Surfing Waves from the Ocean to the River with the Serre-Green-Naghdi Equations

Abstract

A wide variety of hydraulic and coastal flows can be modeled using shallow water theories, where the so-called Serre-Green-Naghdi (SGN) equations constitute a fully nonlinear and weakly dispersive wave theory that has been successfully applied in fluvial and maritime contexts. In the present contribution, we show that SGN models with wave-breaking parameterizations can reproduce challenging nonlinear processes in the surf and swash zones including wave–wave interactions and infragravity wave generation from a narrow-band swell spectrum. The excellent performance of the model motivates us to explore its application to a simplified shallow bar-built river configuration where surf zone–generated infragravity waves may propagate upstream the river. We show that long-wave penetration is controlled by the Froude number over the bar, and that these nonlinear long waves may give rise to a solitonic dynamics. The power spectral density (PSD) signature of free surface time series with a slope of ∼f−1 in the infragravity range is consistent with the latter, as also found in field observations. Transferring of energy into lower frequencies is observed in the numerical experiment as long waves propagate upstream; nevertheless, low-frequency energy also cascades back into the swell energy band. Standard linear Fourier analysis may fail in showing the hidden solitonic dynamics, so nonlinear techniques would need to be applied to fully elucidate the fate of the long-wave energy while propagating upstream of river mouths.