# Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model

## TL;DR

This paper develops a hybrid wave model combining non-linear shallow water and Boussinesq equations, introducing a new wave breaking detection criterion based on free surface Froude number, with a novel discretization approach and extensive benchmarking.

## Contribution

It presents a new physically meaningful wave breaking detection method and integrates it into a hybrid Boussinesq model with a novel finite element discretization strategy.

## Key findings

- The new detection criterion effectively identifies wave breaking events.
- The hybrid model accurately simulates wave propagation, shoaling, and run-up.
- Benchmark tests demonstrate the robustness and advantages of the proposed method.

## Abstract

The aim of the present work is to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sorensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave's crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy, that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.