Unsteady Breaking Waves Research Articles

A new spectral parameter to predict dominant wave breaking based on the JONSWAP spectrum

Finding a robust and universal diagnostic parameter that determines the onset of deep-water breaking and its strength has been challenging in wave breaking research. In this paper, a new diagnostic parameter (wave-breaking spectral parameter, WSP) is defined based on the significant steepness of the spectral peak ϵp that was initially proposed by Banner et al. (2000) and Babanin et al. (2001). In addition, the WSP formula contains the spectral peakedness parameter Qp and spectral width ν for taking into account the effect of spectral shape on the value of breaking threshold. To derive the WSP formula and verify the prediction ability of this diagnostic parameter, multiple wave types including nonbreaking waves, critical waves, and single and multiple breaking wave trains were generated in our experiments. The wave trains were generated using a linear focusing theory with components following the JONSWAP distribution. Results show that the threshold value of the new WSP, which distinguishes single and multiple breaking events from non-breaking waves, is found to be 0.042. Furthermore, WSP is used in isolated deep-water spilling breaking packets, and its prediction accuracy is compared with that of the kinematic criterion parameter Bx. The results show that WSP has advantages in predicting the onset of wave breaking, especially for non-sharp wave conditions. Finally, the spatial variation of the total loss rate of the wave energy flux ΔFbrE¯¯ΔFbE¯¯ in deep-water unsteady breaking waves was investigated. It was found that the new WSP has a strong linear correlation with the total loss rate of the wave energy flux. Considering that WSP can be mathematically expressed directly using the spectrum, WSP will be parameterized into the dissipation term of the spectral wind-wave model in our future work. It is expected that the calculation accuracy of the dissipation term will be improved, and the wave model's prediction accuracy of wave parameters will be enhanced because WSP provides a reliable wave-breaking threshold value and has a strong linear relationship with the dissipation rate.

Breaking-onset, energy and momentum flux in unsteady focused wave packets

Breaking waves on the ocean surface transfer energy and momentum into currents and turbulence. What is less well understood, however, is the associated total loss of wave energy and momentum flux. Further, finding a robust and universal diagnostic parameter that determines the onset of breaking and its strength is still an open question. Derakhti & Kirby (J. Fluid Mech., vol. 761, 2014, pp. 464–506) have recently studied bubble entrainment and turbulence modulation by dispersed bubbles in isolated unsteady breaking waves using large-eddy simulation. In this paper, a new diagnostic parameter ${\it\xi}(t)$ is defined based on that originally proposed by Song & Banner (J. Phys. Oceanogr., vol. 32, 2002, pp. 2541–2558), and it is shown that using a threshold value of ${\it\xi}_{th}=0.05$, the new dynamic criteria is capable of detecting single and multiple breaking events in the considered packets. In addition, the spatial variation of the total energy and momentum flux in intermediate- and deep-water unsteady breaking waves generated by dispersive focusing is investigated. The accuracy of estimating these integral measures based on free surface measurements and using a characteristic wave group velocity is addressed. It is found that the new diagnostic parameter just before breaking, ${\it\xi}_{b}$, has a strong linear correlation with the commonly used breaking strength parameter $b$, suggesting that ${\it\xi}_{b}$ can be used to parameterize the averaged breaking-induced dissipation rate and its associated energy flux loss. It is found that the global wave packet time and length scales based on the spectrally weighted packet frequency proposed by Tian et al. (J. Fluid Mech., vol. 655, 2010, pp. 217–257), are the reasonable estimations of the time and length scales of the carrier wave in the packet close to the focal/break point. A global wave steepness, $S_{s}$, is defined based on these spectrally weighted scales, and its spatial variation across the breaking region is examined. It is shown that the corresponding values of $S_{s}$ far upstream of breaking, $S_{s0}$, have a strong linear correlation with respect to $b$ for the considered focused wave packets. The linear relation, however, cannot provide accurate estimations of $b$ in the range $b<5\times 10^{-3}$. A new scaling law given by $b=0.3(S_{s0}-0.07)^{5/2}$, which is consistent with inertial wave dissipation scaling of Drazen et al. (J. Fluid Mech., vol. 611, 2008, pp. 307–332), is shown to be capable of providing accurate estimates of $b$ in the full range of breaking intensities, where the scatter of data in the new formulation is significantly decreased compared with that proposed by Romero et al. (J. Phys. Oceanogr., vol. 42, 2012, pp. 1421–1444). Furthermore, we examine nonlinear interactions of different components in a focused wave packet, noting interactive effect on a characteristic wave group velocity in both non-breaking and breaking packets. Phase locking between spectral components is observed in the breaking region as well, and subsequently illustrated by calculating the wavelet bispectrum.

Numerical simulations of 2-D steady and unsteady breaking waves

In this work we analyze by means of numerical simulations the features of breaking of two dimensional free surface waves induced by a body or a sloping bottom. The sample cases selected for the simulations characterize different aspects of wave breaking, thus they are supposed to represent rather widely a problem of large interest for ship hydrodynamics and ocean engineering applications. The simulations considered are: wave breaking induced by a fully submerged hydrofoil towed in calm water at constant speed; shallow water waves breaking on a sloping beach in spilling and plunging mode; regular intermediate depth waves breaking gently over a weakly submerged horizontal circular cylinder at a low Keulegan–Carpenter number. Each simulated case is supported by detailed comparisons with experimental data in time and frequency domain. The results presented have been obtained adopting a standard RANS approach. They show a generally good reproduction of the wave breaking characteristics even though it is rather clear that there is a case dependent potential loss of accuracy in the presence of pronounced foamy flow.

Bubble entrainment and liquid–bubble interaction under unsteady breaking waves

AbstractLiquid–bubble interaction, especially in complex two-phase bubbly flow under breaking waves, is still poorly understood. In the present study, we perform a large-eddy simulation using a Navier–Stokes solver extended to incorporate entrained bubble populations, using an Eulerian–Eulerian formulation for a polydisperse bubble phase. The volume-of-fluid method is used for free-surface tracking. We consider an isolated unsteady deep water breaking event generated by a focused wavepacket. Bubble contributions to dissipation and momentum transfer between the water and air phases are considered. The model is shown to predict free-surface evolution, mean and turbulent velocities, and integral properties of the entrained dispersed bubbles fairly well. We investigate turbulence modulation by dispersed bubbles as well as shear- and bubble-induced dissipation, in both spilling and plunging breakers. We find that the total bubble-induced dissipation accounts for more than 50 % of the total dissipation in the breaking region. The average dissipation rate per unit length of breaking crest is usually written as $b{\it\rho}g^{-1}c_{b}^{5}$, where ${\it\rho}$ is the water density, $g$ is the gravitational acceleration and $c_{b}$ is the phase speed of the breaking wave. The breaking parameter, $b$, has been poorly constrained by experiments and field measurements. We examine the time-dependent evolution of $b$ for both constant-steepness and constant-amplitude wavepackets. A scaling law for the averaged breaking parameter is obtained. The exact two-phase transport equation for turbulent kinetic energy (TKE) is compared with the conventional single-phase transport equation, and it is found that the former overpredicts the total subgrid-scale dissipation and turbulence production by mean shear during active breaking. All of the simulations are also repeated without the inclusion of a dispersed bubble phase, and it is shown that the integrated TKE in the breaking region is damped by the dispersed bubbles by approximately 20 % for a large plunging breaker to 50 % for spilling breakers. In the plunging breakers, the TKE is damped slightly or even enhanced during the initial stage of active breaking.

Simultaneous quantitative flow measurement using PIV on both sides of the air–water interface for breaking waves

An experimental method for simultaneously measuring the velocity fields on the air and water side of unsteady breaking waves is presented. The method includes a novel technique for seeding the air flow such that the air velocity can be resolved in the absence of wind. Low density particles that have large Stokes drag and ability to respond to high-frequency flow fluctuations are used to seed the air flow. Multi-camera, multi-laser particle image velocimetry setups are applied to small-scale unsteady breaking waves, yielding fully time-resolved velocity fields. The surface tension of the fluid is altered and controlled to form spilling breaking waves. Results for the velocity and vorticity fields of representative spilling breakers, which show shedding of an air-side vortex and well-documented generation of water-side vorticity, are presented and discussed.

Inertial scaling of dissipation in unsteady breaking waves

Wave dissipation by breaking, or the energy transfer from the surface wave field to currents and turbulence, is one of the least understood components of air–sea interaction. It is important for a better understanding of the coupling between the surface wave field and the upper layers of the ocean and for improved surface-wave prediction schemes. Simple scaling arguments show that the wave dissipation per unit length of breaking crest, ϵl, should be proportional to ρwgc5, where ρw is the density of water, g is the acceleration due to gravity and c is the phase speed of the breaking wave. The proportionality factor, or ‘breaking parameter’ b, has been poorly constrained by experiments and field measurements, although our earlier work has suggested that it should be dependent on measures of the wave slope and spectral bandwidth. In this paper we describe inertial scaling arguments for the energy lost by plunging breakers which predict that the breaking parameter b = β(hk)5/2, where hk is a local breaking slope parameter, and β is a parameter of O(1). This prediction is tested with laboratory measurements of breaking due to dispersive focusing of wave packets in a wave channel. Good agreement is found within the scatter of the data. We also find that if an integral linear measure of the maximum slope of the wave packet, S, is used instead of hk, then b ∝ S2.77 gives better agreement with the data. During the final preparation of this paper we became aware of similar experiments by Banner & Peirson (2007) concentrating on the threshold for breaking at lower wave slopes, using a measure of the rate of focusing of wave energy to correlate measurements of b. We discuss the significance of these results in the context of recent measurements and modelling of surface wave processes.

Laboratory study of the fine structure of short surface waves due to breaking: Two‐directional wave propagation

This study was stimulated by the need to identify the influence of breaking on the evolution of the short surface wave field responsible for microwave scattering. Laboratory measurements of the fine space‐time structure of short gravity‐capillary waves and Ku band scattering at grazing and moderate incidence from spilling and plunging breaking waves in a laboratory wave channel are presented. Unsteady breaking waves are generated by focusing wave groups in space‐time domains. A scanning laser slope gauge was used for measuring capillary‐gravity waves with wavelengths of 2 mm to 10 cm and frequency ranges of up to 150 Hz. A dual polarized (VV, HH) coherent pulsed Ku band scatterometer with a 3 ns temporal resolution was used to simultaneously obtain Doppler spectra of the scattered signals from the breaking area for grazing angles from 6° to 25°. Two‐dimensional filtering in the wave number‐frequency plane as well as plus/minus Doppler spectra were used to separate the direction of propagation of the surface waves within the breaking region. It was found that the breaking splash is the main source of the surface wave generation. The short surface wave slope field produced by breaking could be separated into short (4–8 mm wavelength), fast waves and free gravity‐capillary waves. Both types of waves were found to be co‐ and counterpropagate relative to the dominant wave propagation direction. The Doppler spectrograms reveal the presence of high‐frequency waves at the moment of breaking that can be associated with short, fast waves. The low‐frequency part of the Doppler spectrogram is consistent with the appearance of free waves.

Pulse-to-Pulse Coherent Doppler Measurements of Waves and Turbulence

This paper presents laboratory and field testing of a pulse-to-pulse coherent acoustic Doppler profiler for the measurement of turbulence in the ocean. In the laboratory, velocities and wavenumber spectra collected from Doppler and digital particle image velocimeter measurements compare very well. Turbulent velocities are obtained by identifying and filtering out deep water gravity waves in Fourier space and inverting the result. Spectra of the velocity profiles then reveal the presence of an inertial subrange in the turbulence generated by unsteady breaking waves. In the field, comparison of the profiler velocity records with a single-point current measurement is satisfactory. Again wavenumber spectra are directly measured and exhibit an approximate −5/3 slope. It is concluded that the instrument is capable of directly resolving the wavenumber spectral levels in the inertial subrange under breaking waves, and therefore is capable of measuring dissipation and other turbulence parameters in the upper mixed layer or surface-wave zone.

Energy Dissipation by Breaking Waves

Recent field measurements by Agrawal et al. have provided evidence of a shallow surface mixed layer in which the rate of dissipation due to turbulence is one to two orders of magnitude greater than that in a comparable turbulent boundary layer over a rigid wall. It is shown that predictions by Phillips of the energy lost by breaking surface waves in an equilibrium regime and laboratory measurements by Rapp and Melville of the mixing and turbulence due to breaking together lead to estimates of the enhanced dissipation rate and the thickness of the surface layer consistent with the field measurements. Wave-age-dependent scaling of the dissipation layer is proposed. Laboratory measurements of dissipation rates in both unsteady and quasi-steady breaking waves are examined. It is shown that an appropriately defined dimensionless rate of dissipation in unsteady breaking waves is not constant, but increases with a measure of the wave slope. Differences between dissipation rates in quasi-steady and unsteady breakers are discussed. It is found that measurements of the dissipation rate in unsteady breakers are consistent with independent estimates of the turbulent dissipation. The application of these results to models of dissipation due to breaking and air-sea fluxes is discussed.