Growth and energy dissipation in breaking waves at high wind speeds

The growth rate and energy dissipation of wind-forced breaking waves are investigated using direct numerical simulations of the full air-water Navier-Stokes equations with surface tension. The fully coupled system of airflow, waves, and water column consists of a two-phase boundary layer, where the upper turbulent airflow region drives the growth of a pre-existing narrowband wave field. As the waves reach a critical steepness, they break, dissipating energy into the water column and generating strong near-surface turbulence that enhances vertical mixing. After breaking, the wave field resumes growth under continued wind forcing. By varying the ratio of wind friction velocity to wave phase speed up to unity, we systematically analyze the energy transfers during both growth and breaking phases. First, we show that pressure forcing dominates energy input during growth, while turbulent dissipation controls energy loss during breaking. The wave growth rate scales with $(u_\ast/c)^2$, with strong modulation induced by wave steepness through sheltering effects [1]. Next, we confirm that energy dissipation follows inertial scaling with wave slope at breaking [2], supporting the universality of the breaking process. Following the breaking event, the near-surface vertical profile of turbulence dissipation scales as $z^{-1}$, with its magnitude set by the intensity of the breaking event. Based on this finding, we propose a scaling law for energy dissipation derived from the breaking-induced energy loss that unifies dissipation profiles across different $u_\ast/c$.

[1] Belcher, S. E., and J. C. R. Hunt. "Turbulent shear flow over slowly moving waves." Journal of Fluid Mechanics 251 (1993): 109-148.

[2] Drazen, David A., W. Kendall Melville, and L. U. C. Lenain. "Inertial scaling of dissipation in unsteady breaking waves." Journal of fluid mechanics 611 (2008): 307-332.