Growth rate and energy dissipation in wind-forced breaking waves

TL;DR

This study uses fully resolved direct numerical simulations of the two-phase air–water Navier–Stokes equations to investigate wind-forced breaking waves in high-wind-speed regimes. It independently analyzes energy input during wave growth and energy dissipation during breaking, showing that growth is driven mainly by wind-induced pressure, while breaking transfers energy into underwater turbulence with an inertial dissipation scaling. The authors demonstrate that the non-dimensional growth rate depends on wave slope and wind forcing, consistent with Belcher–Hunt-type sheltering theories, and they confirm a universal breaking-dissipation pattern described by the inertial scaling $b \propto \\mathcal{S}^{5/2}$, where $\\mathcal{S}=(ak)_c$. They further show that post-breaking vertical dissipation profiles collapse when normalized by the breaking-induced energy input, following $\,\\langle\\varepsilon\\rangle(z) \approx A \, (S_{ds}/\\rho_w)/z$ with $A \approx 0.14$, supporting a balanced energy transfer picture $S_{in} \approx S_{ds}$ across growth–break cycles. Overall, the results provide mechanistic insight into air–sea energy exchange under strong winds and offer scalable dissipation laws for improving high-wind wave and upper-ocean turbulence parameterizations.

Abstract

We investigate the energy growth and dissipation of wind-forced breaking waves at high wind speed using direct numerical simulations of the coupled air-water Navier-Stokes equations. A turbulent wind boundary layer drives the growth of a pre-existing narrowband wave field until it breaks, transferring energy into the water column. Under sustained wind forcing, the wave field resumes growth. We separately analyze energy transfers during wave growth and breaking-induced dissipation. Energy transfers are dominated by pressure input during growth and turbulent dissipation during breaking. Wind input during growth is balanced with dissipation during breaking over an entire growing-breaking cycle. The wave growth rate scales with $(u_\ast/c)^2$, modulated by the wave steepness due to sheltering, and the energy dissipation follows the inertial scaling with wave slope at breaking, confirming the universality of the process. Following breaking, near-surface vertical turbulence dissipation profiles scale as $z^{-1}$, with their magnitude controlled by the breaking-induced dissipation.

Figures (3)

• Figure 1: Evolution of the airflow, waves, and underwater currents for $u_\ast/c=0.9$. Top panels show snapshots of (a) first growing ($\omega t=10$, $G_1$), (b) breaking ($\omega t=25$, $B_1$), and (c) second growing ($\omega t=75$, $G_2$) stages under turbulent wind forcing along $x$, with contours of instantaneous streamwise velocity $u$ normalized by $u_\ast$. (d) Time evolution of potential ($E_p$), kinetic ($E_k$), and total ($E_t=E_p+E_k$) energy, all normalized by the initial total energy $E_{t,0}$. Equipartition of $E_t$ is lost due to the breaking event. The instantaneous steepness $a(t)k$ is shown on the right axis (orange curve). (e) Instantaneous streamwise velocity on the $y=0$ plane, normalized by $u_\ast$ (air) and $c$ (water), across the growing and breaking. (f) Vertical profiles of the mean streamwise velocity $\langle u_w\rangle$ in water, normalized by $c$. Profiles at $\omega t = [10,25,50,75]$ are shown in opaque colors, while profiles for $\omega t\in [0,150]$ with reduced transparency. (g) Dissipation rate $\varepsilon(x,y,z)$ on the $y=0$ plane, normalized by $(\rho_a/\rho_w)^{3/2}u_\ast^3/a_0$.
• Figure 2: (a) Wave energy budget (equation \ref{['eqn:amp_t']}) for $u_\ast/c=0.9$. The $y$-axis shows the different terms in eq. 2, $\mathcal{T}$, being $\mathrm{d}E_W/\mathrm{d}t$, the total wind input $S_{in}$, and dissipation $S_d$, each normalized by $\omega E_{W,0}$ with $E_{W,0}=E_W(0)$. (b) Coefficients $\alpha_{p,\nu}$ from Belcher & Hunt theory as a function of $u_\ast/c$ and $ak$, with one data point from the first growth stage and two from the second growing stages. The wave slope $ak$ is colorcoded. The surface roughness $z_0$ is extracted from the mean air velocity profiles, and color-coded. As a reference, the dashed and dot–dashed lines represent the coefficients $\alpha_{p,\nu}$ computed using a fixed roughness of $kz_0=6\times10^{-2}$ and $kz_0=6\times10^{-3}$, using equations \ref{['eqn:alpha_p']} and \ref{['eqn:alpha_nu']}. The symbols follow the same convention as in panels (c)-(d).(c) Non-dimensional growth rate $\overline{\gamma}$ as a function of $u_\ast/c$ computed from the DNS using \ref{['eqn:gamma_inst_2']}. The shaded area shows estimates from equation \ref{['eqn:gamma_BH']} with $\overline{\alpha}_p=31.4$ and $\overline{\alpha}_\nu=4.6$ for $ak$ spanning $\in[0.15,0.33]$. Up triangles are present DNS data; circles are DNS from wu2022revisiting. In (d) we report $\overline{\gamma}$ as in (c) together with other datasets from Large Eddy Simulations yang2013dynamickihara2007relationship, field measurements, and laboratory measurements wu1979experimentalsnyder1981oceanicbuckley2020surface. The orange and the dark red dot-dashed lines show theoretical predictions from equation \ref{['eqn:gamma_BH']} using $ak_{\mathrm{min}}=0.15$ and $ak_{\mathrm{max}}=0.33$, respectively. The gray dashed lines are from plant1982relationship.
• Figure 3: (a) Breaking parameter $b$ versus wave steepness for various $u_\ast/c$. Solid lines show equation \ref{['eqn:b_param']} with $\chi_0=0.25$–$0.95$; vertical dot-dashed lines indicate the range of the critical steepness at breaking $(ak)_c=0.28$–$0.33$; black dot-dashed line is $b=0.4(\mathcal{S}-0.08)^{5/2}$deike2016air. For $u_\ast/c=0.7-0.9$, $b$ is shown for the first ($\blacklozenge$ diamond) and second breaking events ($\bullet$ circle). Experimental data with/without wind banner2007wavedrazen2008inertialgrare2013growth and DNS without wind deike2016airde2018breakingmostert2022high are also included. (b) Time-averaged dissipation profiles for $u_\ast/c=0.9$ during $G_1$, $B_1$, and $G_2$, normalized by $(\rho_a/\rho_w)^{3/2}u_\ast^3/a_0$. (c) Normalized wind input $\overline{S}_{in}=\overline{\tau}_{W,p}c$ over $G_1$ versus breaking-induced dissipation $S_{ds}=\rho_wg^{-1}L_c^{-1}b c^5$ over $B_1$. (d) Dissipation profiles after breaking ($G_2$), normalized by significant wave height $H_s$ and wind input, $\rho_w\langle \varepsilon\rangle(z) H_s/\overline{S}_{in}$. DNS data are in colored lines for different parameters ($u_\ast/c$, Reynolds number). All profiles collapse on a single curve following $z^{-1}$. DNS profiles are compared with field observations (grey, black symbols) from terray1996estimatesdrennan1996oceanicsutherland2015fieldzippel2020measurementsthomson2012wave, where $\overline{S}_{in}$ is estimated from measurement and parameterization of wind input. The best fit lines of the dissipation $\ langle \ varepsilon (z)\rangle$ as a function of the water depth proposed in craig1994modelingterray1996estimatesromero2021representingwu2025turbulence are also reported (dashed, solid, dotted, and dashdot lines). (e) Depth normalized by $a_0$ and dissipation normalized by $\overline{S}_{ds}/(\rho_w a_0)$.