[Paper Review] Implementation of a fully nonlinear Hamiltonian Coupled-Mode Theory, and application to solitary wave problems over bathymetry

Christos Papoutsellis, Andreas Charalampopoulos+1 more|arXiv (Cornell University)|Oct 30, 2017

Coastal and Marine Dynamics62 references3 citations

TL;DR

This paper presents a fully nonlinear, Hamiltonian coupled-mode theory (HCMT) for water waves over arbitrary bathymetry, using an exact semi-separation of variables to derive a new, analytically computed Dirichlet-to-Neumann operator. The method enables efficient, high-accuracy simulation of solitary wave interactions with complex seabed topographies and vertical walls in 2D and 3D, revealing new Bragg scattering effects and wave disintegration/focusing over 3D bathymetries.

ABSTRACT

This paper deals with the implementation of a new, efficient, non-perturbative, Hamiltonian coupled-mode theory (HCMT) for the fully nonlinear, potential flow (NLPF) model of water waves over arbitrary bathymetry, Papoutsellis and Athanassoulis (2017) (arXiv:1704.03276). Applications considered herein concern the interaction of solitary waves with bottom topographies and vertical walls both in two- and three-dimensional environments. The essential novelty is a new representation of the Dirichlet-to-Neumann operator, needed to close the Hamiltonian evolution equations. This representation emerges from the treatment of the substrate kinematical problem by means of exact semi-separation of variables in the irregular fluid domain, established recently by Athanassoulis & Papoutsellis (2017) (https://doi.org/10.1098/rspa.2017.0017). The HCMT ensures an efficient dimensional reduction of the exact NLFP, being able to treat an arbitrary bathymetry as simply as the flat-bottom case. A key point for the efficient implementation of HCMT is the fast and accurate evaluation of the space-time varying coefficients appearing in some of its equations. All varying coefficients are calculated analytically, resulting in a refined version of the theory, characterized by improved accuracy at significantly reduced computational time. This improved version of HCMT is first validated against existing experimental results and other computations, and subsequently applied to new solitary wave-bottom interaction problems. The latter include: i) the investigation of a new type of Bragg scattering effect, appearing when a solitary wave propagates over a seabed with a sinusoidal patch, and ii) the disintegration, focusing and reflection of a solitary wave moving over a three-dimensional bathymetry consisting of parallel banks and troughs, and impinging on a vertical wall.

Motivation & Objective

• To develop a non-perturbative, fully nonlinear Hamiltonian framework for water wave dynamics over arbitrary bathymetry.
• To overcome computational inefficiencies in existing models by enabling dimensional reduction comparable to flat-bottom cases.
• To enable accurate simulation of solitary wave interactions with 3D seabed topographies and vertical walls.
• To investigate novel wave phenomena such as Bragg scattering and wave disintegration over structured bathymetries.
• To validate the model against experimental data and established numerical results.

Proposed method

• The method employs a new representation of the Dirichlet-to-Neumann operator derived via exact semi-separation of variables in irregular fluid domains.
• The theory is formulated within a Hamiltonian framework, ensuring energy conservation and structural fidelity.
• All space-time varying coefficients in the evolution equations are computed analytically, enhancing computational efficiency.
• The approach allows treatment of arbitrary bathymetry with the same computational complexity as the flat-bottom case.
• The model is implemented numerically using a spectral collocation method for spatial discretization and time integration.
• The theory is validated through comparison with experimental data and other numerical simulations.

Experimental results

Research questions

• RQ1 Can a fully nonlinear, Hamiltonian coupled-mode theory be developed that efficiently handles arbitrary bathymetry without perturbative assumptions?
• RQ2 What novel wave phenomena emerge when a solitary wave propagates over a sinusoidal seabed patch?
• RQ3 How does a 3D bathymetry composed of parallel banks and troughs affect the disintegration, focusing, and reflection of a solitary wave?
• RQ4 To what extent does the analytical computation of coefficients improve accuracy and reduce computational cost compared to numerical evaluation?
• RQ5 How well does the HCMT reproduce experimental results for wave-bottom interactions?

Key findings

• The HCMT successfully captures solitary wave interactions with complex bathymetries, including 3D structures, with high accuracy and reduced computational cost.
• The theory reveals a new type of Bragg scattering effect when a solitary wave encounters a sinusoidal seabed patch, indicating periodic wave reflection and modulation.
• Over a 3D bathymetry of parallel banks and troughs, the solitary wave undergoes disintegration, focusing, and complex reflection patterns upon impact with a vertical wall.
• Analytical evaluation of time- and space-varying coefficients leads to significantly improved computational efficiency and accuracy over numerical alternatives.
• The model shows excellent agreement with experimental data and other numerical results, validating its reliability for nonlinear wave problems.
• The method maintains Hamiltonian structure, ensuring long-term energy conservation in simulations.