Wave Energy Growth Research Articles

Wave Buoy Measurements at Short Fetches in the Black Sea Nearshore: Mixed Sea and Energy Fluxes

Wave buoy measurements were carried out near the northeastern Black Sea coast at the natural reserve Utrish in 2020–2021. In total, about 11 months of data records were collected during two stages of the experiment at 600 and 1500 m offshore and depths of 18 and 42 m. The measured waves propagate almost exclusively from the seaward directions. Generally, the waves do not follow the local wind directions, thus, implying a mixed sea state. Nevertheless, dimensionless wave heights and periods appears to be quite close to the previously established empirical laws for the wind-driven seas. The results of the wave turbulence theory are applied for estimates of spectral energy fluxes and their correspondence to the energy flux from the turbulent wind pulsations. These estimates are consistent with today’s understanding of wind–wave interaction. It is shown that the main fraction of the wind energy flux is sent to the direct Kolmogorov–Zakharov cascade to high wave frequencies and then dissipates in small amounts. Less than 1% of the wind energy flux is directed to the low frequency band (the so-called inverse Kolmogorov–Zakharov cascade), thus, providing wave energy growth.

On ST6 Source Terms Model Assessment and Alternative

We present the study of the ST6 balanced set of wind energy input and wave energy dissipation due to wave breaking source terms, offered as the option in operational wave forecasting models and based on theoretical self-similarity analysis and numerical simulation of the wave energy radiative transfer equation. The study relies on the classical limited fetch wind wave excitation problem with constant wind blowing orthogonally to the shoreline toward the open ocean. It is found that the ST6 model exhibits asymptotic quasi self-similar behavior for fetches exceeding ≃25 km, as well as non-universal wave energy growth for shorter fetches, depending on the shoreline wave energy levels. We construct the alternative model PGZ-2 from a self-similar consideration, which reproduces field experimental data almost in the whole fetch span and reduces wave energy evolution dependence on the shoreline energy level. We assert that the PGZ-2 model is more accurate than the ST6 model. Moreover, it has a wider area of applicability.

Revisiting wind wave growth with fully coupled direct numerical simulations

We investigate wind wave growth by direct numerical simulations solving for the two-phase Navier–Stokes equations. We consider the ratio of the wave speed$c$to the wind friction velocity$u_*$from$c/u_*= 2$to 8, i.e. in the slow to intermediate wave regime; and initial wave steepness$ak$from 0.1 to 0.3; the two being varied independently. The turbulent wind and the travelling, nearly monochromatic waves are fully coupled without any subgrid-scale models. The wall friction Reynolds number is 720. The novel fully coupled approach captures the simultaneous evolution of the wave amplitude and shape, together with the underwater boundary layer (drift current), up to wave breaking. The wave energy growth computed from the time-dependent surface elevation is in quantitative agreement with that computed from the surface pressure distribution, which confirms the leading role of the pressure forcing for finite amplitude gravity waves. The phase shift and the amplitude of the principal mode of surface pressure distribution are systematically reported, to provide direct evidence for possible wind wave growth theories. Intermittent and localised airflow separation is observed for steep waves with small wave age, but its effect on setting the phase-averaged pressure distribution is not drastically different from that of non-separated sheltering. We find that the wave form drag force is not a strong function of wave age but closely related to wave steepness. In addition, the history of wind wave coupling can affect the wave form drag, due to the wave crest shape and other complex coupling effects. The normalised wave growth rate we obtain agrees with previous studies. We make an effort to clarify various commonly adopted underlying assumptions, and to reconcile the scattering of the data between different previous theoretical, numerical and experimental results, as we revisit this longstanding problem with new numerical evidence.

A high-order spectral method for effective simulation of surface waves interacting with an internal wave of large amplitude

In this study, we propose a new method for simulating complex surface waves interacting with a large amplitude internal solitary wave. Our model is based on a high-order spectral method for surface waves with the bottom boundary conditions computed from an internal wave solver. The convergence of the model is first tested using an internal wave case without surface waves. We then perform simulations of a canonical nonlinear wave interaction case involving two surface wave components and a prescribed internal wave component. The initial wave energy growth rate of the resonant wave component is found to agree with the analytical value predicted by the perturbation theory. We also use the model to simulate the interaction between surface waves and a weakly nonlinear internal wave, and the result is nearly identical to that calculated using a two-layer model. Finally, we show the application of our model in a multiscale nested modeling framework, where the internal wave parameters are extracted from the mesoscale simulation using a nonhydrostatic ocean model. The evolution of the spatial variations of the surface roughness is captured and the surface wave orbital velocity also changes in space due to the surface motions induced by the internal wave. Our phase-resolved model provides a computationally efficient tool for simulating complex surface wave fields in the background of internal waves of large amplitudes.

Effect of Water Vorticity on Wind-Generated Gravity Waves in Finite Depth

The generation of wind waves at the surface of an established underlying vertically sheared water flow, of constant vorticity, is considered. A particular attention is paid to the role of the vorticity in water on wind-wave generation in finite depth. The present theoretical results are compared with experimental data obtained by Young and Verhagen (Coast Eng 29:47–78, 1996), in the shallow Lake George (Australia), and the least squares fit of these data by Young (Coast Eng 32:181–195, 1997). It is shown that without vorticity in water, there is a deviation between theory and experimental data. However, a good agreement between the theory and the fit of experimental data is obtained when negative vorticity is taken into account. Furthermore, it is shown that the amplitude growth rate increases with vorticity and depth. A limit to the wave energy growth, corresponding to the vanishing of the growth rate, is obtained. The corresponding limiting wave age is derived in a closed form showing its explicit dependence on vorticity and depth. The limiting wave age is found to increase with both vorticity and depth.

Spatial evolution of young wind waves: numerical modelling verified by experiments

Abstract

Miles Theory Revisited with Constant Vorticity in Water of Infinite Depth

The generation of wind waves at the surface of a pre-existing underlying vertically sheared water flow of constant vorticity is considered. Emphasis is put on the role of the vorticity in water on wind-wave generation. The amplitude growth rate increases with the vorticity except for quite old waves. A limit to the wave energy growth is found in the case of negative vorticity, corresponding to the vanishing of the growth rate.

Efficient numerical model of stimulated Raman scattering in optical fibers

The paper proposes a novel efficient numerical model for simulation of spectral and temporal transformation of laser pulses due to interplay of Kerr and Raman nonlinearity and chromatic dispersion in the process of propagation through single-mode optical fibers. The model reproduces qualitatively the spectral shape of Raman gain within the approximation of slowly varying amplitudes using a pair of meshes (for pump and Stokes waves) with a reduced number of points. Nonlinear propagation of 100-ps-long laser pulses along an optical fiber is used as a test bed for the new model. It is shown that the proposed model provides accuracy better than 10% in Stokes wave energy growth speed, while being up to eight times more efficient in memory usage and computation speed compared to the generalized nonlinear Schrödinger equation.

Limited fetch revisited: Comparison of wind input terms, in surface wave modeling

Results pertaining to numerical solutions of the Hasselmann kinetic equation (HE), for wind driven sea spectra, in the fetch limited geometry, are presented. Five versions of source functions, including the recently introduced ZRP model (Zakharov et al., 2012), have been studied, for the exact expression of Snl and high-frequency implicit dissipation, due to wave-breaking. Four of the five experiments were done in the absence of spectral peak dissipation for various Sin terms. They demonstrated the dominance of quadruplet wave–wave interaction, in the energy balance, and the formation of self-similar regimes, of unlimited wave energy growth, along the fetch. Between them was the ZRP model, which strongly agreed with dozens of field observations performed in the seas and lakes, since 1947. The fifth, the WAM3 wind input term experiment, used additional spectral peak dissipation and reproduced the results of a previous, similar, numerical simulation described in Komen et al. (1994), but only supported the field experiments for moderate fetches, demonstrating a total energy saturation at half of that of the Pierson–Moscowits limit. The alternative framework for HE numerical simulation is proposed, along with a set of tests, allowing one to select physically-justified source terms.

Nonlinear saturation of wave packets excited by low-energy electron horseshoe distributions

Horseshoe distributions are shell-like particle distributions that can arise in space and laboratory plasmas when particle beams propagate into increasing magnetic fields. The present paper studies the stability and the dynamics of wave packets interacting resonantly with electrons presenting low-energy horseshoe or shell-type velocity distributions in a magnetized plasma. The linear instability growth rates are determined as a function of the ratio of the plasma to the cyclotron frequencies, of the velocity and the opening angle of the horseshoe, and of the relative thickness of the shell. The nonlinear stage of the instability is investigated numerically using a symplectic code based on a three-dimensional Hamiltonian model. Simulation results show that the dynamics of the system is mainly governed by wave-particle interactions at Landau and normal cyclotron resonances and that the high-order normal cyclotron resonances play an essential role. Specific features of the dynamics of particles interacting simultaneously with two or more waves at resonances of different natures and orders are discussed, showing that such complex processes determine the main characteristics of the wave spectrum's evolution. Simulations with wave packets presenting quasicontinuous spectra provide a full picture of the relaxation of the horseshoe distribution, revealing two main phases of the evolution: an initial stage of wave energy growth, characterized by a fast filling of the shell, and a second phase of slow damping of the wave energy, accompanied by final adjustments of the electron distribution. The influence of the density inhomogeneity along the horseshoe on the wave-particle dynamics is also discussed.

Breaking Waves in Deep and Intermediate Waters

Since time immemorial, surface water waves and their subsequent breaking have been studied. Herein we concentrate on breaking surface waves in deep and intermediate water depths. Progress has been made in several areas, including the prediction of their geometry, breaking onset, and especially energy dissipation. Recent progress in the study of geometric properties has evolved such that we can identify possible connections between crest geometry and energy dissipation and its rate. Onset prediction based on the local wave-energy growth rate appears robust, consistent with experiments, although the application of criteria in phase-resolving, deterministic prediction may be limited as calculation of the diagnostic parameter is nontrivial. Parameterization of the dissipation rate has benefited greatly from synergistic field and laboratory investigations, and relationships among the dynamics, kinematics, and the parameterization of the dynamics using geometric properties are now available. Field efforts continue, and although progress has been made, consensus among researchers is limited, in part because of the relatively few studies. Although direct numerical simulations of breaking waves are not yet a viable option, simpler models (e.g., implementation of an eddy viscosity model) have yielded positive results, particularly with regard to energy dissipation.

Sequential assimilation in the wind wave model for simulations of typhoon events around Taiwan Island

We investigate the effect of data assimilation in the Wind Wave Model (WWM, Hsu et al., 2005) for wind wave simulations of typhoon events using an Optimal Interpolation (OI) method in the coastal waters of Taiwan. The main point of the present study is that the assimilation is conducted for typhoon events with short-time periods around an island. Five real typhoon events were used for numerical assimilation experiments with different combinations of the key parameters: the correlation length, the ratio of the errors between the observation and the prediction, and the number of measuring stations. The retrieved wind velocity is obtained using the wave energy growth curve computed from the WWM. The wave energy dissipation function suggested by Makin and Kudryavtsev (1999) was adopted in the data assimilation. This study shows that assimilation can improve the initial performance of the wave model, but it becomes insignificant after about 12 h. In summary, the OI approach is shown to be a reliable assimilation scheme for the WWM applied to typhoon events in the coastal waters of Taiwan Island.

Weakly turbulent laws of wind-wave growth

The theory of weak turbulence developed for wind-driven waves in theoretical works and in recent extensive numerical studies concludes that non-dimensional features of self-similar wave growth (i.e. wave energy and characteristic frequency) have to be scaled by internal wave-field properties (fluxes of energy, momentum or wave action) rather than by external attributes (e.g. wind speed) which have been widely adopted since the 1960s. Based on the hypothesis of dominant nonlinear transfer, an asymptotic weakly turbulent relation for the total energy ϵ and a characteristic wave frequency ω* was derived The self-similarity parameter αss was found in the numerical duration-limited experiments and was shown to be naturally varying in a relatively narrow range, being dependent on the energy growth rate only.In this work, the analytical and numerical conclusions are further verified by means of known field dependencies for wave energy growth and peak frequency downshift. A comprehensive set of more than 20 such dependencies, obtained over almost 50 years of field observations, is analysed. The estimates give αss very close to the numerical values. They demonstrate that the weakly turbulent law has a general value and describes the wave evolution well, apart from the earliest and full wave development stages when nonlinear transfer competes with wave input and dissipation.

Surface current effects on the fetch‐limited growth of wave energy

To study the fetch‐limited growth of wind wave energy over a region with significant lateral shear of the current field, this study exploited data obtained from two linear phased‐array High Frequency (HF) radar systems. Both the near‐surface currents and wave energy and period were mapped over the highly sheared inshore boundary of the Florida Current. The wave energy growth during two periods when the winds were steady for >12 hours and were directed offshore was computed over a range of fetches from 5 to 45 km. The observed energy growth rates were significantly (∼50 %) lower than predicted by the Donelan et al. (1992) empirical formulation over the high vorticity region. The reduced growth rate was consistent with a shift of the wind stress direction into the current direction due to refraction of the wave field. Dimensionless wave energy increased by as much as 100% over neighboring values when the vertical component of the surface current vorticity was a global minimum. While trapping and enhancement of wave energy is predicted by wave ray theory, it has never before been confirmed within the Florida Current with coincident wave and current measurements.

The growth rate of finite depth wind-generated waves

Fetch limited wind wave experiments traditionally investigate the growth of wave energy with distance from shore. [Donelan, M., Skafel, M., Graber, H., Liu, P., Schwab, D., Venkatesh, S., 1992. On the growth rate of wind-generated waves, Atmos.-Ocean 30, 457–478], have, however, shown that for deep water conditions it is advantageous to investigate the differential growth between points located along the fetch. Based on the extensive data set collected by [Young I.R., Verhagen, L.A., 1996a. The growth of fetch limited waves in water of finite depth, Part I: Total energy and peak frequency, Coastal Eng. 28, 47–78], this approach has been extended to finite depth conditions. The data clearly show that at short fetches the growth rate is comparable to deep water conditions. At longer fetches the finite depth influences increase and the growth rate decreases compared to deep water. Finally the waves approach a depth limited state where the growth rate becomes zero. Based on the observed differential growth between measurement stations, a relationship is developed which can be integrated to yield the development of the total energy with fetch. This relationship is significantly more flexible than previous finite depth growth relationships. Cases in which the water depth and/or wind speed vary with fetch can be investigated. In particular, it is shown that the development of the atmospheric boundary layer with fetch has a significant influence on the observed wave growth. By the inclusion of a realistic relationship for the boundary layer development it is shown that apparently anomalous features of the [Young, I.R., Verhagen, L.A., 1996a. The growth of fetch limited waves in water of finite depth, Part I: Total energy and peak frequency, Coastal Eng. 28, 47–78] data set can be explained.

Three‐dimensional electromagnetic Monte Carlo particle‐in‐cell simulations of critical ionization velocity experiments in space

Although the existence of the critical ionization velocity (CIV) is known from laboratory experiments, no agreement has been reached as to whether CIV exists in the natural space environment. In this paper we move towards more realistic models of CIV and present the first fully three‐dimensional, electromagnetic particle‐in‐cell Monte‐Carlo collision (PIC‐MCC) simulations of typical space‐based CIV experiments. In our model, the released neutral gas is taken to be a spherical cloud traveling across a magnetized ambient plasma. Simulations are performed for neutral clouds with various sizes and densities. The effects of the cloud parameters on ionization yield, wave energy growth, electron heating, momentum coupling, and the three‐dimensional structure of the newly ionized plasma are discussed. The simulations suggest that the quantitative characteristics of momentum transfers among the ion beam, neutral cloud, and plasma waves is the key indicator of whether CIV can occur in space. The missing factors in space‐based CIV experiments may be the conditions necessary for a continuous enhancement of the beam ion momentum. For a typical shaped charge release experiment, favorable CIV conditions may exist only in a very narrow, intermediate spatial region some distance from the release point due to the effects of the cloud density and size. When CIV does occur, the newly ionized plasma from the cloud forms a very complex structure due to the combined forces from the geomagnetic field, the motion induced emf, and the polarization. Hence the detection of CIV also critically depends on the sensor location.

Surface drift effect on wind energy transfer to waves

The effect of surface drift velocity on the energy transfer from the wind to ocean waves and the growth rate of the dominant surface wave was studied analytically. The model analyzed considered first the airflow above the surface wave to be turbulent, the thin drift shear layer to be viscous, and the flow beneath the drift flow to be potential. Then an extension of the analysis was made for the case of a drift shear layer of considerable thickness. The analysis clearly demonstrated the important role that air turbulence and water surface drift play in the wind energy transfer to waves. It was also found that the exponential growth dominated the process of the wind energy transfer to waves. In general, the presence of surface drift increased the wave energy growth rate. The present analysis predicted the existence of a critical value of the normalized surface drift velocity, at which the increment of wave energy growth due to surface drift reached a maximum. When the surface drift exceeded the critical value, there was an adverse effect on the wave energy growth. In the extreme condition in which the surface drift matched the phase speed, it set up the upper limit of the wave energy growth rate. The numerical estimates computed with the theory were consistent with experimental measurements, and with the results obtained with more complex analyses based on numerical models.

Spectral evolution of wind-generated surface gravity waves in a dispersed ice field

The Marginal Ice Zone includes wide areas covered by dispersed ice floes in which wave conditions are significantly affected by the ice. When the wind blows from the solid ice pack, towards the open sea, growing waves are scattered by the floes, and their spectral characteristics modified. To further understand this problem, a model for the evolution of wind waves in a sparse field of ice floes has been developed. The sea state is described by a two-dimensional discrete spectrum. Time-limited wave growth is obtained by numerical integration of the energy balance equation using the exact nonlinear transfer integral. Wave scattering by a single floe is represented in terms of far-field expressions of the diffracted and forced potentials obtained numerically by the Green function method. The combined effect of a homogeneous field of floes on the wave spectrum is expressed in terms of the Foldy–Twersky integral equations under the assumption of single scattering. The results show a strong dependence of the spectrum amplitude and directional properties on the ratio of the ice floe diameter to the wavelength. For a certain range of this parameter, the ice cover appears to be very effective in dispersing the energy; the wave spectrum rapidly tends to isotropy, a tendency which prevents the normal growth of wave energy and the decrease in peak frequency. Therefore, in the Marginal Ice Zone, the ability of an offshore wind to generate a significant wave field is severely limited.

The Response of Wave Directions to Changing Wind Directions

Abstract From the premise that the net growth of wave energy induced by wind is centered around the wind direction, a relaxation model for the response of the main wave direction to changes in the wind direction for young sea states is derived. The time scale of this relaxation model is found to be equal to the time scale of the wave energy growth. A quantitative version of the model, based on universal growth rates of the waves under the local wind is found to be consistent with observations obtained in this study and with a published dataset.

The saturation of electron-cyclotron maser and the time profile of emitted spikes

In this paper the evolution of the hollow beam distribution of energetic electrons giving rise to ECM instability is investigated and the spatial dispersion term is included in the equation of wave energy. The instability causes the growth of wave energy, while the propagation of waves evacuates the electromagnetic energy from the source region. By analysing these two effects spike-like time profiles of waves are obtained. It is found that the saturation time t s of ECM emission and the duration of spikes increase with the decrease of the frequency of solar radio spike emission. The approximate expressions of t s and of the peak wave energy density are derived.