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

TL;DR

This paper addresses modulational dynamics of hydroelastic waves along a compressed ice sheet over deep water by formulating a Hamiltonian potential-flow model that couples ice bending and compression through a Cosserat plate representation. A Hamiltonian Dysthe equation for the envelope is derived using a third-order Birkhoff normal form to remove non-resonant cubic interactions while accounting for resonant triads, with a nonperturbative surface reconstruction built into the normal-form framework. The authors show how ice compression can shift the modulational stability regime from defocusing to focusing, potentially enabling envelope solitons, and validate the model by comparing Dysthe simulations to direct Euler simulations and to NLS approximations. The work provides an energy-conserving higher-order envelope model for hydroelastic waves and offers a robust reconstruction procedure, with implications for ice-cover dynamics and polar engineering applications. The results suggest that incorporating both bending and compression alters the focusing properties and enriches the dynamics beyond classical gravity-water-wave theory, while maintaining a Hamiltonian structure that facilitates accurate long-time predictions.

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.

Figures (11)

• Figure 1: Phase speed $c(k)$ (blue) and group speed $c_g(k)$ (red) as functions of $k$ for (a) $\mathcal{P} = 1$, (b) $\mathcal{P} = 2$, (c) $\mathcal{P} = 5$.
• Figure 2: Linear dispersion relation $\omega^2(k)$ as a function of $k$ for $\mathcal{P} = 0.1$ (blue), $\mathcal{P} = 1$ (red), $\mathcal{P} = 2$ (yellow), $\mathcal{P} = 5$ (green).
• Figure 3: Solution curve $\mathcal{C}^+$ (blue curve) for positive roots $(k_1, k_3)$ of \ref{['tilde-d13=0']} and its neighborhood $\mathcal{C}_\mu^+$ (gray area) for $\mathcal{P} = 1$.
• Figure 4: Location of points $(k_1, k_2)$ and $(k_4-k_2, k_2)$ with $k_4-k_2 >0$ in the case (a) $k_0 = 0.9$ and (b) $k_0=2$ relative to the neighborhood $\mathcal{C}_\mu$. Box 1 represents the set $\mathcal{B}_s (k_0)$ and Box 2 represents the set $\mathcal{B}_c (k_0)$.
• Figure 5: BF instability/stability criterion at $k_{\min}$ for the NLS equation as a function of $\mathcal{P}$.

Theorems & Definitions (19)

• Lemma 3.1
• proof
• Proposition 4.1
• proof
• Proposition 5.1
• proof
• Proposition 5.2
• proof
• Proposition 5.3
• proof