Practical use of a Lie algebra structure in non-dissipative fluid and plasma models

Abstract

A class of Hamiltonian models is considered that both includes many important examples from fluid and plasma theory and admits an elementary and simple abstract formulation. The formalism is illustrated by explicit expressions for both incompressible and compressible ideal magnetohydrodynamics, including a Coriolis force term and, in the incompressible case, allowing for a sharp ideally conducting boundary. The small-amplitude expansion is studied for a general background state, and the abstract setting gives much control of the usually messy algebra. One result is a new general and surprisingly simple symmetric expression for the coupling strength in the resonant three-wave interaction process. It may provide a convenient starting point for wave coupling calculations and seems suitable for the use in conjunction with computer algebra when various concrete models are considered.