Invariants and Geometric Structures in Nonlinear Hamiltonian Magnetic Reconnection

Abstract

Collisionless magnetic reconnection in a two dimensional plasma is analyzed, using a two-fluid model where electron mass and pressure effects are important. Numerical simulations show the formation of current and vorticity layers along two branches crossing at the stagnation point of the plasma flow. These structures are interpreted on the basis of the Hamiltonian Casimirs (conserved fields) of the fluid plasma model. [S0031-9007(98)06155-9] PACS numbers: 52.35.Py, 47.65. + a, 52.65.Kj, 94.30.GmThe problem of magnetic reconnection in collisionless regimes was originally motivated by applications to space plasma processes, such as reconnection events occurring in the Earth magnetotail [1]. Renewed interest in this problem was prompted by the observation of fast relaxations in high temperature laboratory plasmas of thermonuclear interest. One well known example is the so called "sawtooth crash" of the central temperature of a tokamak plasma, which may occur on a time scale short compared to the average electron-ion collision time [2]. Recently it was shown [3,4] that electron inertia may account for the fast time scales observed in the experiments.The first aim of this Letter is to extend this analysis to finite temperature regimes, where electron pressure effects are important. We find that in the early nonlinear phase the growth of the magnetic island is even faster than in the cold electron limit of Ref. [4]. The fast collisionless evolution in this early nonlinear regime is a non-steady-state process, characterized by the formation of increasingly narrower microscales below the electron skin depth. The long term behavior is likely to require kinetic considerations outside the scope of the fluid model adopted in this paper.The second aim is the investigation of the role played by the Casimir invariants. Indeed, it can be shown that the adopted collisionless model admits a Hamiltonian structure [5,6]. While magnetic flux is reconnected in the course of plasma evolution, the conserved fields associated with the Casimirs preserve their initial topology. The nature of collisionless reconnection under these circumstances is entirely different from that of resistive reconnection.The model we consider is an extension of reduced magnetohydrodynamic [7] to a two-fluid description, where electron inertia, associated with the inertial skin depth, d e c͞v pe , and the electron pressure terms are retained in the generalized Ohm's law [8] (a more general model is the four field model of Ref.[9]). More specifically, we adopt an isothermal equation of state for the electrons and disregard diamagnetic effects associated with equilibrium pressure gradients, a valid approximation as long as the electron diamagnetic frequency is small compared with the characteristic growth rate of the reconnection process. On the other hand, we retain the divergence of the electron stress tensor (electron gyroviscosity) in the generalized Ohm's law. The parallel electron compressibility introduces the characteristic scale length, s p T e ͞m i ͞v c...

Fluid vs. kinetic magnetic reconnection with strong guide fields

“…The low-β (β 1), two-dimensional (∂ z =0, wherê z=∇z is the out-of-plane direction), two-field reconnection model [24][25][26]28,40,41 can be derived from the two-fluid magnetised plasma equations 42 assuming uniform temperatures with cold-ions (∇T i =∇T e =0 and T i T e for ion temperature T i ), strong-guide field (|B| B 0 where B=ẑ×∇ψ is the in-plane field, ψ is the flux, and B 0ẑ is the guide-field), an MHD ordering (the electric drift velocity is on the order of the ion thermal speed v E ∼v T i ) and small ion parallel flow (v i v T i ). As is typically done (see e.g.…”

Numerical investigation of a three-dimensional four field model for collisionless magnetic reconnection

“…In this context a description of the plasma behavior can be made considering a simpler two-field description [5] or a more sophisticated four-field description [6]. In particular, the two-field model in [5] has been extensively analyzed in the last decade both in two-dimensional and three-dimensional configurations [7,8,9,10]. In this model the evolution of the magnetic flux and plasma stream function is followed assuming that variations of the magnetic field and of the plasma velocity, along the direction parallel to the guide field, are negligible.…”

Additive Schwarz-based fully coupled implicit methods for resistive Hall magnetohydrodynamic problems

“…Furthermore, other quantities such as vorticity and the current density are expressed as U = ∇ 2 φ and j = −∇ 2 ψ using the stream functions. In this paper we work with a model MHD problem described in [4,18], which also corresponds to the "zero-β" case in [14]:…”