Neutral stability of, and resonances in, a vertically stratified floating ice layer

In this paper we treat analytically and numerically the linear stability of three-dimensional small monochromatic disturbances, i.e., normal modes, in a vertically stratified elastic ice layer of finite thickness and infinite horizontal extension floating on the surface of a vertically stratified water layer of finite depth, and study the responses of the water–ice system to spatially localized harmonic in time perturbations. The viscous effects are neglected in the model, but the influence of gravity is retained and the water is assumed to be compressible. The analytical part of the treatment is an extension of the analysis of Brevdo (Proc. Roy. Soc. London Ser. A 457 (2001) 1951–1971) of a vertically stratified model of the Earthʹs crust. The stability analysis is based on applying an energy-type method to the normal-mode stability problem. In the case when the effect of discontinuity on the ice–water interface of the base normal stresses in the horizontal directions is neglected, we prove that all the normal modes in the model are neutrally stable. When the stress discontinuity on the interface is not neglected, computations lead us to an assessment that in this case the model is neutrally stable as well. Furthermore, the recent analysis of Brevdo (Z. Angew. Math. Phys. 52 (2001) 397–420) proving the existence of a countable unbounded set of resonances in every homogeneous water–ice system is extended to the vertically stratified case. We show that, in the vertically stratified water–ice model, there also exist growing in time resonant responses to localized harmonic in time perturbations with certain – resonant – frequencies. Moreover, in the stratified case, the resonant frequencies are abundant as well, and the set of such frequencies is assessed to be countable and unbounded. This result gives further support to our hypothesis that the ice covers of water basins can be broken by applying a localized oscillatory forcing with a moderate amplitude at a resonant frequency.