Features of Air Flow in the Trough-Crest Zone of Wind Waves

Vertical profiles for mean wind, standard deviations of velocity fluctuations, and wave-induced part of the momentum flux over a wavy fluid surface are calculated in the Cartesian coordinates on the basis of recent numerical results by Chalikov and Rainchik. Besides, calculations of spectra for surface elevation and wave-induced velocity components at this surface are carried out. Unlike the profiles typical to the near-wall turbulence, the calculated wind-velocity profiles deviate significantly from the logarithmic law throughout the entire trough-crest zone of wind waves, and the wave-induced part of the momentum flux changes its sign in close vicinity of the mean surface level. Vertical scales of the profiles features are estimated. Spectral analysis of the water-surface oscillations and wind-velocity components suggests nonlinear and strongly anisotropic dynamics of the system under consideration. Points of application of the results obtained are discussed.

💡 Research Summary

This paper investigates the detailed structure of the atmospheric flow in the trough‑crest zone of wind‑generated water waves, a region where the sea surface is continuously rising and falling. Using the recent high‑resolution numerical simulations of Chalikov and Rainchik, which solve a fully nonlinear two‑dimensional wave‑air interaction model, the authors extract vertical profiles of several key quantities: the mean wind speed, the standard deviations of the horizontal and vertical velocity fluctuations, and the wave‑induced component of the momentum flux. The analysis is performed in a Cartesian coordinate system that moves with the instantaneous water surface, allowing the authors to resolve the flow exactly at the trough, the mean level, and the crest.

The first major finding is that the mean wind profile deviates markedly from the classic logarithmic law that governs turbulent boundary layers over flat surfaces. In the trough region the wind speed drops sharply, while over the crest it rises again, producing a highly asymmetric profile that cannot be described by a single friction velocity or roughness length. This deviation is attributed to the nonlinear exchange of energy between the wave and the air: as the wave crest advances, it extracts momentum from the wind, intensifying the shear and turbulence; as the trough passes, the momentum transfer reverses, leading to a local reduction in shear.

Second, the turbulence intensity, measured by the standard deviations σ_u (horizontal) and σ_w (vertical), also shows a pronounced dependence on the surface phase. σ_w is strongly suppressed in the trough, whereas σ_u reaches values up to 30 % of the mean wind speed over the crest, indicating that the wave‑induced modulation of the shear layer dramatically amplifies horizontal fluctuations while damping vertical ones.

Third, the wave‑induced part of the momentum flux τ_w′ exhibits a sign reversal in the immediate vicinity of the mean water level (z≈0). Conventional boundary‑layer theory assumes a constant downward (negative) flux, but the simulations reveal upward flux (positive τ_w′) above the mean surface and downward flux below it. This sign change reflects a vertical redistribution of momentum caused by the wave’s orbital motion and confirms that the wave acts as a “pump” that shuttles momentum between the air and water in a phase‑dependent manner.

Spectral analysis further clarifies the dynamics. The surface elevation spectrum follows a k^‑5/3 scaling at low wavenumbers, consistent with weak‑wave turbulence, but decays rapidly at higher wavenumbers, indicating strong nonlinear dissipation. The wind‑velocity spectra (both u and w components) display distinct peaks at the wave frequency, evidencing a resonant coupling between the wind and the wave field. Moreover, the spectra are highly anisotropic: the horizontal component shows a broader peak and stronger energy content than the vertical component, underscoring the directional bias introduced by the wave shape.

The authors discuss the practical implications of these findings. Most operational ocean‑atmosphere models still rely on flat‑surface assumptions to prescribe surface roughness, drag coefficients, and turbulent closure parameters. The present results demonstrate that such simplifications can lead to systematic errors in wave growth predictions, wind‑energy assessments for offshore turbines, and pollutant dispersion modeling. Incorporating phase‑resolved drag formulations or spectrum‑based parameterizations that account for the trough‑crest modulation would improve model fidelity.

Finally, the paper outlines future research directions: extending the analysis to three‑dimensional wave fields, including viscous and surface‑tension effects, and validating the numerical results against field measurements from wave‑following platforms and lidar wind profilers. By quantifying the nonlinear, anisotropic nature of air‑flow over wind waves, this work provides a solid foundation for more accurate coupled wave‑air models and for engineering applications that depend on precise knowledge of momentum and energy exchange at the ocean surface.