[Paper Review] On the applicability of the Hasselmann kinetic equation to the Phillips spectrum

A. O. Korotkevich, В. Е. Захаров|arXiv (Cornell University)|Dec 28, 2012

Ocean Waves and Remote Sensing10 references4 citations

TL;DR

This paper investigates the applicability of the Hasselmann kinetic equation to the Phillips spectrum in wind-driven ocean waves. Using large-scale numerical simulations of the Euler equations for potential flow, the authors demonstrate that even under high nonlinearity (where the Phillips spectrum dominates), wave spectral lines remain narrow in the direct cascade region, validating the use of the kinetic equation for modeling wave interactions despite the presence of wave breaking and localized coherent structures.

ABSTRACT

We investigate applicability of the Hasselmann kinetic equation to the spectrum of surface gravity waves at different levels of nonlinearity in the system, which is measured as average steepness. It is shown that even in the case of relatively high average steepness, when Phillips spectrum is present in the system, the spectral lines are still very narrow, at least in the region of direct cascade spectrum. It allows us to state that even in the case of Phillips spectrum the kinetic equation can be applied to the description of the ensembles of ocean waves.

Motivation & Objective

• To assess whether the Hasselmann kinetic equation remains applicable in the presence of the Phillips spectrum, which characterizes high nonlinearity and wave breaking.
• To determine if wave spectral lines remain narrow under conditions where the Phillips spectrum dominates, a key requirement for kinetic theory validity.
• To provide a foundation for constructing a physically justified dissipation term in operational wave prediction models.
• To investigate the coexistence of weak turbulence (described by kinetic equations) and strong turbulence (localized wave breaking) in ocean wave systems.
• To explore the implications of spectral line width for the statistical description of wind-driven wave systems with mixed weak and strong turbulence.

Proposed method

• Numerical simulation of the full Euler equations for incompressible, deep-water potential flow with a free surface using a pseudo-spectral code.
• Computation of Fourier spectra of space-time correlation functions of normal canonical variables to analyze wave spectral line shapes.
• Analysis of frequency spectra of spatial Fourier harmonics to test the narrow-line prediction of weak turbulence theory.
• Use of a high-resolution pseudo-spectral method to resolve both weakly nonlinear wave interactions and localized wave-breaking events.
• Measurement of average steepness as a control parameter to assess nonlinearity levels across simulations.
• Comparison of simulated spectra with theoretical predictions of weak turbulence, including Kolmogorov-Zakharov (KZ) and Phillips spectra.

Experimental results

Research questions

• RQ1 Can the Hasselmann kinetic equation describe wave systems exhibiting the Phillips spectrum, which arises from strong nonlinearity and wave breaking?
• RQ2 Are spectral lines narrow in the direct cascade region when the Phillips spectrum is present, as required for kinetic theory applicability?
• RQ3 Does the presence of localized wave-breaking events (whitecapping) disrupt the narrow spectral line structure predicted by weak turbulence theory?
• RQ4 To what extent can the kinetic equation model wave interactions in a system dominated by strong turbulence phenomena?
• RQ5 What is the relationship between average steepness and the persistence of narrow spectral lines in the presence of wave-breaking dynamics?

Key findings

• Even at relatively high average steepness, where the Phillips spectrum is present, spectral lines in the direct cascade region remain narrow.
• The narrowness of spectral lines persists despite the dominance of wave-breaking processes, indicating that weak turbulence theory remains applicable.
• The results support the use of the Hasselmann kinetic equation for modeling wind-driven ocean wave systems, even in regimes with strong nonlinearity.
• The coexistence of weak turbulence (KZ spectra) and strong turbulence (Phillips spectrum) does not invalidate the kinetic description in the direct cascade region.
• The findings validate the theoretical basis for incorporating a dissipation term in wave prediction models, as the spectral structure remains consistent with kinetic theory.
• Numerical evidence supports that wave-breaking does not destroy the statistical coherence required for kinetic equation modeling, even when the system is far from Gaussian statistics.