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# Communications in Mathematical Sciences

## Volume 16 (2018)

### Number 5

### A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

Pages: 1169 – 1202

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n5.a1

#### Authors

#### Abstract

In geophysics, the shallow water model is a good approximation of the incompressible Navier–Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (*e.g.* the non-hydrostatic or the Green–Naghdi models); the second one by proposing a multilayer version of the shallow water model.

In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to approximate the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity.

#### Keywords

free surface flows, semi-discretisation in space, dispersive models, energy estimates, linear dispersion relations

#### 2010 Mathematics Subject Classification

35B45, 35Q31, 65M08, 76B15

Received 25 July 2017

Received revised 27 January 2018

Accepted 27 January 2018

Published 19 December 2018