The nonlinear evolution of deep-water wave groups, which are initiated by unstable three-wave systems, have been observed in a large wave tank (50 m long, 4.2 m wide, 2.1 m deep), equipped with a programmable, high-resolution wave generator. A large number of experiments were conducted (over 80 cases) for waves 1.0-4.0 m long, initial steepness ∈ = 0.10-0.28, and normalized sideband frequency differences, δω/∈ω, 0.2-1.4. Using an array of eight high-resolution wave wires distributed in range (up to 43 m fetch), spectral evolution was studied in detail including the effect of background disturbances on the evolution. Minimizing those, new observations were made which extend the pioneering work of Lake et al. (1977) and of Melville (1982). Foremost, near recurrence without downshifting was observed without breaking, despite a significant but reversible energy transfer to the lower sideband at peak modulation; complete recurrence was prevented by the spreading of discretized energy to higher frequencies. Strong breaking was found to increase the transfer of energy from the higher to the lower sideband and to render that transfer irreversible. The end state of the evolution following strong breaking is an effective downshifting of the spectral energy, where the lower and the carrier wave amplitudes nearly coincide; the further evolution of this almost two-wave system was not studied here. Breaking during strong modulation was observed not only for the fastest growing initial condition, but over a wide parameter range. An explanation of the sideband behaviour in both the breaking and non-breaking case was given based on wave energy and momentum considerations, including the separate effects of energy and momentum loss due to breaking, and transfer to discretized higher frequencies throughout the spectra. Attention was drawn to the latter, which was almost universally observed.

Laboratory observations of wave group evolution, including breaking effects

Resonant Interaction Theory (RIT)is reviewed in a variety of dynamical settings with an emphasis on the underlying approximations. The elementary interactions of three- and four-wave resonances is examined as well as deterministic and stochastic models for wavefields comprising either a broad or narrow spectrum of waves interacting in multiple and coupled sets. Some special settings in which higher-order resonances have been examined are reviewed together with Resonant Interaction Experiments (RIE) in a variety of dynamical settings with an emphasis on those aspects that are predicted. Experimental investigations on the applicability of Kelvin's dispersion relation and experiments involving gravity, gravity- capillary, and capillary wavetrains are discussed. Results from comparisons between RIT and RIE are briefly summarised and the particular absence of controlled experiments in some applications is noted.

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Resonant interactions among surface water waves

Spatial growth of mechanically generated water waves under the action of wind has been measured in a laboratory wind-wave flume both for pure water and for water containing a surfactant (sodium lauryl sulphate, concentration 2.6 × 10−2%). I n the latter case, no wind waves develop on the surface of the mechanically generated waves as well as on the still water surface for wind speeds up to U10≈ 15 m/s, where U10 is the wind velocity at the height Z = 10 m. Therefore we can study the wind-induced growth of monochromatic waves without the effects of co-existing short wind waves. The mechanically generated waves grew exponentially under the action of the wind, with fetch in both cases. The measured growth rate β for the pure water can be fitted by β/f = 0.34(U*/C)2 0.1 [lsime ] U*/C [lsime ] 1.0, where f is the frequency of the waves, C is the corresponding phase velocity, and U, is the friction velocity obtained from vertical wind profiles. The effect of the wave steepness H/L on the dimensionless growth rate β/f is not clear, but seems to be small. For water containing the surfactant, the measured growth rate is smaller than that for pure water, but the friction velocity of the wind is also small, and the above relation between β/f and U*/C holds approximately if the measured friction velocity U* is used for the relation.

Wind-induced growth of water waves

Wind sea growth and dissipation in a swell-dominated, open ocean environment is investigated to explore the use of wave parameters in air–sea process modeling. Wind, wave, and whitecap observations are used from the Gulf of Alaska surface scatter and air–sea interaction experiment (Critical Sea Test-7, Phase 2), conducted 24 February through 1 March 1992. Wind sea components are extracted from buoy directional wave spectra using an inverted catchment area approach for peak isolation with both wave age criteria and an equilibrium range threshold used to classify the wind sea spectral domain. Dimensionless wind sea energy is found to scale with inverse wave age independently of swell. However, wind trend causes significant variations, such as underdeveloped seas during rising winds. These important effects are neglected in wind-forced air–sea process models. The total rate of wave energy dissipation is conveniently estimated using concepts from the Phillips equilibrium range theory. Replacing wind ...

Wind Sea Growth and Dissipation in the Open Ocean

The dim ensionless parameter u * 2 /vσ can be used widely as a parameter to describe the overall conditions of air-sea boundary processes, where u* is the friction velocity of air, v is the kinematic viscosity of air and s is the spectral peak frequency of the wind waves A critical value of this parameter for the appreciable commencement of breaking of wind waves is 103 Beyond this value, the percentage of waves passing a fixed point that are breaking, α, and the percentage of whitecap coverage, P, are both approximately proportional to this parameter The number concentration of sea-salt particles containing salt in the vicinity of 10−10 g at the 6-m level is also proportional to this parameter The dimensionless roughness length associated with the air flow over water, u*Z0 /V, also correlates better with this parameter than with a parameter which does not contain the spectral peak frequency This gives an approximate relation of z0 σ/u* = 0025, and a corresponding formula for the drag coefficient is proposed The dimensional and physical interpretation of the parameteru */2/vσ is presented

A Parameter Describing Overall Conditions of Wave Breaking, Whitecapping, Sea-Spray Production and Wind Stress

The interaction between mechanically generated regular waves and wind waves is experimentally investigated in a wind-wave tunnel. It is shown that the growth process of regular waves is divided into the four distinct stages as follows: (1) almost independent coexistence of wind waves and regular waves, (2) attenuation of wind waves with simultaneous growth of regular waves, (3) rapid growth of regular waves after disappearance of wind waves, and (4) transition of regular waves to wind waves after the wave breaking. At the second stage there is an apparent relation between the attenuation of wind waves and the growth of regular waves. This fact suggests that there is some strong nonlinear interactions which transfers energy effectively from wind waves to regular waves.

Experimental study on strong interaction between regular waves and wind waves—I

The transition process of mechanically generated regular waves into wind waves is investigated in a wind-wave tunnel. Modulations of characteristic quantities of the waves are examined in both space and time using wave-gauge arrays. The transition begins with the occurrence of wave breaking, and it is associated with the following two processes: (i) the irregularization, i.e. the generation and amplification of a random modulation whose wave height and period are in phase; and (ii) the frequency shift to the lower side by the mutual coalescence of waves through the amplification of the modulation, the coalescence occurring at the troughs of the period modulation. The modulational properties under the effect of wind with wave breaking are thus in part different from the theoretical prediction of modulational instability of a freely travelling wavetrain. It is concluded that the mechanism of the transition is the modulational instability coupled with the wind effects.

Transition of mechanically generated regular waves to wind waves under the action of wind

The fine structure of wind-wave spectra in a wind-wave tunnel is investigated in connection with the process of nonlinear interaction among spectral components, where each spectrum is calculated from about one hour of data measured with 8 wave gauges arranged at a distance of 25 cm. It is found that there are several significant spikes, or primary modes, around the spectral maximumf
0. Among such primary modes, three primary modes including the spectral maximum satisfy the resonance condition 2f
0=f
1+f
2, wheref
1 andf
2 are frequencies of the higher and lower frequency modes adjacent tof
0, while the frequency width between adjacent modes,f
1−f
0 orf
0−f
2, is nearly equal to 0.1f
0. In the process of growth of the spectra with fetch, the energy density at the lower frequencyf
2 increases accompanied with an energy decrease at the higher frequencyf
1, and the modef
2 replaces the modef
0 as the spectral maximum. This evolutional process is repeated several times in the range of the present experimental condition. These results suggest that the discrete primary modes are not due to statistical error, but rather due to some wave-wave interaction.

Nonlinear Properties of Laboratory Wind Waves at Energy Containing Frequencies Part 2. Detailed Structures of Power Spectra and their Evolution with Fetch

Nonlinear properties of wind waves in a wind-wave tunnel are investigated by measuring the probability density distribution of surface elevation. The surface elevation distribution of raw records are found to have a positive skewness (K~=0.21 to 0.43) and a negative kurtosis (K~=-0 .74 to -0.41) with magnitude depending of fetch and wind speed. The values of skewness are in qualitative agreement with a prediction of the weak interaction theory for a random wave field incorporating the effects of second harmonics (Tayfun, 1980), but the values of kurtosis are different in sign from the prediction. To examine the nonlinear properties of energy containing components, higher harmonic components are excluded from the wave records by using a kind of a band-pass filter. The surface elevation distributions of the filtered waves show a sharp decrease in skewness (Ka'=0), but the distributions remain highly non-Gaussian with a large negative kurtosis almost independent of the fetch and wind speed (K.lt=-0.66). It is concluded that the negative kurtosis is due to the non-random character of the phase and amplitude among the energy containing components, and that nonlinear interactions occur amongst the energy containing frequencies. I. Introduct ion W i n d waves have long been regarded as a sum of l inear i rrotat ional waves w i th r andom phases to a first order approximation. Higher order nonl inear corrections to such a random wave field were first made by Tick (1959), Phillips (1960), Hasse lmann (1962) and others. They calculated the spectral energy of bound waves and nonl inear energy t ransfer among component waves t h rough resonant interactions. Fu r the r studies on wind wave dynamics along the lines of the above spectral approach have been intensively made, and it has been suggested tha t nonl inear energy t ransfer plays a dominan t role in the g r o w t h of wind waves (Hasselman et al . , 1973), and t h a t bound waves are dominan t at higher frequencies (Masuda et al . , 1979; Mitsuyasu et al . , 1979). Recently, however , several new characterist ics of wind waves have been found, and some * Received 22 December 1982; in revised form 29 June 1983; accepted 6 September I983. Geophysical Institute, Faculty of Science, Tohoku University, Sendai 980, Japan. Present address: Maizuru Marine Observatory, Shimofukui, Maizuru 624, Japan. authors have a t t empted to develop nonspectral models in which the effects of nonl inear i ty dominates. After the finding tha t the phase velocity of spectral components does not follow the l inear wate r wave theory (Yefimov et al . , 1972; Ramamonj iar isoa , 1974), Lake and Yuen (1978) proposed a model based on the dynamics of a coherent nonl inear wa te r wave. In this model, the spectral components o ther than a single carrier wave are regarded to be non-dispersive bound waves of the carrier wave. After the finding of a strongly vortical region at individual wave crests (Toba et al . , 1975; Okuda et al . , 1977; Okuda, 1982), Toba (1978) considered t ha t the wind wave is s trongly nonl inear phenomenon coupled wi th the wind induced rotat ional flow. As to the h igher frequencies, say f > l . 5 f p where f , is a spectral peak frequency, it has been confirmed by various approaches tha t spectral component waves are bound waves corre lated wi th energy conta ining components . On the o ther hand, in regard to the energy conta ining components , several distinct view points have been proposed as out l ined above: l inear waves (spectral approach), nonl inear coherent

Nonlinear properties of laboratory wind waves at energy containing frequencies - Part 1. Probability density distribution of surface elevation

The evolution of mechanically generated regular waves under the wind has been studied using two wind-wave tanks of different dimensions. There are four distinct stages: (1) the coexistence of local wind waves with unaffected regular waves, (2) the attenuation of wind waves and the simultaneous growth of regular waves, (3) the rapid growth of regular waves after the disappearance of wind wave peak, and (4) the transition of regular waves to wind waves accompanied by a frequency shift to longer periods. In Stage 2 the existence is suggested of some strong nonlinear interactions which effectively transfer energy from the wind to regular waves under a condition where the peak frequency of the local wind waves is close to that of the second harmonics of the regular wave. In Stage 4, the modulation of the regular wave, the amplification of the modulation with its irregularities, and the mutual coalescence of waves, seem essential.

Mitsuhiko Hatori

Papers

Experimental study on strong interaction between regular waves and wind waves—I

Transition of mechanically generated regular waves to wind waves under the action of wind

Nonlinear Properties of Laboratory Wind Waves at Energy Containing Frequencies Part 2. Detailed Structures of Power Spectra and their Evolution with Fetch

Nonlinear properties of laboratory wind waves at energy containing frequencies - Part 1. Probability density distribution of surface elevation

Experimental Study of Elementary Processes in Wind-Waves Using Wind over Regular Waves