A multiscale model for weakly nonlinear shallow water waves over periodic bathymetry

TL;DR

The study develops a multiscale homogenization of the shallow-water (Saint-Venant) system with periodic bathymetry to derive high-order, constant-coefficient dispersive models. Through a systematic averaging framework, the authors obtain a hierarchy of effective equations, analyze their dispersion and traveling-wave solutions, and compare them to direct variable-bathymetry simulations. The resulting dispersive homogenized models reproduce solitary-wave trains and related dynamics while offering substantial computational speedups (up to roughly two orders of magnitude) relative to full shallow-water simulations. The work provides a robust methodological pathway for studying nonlinear waves in periodically structured shallow-water environments and demonstrates practical benefits for simulations and analysis.

Abstract

We study the behavior of shallow water waves over periodically-varying bathymetry, based on the first-order hyperbolic Saint-Venant equations. Although solutions of this system are known to generally exhibit wave breaking, numerical experiments suggest a different behavior in the presence of periodic bathymetry. Starting from the first-order variable-coefficient hyperbolic system, we apply a multiple-scale perturbation approach in order to derive a system of constant-coefficient high-order partial differential equations whose solution approximates that of the original system. The high-order system turns out to be dispersive and exhibits solitary-wave formation, in close agreement with direct numerical simulations of the original system. We show that the constant-coefficient homogenized system can be used to study the properties of solitary waves and to conduct efficient numerical simulations.

Figures (9)

• Figure 1: Depiction of notational conventions used in this work \newlabelfig:scenario0
• Figure 1: Non-dimensional dispersion relations corresponding to the three forms of the linearized homogenized systems \ref{['o4avg-ttt']}, \ref{['o4avg']}, and \ref{['o4avg-xxt']}. \newlabelfig:disp_rel_30
• Figure 1: Comparison of homogenized and direct solutions, with bathymetry and initial data given by \ref{['scenario_a']}. The surface elevation $\eta - \eta^0$ is shown, with the $x$-axis shifted to show the wave structure at each time. \newlabelfig:scenario_a0
• Figure 2: Evolution of an initial Gaussian pulse over periodic bathymetry. The surface elevation is shown, measured in meters. For comparison, the dashed blue line shows the solution for flow over a flat bottom. \newlabelfig:example10
• Figure 2: Typical potential for the "motion" of the particle. The first panel represents a typical potential $U(\xi)$ of Eq. \ref{['fig:potential']}, while the second panel is a closeup of the first one in the area near the origin. If the total energy is less than zero and the initial position of the particle is located between $0$ and $\eta_C$ (second panel), then the motion is periodic. This case is described by the red curve, which corresponds to a "particle" oscillating between points $A$ and $B$. $\eta_A$ and $\eta_B$ represent the minimum and the maximum of the periodic traveling wave, while the period is twice the "time" necessary to go from $A$ to $B$. If the level of the energy approaches zero from below, then the period becomes longer and longer, up to the limit situation corresponding to the dashed line. In such a case the solution describes a solitary wave traveling with speed $V$. The third panel displays the trajectory in phase space. The red energy level corresponds to the periodic solution which oscillates between $A$ and $B$. The dashed line represents the separatrix, whose right lobe corresponds to a solitary wave, which is reported in the right panel. Finally, the last panel shows the shape of the solitary wave as a function of $\xi=x-Vt$.

Theorems & Definitions (32)

• Remark 1
• Proposition 1
• Proof 1
• Proposition 2
• Proof 2
• Corollary 1
• Proposition 3
• Proof 3
• Lemma 1
• Proof 4