[Paper Review] On the kurtosis of ocean waves in deep water

Francesco Fedele|arXiv (Cornell University)|Dec 28, 2014

Ocean Waves and Remote Sensing8 citations

TL;DR

This paper refines Janssen's (2003) formulation for dynamic excess kurtosis in weakly nonlinear deep-water gravity waves, deriving an analytical solution for narrowband directional waves with Gaussian spectra. It shows that kurtosis peaks initially at a critical time scale τ_c ≈ 0.13 / (uσ_θ), reaching positive maxima in focusing regimes or negative minima in defocusing ones, before decaying to zero as the system approaches quasi-equilibrium dominated by bound harmonics.

ABSTRACT

In this paper, we revisit Janssen's (2003) formulation for the dynamic excess kurtosis of weakly nonlinear gravity waves at deep water. For narrowband directional spectra, the formulation is given by a sixfold integral that depends upon the Benjamin-Feir index and the parameter $R=\sigma_{ heta}^{2}/2 u^{2}$, a measure of short-crestedness for the dominant waves with $ u$ and $\sigma_{ heta}$} denoting spectral bandwidth and angular spreading. Our refinement leads to a new analytical solution for the dynamic kurtosis of narrowband directional waves described with a Gaussian type spectrum. For multidirectional or short-crested seas initially homogenous and Gaussian, in a focusing (defocusing) regime dynamic kurtosis grows initially, attaining a positive maximum (negative minimum) at the intrinsic time scale \[ au_{c}= u^{2}\omega_{0}t_{c}=1/\sqrt{3R},\qquad\mathrm{or}\qquad t_{c}/T_{0}\approx0.13/ u\sigma_{ heta}, \] where $\omega_{0}=2\pi/T_{0}$ denotes the dominant angular frequency. Eventually the dynamic excess kurtosis tends monotonically to zero as the wave field reaches a quasi-equilibrium state characterized with nonlinearities mainly due to bound harmonics. Quasi-resonant interactions are dominant only in unidirectional or long-crested seas where the longer-time dynamic kurtosis can be larger than that induced by bound harmonics, especially as the Benjamin-Feir index increases. Finally, we discuss the implication of these results on the prediction of rogue waves.

Motivation & Objective

• To refine Janssen's (2003) formulation for dynamic excess kurtosis in weakly nonlinear deep-water waves.
• To derive an analytical solution for the dynamic kurtosis of narrowband directional waves with Gaussian-type spectra.
• To investigate the time evolution of kurtosis in multidirectional or short-crested seas under varying spectral parameters.
• To assess the role of quasi-resonant interactions versus bound harmonics in shaping kurtosis dynamics.
• To evaluate implications for rogue wave prediction based on kurtosis evolution and spectral characteristics.

Proposed method

• Reformulates Janssen's sixfold integral expression for dynamic excess kurtosis using a Gaussian-type directional spectrum.
• Introduces the parameter R = σ_θ² / (2u²) to quantify short-crestedness of dominant waves, combining spectral bandwidth u and angular spreading σ_θ.
• Derives an analytical solution for dynamic kurtosis in narrowband directional seas, enabling closed-form analysis of time evolution.
• Identifies the intrinsic time scale τ_c = u²ω₀t_c = 1/√(3R), which determines the peak of dynamic kurtosis.
• Analyzes the transition to quasi-equilibrium, where excess kurtosis decays monotonically to zero due to bound harmonic nonlinearities.
• Compares dominance of quasi-resonant interactions (in unidirectional seas) versus bound harmonics (in multidirectional seas) in shaping kurtosis.

Experimental results

Research questions

• RQ1 How does the dynamic excess kurtosis of weakly nonlinear deep-water waves evolve over time in narrowband directional seas?
• RQ2 What is the role of the parameter R = σ_θ² / (2u²) in determining the timing and magnitude of kurtosis peaks?
• RQ3 In what conditions do quasi-resonant interactions dominate over bound harmonic effects in kurtosis generation?
• RQ4 How does the time scale τ_c = 1/√(3R) relate to the observed peak of dynamic kurtosis in focusing and defocusing regimes?
• RQ5 To what extent does the kurtosis in multidirectional seas remain elevated compared to unidirectional seas, and how does this affect rogue wave likelihood?

Key findings

• The dynamic excess kurtosis reaches a positive maximum in focusing regimes or a negative minimum in defocusing regimes at the intrinsic time scale τ_c ≈ 0.13 / (uσ_θ).
• The peak kurtosis magnitude increases with the Benjamin-Feir index, especially in unidirectional or long-crested seas where quasi-resonant interactions dominate.
• After the peak, dynamic excess kurtosis decays monotonically to zero as the wave field approaches a quasi-equilibrium state dominated by bound harmonics.
• In multidirectional or short-crested seas, the influence of quasi-resonant interactions is suppressed, and kurtosis is primarily governed by bound harmonic nonlinearities.
• The analytical solution for Gaussian-type spectra enables precise prediction of kurtosis evolution, particularly around the critical time τ_c.
• The results suggest that kurtosis dynamics are most pronounced in seas with moderate to high directional spreading and spectral bandwidth, with implications for rogue wave risk assessment.