A Hamiltonian Dysthe equation for hydroelastic waves in a compressed ice sheet
Guyenne P. Kairzhan A. Sulem C.
6 January 2025Cambridge University Press
Journal of Fluid Mechanics
2025#1002
Nonlinear hydroelastic waves along a compressed ice sheet lying on top of a two-dimensional fluid of infinite depth are investigated. Based on a Hamiltonian formulation of this problem and by applying techniques from Hamiltonian perturbation theory, a Hamiltonian Dysthe equation is derived for the slowly varying envelope of modulated wavetrains. This derivation is further complicated here by the presence of cubic resonances for which a detailed analysis is given. A Birkhoff normal form transformation is introduced to eliminate non-resonant triads while accommodating resonant ones. It also provides a non-perturbative scheme to reconstruct the ice-sheet deformation from the wave envelope. Linear predictions on the modulational instability of Stokes waves in sea ice are established, and implications for the existence of solitary wave packets are discussed for a range of values of ice compression relative to ice bending. This Dysthe equation is solved numerically to test these predictions. Its numerical solutions are compared with direct simulations of the full Euler system, and very good agreement is observed.
elastic waves , Hamiltonian theory , surface gravity waves
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Department of Mathematical Sciences, University of Delaware, Newark, 19716, DE, United States
Department of Mathematics, Nazarbayev University, 010000, Kazakhstan
Department of Mathematics, University of Toronto, M5S2E4, ON, Canada
Department of Mathematical Sciences
Department of Mathematics
Department of Mathematics
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