Stability of Periodic Travelling Flexural-Gravity Waves in Two Dimensions

TL;DR

This paper investigates the stability of periodic flexural-gravity waves under ice sheets by comparing linear and nonlinear ice models, deriving a nonlinear Schrödinger equation, and validating stability predictions through numerical methods.

Contribution

It introduces a reformulation of Euler's equations for flexural-gravity waves, derives a nonlinear Schrödinger equation for modulational instability, and compares stability regimes of different ice models.

Findings

Different models exhibit distinct stability regimes at high flexural rigidity.

Numerical methods confirm high-frequency instabilities are similar across models.

Asymptotic analysis aligns well with numerical stability computations.

Abstract

In this work, we solve the Euler's equations for periodic waves travelling under a sheet of ice using a reformulation introduced in Ablowitz et al. (2006). These waves are referred to as flexural-gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrodinger equation that describes the modulational instability of periodic travelling waves. We compare this asymptotic result with numerical computation of stability using the Fourier-Floquet-Hill method and show how well these agree. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are used to go beyond the modulational instability and show high frequency instabilities that are the same for both models for ice in the regime examined.