# Fully nonlinear Boussinesq equations for long wave propagation and run-up in sloping channels with parabolic cross sections

## Abstract

A general framework for derivation of long wave equations in narrow channels, and their transformation to Lagrangian coordinates is briefly established. Then, fully nonlinear Boussinesq equations are derived for channels of parabolic cross sections. The simplified version with normal nonlinearity is compared with corresponding models from the literature, and propagation properties are discussed. A Lagrangian run-up model is adapted to the fully nonlinear set. This model is tested by means of controlled residues and by a well-controlled comparison to exact analytic solutions from the literature. Then, run-up of solitary waves in simple geometries is simulated and compared to a semi-analytic solution that is derived for propagation and run-up in a composite channel. The dispersive model retains the higher run-up height in a parabolic channels, as reported in the recent literature for NLSW solutions, as compared to a rectangular channel.

## Keywords

Run-up Parabolic channel Boussinesq equations## Notes

### Acknowledgments

The subject presented was first suggested after a talk by Efim Pelinovsky back in 2011. Unfortunately, it took a while before the present study materialized. The author is also grateful to Patricio Winckler Grez for helpful communication. This work has been supported by the Norwegian Research Council under project no. 205184.

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