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Michael L. Banner

Abstract

Equilibrium spectral behavior for ocean gravity wind waves has been investigated actively over the past three decades, yet fundamental problems remain in reconciling theory with observations. Predicted equilibrium spectral forms from physical models proposed recently by Kitaigorodskii and by Phillips are examined in the light of wavenumber and frequency spectra reported by several investigators. While frequency domain observations appear to support the model predictions, observed wavenumber spectra are found to differ both in the spectral dependence on wavenumber and on the wind speed.

Based on observed wavenumber and frequency spectra for fetch-limited condition a model is proposed for the form of the directional wavenumber spectrum slice in the dominant wave direction. Reduced wavenumber and frequency spectra are calculated from this model, assuming an empirical spectral directional spreading function and the linear gravity wave dispersion relation. These calculations reveal the underlying influences which shape these reduced spectra. In the energy containing subrange, just above the spectral peak, the dominant influence shaping these spectra is the variation of the directional spreading function with distance from the spectral peak. For frequency spectra, at higher frequencies, the model calculations predict that the range of observed frequency spectral dependences is due primarily to the Doppler shifting from advection of the shorter waves by the orbital motion of the dominant waves, with possible additional influences of wind drift and ambient currents.

Combining these results, composite calculated frequency spectra and one-dimensional wavenumber spectra show close correspondence with measured field spectra. In addition to clarifying the key processes that shape different regimes in the frequency spectrum, a refinement of the bounds of the gravity equilibrium subrange is proposed.

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Jin-Bao Song and Michael L. Banner

Abstract

Wave breaking in the open ocean and coastal zones remains an intriguing yet incompletely understood process, with a strong observed association with wave groups. Recent numerical study of the evolution of fully nonlinear, two-dimensional deep water wave groups identified a robust threshold of a diagnostic growth-rate parameter that separated nonlinear wave groups that evolved to breaking from those that evolved with recurrence. This paper investigates whether these deep water wave-breaking results apply more generally, particularly in finite- water-depth conditions. For unforced nonlinear wave groups in intermediate water depths over a flat bottom, it was found that the upper bound of the diagnostic growth-rate threshold parameter established for deep water wave groups is also applicable in intermediate water depths, given by k 0 h ≥ 2, where k 0 is the mean carrier wavenumber and h is the mean depth. For breaking onset over an idealized circular arc sandbar located on an otherwise flat, intermediate-depth (k 0 h ≥ 2) environment, the deep water breaking diagnostic growth rate was found to be applicable provided that the height of the sandbar is less than one-quarter of the ambient mean water depth. Thus, for this range of intermediate-depth conditions, these two classes of bottom topography modify only marginally the diagnostic growth rate found for deep water waves. However, when intermediate- depth wave groups (k 0 h ≥ 2) shoal over a sandbar whose height exceeds one-half of the ambient water depth, the waves can steepen significantly without breaking. In such cases, the breaking threshold level and the maximum of the diagnostic growth rate increase systematically with the height of the sandbar. Also, the dimensions and position of the sandbar influenced the evolution and breaking threshold of wave groups. For sufficiently high sandbars, the effects of bottom topography can induce additional nonlinearity into the wave field geometry and associated dynamics that modifies the otherwise robust deep water breaking-threshold results.

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Michael L. Banner and Jin-Bao Song

Abstract

Part I of this study describes the authors' findings on a robust threshold variable that determines the onset of breaking for unforced, irrotational deep water waves and proposes a means of predicting the strength of breaking if the breaking threshold is exceeded. Those results were based on a numerical study of the unforced evolution of fully nonlinear, two-dimensional inviscid wave trains and highlight the fundamental role played by the nonlinear wave group dynamics. In Part II the scope of these calculations is extended to investigate the additional influence of wind forcing and background shear on the evolution to breaking.

Using the methodology described in Part I, the present study focuses on the influence of wind forcing and vertical shear on long-term evolution toward breaking or recurrence of the maximum of the local energy density within a wave group. It investigates the behavior of a dimensionless local growth rate parameter that reflects the mean energy flux to the energy maximum in the wave group and provides a clearer physical interpretation of the evolution toward recurrence or breaking. Typically, the addition of the wind forcing and surface layer shear results in only small departures from the irrotational, unforced cases reported in Part I. This indicates that nonlinear hydrodynamic energy fluxes within wave groups still dominate the evolution to recurrence or breaking even in the presence of these other mechanisms. Further, the calculations confirm that the breaking threshold for this growth rate found for unforced irrotational wave groups in Part I is also applicable for cases with wind forcing and shear typical of open ocean conditions. Overall, this approach provides an earlier and more decisive indicator for the onset of breaking than previously proposed breaking thresholds and suggests a foundation for predicting the strength of breaking events.

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Jin-Bao Song and Michael L. Banner

Abstract

Finding a robust threshold variable that determines the onset of breaking for deep water waves has been an elusive problem for many decades. Recent numerical studies of the unforced evolution of two-dimensional nonlinear wave trains have highlighted the complex evolution to recurrence or breaking, together with the fundamental role played by nonlinear intrawave group dynamics. In Part I of this paper the scope of two-dimensional nonlinear wave group calculations is extended by using a wave-group-following approach applied to a wider class of initial wave group geometries, with the primary goal of identifying the differences between evolution to recurrence and to breaking onset. Part II examines the additional influences of wind forcing and background shear on these evolution processes.

The present investigation focuses on the long-term evolution of the maximum of the local energy density along wave groups. It contributes a more complete picture, both long-term and short-term, of the approach to breaking and identifies a dimensionless local average growth rate parameter that is associated with the mean convergence of wave-coherent energy at the wave group maximum. This diagnostic growth rate appears to have a common threshold for all routes to breaking in deep water that have been examined and provides an earlier and more decisive indicator for the onset of breaking than previously proposed breaking thresholds. The authors suggest that this growth rate may also provide an indicative measure of the strength of wave breaking events.

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Peter D. Craig and Michael L. Banner

Abstract

Until recently, measurements below the ocean surface have tended to confirm “law of the wall” behavior, in which the velocity profile is logarithmic, and energy dissipation decays inversely with depth. Recent measurements, however, show a sublayer, within meters of the surface, in which turbulence is enhanced by the action of surface waves. In this layer, dissipation appears to decay with inverse depth raised to a power estimated between 3 and 4.6. The present study shows that a conventional model, employing a “level 2½” turbulence closure scheme predicts near-surface dissipation decaying as inverse depth to the power 3.4. The model shows agreement in detail with measured profiles of dissipation. This is despite the fact that empirical constants in the model are determined for situations very different from this near-surface application. The action of breaking waves is modeled by a turbulent kinetic energy input at the surface. In the wave-enhanced layer, the downward flux of turbulent kinetic energy balances its dissipation. The model produces analytic descriptions for the depth of the layer, and for profiles of velocity, turbulent kinetic energy, and dissipation. The surface roughness length (in the water) is a critical parameter in the solutions. There are indications of a relationship between the roughness length and surface wave parameter such as the amplitude or inverse wavenumber. Roughness lengths at least up to 1 m appear to be feasible.

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Michael L. Banner, Christopher J. Zappa, and Johannes R. Gemmrich

Abstract

There has been a recent upsurge in interest in quantifying kinematic, dynamic, and energetic properties of wave breaking in the open ocean, especially in severe sea states. The underpinning observational and modeling framework is provided by the seminal paper of O. M. Phillips. In this note, a fundamental issue contributing to the scatter in results between investigators is highlighted. This issue relates to the choice of the independent variable used in the expression for the spectral density of the mean breaking crest length per unit area. This note investigates the consequences of the different choices of independent variable presently used by various investigators for validating Phillips model predictions for the spectral density of the breaking crest length per unit area and the associated spectral breaking strength coefficient. These spectral measures have a central role in inferring the associated turbulent kinetic energy dissipation rate and the momentum flux to the upper ocean from breaking wave observations.

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Johannes R. Gemmrich, Michael L. Banner, and Chris Garrett

Abstract

Video observations of the ocean surface taken from aboard the Research Platform FLIP reveal the distribution of the along-crest length and propagation velocity of breaking wave crests that generate visible whitecaps. The key quantity assessed is Λ(c)dc, the average length of breaking crests per unit area propagating with speeds in the range (c, c + dc). Independent of the wave field development, Λ(c) is found to peak at intermediate wave scales and to drop off sharply at larger and smaller scales. In developing seas breakers occur at a wide range of scales corresponding to phase speeds from about 0.1 c p to c p, where c p is the phase speed of the waves at the spectral peak. However, in developed seas, breaking is hardly observed at scales corresponding to phase speeds greater than 0.5 c p. The phase speed of the most frequent breakers shifts from 0.4 c p to 0.2 c p as the wave field develops. The occurrence of breakers at a particular scale as well as the rate of surface turnover are well correlated with the wave saturation. The fourth and fifth moments of Λ(c) are used to estimate breaking-wave-supported momentum fluxes, energy dissipation rate, and the fraction of momentum flux supported by air-entraining breaking waves. No indication of a Kolmogorov-type wave energy cascade was found; that is, there is no evidence that the wave energy dissipation is dominated by small-scale waves. The proportionality factor b linking breaking crest distributions to the energy dissipation rate is found to be (7 ± 3) × 10−5, much smaller than previous estimates.

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Michael L. Banner, Alexander V. Babanin, and Ian R. Young

Abstract

The breaking probability is investigated for the dominant surface waves observed in three geographically diverse natural bodies of water: Lake Washington, the Black Sea, and the Southern Ocean. The breaking probability is taken as the average number of breaking waves passing a fixed point per wave period. The data covered a particularly wide range of dominant wavelengths (3–300 m) and wind speeds (5–20 m s−1). In all cases, the wave breaking events were detected visually. It was found that the traditional approach of relating breaking probability to the wind speed or wave age provided reasonable correlations within individual datasets, but when the diverse datasets are combined, these correlations are significantly degraded.

Motivated by the results of recent computational studies of breaking onset in wave groups, the authors investigated the hypothesis that nonlinear hydrodynamic processes associated with wave groups are more fundamental to the process of breaking than previously advocated aerodynamic properties, such as the wind speed or wave age. Further, these computational studies suggest that the significant wave steepness is an appropriate parameter for characterizing the nonlinear group behavior.

Based on this approach, analysis of the data revealed that the probability of dominant wave breaking is strongly correlated with the significant wave steepness for the broad range of wave conditions investigated. Of particular interest is a threshold of this parameter below which negligible dominant wave breaking occurs. Once this threshold is exceeded, a near-quadratic dependence of the breaking probability on the significant wave steepness was observed, with a correlation coefficient of 0.78. The inclusion of parameters representing the secondary influence of wind forcing and background current shear improved the correlation only marginally to 0.81.

The applicability of the breaking probability dependence found for the dominant waves was investigated for higher-frequency bins up to twice the spectral peak frequency f p. The Black Sea data were used for this analysis, in which shorter breaking wave statistics were also measured. It was found that the maximum of the composite breaking frequency distribution gradually shifts from about 1.6f p for lower values of the peak steepness parameter to f p for higher values of this parameter. The breaking probability in a comparable higher frequency band has a similar dependence on significant steepness to that found for the dominant waves.

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Jose Henrique G. M. Alves and Michael L. Banner

Abstract

A new formulation of the spectral dissipation source term Sds for wind-wave modeling applications is investigated. This new form of Sds is based on a threshold behavior of deep-water wave-breaking onset associated with nonlinear wave-group modulation. It is expressed in terms of the azimuth-integrated spectral saturation, resulting in a nonlinear dependence of dissipation rates on the local wave spectrum. Validation of the saturation-based Sds is made against wave field parameters derived from observations of fetch-limited wind-wave evolution. Simulations of fetch-limited growth are made with a numerical model featuring an exact nonlinear form of the wave–wave-interactions source term Snl. For reference, the performance of this saturation-based Sds is compared with the performance of the wave-dissipation source-term parameterization prescribed for the Wave Modeling Project (WAM) wind-wave model. Calculations of integral spectral parameters using the saturation-based model for Sds agree closely with fetch-limited observations. It is also shown that the saturation-based Sds can be readily adjusted to accommodate several commonly used parameterizations of the wind input source term Sin. Also, this new form of Sds provides greater flexibility in controlling the shape of the wave spectrum in the short gravity-wave region.

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Michael L. Banner, Johannes R. Gemmrich, and David M. Farmer

Abstract

Recent numerical model studies of nonlinear deep water wave group evolution suggest that wave breaking onset is associated primarily with a threshold behavior linked to the nonlinear wave group hydrodynamics. Motivated by these findings, a recently published probability analysis of observed dominant ocean wind wave breaking events reported a threshold behavior using the significant wave steepness as a measure of the mean nonlinearity of these waves. The present study investigates whether a similar threshold dependence in terms of an appropriate spectral measure of wave steepness, the spectral saturation, may be found for the breaking probability of shorter wind waves above the spectral peak. Extensive data records of open ocean whitecap breaking wave occurrences for wind speeds up to 18 m s−1 were analyzed for breaking probability dependence on spectral saturation in spectral bands with center frequencies ranging from 1 to 2.48 times the spectral peak frequency. Results are based on the measured ratio of passage rates past a fixed point of breaking crests to total crests for different wave scales. An extension of the zero-crossing method for counting wave crests was developed. Using this method the authors found that in any spectral subrange within the observed range of frequencies, a strong correlation exists between breaking probability and an appropriate mean spectral steepness parameter and that this correlation is characterized by a robust threshold behavior, just as was reported previously for the spectral peak waves. Further, to offset the influence of increasing directional spreading of the waves above the spectral peak frequency, an empirical directional spreading function was used to normalize the azimuth-integrated spectral saturation. Under this normalization, the spectral saturation threshold for breaking onset appears to have a common level over the frequency range investigated. This study also examined the correlation of breaking probability with spectral peak wave age. The low correlation found for all spectral ranges investigated suggests that nonlinear wave hydrodynamics are more important than wind forcing for the breaking of these wind waves.

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