A Continuous Non-ergodic Theory for the Wave Set-up

TL;DR

This paper develops a continuous, slope-dependent theoretical model to accurately compute wave-induced set-down and set-up in finite-depth shoaling, accounting for wave inhomogeneities and phase distributions.

Contribution

It introduces a novel non-ergodic, analytical approach to model wave set-up and set-down considering wave inhomogeneities and phase effects in finite-depth shoaling.

Findings

Provides a slope-dependent analytical formula for mean water level.

Achieves continuous modeling of wave set-up across all water depths.

Links slope magnitude to phase distribution in wave fields.

Abstract

Inhomogeneities in the wave field due to wave groups, currents, and shoaling among other ocean processes can affect the mean water level. In this work, the classical and unsolved problem of continuously computing the set-down and the following set-up induced by wave breaking on a shoal of constant finite slope is tackled. This is possible by using available theoretical knowledge on how to approximate the distribution of wave random phases in finite depth. Then, the non-homogeneous spectral analysis of the wave field allows the computation of the ensemble average by means of the phase distribution and the inversion of the integral of the second moment for the special case of a shoaling process with uniform phase distribution. In doing so, I am able to obtain a direct effect of the slope magnitude on the phases distribution. Therefore, an analytical and slope-dependent mean water level with continuity over the entire range of water depth is provided.