A Hamiltonian Dysthe equation for hydroelastic waves in a compressed ice sheet

TL;DR

This paper derives a Hamiltonian Dysthe equation for hydroelastic waves in compressed ice sheets, analyzes resonances, predicts modulational instability, and validates results through numerical simulations.

Contribution

It introduces a novel Hamiltonian Dysthe equation for hydroelastic waves in ice sheets, incorporating resonances and enabling reconstruction of ice deformation from wave envelopes.

Findings

Linear modulational instability predictions for sea ice waves.

Numerical solutions agree well with full Euler simulations.

Implications for solitary wavepackets in ice-covered waters.

Abstract

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 wavepackets 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 to direct simulations of the full Euler system, and very good agreement is observed.