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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: [email protected]
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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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