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Exponential neutral stability of a floating ice layer. (English) Zbl 0909.73022
Summary: Linear stability of two-dimensional monochromatic waves, i.e., normal modes, in a homogeneous elastic ice layer of finite thickness and infinite horizontal extension, floating on the surface of a water layer of finite depth, is treated analytically. The water is assumed to be compressible but the viscous effects are neglected in the model. The treatment is an extension of the two-step analysis in L. Brevdo [Acta Mech. 117, 71-80 (1996; Zbl 0873.73021)] for a homogeneous waveguide overlaying a rigid half-space.
First, we apply an energy-type method and show that, for real wavenumbers \(k\), \(\omega^2\) is real, where \(\omega\) is a frequency. Further, to exclude purely imaginary frequencies for real \(k\), we use the dispersion relation function of the problem \(D (k, \omega)\) and show that for \(\omega= is\), \(s\in \mathbb{R}^+\), the equation \(D(k,is)=0\) does not have real roots in \(k\). Hence, due to the symmetry, all normal modes in this model are neutrally stable. This result on the one hand provides a theoretical support for the physical relevance of the model and on the other hand points to a possibility of resonant algebraically growing responses to localized harmonic in time perturbations.
It is also shown that an unstable vertically stratified ice layer is always absolutely unstable. Based on this result, a conjecture is made concerning a possible mechanism of spontaneous ice breaking as a consequence of emergence of absolute instability, which is caused by a weather induced appearance of an unstable ice stratification.

74J10 Bulk waves in solid mechanics
74G99 Equilibrium (steady-state) problems in solid mechanics
74H99 Dynamical problems in solid mechanics
86A05 Hydrology, hydrography, oceanography
86A40 Glaciology
35Q72 Other PDE from mechanics (MSC2000)
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