scholarly journals Hamiltonian structure for two-dimensional extended Green–Naghdi equations

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160127 ◽  
Author(s):  
Yoshimasa Matsuno
Keyword(s):  
Dispersion Relation ◽  
Gravity Waves ◽  
Hamiltonian Structure ◽  
Bottom Topography ◽  
Higher Order ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Linear Dispersion ◽  
Surface Gravity Waves ◽  
Linear Dispersion Relation

The two-dimensional Green–Naghdi (GN) shallow-water model for surface gravity waves is extended to incorporate the arbitrary higher-order dispersive effects. This can be achieved by developing a novel asymptotic analysis applied to the basic nonlinear water wave problem. The linear dispersion relation for the extended GN system is then explored in detail. In particular, we use its characteristics to discuss the well-posedness of the linearized problem. As illustrative examples of approximate model equations, we derive a higher-order model that is accurate to the fourth power of the dispersion parameter in the case of a flat bottom topography, and address the related issues such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom on the basis of this model. The original Green–Naghdi (GN) model is then briefly described in the case of an uneven bottom topography. Subsequently, the extended GN system presented here is shown to have the same Hamiltonian structure as that of the original GN system. Last, we demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

scholarly journals Measurement of the dispersion relation for random surface gravity waves

2015 ◽  
Vol 766 ◽  
pp. 326-336 ◽  
Author(s):  
Tore Magnus A. Taklo ◽  
Karsten Trulsen ◽  
Odin Gramstad ◽  
Harald E. Krogstad ◽  
Atle Jensen
Keyword(s):  
Dispersion Relation ◽  
Gravity Waves ◽  
Laboratory Experiments ◽  
Surface Gravity ◽  
Linear Dispersion ◽  
Random Surface ◽  
Zakharov Equation ◽  
Surface Gravity Waves ◽  
Linear Dispersion Relation ◽  
Jonswap Spectrum

AbstractWe report laboratory experiments and numerical simulations of the Zakharov equation, designed to have sufficient resolution in space and time to measure the dispersion relation for random surface gravity waves. The experiments and simulations are carried out for a JONSWAP spectrum and Gaussian spectra of various bandwidths on deep water. It is found that the measured dispersion relation deviates from the linear dispersion relation above the spectral peak when the bandwidth is sufficiently narrow.

scholarly journals On dispersion of directional surface gravity waves

2017 ◽  
Vol 812 ◽  
pp. 681-697 ◽  
Author(s):  
Tore Magnus A. Taklo ◽  
Karsten Trulsen ◽  
Harald E. Krogstad ◽  
José Carlos Nieto Borge
Keyword(s):  
Gravity Waves ◽  
Nonlinear Evolution ◽  
Laboratory Data ◽  
Surface Gravity ◽  
Linear Dispersion ◽  
Marine Research ◽  
Surface Gravity Waves ◽  
Linear Dispersion Relation ◽  
Directional Waves ◽  
Crest Length

Using a nonlinear evolution equation we examine the dependence of the dispersion of directional surface gravity waves on the Benjamin–Feir index (BFI) and crest length. A parameter for describing the deviation between the dispersion of simulated waves and the theoretical linear dispersion relation is proposed. We find that for short crests the magnitude of the deviation parameter is low while for long crests the magnitude is high and depends on the BFI. In the present paper we also consider laboratory data of directional waves from the Marine Research Institute of the Netherlands (MARIN). The MARIN data confirm the simulations for three cases of BFI and crest length.

scholarly journals Scattering of surface gravity waves by bottom topography with a current

2007 ◽  
Vol 576 ◽  
pp. 235-264 ◽  
Author(s):  
FABRICE ARDHUIN ◽  
RUDY MAGNE
Keyword(s):  
Gravity Waves ◽  
Bottom Topography ◽  
Wave Spectrum ◽  
Action Spectrum ◽  
Wave Action ◽  
Surface Gravity ◽  
Small Scale ◽  
Surface Gravity Waves ◽  
Reflected Wave ◽  
A Current

A theory is presented that describes the scattering of random surface gravity waves by small-amplitude topography, with horizontal scales of the order of the wavelength, in the presence of an irrotational and almost uniform current. A perturbation expansion of the wave action to order η2 yields an evolution equation for the wave action spectrum, where η = max(h)/H is the small-scale bottom amplitude normalized by the mean water depth. Spectral wave evolution is proportional to the bottom elevation variance at the resonant wavenumbers, representing a Bragg scattering approximation. With a current, scattering results from a direct effect of the bottom topography, and an indirect effect of the bottom through the modulations of the surface current and mean surface elevation. For Froude numbers of the order of 0.6 or less, the bottom topography effects dominate. For all Froude numbers, the reflection coefficients for the wave amplitudes that are inferred from the wave action source term are asymptotically identical, as η goes to zero, to previous theoretical results for monochromatic waves propagating in one dimension over sinusoidal bars. In particular, the frequency of the most reflected wave components is shifted by the current, and wave action conservation results in amplified reflected wave energies for following currents. Application of the theory to waves over current-generated sandwaves suggests that forward scattering can be significant, resulting in a broadening of the directional wave spectrum, while back-scattering should be generally weaker.

Surface gravity waves over a two-dimensional random seabed

2002 ◽  
Vol 66 (1) ◽  
Author(s):  
Jørgen H. Pihl ◽  
Chiang C. Mei ◽  
Matthew J. Hancock
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Surface Gravity Waves

scholarly journals Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves

2012 ◽  
Vol 370 (1964) ◽  
pp. 1572-1586 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Gravity Waves ◽  
Finite Amplitude ◽  
Numerical Results ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Surface Gravity Waves ◽  
Stokes Waves ◽  
Irrotational Motion ◽  
Dimensional Surface

The velocity and other fields of steady two-dimensional surface gravity waves in irrotational motion are investigated numerically. Only symmetric waves with one crest per wavelength are considered, i.e. Stokes waves of finite amplitude, but not the highest waves, nor subharmonic and superharmonic bifurcations of Stokes waves. The numerical results are analysed, and several conjectures are made about the velocity and acceleration fields.

On modifications of the Zakharov equation for surface gravity waves

1984 ◽  
Vol 143 ◽  
pp. 47-67 ◽  
Author(s):  
Michael Stiassnie ◽  
Lev Shemer
Keyword(s):  
Integral Equation ◽  
Gravity Waves ◽  
Higher Order ◽  
Class I ◽  
Surface Gravity ◽  
Stokes Wave ◽  
Zakharov Equation ◽  
Infinite Depth ◽  
Surface Gravity Waves ◽  
New Equation

The Zakharov integral equation for surface gravity waves is modified to include higher-order (quintet) interactions, for water of constant (finite or infinite) depth. This new equation is used to study some aspects of class I (4-wave) and class II (5-wave) instabilities of a Stokes wave.

scholarly journals Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth

2018 ◽  
Vol 844 ◽  
pp. 491-518 ◽  
Author(s):  
Didier Clamond ◽  
Denys Dutykh
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Fast Computation ◽  
Surface Gravity Waves ◽  
Irrotational Motion ◽  
Arbitrary Precision ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral ◽  
Dimensional Surface

This paper describes an efficient algorithm for computing steady two-dimensional surface gravity waves in irrotational motion. The algorithm complexity is $O(N\log N)$, $N$ being the number of Fourier modes. This feature allows the arbitrary precision computation of waves in arbitrary depth, i.e. it works efficiently for Stokes, cnoidal and solitary waves, even for quite large steepnesses, up to approximately 99 % of the maximum steepness for all wavelengths. In particular, the possibility to compute very long (cnoidal) waves accurately is a feature not shared by other algorithms and asymptotic expansions. The method is based on conformal mapping, the Babenko equation rewritten in a suitable way, the pseudo-spectral method and Petviashvili iterations. The efficiency of the algorithm is illustrated via some relevant numerical examples. The code is open source, so interested readers can easily check the claims, use and modify the algorithm.

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