Internal gravity waves, boundary integral equations and radiation conditions (original) (raw)
Related papers
Internal Gravity Waves and Hyperbolic Boundary-Value Problems
Three-dimensional time-harmonic internal gravity waves are generated by oscillating a bounded object in an unbounded stratified fluid. Energy is found in conical wave beams. The problem is to calculate the wave fields for an object of arbitrary shape. It can be formulated as a hyperbolic boundary-value problem. The following aspects are discussed: reduction to boundary integral equations; single-layer and double-layer potentials; estimation of far fields and radiation conditions. The problem is complicated because the group and phase velocities are orthogonal. In addition, singular boundary integrals arise: their integrands are infinite along a certain curve (not just at a point) on the boundary, and this happens even when the field point is off the boundary (but within one of the conical wave beams).
Generation of internal gravity waves by an oscillating horizontal disc
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011
Internal gravity waves are generated in a stratified fluid by arbitrary forced oscillations of a horizontal disc. The wave fields are calculated in both the time domain and the frequency domain. In the time domain, an initial-value problem is solved using Laplace transforms; causality is imposed. In the frequency domain (time-harmonic oscillations), a radiation condition is imposed: a plane-wave (Fourier) decomposition is used in which waves with outgoing group velocity are selected. It is shown that both approaches lead to the same solution, once transient effects are ignored. Then, a method is given for calculating the far-field, using asymptotic approximations of double integrals. It is shown that the total energy flux is outwards, for arbitrary forcings of the disc. Further investigations of energy transport are made with a view to clarifying the nature of radiation conditions in the frequency domain.
Solitary flexural–gravity waves in three dimensions
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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’.
Flexural gravity wave problems in two-layer fluids
Wave Motion, 2008
Expansion formulae for flexural gravity wave problems in two-layer fluids are developed in both the cases of water of finite and infinite depths. The developed expansion formulae are applied to (i) derive the line source flexural gravity wave potentials in the presence of floating ice sheet of finite thickness and (ii) investigate the scattering of ice-coupled waves by a narrow crack in an infinite floating ice sheet. Both the problems are analyzed in two dimensions in a two-layer fluid having an interface in case of finite and infinite depths separately. Relations based on Green's identity are derived for the reflection and transmission coefficients in surface and interface modes. Effect of the density ratio and the position of interface on the reflection and transmission coefficients and surface and interface elevations in the scattering problem is analyzed.
Gravity waves under nonuniform pressure over a free surface. Exact solutions
Fluid Dynamics, 2013
Plane periodic oscillations of an infinitely deep fluid are studied in the case of a nonuniform pressure distribution over its free surface. The fluid flow is governed by an exact solution of the Euler equations in the Lagrangian variables. The dynamics of an oscillating standing soliton are described, together with the scenario of the soliton evolution and the birth of a wave of an anomalously large amplitude against the background of the homogeneous Gerstner undulation (freak wave model). All the flows are nonuniformly vortical.
Journal of Computational Physics, 1996
The numerical simulation of nonlinear gravity waves propagating at the surface of a perfect fluid is now usually solved by totally For this kind of equations in unbounded domains, exact nonlinear time-domain numerical models in two dimensions, and absorbing boundary conditions nonlocal in both space and this approach is being extended to three dimensions. The original time [3], or partially nonlocal [4, 5] has been developed initial boundary value problem is posed in an unbounded region, for FEM solvers. Following Engquist and Majda [6], some extending horizontally up to infinity to model the sea. Its numerical authors have devised higher order approximate local solution requires truncating the domain at a finite distance. Unfortunately, no exact nonreflecting boundary condition on the truncating NRBC, in order to improve the results obtained with the surface exists in this time-domain formulation. The proposed stratclassical first-order Sommerfeld condition [7-9]. Using egy is based on the coupling of two previously known methods in similar techniques, Bayliss et al. [10] derived high order order to benefit from their different, and complementary, band-NRBC for time independent elliptic problems in extewidth: the numerical ''beach,'' very efficient in the high frequency rior region. range; and a piston-like Neumann condition, asymptotically ideal for low frequencies. The coupling method gives excellent results The mathematical modelling of the propagation of in the whole range of frequencies of interest and is as easy to free surface gravity waves leads to an initial boundary implement in nonlinear as in linear versions. One of its major advanvalue problem (IBVP) posed in a domain bounded by tages is that it does not require any spectral knowledge of the a moving unknown free surface on which a nonlinear incident waves. ᮊ 1996 Academic Press, Inc. boundary condition has to be satisfied. In that context, even after linearizing that condition which leads to posing the problem in a fixed time independent domain, no 139
Nonlinear interaction of internal and surface gravity waves in a two-layer fluid with free surface
Journal of Mathematical Sciences, 2010
UDC 532.59 A new nonlinear model of the propagation of wave packets in the system "liquid layer with solid bottom-liquid layer with free surface" is considered. With the use of the method of multiple-scale expansions, the first three linear approximations of the nonlinear problem are obtained. Solutions of problem of the first approximation are constructed and analyzed in detail. It is shown that there exist internal and surface components of the wave field, and their interaction is analyzed.
On multisolitonic decay behavior of internal gravity waves
We claim that changes of scales and fine-structure could increase from multisoliton behavior of internal waves dynamics and, further, in the so-called "wave mixing". We consider initial-boundary problems for Euler equations with a stratified background state that is valid for internal water waves. The solution of the problem we search in the waveguide mode representation for a current function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm-Liouville problem for the vertical variable. The Cauchy problem is solved for initial conditions with realistic geometry and magnitude. We choose the geometry and dimensions of the McEwan experiment with the stratification of constant buoyancy frequency. The horizontal profile is defined by numerical solutions of a coupled Korteweg-de Vries system. The numerical scheme is proved to be convergent, stable and tested by means of explicit solutions for integrable case of the system. To...