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Wave modulation: the geometry, kinematics, and dynamics of surface-wave packets

Published online by Cambridge University Press:  19 August 2016

N. E. Pizzo  and
W. Kendall Melville
N. E. Pizzo*
Affiliation:
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0213, USA
W. Kendall Melville
Affiliation:
Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0213, USA
*
†Email address for correspondence: npizzo@ucsd.edu
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Abstract

We examine the geometry, kinematics, and dynamics of weakly nonlinear narrow-banded deep-water wave packets governed by the modified nonlinear Schrödinger equation (Dysthe, Proc. R. Soc. Lond. A., vol. 369, 1979, pp. 105–114; MNLSE). A new derivation of the spatial MNLSE, by a direct application of Whitham’s method, elucidates its variational structure. Using this formalism, we derive a set of conserved quantities and moment evolution equations. Next, by examining the MNLSE in the limit of vanishing linear dispersion, analytic solutions can be found. These solutions then serve as trial functions, which when substituted into the moment evolution equations form a closed set of equations, allowing for a qualitative and quantitative examination of the MNLSE without resorting to numerically solving the full equation. To examine the theory we consider initially symmetric, chirped and unchirped wave packets, chosen to induce wave focusing and steepening. By employing the ansatz for the trial function discussed above, we predict, a priori, the evolution of the packet. It is found that the speed of wave packets governed by the MNLSE depends on their amplitude, and in particular wave groups speed up as they focus. Next, we characterize the asymmetric growth of the wave envelope, and explain the steepening of the forward face of the initially symmetric wave packet. As the packet focuses, its variance decreases, as does the chirp of the signal. These theoretical results are then compared with the numerical predictions of the MNLSE, and agreement for small values of fetch is found. Finally, we discuss the results in the context of existing theoretical, numerical and laboratory studies.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Mathematical Foundations: Variational methods Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 803 , 25 September 2016 , pp. 292 - 312
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Copyright
© 2016 Cambridge University Press 

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References

Ablowitz, M. J. & Segur, H. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92 (04), 691–715.Google Scholar
Agrawal, G. P. 2007 Nonlinear Fiber Optics. Academic.Google Scholar
Akylas, T. 1989 Higher-order modulation effects on solitary wave envelopes in deep water. J. Fluid Mech. 198, 387–397.Google Scholar
Anderson, D. 1983 Variational approach to nonlinear pulse propagation in optical fibers. Phys. Rev. A 27 (6), 3135–3145.Google Scholar
Anderson, D. & Lisak, M. 1983 Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides. Phys. Rev. A 27 (3), 1393–1398.CrossRefGoogle Scholar
Banner, M. L. & Peirson, W. L. 2007 Wave breaking onset and strength for two-dimensional deep-water wave groups. J. Fluid Mech. 585 (1), 93–115.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27 (03), 417–430.Google Scholar
Chereskin, T. K. & Mollo-Christensen, E. 1985 Modulational development of nonlinear gravity-wave groups. J. Fluid Mech. 151, 337–365.Google Scholar
Chu, V. H. & Mei, C. C. 1970 On slowly-varying Stokes waves. J. Fluid Mech. 41 (04), 873–887.Google Scholar
Clamond, D., Francius, M., Grue, J. & Kharif, C. 2006 Long time interaction of envelope solitons and freak wave formations. Eur. J. Mech. (B/Fluids) 25 (5), 536–553.Google Scholar
Drazen, D. A. & Melville, W. K. 2009 Turbulence and mixing in unsteady breaking surface waves. J. Fluid Mech. 628, 85–119.Google Scholar
Drazen, D. A., Melville, W. K. & Lenain, L. 2008 Inertial scaling of dissipation in unsteady breaking waves. J. Fluid Mech. 611, 307–332.Google Scholar
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369 (1736), 105–114.Google Scholar
Dysthe, K. B., Trulsen, K., Krogstad, H. E. & Socquet-Juglard, H. 2003 Evolution of a narrow-band spectrum of random surface gravity waves. J. Fluid Mech. 478, 1–10.Google Scholar
Fedorov, A. V. & Melville, W. K. 1998 Nonlinear gravity–capillary waves with forcing and dissipation. J. Fluid Mech. 354, 1–42.Google Scholar
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond. A 289 (1361), 373–404.Google Scholar
Goldman, M. V. & Nicholson, D. R. 1978 Virial theory of direct Langmuir collapse. Phys. Rev. Lett. 41 (6), 406.Google Scholar
Gordon, J. P. 1986 Theory of the soliton self-frequency shift. Opt. Lett. 11 (10), 662–664.Google Scholar
Gramstad, O. & Trulsen, K. 2011 Hamiltonian form of the modified nonlinear Schrödinger equation for gravity waves on arbitrary depth. J. Fluid Mech. 670, 404–426.Google Scholar
Janssen, P. A. E. M. 1983 On a fourth-order envelope equation for deep-water waves. J. Fluid Mech. 126, 1–11.Google Scholar
Kit, E. & Shemer, L. 2002 Spatial versions of the Zakharov and Dysthe evolution equations for deep-water gravity waves. J. Fluid Mech. 450, 201–205.Google Scholar
Lighthill, M. J. 1965 Contributions to the theory of waves in non-linear dispersive systems. IMA J. Appl. Math. 1 (3), 269–306.Google Scholar
Lo, E. & Mei, C. C. 1985 A numerical study of water–wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395–416.Google Scholar
Lo, E. Y.1985 Long-time evolution of surface waves in coastal waters. Department of Civil Engineering, Massachusetts Institute of Technology.Google Scholar
Longuet-Higgins, M. S. 1974 Breaking waves in deep or shallow water. In Proceedings of the 10th Conf. on Naval Hydrodynamics, pp. 597–605. MIT Press.Google Scholar
Longuet-Higgins, M. S. 1995 Parasitic capillary waves: a direct calculation. J. Fluid Mech. 301, 79–107.Google Scholar
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27 (02), 395–397.Google Scholar
McIntyre, M. E. 1981 On the wave momentum myth. J. Fluid Mech. 106, 331–347.Google Scholar
Melville, W. K. 1983 Wave modulation and breakdown. J. Fluid Mech. 128, 489–506.Google Scholar
Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165–185.Google Scholar
Melville, W. K. & Fedorov, A. V. 2015 The equilibrium dynamics and statistics of gravity–capillary waves. J. Fluid Mech. 767, 449–466.Google Scholar
Miles, J. W. 1977 On Hamilton’s principle for surface waves. J. Fluid Mech. 83 (01), 153–158.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Pierson, W. Jr., Donelan, M. A. & Hui, W. H. 1992 Linear and nonlinear propagation of water wave groups. J. Geophys. Res. Oceans 97 (C4), 5607–5621.Google Scholar
Pizzo, N. E., Deike, L. & Melville, W. K. 2016 Current generation by deep-water wave breaking. J. Fluid Mech. (in press).Google Scholar
Pizzo, N. E. & Melville, W. K. 2013 Vortex generation by deep-water breaking waves. J. Fluid Mech. 734, 198–218.Google Scholar
Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Phil. Trans. R. Soc. Lond. A 331, 735–800.Google Scholar
Robinson, P. A. 1997 Nonlinear wave collapse and strong turbulence. Rev. Mod. Phys. 69 (2), 507.Google Scholar
Shemer, L., Kit, E. & Jiao, H. 2002 An experimental and numerical study of the spatial evolution of unidirectional nonlinear water–wave groups. Phys. Fluids 14 (10), 3380–3390.Google Scholar
Su, M.-Y., Bergin, M., Marler, P. & Myrick, R. 1982a Experiments on nonlinear instabilities and evolution of steep gravity-wave trains. J. Fluid Mech. 124, 45–72.Google Scholar
Su, M.-Y., Bergin, M., Marler, P. & Myrick, R. 1982b Experiments on nonlinear instabilities and evolution of steep gravity wave trains. J. Fluid Mech. 124, 45–72.Google Scholar
Sulem, C. & Sulem, P.-L. 1999 The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse, vol. 139. Springer.Google Scholar
Tao, T. 2006 Nonlinear Dispersive Equations: Local and Global Analysis, vol. 106. American Mathematical Society.Google Scholar
Titchmarsh, E. C. 1948 Introduction to the Theory of Fourier Integrals. Clarendon Press.Google Scholar
Trulsen, K. 1998 Crest pairing predicted by modulation theory. J. Geophys. Res. Oceans 103 (C2), 3143–3147.Google Scholar
Trulsen, K. 1999 Wave kinematics computed with the nonlinear Schrödinger method for deep water. J. Offshore Mech. Arctic Engng 121 (2), 126–130.Google Scholar
Trulsen, K. 2006 Weakly Nonlinear and Stochastic Properties of Ocean Wave Fields. Application to an Extreme Wave Event. Springer.Google Scholar
Trulsen, K. & Dysthe, K. B. 1997 Frequency downshift in three-dimensional wave trains in a deep basin. J. Fluid Mech. 352, 359–373.Google Scholar
Trulsen, K., Kliakhandler, I., Dysthe, K. B. & Velarde, M. G. 2000 On weakly nonlinear modulation of waves on deep water. Phys. Fluids 12 (10), 2432–2437.Google Scholar
Whitham, G. B. 1965 A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22 (02), 273–283.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22 (67), 229.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.Google Scholar