Spectral Energy Dissipation due to Surface Wave Breaking

Abstract

A semiempirical determination of the spectral dependence of the energy dissipation due to surface wave breaking is presented and then used to propose a model for the spectral dependence of the breaking strength parameter b, defined in the O. M. Phillips's statistical formulation of wave breaking dynamics. The determination of the spectral dissipation is based on closing the radiative transport equation for fetch-limited waves, measured in the Gulf of Tehuantepec Experiment, by using the measured evolution of the directional spectra with fetch, computations of the four-wave resonant interactions, and three models of the wind input source function. The spectral dependence of the breaking strength is determined from the Kleiss and Melville measurements of the breaking statistics and the semiempirical spectral energy dissipation, resulting in ö = b(k, Cplu^), where k is the wavenumber and the parametric dependence is on the wave age, Cplu¡^. Guided by these semiempirical results, a model for b{k, Cplu^,) is proposed that uses laboratory data from a variety of sources, which can be represented by i = a{S -So)", where 5 is a measure of the wave slope at breaking, a is a constant, 5o is a threshold slope for breaking, and 2.5 < n <3 is a power law consistent with inertial wave dissipation scaling and laboratory measurements. The relationship between b{S) in the laboratory and b{k) in the field is based on the relationship between the saturation and mean square slope of the wave field. The results are discussed in the context of wind wave modeling and improved measurements of breaking in the field.

Air entrainment and bubble statistics in breaking waves

“…Figure 5b shows b as a function of the initial wave slope S in the DNS, and we observe a very good agreement for strong plunging waves with the semi-empirical result given by Eq. 3.4 initially derived from laboratory data (Drazen et al 2008;Romero et al 2012), see also data from Grare et al (2013) and two-dimensional numerical simulations from Deike et al (2015). This confirms that the present three-dimensional DNS captures the dissipative scales of the breaking wave process.…”

“…This inertial model, based on a simple physical argument for strong plunging waves, has been confirmed through extensive experimental studies and modeling, well beyond the region of validity of the initial hypothesis (Romero et al 2012;Grare et al 2013;Pizzo & Melville 2013;Melville & Fedorov 2015;Deike et al 2015). Note that proportionality between the initial slope and the slope at breaking is also true in our DNS of steep Stokes waves, and following the definition of Drazen et al (2008) where h is the vertical distance the breaking wave toe travels before impact, hk = −0.05 + S for both the 3D DNS presented here and the 2D results presented in Deike et al (2015).…”

“…This numerical modeling study of entrainment by breaking waves has shown that the prospects are strong for being able to follow the example of Romero et al (2012) who modeled the dynamics of breaking using the inertial scaling of dissipation due to breaking (Drazen et al 2008) along with field measurements of the kinematics to improve ocean wave modeling. In this case, we foresee the possibility of using the results of this paper, along with field measurements of breaking to improve the models of air entrainment, and ultimately air-sea gas transfer (c.f.…”

“…Drazen et al (2008) have shown that the slope at breaking hk is approximatively proportional to the linearly predicted maximum slope, S, of a focusing packet. Romero et al (2012) derived the following The air cavity has started to collapse and is connected to the main surface by thin filaments of air. Bubbles of various sizes start to be visible.…”

“…r m = 10 mm is estimated directly from the data as the end of the bubble cascade. l can be estimated from the wave packet parameters used by Deane & Stokes (2002), the wave slope S, the central wavenumber and frequency, and then using the semi-empirical relationship from Romero et al (2012), Eq. 3.4.…”