Resonant destabilization of a floating homogeneous ice layer

Abstract.

We show that a homogeneous elastic ice layer of finite thickness and infinite horizontal extension floating on the surface of a homogeneous water layer of finite depth possesses a countable unbounded set of of resonant frequencies. The water is assumed to be compressible, the viscous effects are neglected in the model. Responses of this water-ice system to spatially localized harmonic in time perturbations with the resonant frequencies grow at least as $\sqrt{t}$ in the two-dimensional (2-D) case and at least as lnt in the three-dimensional (3-D) case, when time $t\to\infty.$ The analysis is based on treating the 3-D linear stability problem by applying the Laplace-Fourier transform and reducing the consideration to the 2-D case. The dispersion relation for the 2-D problem ${D}(k,\omega) = 0,$ obtained previously by Brevdo and Il'ichev [10], is treated analytically and also computed numerically. Here k is a wavenumber, and $\omega$ is a frequency. It is proved that the system ${D}(k,\omega) = 0, {D}_k(k,\omega) = 0$ possesses a countable unbounded set of roots $(k, \omega) = (0,\omega_n), n\in\Bbb Z$ with $\rm{Im}\ \omega_n = 0.$ Then the analysis of Brevdo [6], [7], [8], [9], which showed the existence of resonances in a homogeneous elastic waveguide, is applied to show that similar resonances exist in the present water-ice model. We propose a resonant mechanism for ice-breaking. It is based on destabilizing the floating ice layer by applying localized harmonic perturbations, with a moderate amplitude and at a resonant frequency.