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  • Cambridge University Press  (10)
  • 1
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    Long-time behaviour of a random inhomogeneous field of weakly nonlinear surface gravity waves (1983)
    Cambridge University Press
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    Publication Date: 1983-08-01
    Description: In this paper we investigate nonlinear interactions of narrowband, Gaussian-random, inhomogeneous wavetrains. Alber studied the stability of a homogeneous wave spectrum as a function of the width a of the spectrum. For vanishing bandwidth the deterministic results of Benjamin & Feir on the instability of a uniform wavetrain were rediscovered whereas a homogeneous wave spectrum was found to be stable if the bandwidth is sufficiently large. Clearly, a threshold for instability is present, and in this paper we intend to study the long-time behaviour of a slightly unstable modulation by means of a multiple-timescale technique. Two interesting cases are found. For small but finite bandwidth a the amplitude of the unstable modulation shows initially an overshoot, followed by an oscillation around the time-asymptotic value of the amplitude. This oscillation damps owing to phase mixing except for vanishing bandwidth because then the well-known Fermi-Pasta-Ulam recurrence is found. For large bandwidth, however, no overshoot is found since the damping is overwhelming. In both cases the instability is quenched because of a broadening of the spectrum. © 1983, Cambridge University Press. All rights reserved.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 2
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    On a fourth-order envelope equation for deep-water waves (1983)
    Cambridge University Press
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    Publication Date: 1983-01-01
    Description: The ordinary nonlinear Schrödinger equation for deep-water waves (found by a perturbation analysis to 0(є3) in the wave steepness e) compares unfavourably with the exact calculations of Longuet-Higgins (1978) for є ~ 0.10. Dysthe (1979) showed that a significant improvement is found by taking the perturbation analysis one step further to 0(є4). One of the dominant new effects is the wave-induced mean flow. We elaborate the Dysthe approach by investigating the effect of the wave-induced flow on the long-time behaviour of the Benjamin—Feir instability. The occurrence of a wave-induced flow may give rise to a Doppler shift in the frequency of the carrier wave and therefore could explain the observed down-shift in experiment (Lake et al. 1977). However, we present arguments why this is not a proper explanation. Finally, we apply the Dysthe equations to a homogeneous random field of gravity waves and obtain the nonlinear energy-transfer function recently found by Dungey & Hui (1979). © 1983, Cambridge University Press. All rights reserved.
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  • 3
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    The period-doubling of gravity–capillary waves (1986)
    Cambridge University Press
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    Publication Date: 1986-11-01
    Description: In this paper an attempt is made to explain the period-doubling of wind-generated gravity-capillary waves as observed in the experiment of Choi (1977). It is conjectured that period-doubling is closely related to the phenomenon of second-harmonic resonance. In order to obtain a simple dynamical model, results of McGoldrick (1970) and Simmons (1969) are extended to include the effect of wind input and shear in the current. For pure gravity-capillary waves (no wind, no current) the condition for energy transfer from the second harmonic to the fundamental wave of Chen & Saffman (1979) is recovered. We also discuss the effect of wind and we find that wind input gives rise to a very sudden period-doubling. Qualitative agreement with experiment is obtained. © 1986, Cambridge University Press. All rights reserved.
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  • 4
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    An analytic model of the generation of surface gravity waves by turbulent air flow (1992)
    Cambridge University Press
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    Publication Date: 1992-03-01
    Description: Turbulent air flow over a surface gravity wave of small amplitude is studied on the basis of a family of first-order closure models, of which the eddy viscosity model and Prandtl's mixing-length model are members. Results are obtained by the method of matched asymptotic expansions in three layers. The problem is modelled by taking into account the combined effects of turbulence and molecular viscosity, which accommodates a proper imposition of the boundary conditions at the wave surface. The detailed structure of the various wave-induced field variables throughout the flow is then investigated. In addition, it is found that the growth rate of the waves by wind depends on the turbulence model. In particular, the more sensitively the mixing length depends on the shear in the mean air flow, the higher the growth rate. The validity of the results we obtain is restricted to small drag coefficient and small phase speed. Comparisons are made with other theoretical studies and with recent laboratory and field observations. © 1992, Cambridge University Press. All rights reserved.
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  • 5
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    On some consequences of the canonical transformation in the Hamiltonian theory of water waves (2009)
    Cambridge University Press
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    Publication Date: 2009-09-18
    Description: We discuss some consequences of the canonical transformation in the Hamiltonian theory of water waves (Zakharov, J. Appl. Mech. Tech. Phys., vol. 9, 1968, pp. 190194). Using Krasitskii's canonical transformation we derive general expressions for the second-order wavenumber and frequency spectrum and the skewness and the kurtosis of the sea surface. For deep-water waves, the second-order wavenumber spectrum and the skewness play an important role in understanding the so-called sea-state bias as seen by a radar altimeter. According to the present approach but in contrast with results obtained by Barrick & Weber (J. Phys. Oceanogr., vol. 7, 1977, pp. 1121), in deep water second-order effects on the wavenumber spectrum are relatively small. However, in shallow water in which waves are more nonlinear, the second-order effects are relatively large and help to explain the formation of the observed second harmonics and infra-gravity waves in the coastal zone. The second-order effects on the directional-frequency spectrum are as a rule more important; in particular it is shown how the Stokes-frequency correction affects the shape of the frequency spectrum, and it is also discussed why in the context of the second-order theory the mean-square slope cannot be estimated from time series. The kurtosis of the wave field is a relevant parameter in the detection of extreme sea states. Here, it is argued that in contrast perhaps to one's intuition, the kurtosis decreases while the waves approach the coast. This is related to the generation of the wave-induced current and the associated change in mean sea level. © 2009 Copyright Cambridge University Press.
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  • 6
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    Asymptotics for the long-time evolution of kurtosis of narrow-band ocean waves (2018)
    Cambridge University Press
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    Publication Date: 2018-11-26
    Description: In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863-884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25-36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events. © 2018 Cambridge University Press.
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  • 7
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    On a random time series analysis valid for arbitrary spectral shape (2014)
    Cambridge University Press
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    Publication Date: 2014-10-23
    Description: While studying the problem of predicting freak waves it was realized that it would be advantageous to introduce a simple measure for such extreme events. Although it is customary to characterize extremes in terms of wave height or its maximum it is argued in this paper that wave height is an ill-defined quantity in contrast to, for example, the envelope of a wave train. Also, the last measure has physical relevance, because the square of the envelope is the potential energy of the wave train. The well-known representation of a narrow-band wave train is given in terms of a slowly varying envelope function ρ and a slowly varying frequency ω = -∂φ/∂t where φ is the phase of the wave train. The key point is now that the notion of a local frequency and envelope is generalized by also applying the same definitions for a wave train with a broad-banded spectrum. It turns out that this reduction of a complicated signal to only two parameters, namely envelope and frequency, still provides useful information on how to characterize extreme events in a time series. As an example, for a linear wave train the joint probability distribution of envelope height and period is obtained and is validated against results from a Monte Carlo simulation. The extension to the nonlinear regime is, as will be seen, fairly straightforward. © Cambridge University Press 2014.
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  • 8
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    Quasilinear approximation for the spectrum of wind-generated water waves (1982)
    Cambridge University Press
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    Publication Date: 1982-04-01
    Description: According to Miles’ theory of wind-wave generation, water waves grow if the curvature of the wind profile at the critical height is negative. As a result, the wind profile changes in time owing to the transfer of energy to the waves. In thequasilinear approximation (where the interaction of the waves with one another is neglected) equations for the coupled air-water system are obtained by means of a multiple-time-scale analysis. In this way the validity of Miles’ calculationsis extended, thereby allowing a study of the large-time behaviour.While the water waves grow owing to the energy transfer from the air flow, the waves in turn modify the flow in such a way that for large times the curvature of the velocity profile vanishes. The amplitude of the waves is then limited because the energy transfer is quenched.In the high-frequency range the asymptotic wave spectrum is given by a ‘ - 4’ law in the frequency domain rather than the ‘ classical ‘ ‘ - 5 ‘ law. © 1982, Cambridge University Press. All rights reserved.
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  • 9
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    On phase velocity and growth rate of wind-induced gravity-capillary waves (1985)
    Cambridge University Press
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    Publication Date: 1985-12-01
    Description: The generation and growth of gravity—capillary waves (A « 1 cm) by wind are reconsidered using linear instability theory to describe the process. For all friction velocities we solve the resulting Orr—Sommerfeld equation using asymptoticmethods. New elements in our theory, compared with the work of Benjamin (1959) andMiles (1962), aremore stress onmathematical rigour and the incorporation of the wind-induced shear current. We find that the growth rate of the initial wavelets, the first waves to be generated by the wind, is proportional to u. We also study the effect of changes in the shape of the profiles of wind and wind-induced current. In doing this we compare results ofMiles (1962), Larson & Wright (1975), Valenzuela (1976), Kawai (1979), Plant & Wright (1980) and our study. We find that the growth rate is very sensitive to the shape of the wind profile while the influence of changes in the current profile ismuch smaller. To determine correctly the phase velocity, the value of current and current shear at the interface are very important,muchmore so than the shape of either wind or current profile. © 1985, Cambridge University Press. All rights reserved.
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  • 10
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    The initial evolution of gravity-capillary waves (1987)
    Cambridge University Press
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    Publication Date: 1987-11-01
    Description: In this paper we discuss the initial evolution of wind-generated, gravity-capillary waves by means of a dynamical model that includes the effects of wind input, viscous dissipation and three-wave interactions. In particular, we study the generation of the initial wavelets by wind and the subsequent migration of the peak of the spectrum to lower wavenumbers. Under certain conditions a sudden migration of the peak wavenumber is found. It is argued that this sudden migration is related to the phenomenon of second-harmonic resonance. We also observe that during the generation of the initial wavelets by wind, nonlinear three-wave interactions may be important. Therefore, the experimental determination of the growth rate of the waves by wind by just analysing the time series of the surface elevation (as is done by e.g. Kawai 1979 and Plant & Wright 1977) might be in error. © 1987, Cambridge University Press. All rights reserved.
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