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The use of these asymptotic models in such configurations is of course far from being rigorously justified. 2. Related frameworks. We are interested here in applications to coastal oceanography where the typical scale of interest is the size of the wave or the size of the beach. Shallow water models are also derived in oceanography for much larger scales for which rotation effects should be included. If we identify the vertical axis with the direction of gravity, we must consider that the equations are written in a rotating frame, and therefore add a Coriolis force.

See [17, 264, 152, 153]. In the paper by Craig et al. [105], another approach is used. 4), and (Hamiltonian) asymptotic models are deduced from the homogenized Hamiltonian. The periodic case is addressed in [105], while random topographies are treated in [43]. None of these two approaches brings a complete justification of the homogenized models they propose: The shallow water models used as a starting point in the first approach are a priori not valid for rapidly varying topographies, and the homogenization limit performed on the Hamiltonian in the second approach is not fully justified.

3, the choice of a scaling length for the vertical variable is purely technical. , z z = . 1), namely, √ √ − μ + β μb ≤ z ≤ ζ , ΔγX ,z Φ = 0 for Φ|z = ζ =ψ, ∂n Φ|z =−√μ+β√μb = 0, ΔγX ,z = ∂z2 + γ 2 ∂y2 + ∂z2 (if d = 2). Because of the change in the scaling where of z, the dependence on μ no longer appears in the Laplace operator, but in the bottom parametrization. Both formulations are of course completely equivalent in the sense that they give the same expression for G[ζ, b]ψ. While the first scaling is more convenient to study shallow water regimes, the second scaling is the relevant one to handle the case of infinite depth.