Hamiltonian-Formulation of Low-Frequency, Nonlinear Plasma Dynamics

In this paper we present a set of equations that governs the linear and nonlinear evolution of plasma phenomena with frequencies below the ion cyclotron and the magneto-sonic and above the ion-acoustic frequencies. Finite electron mass and ion gyroradius effects are taken into account. The spatial scales of the phenomena may range from MI-ID scales down to the inertia electron skin depth. In a high-temperature plasma, this skin depth is smaller than the gyro-radius of a thermal ion. This set describes Alfven and drift vortices, magnetic islands and current sheets. These equations can be cast in (noncanonical) Hamiltonian form. It is shown that infinite sets of conserved quantities (Casimirs) exist that reduce to the Casimirs of 2-D reduced MHD in the appropriate limit. Sufficient conditions for stability are discussed on the basis of the second variation, at constant Casimirs, of the Hamiltonian functional.