Tsunami propagation in an ice-covered sea

Tsunami propagation in an ice-covered sea

 

Yakovlev V.V., Tkachenko V.O., Bondar V.V., Goncharenko T.B.

Institute of Hydromechanics, of  National Academy of Sciences of Ukraine

 

When a tsunami wave propagates into a sea area covered with solid ice, the part of the wave that has passed under the ice cover will be affected by the elastic properties of the ice sheet. These properties radically change the nature of the tsunami wave propagation.

A long-wave nonlinear dispersion model describing the propagation of tsunami flexural-gravity waves in a continuous ice sheet floating on the sea surface is constructed by expanding the initial three-dimensional problem of hydroelastic oscillations of the system "elastic plate – layer of ideal incompressible fluid of variable depth" in a small parameter. The model takes into account the effects of nonlinear dispersion of fluid, as well as inertia, elasticity and geometrically nonlinear deflection of the plate. Based on the obtained equations, a hierarchical sequence of simpler models is constructed, generalizing the equations of Peregrine, Boussinesque and Korteweg-de Vries, known from the theory of surface waves, to the case of flexural-gravity waves. In the particular case of the generalized Korteweg-de Vries equation, which describes the propagation of tsunami waves, exact solutions are constructed and analyzed, describing the propagation of solitons and cnoidal waves in the sea covered with solid ice. It is shown that flexural-gravity tsunami waves have some mirror properties compared to tsunami waves on water. In relation to the soliton, this means that without changing the shape, a depression, not a hump, as in the case of a tsunami wave on water, propagates, and the speed of its propagation decreases with increasing amplitude, not increases. In addition, the characteristics of flexural-gravity tsunami waves are determined by the amplitude and dispersion of the flexural rigidity of the plate and do not depend on the dispersion of water and the inertial properties of the ice cover.

For the generalized Korteweg-de Vries equation, to which the model of tsunami wave propagation in the sea covered with solid ice is reduced, the regions of variation of the physical parameters of the problem are identified, where different types of soliton-like solutions of this equation can exist. The nature of the eigenvalues for different ratios of the physical parameters of the problem is investigated and the region of variation of the parameters is determined, in which stationary solutions of the classical solitary wave type can take place.