[Paper Review] Modeling shallow water waves

David Lannes|HAL (Le Centre pour la Communication Scientifique Directe)|Jan 27, 2020

Coastal and Marine Dynamics6 citations

TL;DR

This paper presents a unified asymptotic derivation of shallow water wave models, including nonlinear shallow water, Boussinesq, and Serre-Green-Naghdi equations, by analyzing turbulent and non-hydrostatic pressure terms in vertically integrated Euler equations. It extends these models to include vorticity effects and topography, offering new formulations for multi-layer systems and rotational flows, with applications to wave breaking and strong current interactions.

ABSTRACT

We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the "turbulent" and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre-Green-Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe-Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension $d=1$; among them are the KdV, BBM, Camassa-Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.

Motivation & Objective

• To provide a systematic and unified derivation of major shallow water wave models used in coastal oceanography.
• To extend classical models (e.g., NSW, Boussinesq, SGN) to include vorticity and non-flat topography for improved physical realism.
• To address open mathematical problems in wave breaking, boundary conditions, and singularity formation in nonlinear dispersive systems.
• To introduce new multi-layer and Isobe-Kakinuma-type models with reduced numerical complexity and enhanced vertical resolution.
• To explore the role of vorticity in wave dynamics, particularly in modeling wave breaking and propagation over strong currents.

Proposed method

• Derives the water wave equations from the free-surface Euler equations via vertical integration, identifying turbulent and non-hydrostatic pressure terms.
• Applies asymptotic expansions in the shallow water regime (small aspect ratio μ) to systematically derive approximate models.
• Uses weak nonlinearity (ε ≪ 1) and higher-order expansions to derive Boussinesq and Serre-Green-Naghdi models with dispersive effects.
• Introduces multi-layer formulations to better capture vertical velocity and pressure structure without high-order derivatives.
• Generalizes formulations to include non-flat topography using bottom slope β and depth-dependent corrections.
• Incorporates vorticity via a strength parameter α, with special treatment for α = 1/2 to model strong rotational effects.

Experimental results

Research questions

• RQ1 How can the full range of shallow water models be derived from a unified asymptotic framework based on turbulent and non-hydrostatic terms?
• RQ2 What are the mathematical and physical implications of including vorticity in nonlinear shallow water and Boussinesq-type models?
• RQ3 How do topographic variations affect the derivation and structure of shallow water models, particularly in the Serre-Green-Naghdi and multi-layer formulations?
• RQ4 What are the limitations and potential of scalar models (e.g., KdV, Camassa-Holm) in describing unidirectional wave propagation?
• RQ5 Can rotational shallow water models with vorticity provide a mathematically justifiable framework for wave breaking phenomena?

Key findings

• The nonlinear shallow water (NSW) equations emerge as the leading-order approximation when both turbulent and non-hydrostatic terms are neglected.
• The Boussinesq systems are derived under weak nonlinearity and retain dispersive effects through a specific non-hydrostatic pressure correction.
• The Serre-Green-Naghdi (SGN) model is obtained without the weak nonlinearity assumption, capturing higher-order dispersive and nonlinear effects.
• Multi-layer extensions of NSW and SGN models improve vertical resolution and reduce numerical stiffness compared to high-order single-layer models.
• The Isobe-Kakinuma model is shown to be a viable alternative for enhanced accuracy without introducing high-order derivatives.
• Vorticity effects are incorporated via a strength parameter α, with α = 1/2 leading to a distinct scaling that captures strong rotational flows and wave breaking dynamics.