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Optimal Boussinesq model for shallow-water waves interacting with a microstructure.

Josselin Garnier¹, Roberto A Kraenkel, André Nachbin

¹Laboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France. garnier@math.jussieu.fr

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|November 13, 2007

Summary

This study optimizes reduced models for water wave propagation over periodic topography. An optimal parameter choice matches reduced models to full potential theory, enhancing understanding of microstructured wave dynamics.

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Area of Science:

Fluid dynamics
Applied mathematics
Wave propagation

Background:

Water wave propagation is complex, especially with varying bottom topography.
Periodic microstructured topography introduces challenges in modeling wave behavior.
Existing models may not fully capture the interplay of dispersion and nonlinearity.

Purpose of the Study:

To analyze water wave propagation in a long-wave regime over short-scale periodic topography.
To compare a full potential theory model with reduced Boussinesq systems.
To derive and optimize effective Korteweg-de Vries (KdV) equations for such systems.

Main Methods:

Multiscale asymptotic analysis of fluid dynamics models.
Parametric study of reduced Boussinesq systems.
Derivation of effective Korteweg-de Vries (KdV) equations.

Main Results:

Explicit expressions for effective KdV equation coefficients were obtained.
An optimal parameter for reduced models was identified to match full model limits.
Nonlinearity is enhanced by rough bottoms, with dispersion effects varying based on topography period.

Conclusions:

Reduced Boussinesq systems can accurately model water waves over periodic microstructured topography when optimally parametrized.
The choice of evaluation depth parameter is crucial for model accuracy.
Rough bottoms significantly alter wave propagation characteristics, affecting both nonlinearity and dispersion.