Envelope Equations for Three-Dimensional Gravity and Flexural-Gravity Waves Based on a Hamiltonian Approach
A Hamiltonian formulation for three-dimensional nonlinear flexural-gravity waves propagating at the surface of an ideal fluid covered by a thin ice sheet is presented. This is accomplished by introducing the Dirichlet–Neumann operator which reduces the original Laplace problem to a lower-dimensional system involving quantities evaluated at the fluid-ice interface alone. The ice-sheet model is based on the special Cosserat theory for hyperelastic shells, which yields a conservative and nonlinear expression for the bending force. By applying a Hamiltonian perturbation approach suitable for such a formulation, weakly nonlinear envelope equations for small-amplitude waves are derived. The various steps of this formal derivation are discussed including the modulational Ansatz, canonical transformations and expansions of the Hamiltonian. In particular, the contributions from higher harmonics are examined. Both cases of finite and infinite depth are considered, and comparison with direct numerical simulations is shown.
This work was partially supported by the Simons Foundation through grant No. 246170. The author thanks Walter Craig, Emilian Părău and Catherine Sulem for fruitful discussions. Part of this work was carried out while the author was visiting the Isaac Newton Institute for Mathematical Sciences during the Theory of Water Waves program in the summer 2014.
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