L. Brevdo et A. Ilichev, EXPONENTIAL NEUTRAL STABILITY OF A FLOATING ICE LAYER, Zeitschrift fur angewandte Mathematik und Physik, 49(3), 1998, pp. 401-419
Linear stability of two-dimensional monochromatic waves, i. e., normal
modes, in a homogeneous elastic ice layer of finite thickness and inf
inite horizontal extension floating on the surface of a water layer of
finite depth is treated analytically. The water is assumed to be comp
ressible but the viscous effects are neglected in the model. The treat
ment is an extension of the two-step analysis in Brevdo [4] for a homo
geneous waveguide overlaying a rigid half space. First, we apply an en
ergy-type method and show that, for real wavenumbers k, omega(2) is re
al, where omega is a frequency. Fur ther, to exclude purely imaginary
Frequencies for real k, omega(2) use the dispersion relation function
of the problem D(k, omega) and show that for omega = is, s is an eleme
nt of R+, the equation D(k, is) = 0 does not have real roots in k. Hen
ce, 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 vert
ically stratified ice layer is always absolutely unstable. Based on th
is result, a conjecture is made concerning a possible mechanism of spo
ntaneous ice breaking as a consequence of emergence of absolute instab
ility, which is caused by a weather induced appearance of an unstable
ice stratification.