Can Short‐Wave Nonlinearity Affect the Prediction of Wave Setup?

Based on high-resolution laboratory data of instantaneous surface elevation and fluid velocity associated with the propagation of short waves over a gently sloping beach (GLOBEX project), the present work compares two methods for predicting the wave setup. A one-dimensional (cross-shore) model is considered to solve the balance equation that links the pressure gradient induced by the mean surface elevation profile with the wave radiation stress (including wave roller) and bottom stress profiles. When compared to measurements and using a wave-by-wave approach, the mean elevation appears to be significantly better predicted with a nonlinear approach based on stream function theory than with the commonly used linear approach based on Airy wave theory. At the shoreline, the linear method overestimates the wave setup by at least about (Formula presented.) while this error is globally reduced by a factor 2 to 4 in the nonlinear case. In the framework of this study, the combined contribution of bottom stress and wave roller to the wave setup appears secondary since it accounts for about (Formula presented.) of the predicted setup at the shoreline. Alternative computational methods are also considered to model the wave setup. In the linear case, using a spectral instead of the wave-by-wave approach greatly improves the wave setup predictions. This improvement is related to the disparity between the representative wave height (and thus the wave energy) obtained from spectral and wave-by-wave analysis when short-wave nonlinearity becomes significant.