Trichtchenko, O; Milewski, P; Parau, E; Vanden-Broeck, J; (2019) Stability of periodic travelling flexural-gravity waves in two dimensions. Studies in Applied Mathematics , 142 (1) pp. 65-90. 10.1111/sapm.12233.

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Abstract

In this work, we solve the Euler’s equations for periodic waves travelling under a sheet of ice using a reformulation introduced in [1]. These waves are referred to as flexural-gravity waves. We compare and contrast two models for the e↵ect 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 Schr¨odinger equation that describes the modulational instability of periodic travelling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier-Floquet-Hill method to show they agree qualitatively. We show that di↵erent models have di↵erent stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyse high frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.

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