Periodic Patterns, Linear Instability, Symplectic Structure and Mean-Flow Dynamics for Three-Dimensional Surface Waves

Space and time periodic waves at the two-dimensional surface of an irrotational inviscid fluid of finite depth are considered. The governing equations are shown to have a new formulation as a generalized Hamiltonian system on a multisymplectic structure where there is a distinct symplectic operator corresponding to each unbounded space direction and time. The wave-generated mean flow in this framework has an interesting characterization as drift along a group orbit. The theory has interesting connections with, and generalizations of, the concepts of action, action flux, pseudofrequency and pseudowavenumber of the Whitham theory. The multisymplectic structure and novel characterization of mean flow lead to a new constrained variational principle for all space and time periodic patterns on the surface of a finite-depth fluid. With the additional structure of the equations, it is possible to give a direct formulation of the linear stability problem for three-dimensional travelling waves. The linear instability theory is valid for waves of arbitrary amplitude. For weakly nonlinear waves the linear instability criterion is shown to agree exactly with the previous results of Benney-Roskes, Hayes, Davey-Stewartson and Djordjevic-Redekopp obtained using modulation equations. Generalizations of the instability theory to study all periodic patterns on the ocean surface are also discussed.