A geometrical Green-Naghdi type system for dispersive-like waves in prismatic channels

TL;DR

This work derives a fully nonlinear 1D geometrical Green-Naghdi (gGN) model by transverse averaging of the 2D Saint-Venant equations with bathymetric variation, capturing dispersive-like effects solely due to geometry. The model retains a Lagrangian structure and exact energy balance, supports exact travelling-wave solutions (solitary and composite waves), and admits two numerically friendly reformulations: an elliptic–hyperbolic split and a hyperbolic relaxation approach. Dispersion is governed by a single geometric parameter χ, computable for common cross-sections (triangular, trapezoidal), and the framework is validated through analytical solitons, 2D SHW comparisons, and Treske’s trapezoidal experiments, showing good agreement in the low-Froude regime where geometrical dispersion dominates. The results highlight the existence and significance of geometrical dispersive waves in prismatic channels and provide robust numerical tools for simulating bore propagation and wave breaking tendencies in engineering and environmental contexts, with potential for coupling to horizontal dispersion and non-hydrostatic effects.

Abstract

We consider 2D free surface gravity waves in prismatic channels with bathymetric variations uniquely in the transverse direction. Starting from the Saint-Venant equations (shallow water equations) we derive a 1D transverse averaged model describing dispersive effects solely related to variations of the channel topography. These effects have been demonstrated in Chassagne et al. JFM 2019 to be predominant in the propagation of bores with Froude numbers below a critical value of about 1.15. The model proposed is fully nonlinear, Galilean invariant, and admits a variational formulation under natural assumptions about the channel geometry. It is endowed with an exact energy conservation law, and admits exact travelling wave solutions. Our model generalizes and improves the linear equations proposed by Chassagne et al. JFM 2019, as well as Quezada de Luna and Ketcheson JFM 2021. The system is recast in two useful forms appropriate for its numerical approximations, whose properties are discussed. Numerical results allow to verify against analytical solutions the implementation of these formulations, and validate our model against fully 2D nonlinear shallow water simulations, as well as the famous experiments by Treske J. Hyd. Res. 1994.

Figures (21)

• Figure 1: Channel geometry. (a) longitudinal axis $x$, transverse axis $y$, vertical axis $z$. (b) triangular section with ${\it l}={\it l}_1+{\it l}_2$, and $b(0)=b({\it l})=b_0, \; b({\it l}_1)=0$. (c) trapezoidal section with ${\it l}={\it l}_1 + {\it l}_2 +{\it l}_3$ and $b(0)=b({\it l})=b_0, \; b({\it l}_1)= b({\it l}_1+{\it l}_2)=0$.
• Figure 2: Dispersion relation for trapezoidal sections. (a) notation. (b) phase speed for a deep channel with steep slopes. (c) shallow channel with mild slopes. Simplified gGN model in blue, linear model of Chassagne in red.
• Figure 3: Lemoine analogy for the experiments by Treske. Comparison of the theoretical predictions using the dispersion relation of the simplified gGN model (blue), with the one of Chassagne (red), and the data by Treske (symbols).
• Figure 4: Solitary waves obtained for values of $\chi\in\{0.2,\,0.4,\,0.6\}\text{m}^4$, and non-linearity $\epsilon=0.2$ (a), and $\epsilon=0.5$ (b).
• Figure 5: Composite travelling wave solution obtained for $\chi=0.4$ m$^4$, setting $\tau_1=1$ m$^{-1}$, $\tau_2=1.3$ m$^{-1}$, and $\tau_3=1.301$$m^{-1}$. (a) solution on the whole domain [-350, 2100] m. (b) zoom of the regularized jump and first peak.

Theorems & Definitions (3)

• Proposition 1: Lagrangian structure for symmetric and quasi-symmetric channel cross-sections
• Proposition 2: Lagrangian structure for wide channel cross-sections
• Proposition 3: Coercivity of the elliptic operator