Stabilization of uni-directional water wave trains over an uneven bottom
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ORIGINAL PAPER
Stabilization of uni-directional water wave trains over an uneven bottom Andrea Armaroli · Alexis Gomel · Amin Chabchoub · Maura Brunetti · Jérôme Kasparian
Received: 25 February 2020 / Accepted: 9 July 2020 © The Author(s) 2020
Abstract We study the evolution of nonlinear surface gravity water wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrödinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework is the theory of autoresonance in nonlinear systems and leads to a quasi-frozen state. Three main stages are identified: amplification from small sideband amplitudes, separatrix crossing and adiabatic conversion to orbits oscillating around an elliptic fixed point. Analytical estimates on the three stages are obtained from the low-dimensional approxiA. Armaroli · A. Gomel · M. Brunetti · J. Kasparian (B) Institute for Environmental Sciences, Université de Genève, Boulevard Carl-Vogt 66, 1205 Genève, Switzerland e-mail: [email protected] A. Armaroli · A. Gomel · M. Brunetti · J. Kasparian GAP, Université de Genève, Chemin de Pinchat 22, 1227 Carouge, Switzerland A. Chabchoub Centre for Wind, Waves and Water, School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia A. Chabchoub Marine Studies Institute, The University of Sydney, Sydney, NSW 2006, Australia
mation and validated by NLSE simulations. Our result will contribute to understand the dynamical stabilization of nonlinear wave packets and the persistence of large undulatory events in hydrodynamics and other nonlinear dispersive media. Keywords Surface gravity waves · Nonlinear waves · Modulational instability · Nonlinear Schrödinger equation · Classical mechanics · Separatrix crossing
List of symbols 1
ω Angular frequency in m− 2 κ Wavenumber-depth product (ad.) ε Wave steepness σ Depth correction factor Group velocity cg β Dispersion coefficient γ Nonlinear coefficient γ˜ Effective nonlinear coefficient Shoaling coefficient μ0 U (ξ, τ ) Complex envelope of surface elevation V (ξ, τ ) Shoaling-corrected U Carrier amplitude V0 α Three-wave parameter Ω Modulation detuning Cut-off MI detuning ΩC Peak MI detuning ΩM ψ Relative sideband phase η Conversion rate to sidebands
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(ψ˜ i , η˜ i ) Fixed points of the three-wave system H (ξ ) Hamiltonian function of the three-wave system H (X ) Hamiltonian function of the three-wave system (w.r.t. the reduced variable X ) Hmin Value of the Hamiltonian function at centers Slope of α in the linear stage Δα i Slope of α in the intermediate stage Δα t Slope of α in the adiabatic stage Δα f Crossing into MI sideband X∗ Start of a
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