  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.
