Rectification of a deep water model for surface gravity waves

TL;DR

This paper analyzes a quadratic nonlinear model for deep water waves, identifies its ill-posedness issues, and proposes a regularization method that ensures well-posedness and preserves key physical and numerical properties.

Contribution

It introduces a regularization technique for the quadratic water wave model, providing the first rigorous justification for filtering practices in numerical simulations.

Findings

The quadratic model is likely ill-posed in finite regularity spaces.

Regularizing operators restore well-posedness without losing Hamiltonian structure.

Numerical simulations support the effectiveness of the regularization approach.

Abstract

In this work we discuss an approximate model for the propagation of deep irrotational water waves, specifically the model obtained by keeping only quadratic nonlinearities in the water waves system under the Zakharov/Craig-Sulem formulation. We argue that the initial-value problem associated with this system is most likely ill-posed in finite regularity spaces, and that it explains the observation of spurious amplification of high-wavenumber modes in numerical simulations that were reported in the literature. This hypothesis has already been proposed by Ambrose, Bona, and Nicholls [4] but we identify a different instability mechanism. On the basis of this analysis, we show that the system can be "rectified". Indeed, by introducing appropriate regularizing operators, we can restore the well-posedness without sacrificing other desirable features such as a canonical Hamiltonian structure, cubic accuracy as an asymptotic model, and efficient numerical integration. This provides a first rigorous justification for the common practice of applying filters in high-order spectral methods for the numerical approximation of surface gravity waves. While our study is restricted to a quadratic model, we believe it can be generalized to any order and paves the way towards the rigorous justification of a robust and efficient strategy to approximate water waves with arbitrary accuracy. Our study is supported by detailed and reproducible numerical simulations.