Breaking-induced energy dissipation of surface gravity waves at varying scales and co-flowing wind stresses

Breaking-induced energy dissipation is studied for individual unsteady breaking waves using laboratory measurements of unidirectional surface gravity wave groups across a range of wave scales and wind stresses. A refined framework to estimate breaking-induced dissipation $\Delta E_{br}$ is proposed that accounts for background dissipation from non-breaking processes. Using this framework, we show that variations in wave scale primarily influence breaking energetics, such as fractional dissipation $\Delta E_{br}/E_0$ and dissipation rate $\epsilon_b$, by modifying the breaking onset threshold. Also, co-flowing wind systematically reduces both $\Delta E_{br}/E_0$ and $\epsilon_b$ relative to unforced conditions, as wind-forced waves break earlier with reduced crest forward-leaning. Exploiting the crest-front steepness at incipient breaking $\mathcal{S}_{\text{front}}(t_b)$ to characterise breaking onset and local crest geometry, we formulate a scaling for $\epsilon_b$ based on this local measure. This then yields $\Delta E_{br}/E_0 \propto \beta^{*}\,\mathcal{S}_b\,(\tau_b/T_b)$, where $\beta^{*}$ is crest forward leaning, $\mathcal{S}_b$ local steepness, and $\tau_b/T_b$ non-dimensional breaking duration. This scaling highlights the important roles of crest asymmetry and breaking duration in setting the breaking energy dissipation. Finally, we consider the breaking strength parameter $b$ by assessing existing steepness-based scaling laws, and relate $b$ to $\mathcal{S}_{\text{front}}(t_b)$, yielding an approximately linear dependence once the breaking-onset threshold is considered.