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On wave action and phase in the non-canonical Hamiltonian formulation

Vanneste, J. and Shepherd, T. G. (1999) On wave action and phase in the non-canonical Hamiltonian formulation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 455 (1981). pp. 3-21. ISSN 1364-5021

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To link to this article DOI: 10.1098/rspa.1999.0301

Abstract/Summary

The long time–evolution of disturbances to slowly–varying solutions of partial differential equations is subject to the adiabatic invariance of the wave action. Generally, this approximate conservation law is obtained under the assumption that the partial differential equations are derived from a variational principle or have a canonical Hamiltonian structure. Here, the wave action conservation is examined for equations that possess a non–canonical (Poisson) Hamiltonian structure. The linear evolution of disturbances in the form of slowly varying wavetrains is studied using a WKB expansion. The properties of the original Hamiltonian system strongly constrain the linear equations that are derived, and this is shown to lead to the adiabatic invariance of a wave action. The connection between this (approximate) invariance and the (exact) conservation laws of pseudo–energy and pseudomomentum that exist when the basic solution is exactly time and space independent is discussed. An evolution equation for the slowly varying phase of the wavetrain is also derived and related to Berry's phase.

Item Type:Article
Refereed:Yes
Divisions:No Reading authors. Back catalogue items
Faculty of Science > School of Mathematical and Physical Sciences > Department of Meteorology
ID Code:32853
Uncontrolled Keywords:Hamiltonian system; wave action; adiabatic invariant; WKB solution; slowly varying wave; Poisson structure
Publisher:Royal Society Publishing

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