A Terrain-Following Boussinesq System

Nachbin, André

doi:10.1137/s0036139901397583

Cited by 59 publications

(101 citation statements)

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“…In particular the Fourier analysis presented naturally suggests the use of FFT based methods to generate efficient numerical schemes. The formulation presented here also extends a, recently published [23], weakly dispersive terrain-following Boussinesq system. A regime of interest is pulse shaped surface gravity waves interacting with rapidly varying, disordered, topographies.…”

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confidence: 62%

“…New nonlinear evolution equations are derived that generalize the system by Matsuno [16] and a terrain-following Boussinesq system by Nachbin [23]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth.…”

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confidence: 99%

“…Matsuno develops a method based on the theory of complex functions and a systematic perturbation theory with respect to the steepness parameter ε ≡ a/ℓ (a is the amplitude scale and ℓ is a wavelength scale). We start by using dimensionless parameters such as in Whitham's book [30] and in [23]: let α = a/h o be the nonlinearity parameter (with the typical, say average depth denoted by h o ) and β = h 2 o /ℓ 2 be the dispersion parameter. As pointed out in [16] the formulation naturally suggests combining these parameters as α √ β ≡ ε.…”

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confidence: 99%

“…Boussinesq-type models are of great interest in Geophysical applications. Recently, using a power series expansion for the velocity potential, Nachbin [23] derived a terrainfollowing Boussinesq system valid for weakly dispersive, weakly nonlinear waves in the presence of a rapidly varying topography. This model was used to study long waves interacting with disordered (randomly varying) topographies [17,18,10,12].…”

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confidence: 99%

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Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator

Artiles¹,

Nachbin²

2004

Methods and Applications of Analysis

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Abstract. New nonlinear evolution equations are derived that generalize the system by Matsuno [16] and a terrain-following Boussinesq system by Nachbin [23]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters [1].

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confidence: 62%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator

Artiles¹,

Nachbin²

2004

Methods and Applications of Analysis

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“…The interaction of surface waves with mild or tough bottoms has always attracted the particular attention of researchers [4,14,26,44]. There are however few studies which attempt to include the bottom curvature effect into the classical Saint-Venant [16,53] or Savage-Hutter 1 [31,49] equations.…”

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confidence: 99%

Modified shallow water equations for significantly varying seabeds

Dutykh

Clamond²

2016

Applied Mathematical Modelling

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Abstract. In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.

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An iterative method to compute conformal mappings and their inverses in the context of water waves over topographies

Flamarion

Ribeiro-Jr

2021

Numerical Methods in Fluids

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An iterative numerical method to compute the conformal mapping in the context of propagating water waves over uneven topographies is investigated.The map flattens the fluid domain onto a canonical strip in which computations are performed. The accuracy of the method is tested by using the MAT-LAB Schwarz-Christoffel toolbox mapping as a benchmark. Besides, we give a numerical alternative to compute the inverse of the conformal map.

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A Terrain-Following Boussinesq System

Cited by 59 publications

References 9 publications

Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator

Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator

Modified shallow water equations for significantly varying seabeds

An iterative method to compute conformal mappings and their inverses in the context of water waves over topographies

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