by the system are time-dependent responses to a forcing with a described weight distribution, moving at a time-dependent velocity. The model is versatile enough to allow the study of a wide range of situations including the motion of a combination of point loads and loads of arbitrary shape. Numerical solutions of the system are compared to data from a number of field campaigns on ice-covered lakes, and good agreement between the deflectometer records and the numerical simulations is observed in most cases. Consideration is also given to waves generated by an accelerating or decelerating load, and it is shown that a decelerating load may trigger a wave response with a far greater amplitude than a load moving at constant celerity