Periodic Patterns, Linear Instability, Symplectic Structure and MeanFlow Dynamics for ThreeDimensional Surface Waves
Abstract
Space and time periodic waves at the twodimensional 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 wavegenerated 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 finitedepth fluid. With the additional structure of the equations, it is possible to give a direct formulation of the linear stability problem for threedimensional 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 BenneyRoskes, Hayes, DaveyStewartson and DjordjevicRedekopp obtained using modulation equations. Generalizations of the instability theory to study all periodic patterns on the ocean surface are also discussed.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 March 1996
 Bibcode:
 1996RSPSA.354..533B