D. Lannes, Nonlinearity 33 (2020), R1 Download

We review here the derivation of many of the most important models that
appear in the literature (mainly in coastal oceanography) for the description
of waves in shallow water. We show that these models can be obtained
using various asymptotic expansions of the ‘turbulent’ and non-hydrostatic
terms that appear in the equations that result from the vertical integration of
the free surface Euler equations. Among these models are the well-known
nonlinear shallow water (NSW), Boussinesq and Serre–Green–Naghdi (SGN)
equations for which we review several pending open problems. More recent
models such as the multi-layer NSW or SGN systems, as well as the Isobe–
Kakinuma equations are also reviewed under a unified formalism that should
simplify comparisons. We also comment on the scalar versions of the various
shallow water systems which can be used to describe unidirectional waves in
horizontal dimension *d *= 1; among them are the KdV, BBM, Camassa–Holm
and Whitham equations. Finally, we show how to take vorticity effects into
account in shallow water modeling, with specific focus on the behavior of the
turbulent terms. As examples of challenges that go beyond the present scope
of mathematical justification, we review recent works using shallow water
models with vorticity to describe wave breaking, and also derive models for
the propagation of shallow water waves over strong currents.