Stability of periodic traveling flexural‐gravity waves in two dimensions

Abstract

In this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. 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 Schrödinger equation that describes the modulational instability of periodic traveling 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 different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze 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.

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

“…Experimental observation of moving vehicles over a frozen body of water demonstrate that typically a wide range of wavelengths are observed while the amplitude of the oscillations remains small 46 . However, nonlinear effects, albeit weak, likely play a role in the wave evolution and various works demonstrate that nonlinearity has a marked effect on hydroelastic wave dynamics both for localized disturbances 47,48 and periodic waves 49,50 . Recently, the Whitham equation with quadratic nonlinearity was investigated in this context to capture weak nonlinearity and full linear wave dispersion 4 capturing qualitative features of solutions computed in the aforementioned studies.…”

Nonlinear hydroelastic waves on a linear shear current at finite depth

“…A Stokes wave solution to the vor-NLS can be written as Perturbing the solution, as in Trichtchenko et al. (2019), by a complex function of magnitude where takes the form results in the following condition for non-trivial solutions (Craik 1988) For , is complex and the Stokes wave solution is unstable. Since is arbitrary, the condition indicates instability for sufficiently small , and the associated growth rate of instability reads The maximum growth rate is attained when The range of unstable modulational wavenumbers is where .…”

“…Stability of two-dimensional hydroelastic periodic waves was investigated by Trichtchenko et al. (2019) via asymptotic analysis for modulational instability and linear spectral analysis using the Fourier–Floquet–Hill method. Besides, numerical computations were used to analyse high-frequency instabilities in addition to the modulational instability.…”

Progressive flexural–gravity waves with constant vorticity

“…Since then, nonlinear flexural-gravity waves with the Toland elastic model have attracted intensive attention. Of interest are the following works: Guyenne & Pȃrȃu (2012, 2014 searched for hydroelastic solitary waves for the full Euler equations using the boundary integral method and performed unsteady simulations by truncating the Dirichlet-Neumann operator in arbitrary depth; Gao & Vanden-Broeck (2014) investigated the elevation generalised solitary waves in finite depth; Gao, Wang & Vanden-Broeck (2016) studied the stability and dynamics of solitary waves for the fully nonlinear equations via a time-dependent conformal mapping technique; and Trichtchenko et al (2019) carried out the linear spectral analysis for periodic waves using the Fourier-Floquet-Hill method and compared the results with those obtained by a modulational instability analysis.…”