Full Nonlinearity in Weakly Dispersive Boussinesq Models: Luxury or Necessity

Boussinesq-type (BT) models play a major role in coastal engineering applications. Due to their asymptotic nature, these models have lent themselves to constant study and improvement based on the inclusion/removal of asymptotically small terms. In this work, we go back to the fundamental question of whether full nonlinearity in weakly dispersive BT models is a necessity. We reconsider the tests first used in the literature to address this issue, as well as a number of more demanding issues. We also consider different families of weakly nonlinear BT models, with different shoaling characteristics, especially when nonlinear waves are involved. Our study allows us to point out that for many cases, it is quite hard to conclude whether full nonlinearity is really necessary. There are a few discriminant cases, which are unfortunately not those mostly used in the literature proposing new models or new numerical methods.