An analytical framework for wave-coherent airflow over fast surface waves
Abstract
Classical theories attribute wind-wave momentum transfer to a critical layer, where the mean wind speed equals the wave phase speed. For waves that travel faster than the near-surface wind, this layer may lie outside the wave boundary layer or be absent altogether. We develop an analytical description of the wave-coherent airflow in this regime for a prescribed mean wind, using a Wentzel--Kramers--Brillouin--Jeffreys (WKBJ) approximation to the viscous Orr--Sommerfeld equation in wave-following coordinates. The solution combines an outer Rayleigh mode, which carries the dominant wave-coherent motion, with a thin viscous Stokes layer that supplies the phase shift producing the pressure form stress. At leading order, the form stress scales with the inverse square root of the wave Reynolds number, with a coefficient set by an integral of the outer response, and transfers momentum from the waves to the air. For an exponential mean profile, the WKBJ approximation gives closed-form expressions for the wave-coherent response. To accommodate the rapid near-surface variation of realistic wall-bounded profiles, we develop a long-wave expansion near the surface and match it to the outer WKBJ mode (LW--WKBJ). For wall-modelled simulations, whose first resolved level lies above the viscous sublayer, we replace the molecular viscosity in the Stokes layer with a constant-stress effective viscosity. Using mean profiles from direct numerical simulations and wave-phase-resolved large-eddy simulations, the LW--WKBJ framework closely reproduces the dominant wave-coherent velocity and pressure obtained from numerical solutions and available simulation data. The predicted form-stress coefficients also agree well with the available LES estimates.
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BibTeX
@misc{aiyer2026analytical,
title = {{An analytical framework for wave-coherent airflow over fast surface waves}},
author = {Aditya K. Aiyer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40933},
eprint = {2610.40933},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40933},
}
RIS
TY - UNPB
TI - An analytical framework for wave-coherent airflow over fast surface waves
AU - Aditya K. Aiyer
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40933
UR - https://arxiv.org/abs/2610.40933
AB - Classical theories attribute wind-wave momentum transfer to a critical layer, where the mean wind speed equals the wave phase speed. For waves that travel faster than the near-surface wind, this layer may lie outside the wave boundary layer or be absent altogether. We develop an analytical description of the wave-coherent airflow in this regime for a prescribed mean wind, using a Wentzel--Kramers--Brillouin--Jeffreys (WKBJ) approximation to the viscous Orr--Sommerfeld equation in wave-following coordinates. The solution combines an outer Rayleigh mode, which carries the dominant wave-coherent motion, with a thin viscous Stokes layer that supplies the phase shift producing the pressure form stress. At leading order, the form stress scales with the inverse square root of the wave Reynolds number, with a coefficient set by an integral of the outer response, and transfers momentum from the waves to the air. For an exponential mean profile, the WKBJ approximation gives closed-form expressions for the wave-coherent response. To accommodate the rapid near-surface variation of realistic wall-bounded profiles, we develop a long-wave expansion near the surface and match it to the outer WKBJ mode (LW--WKBJ). For wall-modelled simulations, whose first resolved level lies above the viscous sublayer, we replace the molecular viscosity in the Stokes layer with a constant-stress effective viscosity. Using mean profiles from direct numerical simulations and wave-phase-resolved large-eddy simulations, the LW--WKBJ framework closely reproduces the dominant wave-coherent velocity and pressure obtained from numerical solutions and available simulation data. The predicted form-stress coefficients also agree well with the available LES estimates.
ER -