Hamiltonian derivation of the Charney-Hasegawa-Mima equation

When dissipative terms are dropped, all of the important models of plasma physics are described by partial differential equations that possess Hamiltonian form in terms of noncanonical Poisson brackets. For example, this is the case for ideal magnetohydrodynamics, the Vlasov–Maxwell equations, and other systems see Refs. 7–9 for review . Among these, there exist several reduced fluid models whose Hamiltonian structure has been derived a posteriori. These include the four-field model for tokamak dynamics of Hazeltine et al., models for collisionless magnetic reconnection derived and investigated by Schep et al., Kuvshinov et al., and Tassi et al.; and the recent gyrofluid model of Waelbroeck et al. The noncanonical Hamiltonian formulation has also been adopted to investigate the electron temperature gradient driven mode and convective-cell formation in plasma fluid systems. In addition to these fluid models, the Hamiltonian structure of kinetic and reduced kinetic equations has also been highlighted, for example, in guiding-center theory and gyrokinetics see Refs. 17–21 for review . This Hamiltonian form originates from the Hamiltonian and action principle forms of the basic electromagnetic interaction, i.e., the Hamiltonian form possessed by the equations that describe a system of charged particles coupled to Maxwell’s equations see, e.g., Ref. 9 for discussion . It is now well established that there exist numerous advantages of such a Hamiltonian formulation, among which are the identification of conserved quantities that are important for the verification of numerical codes , the study of stability, the use of techniques for Hamiltonian systems like averaging and perturbation theory, etc. Here we perform a perturbative derivation within the noncanonical Hamiltonian context, which means the Poisson bracket as well as the Hamiltonian must be expanded. In a nutshell, a Hamiltonian system is a system whose dynamics of any observable F depending on a finite or infinite number of variables can be written using a Hamiltonian scalar function H and a Poisson bracket · , · as F