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In this paper, we study the particle dynamics of nonlinear flexural-gravity waves propagating in a finite water depth, which is the interaction between ice sheets and water flows. The nonlinear deformation of a floating elastic sheet is modeled by the Cosserat shell theory. The theoretical analysis is performed by a uniform asymptotic perturbation expansion. A third-order explicit parametric solution of particle trajectories in Lagrangian coordinates is presented. Taking the time average of particle motion, the mass transport velocity and the Lagrangian surface setup are also derived. Numerical simulations are computed. The influences of flexural rigidity on the water particle orbits and the mass transport velocity of nonlinear flexural-gravity waves are first discussed.

International audience; In this Brief Research Report, we have shown, within the framework of the vor-NLS equation in deep water, that the Peregrine breather travelling at the free surface of a shear current of slowly varying vorticity may transform into grey solitons. Contribution to the field Based on numerical simulations of the nonlinear Schrödinger equation in infinite depth we conjecture the existence of grey solitons at the sea surface of slowly varying vortical current. This occurrence of dark solitons is due to the transformation of the Peregrine breather traveling at the free surface of the varying vortical current. Ethics statements Studies involving animal subjects Generated Statement: No animal studies are presented in this manuscript. Studies involving human subjects Generated Statement: No human studies are presented in this manuscript. Inclusion of identifiable human data Generated Statement: No potentially identifiable human images or data is presented in this study.

Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p = 0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor's series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.