Wind Wave Spectrum Research Articles

Sea Surface Drag Fluctuation in Relation to the Saturation Ratio of Wind Wave Spectrum

A new linear relation between sea surface drag coefficient and the saturation ratio of wind wave spectrum was obtained. The theory is based on the 3/2 power law of wave heights and periods, which was proposed by Toba (1972). The saturation ratio of wind wave spectrum is calculated by using spectrum analysis in principle, however a new parameter derived by the 3/2 power law gives us a saturation ratio from significant wave height and period instead of spectrum analysis. It is found that the negative correlation between sea surface drag and saturation ratio of wind wave spectrum exists. The relation can be explained that the saturated wind wave conditions reduce the momentum exchange from wind to wave, so the sea surface drag reduce in proportion to the momentum exchange.

Computation of wind-wave spectra in coastal waters with SWAN on unstructured grids

Abstract An unstructured-grid procedure for SWAN is presented. It is a vertex-based, fully implicit, finite difference method which can accommodate unstructured meshes with a high variability in geographic resolution suitable for representing complicated bottom topography in shallow areas and irregular shoreline. The numerical solution is found by means of a point-to-point multi-directional Gauss–Seidel iteration method requiring a number of sweeps through the grid. The approach is stable for any time step while permitting local mesh refinements in areas of interest. A number of applications are shown to verify the correctness and numerical accuracy of the unstructured version of SWAN.

Relating the Drag Coefficient and the Roughness Length over the Sea to the Wavelength of the Peak Waves

Abstract The standard 10-m reference height for computing the drag coefficient over the sea is admittedly arbitrary. The literature contains occasional suggestions that a scaling length based on the wavelength of the peak waves λp is a more natural reference height. Attempts to confirm this hypothesis must be done carefully, however, because of the potential for fictitious correlation between nondimensional dependent and independent variables. With the DMAJ dataset as an example, this study reviews the issue of fictitious correlation in analyses that use λp/2 as the reference height for evaluating the drag coefficient and that use kp (=2π/λp) as a scale for the roughness length z0. (The DMAJ dataset is a compilation of four individual datasets; D, M, A, and J, respectively, identify the lead authors of the four studies: Donelan, Merzi, Anctil, and Janssen.) This dataset has been used in several previous studies to evaluate the dependence of kpz0 and the drag coefficient evaluated at λp/2 on the nondimensional wave parameter ω* = ωpu*/g. Here ωp is the radian frequency of the peak in the wind–wave spectrum, u* is the friction velocity, and g is the acceleration of gravity. Because the DMAJ dataset does not, however, include independent measurements of λp and ωp, λp had to be inferred from measurements of ωp through the wave dispersion relation. The presence of ωp in both the dependent and independent variables, therefore, exacerbates the fictitious correlation. One conclusion, thus, is that using λp to formulate the drag coefficient and the nondimensional roughness length as functions of a nondimensional variable that includes ωp requires a dataset with independent measurements of λp and ωp.

The form of the asymptotic depth-limited wind-wave spectrum: Part III — Directional spreading

The directional spreading of both the wavenumber and frequency spectra of finite-depth wind generated waves at the asymptotic depth limit are examined. The analysis uses the Wavelet Directional Method, removing the need to assume a form for the dispersion relationship. The paper shows that both the wavenumber and frequency forms are narrowest at the spectral peak and broaden at wavenumbers (frequencies) both above and below the peak. The directional spreading of the wavenumber spectrum is bi-modal above the spectral peak. In contrast, the frequency spectrum is uni-modal. This difference is shown to be the result of energy in the wind direction being displaced from the linear dispersion shell. A full parametric relationship for the directional spreading of the wavenumber spectrum is developed. The analysis clearly shows that typical dispersion relationships are questionable at high frequencies and that such effects can be significant. This result supports greater attention being focussed on the routine recording of wavenumber spectra, rather than frequency spectra.

A Two-Scale Approximation for Efficient Representation of Nonlinear Energy Transfers in a Wind Wave Spectrum. Part II: Application to Observed Wave Spectra

Abstract In Part I of this series, a new method for estimating nonlinear transfer rates in wind waves, based on a two-scale approximation (TSA) to the full Boltzmann integral (FBI) for quadruplet wave–wave interactions, was presented, and this new method was tested for idealized spectral data. Here, the focus is on comparisons of the TSA and the discrete interaction approximation (DIA) with the FBI for observed wave spectra from field measurements. Observed wave spectra are taken from a wave gauge array in Currituck Sound and a directional waverider off the coast near the Field Research Facility at Duck, North Carolina. Results show that the TSA compares much more favorably to the FBI than does the DIA, even for cases in which the parametric component of the formulation does not capture the spectral energy distribution very well. These results remain valid for the TSA estimates when the FBI results are significantly affected by the directional distribution of energy. It is also shown that although nonlinear transfers are substantially weaker in swell portions of the spectrum these interactions contribute significantly to the spectral evolution and net energy balance in long-distance swell propagation.

Wave-Induced Wind in the Marine Boundary Layer

AbstractRecent field observations and large-eddy simulations have shown that the impact of fast swell on the marine atmospheric boundary layer (MABL) might be stronger than previously assumed. For low to moderate winds blowing in the same direction as the waves, swell propagates faster than the mean wind. The momentum flux above the sea surface will then have two major components: the turbulent shear stress, directed downward, and the swell-induced stress, directed upward. For sufficiently high wave age values, the wave-induced component becomes increasingly dominant, and the total momentum flux will be directed into the atmosphere. Recent field measurements have shown that this upward momentum transfer from the ocean into the atmosphere has a considerable impact on the surface layer flow dynamics and on the turbulence structure of the overall MABL. The vertical wind profile will no longer exhibit a logarithmic shape because an acceleration of the airflow near the surface will take place, generating a low-level wave-driven wind maximum (a wind jet). As waves propagate away from their generation area as swell, some of the wave momentum will be returned to the atmosphere in the form of wave-driven winds.A model that qualitatively reproduces the wave-following atmospheric flow and the wave-generated wind maximum, as seen from measurements, is proposed. The model assumes a stationary momentum and turbulent kinetic energy balance and uses the dampening of the waves at the surface to describe the momentum flux from the waves to the atmosphere. In this study, simultaneous observations of wind profiles, turbulent fluxes, and wave spectra during swell events are presented and compared with the model. In the absence of an established model for the linear damping ratio during swell conditions, the model is combined with observations to estimate the wave damping. For the cases in which the observations showed a pronounced swell signal and almost no wind waves, the agreement between observed and modeled wind profiles is remarkably good. The resulting attenuation length is found to be relatively short, which suggests that the estimated damping ratios are too large. The authors attribute this, at least partly, to processes not accounted for by the model, such as the existence of an atmospheric background wind. In the model, this extra momentum must be supplied by the waves in terms of a larger damping ratio.

Determining the influence of wind-wave breaking on the dissipation of the turbulent kinetic energy in the upper ocean and on the dependence of the turbulent kinetic energy on the stage of wind-wave development

New experimental data that make it possible to explain and predict the observed variability of turbulent-energy dissipation in the upper ocean are discussed. For this purpose, the dependence of the energy dissipation rate of breaking wind waves on their propagation velocity (see [1]) is used. The turbulent-energy dissipation values obtained earlier in [2, 3] by a direct method are compared to the results of radar measurements of individual breaking events presented in [1]. On the basis of this comparison, a strong dependence of the turbulent-energy dissipation value on the stage of wind-wave development, which is characterized by the ratio U a /c p (U a is the wind speed and c p is the phase speed of the peak of the wind-wave spectrum) is confirmed. This dependence was found earlier purely empirically. Moreover, it is shown that the theoretically obtained dependence (c p /U a )4, does not contradict the available empirical data. The results of this study opens possibilities for scientifically substantiated calculations of greenhouse-gas exchange (specifically, CO2 exchange between the ocean and the atmosphere).

Effects of three-wave interactions in the gravity-capillary range of wind waves

The formation of the spectrum of short wind waves from the gravity-capillary and capillary ranges under the effect of three-wave interactions is considered. In order to determine the spectrum, the kinetic equation for wave packets is integrated to the point where the solution is established. Three-wave interactions are described by a collision integral without introducing any additional assumptions simplifying the problem. This calculation procedure reproduces the Zakharov-Filonenko theoretical spectra, which correspond to the cases of energy equipartition and the inertial range. It is shown that the main role of three-wave interactions lies in the energy transfer from the range of short gravity waves to waves with shorter wavelengths. This transfer is accomplished both locally in the Fourier space and as a result of interactions between short and long waves. Its characteristic features are the formation of a dip on the curvature spectrum in the region of a minimum phase velocity of waves and the formation of a secondary peak in the capillary range. The dip is filled and disappears as the wind speed increases. Taking into account the interaction between short and long waves increases the spectrum in the capillary range several times, and the balance between energy input from long waves and viscous dissipation is established in the capillary range. The energy sink caused by three-wave interactions, viscous dissipation, and wind forcing cannot give the stability of the spectrum of short gravity waves.

The role of wind waves in dynamics of the air-sea interface

Wind waves are considered as an intermediate small-scale dynamic process at the air-sea interface, which modulates radically middle-scale dynamic processes of the boundary layers in water and air. It is shown that with the aim of a quantitative description of the impact said, one can use the numerical wind wave models which are added with the blocks of the dynamic atmosphere boundary layer (DABL) and the dynamic water upper layer (DWUL). A mathematical formalization for the problem of energy and momentum transfer from the wind to the upper ocean is given on the basis of the well known mathematical representations for mechanisms of a wind wave spectrum evolution. The problem is solved quantitatively by means of introducing special system parameters: the relative rate of the wave energy input, I RE , and the relative rate of the wave energy dissipation, D RE . For two simple wave-origin situations, the certain estimations for values of I RE and D RE are found, and the examples of calculating an impact of a wind sea on the characteristics of both the boundary layer of atmosphere and the water upper layer are given. The results obtained permit to state that the models of wind waves of the new (fifth) generation, which are added with the blocks of the DABL and the DWUL, could be an essential chain of the general model describing the ocean-atmosphere circulation.

Nonlinear High Wave Groups in Bimodal Sea States

Nonlinear effects for wave groups in bimodal sea states are investigated. The linear free surface displacement is obtained by applying the quasi-determinism theory of the highest waves, with the spectrum given by the superposition of two JONSWAP spectra. Linear groups are modeled as the sum of two groups produced by each unimodal spectrum. Nonlinear effects are given as second-order contribution to the linear free surface displacement. The structure of wave groups is then analyzed for some bimodal spectra. The linear component of wave groups shows a strong variation with respect to the classical structure with unimodal spectra, particularly in mixed seas. It is observed that the nonlinearity gives small contributions for swell dominated seas. For wind dominated sea states, the main nonlinear contribution is given by the wind wave spectral components. For mixed sea states, the interaction among the components of the wind wave spectrum and the components of the swells could give a meaningful contribution to the nonlinear profile. The strongest nonlinear effects occur when the two peaks of the spectrum are close to each other. Finally, the results are validated by means of Monte Carlo simulations of nonlinear sea waves.

Modification of SAR spectra associated with surface wind fields in the sea off the Kii Peninsula: A case study

Using surface wave parameters and a high-resolution surface wind field derived from Synthetic Aperture Radar (SAR) image mode data, we have investigated the spatial modification of SAR spectra. We found a surface wind front, formed by sheltering effect of the Kii Mountains, separating high and low wind-speed regions in a sea area of an European Remote-Sensing Satellite (ERS) SAR image off the Kii Peninsula. A swell system propagating westward dominates in the whole sea area covered by the SAR image. The wavelength retrieved from the SAR spectra in the sheltered (non-sheltered) region is longer (shorter). Since the distributions of surface wave parameters and surface wind speed are so well correlated, it can be considered that the SAR spectra are modified differently by the sheltered/non-sheltered surface winds. In order to examine the phenomena observed on the SAR image we have estimated the wind-wave SAR spectrum using the SAR surface winds, a wind-wave spectrum model and a SAR wave imaging model. We assume that the SAR spectrum related to the swell is homogeneous in the area imaged by SAR, and that the SAR spectrum of the wind-wave components causes the observed SAR spectra modification. Differences between the observed SAR spectra and the estimated SAR spectra in the sheltered and non-sheltered regions agree well with each other. In the present case, it can be concluded that the observed SAR spectra can be regarded as a linear combination of the wind-wave SAR spectra and the swell SAR spectra.

Neural network parameterisation of the mapping of wave spectra onto nonlinear four-wave interactions

A new approach to parameterise the exact nonlinear interaction source ( S nl ) term for wind wave spectra is presented. Discrete wave spectra are directly mapped onto the corresponding S nl -terms using a neural net (NN). The NN was trained with modelled wave spectra varying from single mode spectra to highly complex ones. The specification of training data was based on a classification of the wave spectra by cluster analysis. In course of the structuring of the NN the intrinsic dimensionality of the spectra was estimated with an auto-associative neural net (AANN). The AANN might be used for a scope check of the method.

The form of the asymptotic depth-limited wind-wave spectrum: Part II — The wavenumber spectrum

Data from a spatial array of wave gauges is analysed using the Wavelet Directional Method (WDM) to directly determine the wavenumber spectrum. The data shows that the asymptotic depth-limited wavenumber spectrum can be represented as a two-parameter form, which is far simpler than the corresponding frequency spectrum. The WDM analysis shows that there are significant nonlinear processes active in the finite depth water, which results in energy being “smeared” across a range of wavenumbers and frequencies around the standard dispersion shell. As a result, the wavenumber spectrum has much less peak enhancement than seen in the frequency spectrum obtained with standard Fourier analysis. In addition, the wavenumber spectrum does not have the clear harmonic previously observed in the finite depth frequency spectrum. This result demonstrates that the harmonic is nonlinearly phase-locked to the spectral peak.

A Two-Scale Approximation for Efficient Representation of Nonlinear Energy Transfers in a Wind Wave Spectrum. Part I: Theoretical Development

Abstract A new method for estimating the transfer rates in wind wave spectra is derived and tested, based on a two-scale approximation (TSA) to the total integral for quadruplet wave–wave interactions. Comparisons of this new estimation method to the full integral are given for several idealized spectra, including Joint North Sea Wave Project spectra with different peakednesses, a finite depth case, and cases with perturbations added to underlying parametric spectra. In particular, these comparisons show that the TSA is a significant improvement over the discrete interaction approximation (DIA) in deep water and an even greater improvement in shallow water.

Quasi-linear model of interaction of surface waves with strong and hurricane winds

A quasi-linear model for determining the aerodynamic drag coefficient of the sea surface and the growth rate of surface waves under a hurricane wind is proposed. The model explains the reduction (stabilization) in the drag coefficient during hurricane winds. This model is based on the solution of the Reynolds equations in curvilinear coordinates with the use of the approximation of the eddy viscosity, which takes into account the presence of the viscous sublayer. The profile of the mean wind velocity is found with consideration for nonlinear wave stresses (wave momentum flux), whereas wave disturbances induced in air by waves on the water surface are determined in the context of linear equations. The model is verified by comparing the calculation results with experimental data for a wide range of wind velocities. The growth rate and drag coefficient for hurricane winds are calculated both with and without consideration for the shortwave portion of the windwave spectrum. On the basis of calculations with the quasi-linear model, a simple parametrization is proposed for the drag coefficient and the growth rate of surface waves during hurricane winds. This model is convenient for use in models of forecasting winds and waves.

An analytical algorithm with a wave age factor for altimeter wind speed retrieval

Based on the specular reflection theory of electromagnetic waves at rough sea surface and the wind wave spectrum model with a wave age factor, the sea surface wind speeds are retrieved from the normalized radar backscatter cross‐section (NRCS) measured by TOPEX/Poseidon (T/P) Ku‐band altimeter using the mean square slope (MSS) calculated from the spectrum models of the wind waves and the gravity‐capillary waves. A relationship between wave age and non‐dimensional wave height is applied to compute the wave age factor using the significant wave height (SWH) and wind speeds obtained from buoy or altimeter simultaneously. The study indicates that the wave age factor has a significant impact on the retrieval of altimeter wind speed. Compared with the operational algorithm for retrieving altimeter wind speed, the wind speed retrieved from the new analytical algorithm based on the wind wave spectrum model with the wave age factor, proposed in this study, can match the buoy measurements better. The effects of the wave age factor on altimeter wind speed retrieval are also shown quantitatively through a series of experiments and measurements. The comparison with the operational algorithm indicates that both the bias and root mean square error (RMSE) between wind speeds retrieved by the proposed analytical algorithm and those observed by the buoy decrease significantly. In the Gulf of Mexico, with the new analytical algorithm, more accurate altimeter wind speeds are retrieved.

Earth-Viewing L-Band Radiometer Sensing of Sea Surface Scattered Celestial Sky Radiation—Part I: General Characteristics

The ldquogalactic glitterrdquo phenomenon at L-band, i.e., the scattering of celestial sky radiation by the rough ocean surface, is examined here as a potential source of error for sea surface salinity (SSS) remote sensing. We begin by considering the transformations that must be applied to downwelling celestial noise in order to compute the eventual impact on the antenna temperature. Then, outside the context of any particular measurement system, we use approximate scattering models along with a model for the equilibrium wind wave spectrum to examine how the scattered signal at the surface might depend on the geophysical conditions and scattering geometry. It is found that, when the specular point lies far away from the galactic plane, where the incident celestial brightness is uniform, sea surface roughness has a negligible impact on the glitter. At such a point, variations in both the orientation of the incidence plane and the wind direction relative to the scattering azimuth have negligible impact. By contrast, when the specular point lies in the vicinity of a localized maximum of brightness, scattering by the roughened ocean surface may reduce the glitter by more than 30%, as compared to a perfectly flat surface, and the glitter amplitude may vary by up to 0.7 K with variations in wind direction and by up to 0.5 K with variations in incidence plane orientation. It is shown that accounting for the roughness impact on celestial noise contamination is of particular concern for the remote sensing of SSS.

Correction to a Generalized TMA Spectrum and Proposal of Highly Accurate Approximation Expressions for its Integral Quantities

We present a corrected form of a universal frequency spectrum of wind waves in finite depth water under ideal generation condition made by Yamaguchi (1988). It is a generalized form (G-TMA) of the TMA spectrum proposed by Bouws et al.(1985) as an extension of the JONSWAP spectrum in deep water. The G-TMA spectrum covers other universal deep-water spectra such as the Donelan spectrum and shallow water extensions such as the FRF spectrum. We indicate that the supposed peak frequency fp in the G-TMA spectrum is actually not the peak frequency. This is remedied in our generalized form, with a non-trivial term. In addition, we present rather accurate approximations of spectral moments in terms of total wave energy and various average frequencies.

The equilibrium range of wind wave spectra: an explanation based on white noise

Laboratory experiments and field observations show that the equilibrium range of wind wave spectra presents a −4 power law when it is scaled properly. This feature has been attributed to energy balance in spectral space by many researchers. In this paper we point out that white noise on an oscillation system can also lead to a similar inverse power law in the corresponding displacement spectrum, implying that the −4 power law for the equilibrium range of wind wave spectra may probably only reflect the randomicity of the wind waves rather than any other dynamical processes in physical space. This explanation may shed light on the mechanism of other physical processes with spectra also showing an inverse power law, such as isotropic turbulence, internal waves, etc.

Wind wave spectral observations in Currituck Sound, North Carolina

We examine a set of 1626 high‐resolution frequency‐direction wind wave spectra and collocated winds collected during a 7‐month period at a site in Currituck Sound, North Carolina, in terms of one‐dimensional spectral structure and directional distribution functions. The data set includes cases of shore‐normal winds in broad‐fetch conditions as well as winds oblique to the basin geometry, with all fetches of order 10 km or less. Using equilibrium‐range scaling, all one‐dimensional spectra have a spectral peak region, an equilibrium range of finite bandwidth following an f−4 slope at slightly higher frequencies, and a high‐frequency tail that falls off more rapidly than f−4. For shore‐normal winds, spectral peakedness appears to be high and approximately constant for young waves, low and approximately constant for old waves, and steeply graded for intermediate inverse wave ages in the range 1.0 < u10/cp < 1.7. Equilibrium‐range bandwidth seems to be narrow for young waves and increases with increasing wave age. Directional distribution functions in shore‐normal winds are symmetric about the wind direction, narrow at spectral peaks, and broad at high frequencies with distinct directionally bimodal peaks, consistent with other observations. In oblique‐wind cases, directional distribution functions are asymmetric and directionally sheared in spectral peak regions, with peak directions aligned with longer fetch directions. At high frequencies, directional distributions are more nearly symmetric about the wind direction. One‐dimensional spectra tend to have reduced spectral peakedness and highly variable equilibrium‐range bandwidths in oblique‐wind conditions, clearly indicating a more complex balance of source terms in these cases than in the more elementary situation of shore‐normal winds. These complications are not without consequence in wave modeling, as any bounded or semibounded lake or estuary will be subject to oblique winds, and current operational models do not deal well with conditions like those we find here.