Mean flow modelling in high-order nonlinear Schr\"odinger equations

TL;DR

This paper derives a precise mean flow expression for high-order nonlinear Schrödinger equations, improving wave evolution modeling in various water depths and enhancing predictions of fluid particle trajectories.

Contribution

It introduces an accurate mean flow formula at finite depth and develops fourth-order envelope equations including mean flow effects for better wave group simulations.

Findings

Derived mean flow expression matches deep-water limit at third order

Provided fourth-order envelope equations with mean flow for unidirectional waves

Improved modeling accuracy for water wave experiments across depths

Abstract

The evaluation and consideration of the mean flow in wave evolution equations are necessary for the accurate prediction of fluid particle trajectories under wave groups, with relevant implications in several domains, from the transport of pollutants in the ocean, to the estimation of energy and momentum exchanges between the waves at small scales and the ocean circulation at large scale. We derive an expression of the mean flow at finite water depth which, in contrast to other approximations in the literature, accurately accords with the deep-water limit at third order in steepness, and is equivalent to second-order formulations in intermediate water. We also provide envelope evolution equations at fourth order in steepness for the propagation of unidirectional wave groups either in time or space that include the respective mean flow term. The latter, in particular, is required for accurately modelling experiments in water wave flumes in arbitrary depths.