Hamiltonian formulation and analysis of a collisionless fluid reconnection model

Abstract

The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented. The formulation is noncanonical with a corresponding Lie-Poisson bracket. The bracket is used to obtain new independent families of invariants, so-called Casimir invariants, three of which are directly related to Lagrangian invariants of the system. The Casimirs are used to obtain a variational principle for equilibrium equations that generalize the Grad-Shafranov equation to include flow. Dipole and homogeneous equilibria are constructed. The linear dynamics of the latter is treated in detail in a Hamiltonian context: canonically conjugate variables are obtained; the dispersion relation is analyzed and exact thresholds for spectral stability are obtained; the canonical transformation to normal form is described; an unambiguous definition of negative energy modes is given; and thresholds sufficient for energy-Casimir stability are obtained. The Hamiltonian formulation also is used to obtain an expression for the collisionless conductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode.

Role of singular layers in the plasma response to resonant magnetic perturbations

“…We describe the response of the plasma in the resonant layer by using the "fourfield" model of Hazeltine et al [68,69] We note that the the four-field model uses a finite Larmor radius (FLR) expansion for the ion dynamics (kρ i 1, where ρ i is the ion Larmor radius). In the low-collisionality regime of the DIII-D experiments, however, the resonant layers are narrower than the Larmor radius.…”

A compressible Hamiltonian electromagnetic gyrofluid model

“…To facilitate comparison to earlier models including the cold-ion version of the four-field model [31] and the incompressible (3-field) gyrofluid model, [25] we rewrite the dynamical equations as…”

“…Grasso et al [33] have shown that the mixing of such Lagrangian quantities by the convecting flows is an important ingredient in rapid collisionless reconnection. In this respect, it is worth noticing that a limit of this model [31], obtained by neglecting the ion gyroradius, possesses a Poisson bracket of a different type (the sum of direct and semidirect products), leading to only three Lagrangian invariants, two of which undergoing phase mixing.…”

On Hamiltonian and Action Principle Formulations of Plasma Dynamics

“…One of the early examples is reduced MHD [6], but many models for many purposes have been obtained over the years, e.g. [7,8,9,10,11,12,13,14,15,16]. Sometimes models like these arise from systematic asymptotic expansion, while in other cases they have been proposed in a more ad hoc manner.…”