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Long-time behaviour of a random inhomogeneous field of weakly nonlinear surface gravity waves

Published online by Cambridge University Press:  20 April 2006

Peter A. E. M. Janssen
Peter A. E. M. Janssen
Affiliation:
Department of Mechanical Engineering, University of Rochester, Rochester, NY 14627 Permanent address: Department of Oceanography, Royal Netherlands Meteorological Institute, De Bilt, Holland.
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Abstract

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 σ 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 – 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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 133 , August 1983 , pp. 113 - 132
Copyright
© 1983 Cambridge University Press

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References

Alber, I. E. 1978 The effects of randomness on the stability of two-dimensional surface wavetrains Proc. R. Soc. Lond. A363, 545–546.Google Scholar
Alber, I. E. & Saffman, P. G. 1978 Stability of random nonlinear deep water waves with finite bandwidth spectra. TRW Defense and Space Systems Group Rep. no. 31326-6035-RU-00.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory J. Fluid Mech. 27, 417–430.Google Scholar
Case, K. M. 1959 Plasma oscillations Ann. Phys. (N. Y.) 7, 349.Google Scholar
Crawford, D. R., Saffman, P. G. & Yuen, H. C. 1980 Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves Wave Motion 2, 1–16.Google Scholar
Drazin, P. G. 1970 Kelvin-Helmholtz instability of finite amplitude J. Fluid Mech. 42, 321.Google Scholar
Fermi, E., Pasta, J. & Ulam, S. 1940 Studies of nonlinear problems. In Collected papers of Enrico Fermi, vol. 2, p. 978. University of Chicago Press, 1962.
Gelfand, I. M. & Shilov, G. E. 1964 Generalized Functions, vol. 1, Academic.
Hasegawa, A. 1975 Dynamics of an ensemble of plane waves in nonlinear dispersive media Phys. Fluids 18, 77–79.Google Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory J. Fluid Mech. 12, 481.Google Scholar
Hasselmann, K. 1963 On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems, wave-particle analogy, irreversibility J. Fluid Mech. 15, 273.Google Scholar
Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright, D. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruseman, P., Meerburg, A., MÜLLER, P., Olbers, D. J., Richter, K., Sell, W. & Walden, H. 1973 Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Deutsche Hydrogr. Z., Suppl. A(80), no. 12.Google Scholar
Infeld, E. 1981 Quantitative theory of the Fermi-Pasta-Ulam recurrence in the nonlinear Schrödinger equation Phys. Rev. Lett. 47, 717–718.Google Scholar
Janssen, P. A. E. M. 1981 Modulational instability and the Fermi-Pasta-Ulam recurrence Phys. Fluids 24, 23–26.Google Scholar
Janssen, P. A. E. M. & Rasmussen, J. 1981 Limit cycle behavior of the bump-on-tail instability Phys. Fluids 24, 268–273.Google Scholar
Kampen, N. G. Van 1955 On the theory of stationary waves in plasmas Physica 21, 949–963.Google Scholar
Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.Google Scholar
Landau, L. 1946 On the vibrations of the electronic plasma. J. Phys. (USSR) 10, 25–34.Google Scholar
Longuet-Higgins, M. S. 1976 On the nonlinear transfer of energy in the peak of a gravity-wave spectrum: a simplified model. Proc. R. Soc. Lond. A347, 311–328.Google Scholar
Nayfeh, A. H. & Saric, W. C. 1972 Nonlinear waves in a Kelvin-Helmholtz flow J. Fluid Mech. 55, 311.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1 J. Fluid Mech. 9, 193–217.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean, 2nd edn. Cambridge University Press.
Simon, A. 1968 Nonlinear transport in weakly ionized plasma Phys. Fluids 11, 1181.Google Scholar
Simon, A. & Rosenbluth, M. N. 1976 Single-mode saturation of the bump-on-tail instability: immobile ions. Phys. Fluids 19, 1567–1580.Google Scholar
Tappert, F. D. 1971 Derivation of the collisionless wave kinetic equation SIAM Rev. 13, 281–282.Google Scholar
Watson, K. M. & West, B. J. 1975 A transport equation description of nonlinear ocean surface wave interactions J. Fluid Mech. 70, 815–826.Google Scholar
Wigner, E. 1932 On the quantum correction for thermodynamic equilibrium Phys. Rev. 40, 749–759.Google Scholar
Willebrand, J. 1975 Transport in a nonlinear and inhomogeneous random gravity wave field J. Fluid Mech. 70, 113–126.Google Scholar
Yuen, H. C. & Ferguson, W. E. 1978 Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrödinger equation Phys. Fluids 21, 1275–1278.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid Zh. Prikl. Mekh. Tekh. Fiz. 9, 86–94. [English transl. in J. Appl. Mech. Tech. Phys. 9, 190–194.]Google Scholar