D. Grasso

Recent advances in collisionless magnetic reconnection

One of the recurring problems in magnetic reconnection is the identification of the appropriate generalized Ohms law. In weakly collisional plasmas with a strong magnetic guide field component, a fluid model may be adopted, where electron inertia and the electron pressure gradient play important roles. In the absence of collisions, electron inertia provides the mechanism for magnetic field-line breaking. Electron compressibility alters significantly the structure of the reconnection region and allows for faster reconnection rates, which are consistent with the fast relaxation times of sawtooth oscillations in tokamak plasmas. The Hall term may also become important when the guide field is weak. The very possibility of nonlinear, irreversible magnetic reconnection in the absence of dissipation is addressed. We show that in a collisionless plasma, magnetic islands can grow and reach a saturated state in a coarse-grained sense. Magnetic energy is transferred to kinetic energy in smaller and smaller spatial scale lengths through a phase mixing process. The same model is then applied to the interpretation of driven reconnection events in the vicinity of a magnetic X-line observed in the VTF experiment at MIT. The reconnection is driven by externally induced plasma flows in a background magnetic configuration that has a hyperbolic null in the reconnection plane and a magnetic guide field component perpendicular to that plane. In the limit where the guide field is strong, assuming the external drive to be sufficiently weak for a linear approximation to hold, a dynamic evolution of the system is obtained which does not reach a stationary state. The reconnection process develops in two phases: an initial phase, whose characteristic rate is a fraction of the Alfven frequency, and a later one, whose rate is determined by the electron collision frequency.

Hamiltonian magnetic reconnection

Magnetic reconnection in two dimensional (2D), collisionless, non-dissipative regimes is investigated analytically and numerically by means of a finite difference code in the nonlinear regime where the island size becomes macroscopic. The cross-shaped structure of the reconnection region, originally reported by Cafaro et al (1998 Phys. Rev. Lett. 80 20) is analysed as a function of the ratio between the ion sound Larmor radius and the inertial skin depth. This cross shape structure is found to survive in the presence of weak dissipation. Further insight on the quasi-explosive behaviour of the magnetic island width as a function of time and on the spatial structure of the perturbed current density is provided. We confirm that the amount of reconnected flux becomes of order unity on the time scale of the inverse linear growth rate. Results in the collisionless limit are interpreted on the basis of the Hamiltonian properties of the adopted collisionless, 2D, fluid model. Thus, collisionless reconnection is a fast, non-steady-state process, fundamentally different from 2D resistive magnetic reconnection, of which the Sweet-Parker model is the classic paradigm.

Hamiltonian formulation and analysis of a collisionless fluid reconnection model

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 thresholdssufficientforenergy-Casimirstabilityareobtained. TheHamiltonian formulationisalsousedtoobtainanexpressionforthecollisionlessconductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode. (Some figures in this article are in colour only in the electronic version)

Aspects of three-dimensional magnetic reconnection

The nonlinear behavior of reconnecting modes in three spatial dimensions (3D) is investigated, on the basis of a collisionless fluid model in slab geometry, assuming a strong constant guide field in one direction. Unstable modes in the so-called large Δ′ regime are considered. Single helicity modes, i.e., modes with the same orientation with respect to the guide field, depending on all three spatial coordinates correspond to “oblique” modes with, in general, mixed parity around the corresponding resonant magnetic surface, giving rise to a nonlinear drift of the magnetic island X point. The nonlinear coupling of initial perturbations with different helicities introduces additional helicities that evolve in time in agreement with quasilinear estimates, as long as their amplitudes remain relatively small. Magnetic field lines become stochastic when islands with different helicities are present. Basic questions such as the proper definition of the reconnection rate in 3D are addressed.

Extended theory of the Taylor problem in the plasmoid-unstable regime

A fundamental problem of forced magnetic reconnection has been solved taking into account the plasmoid instability of thin reconnecting current sheets. In this problem, the reconnection is driven by a small amplitude boundary perturbation in a tearing-stable slab plasma equilibrium. It is shown that the evolution of the magnetic reconnection process depends on the external source perturbation and the microscopic plasma parameters. Small perturbations lead to a slow nonlinear Rutherford evolution, whereas larger perturbations can lead to either a stable Sweet-Parker-like phase or a plasmoid phase. An expression for the threshold perturbation amplitude required to trigger the plasmoid phase is derived, as well as an analytical expression for the reconnection rate in the plasmoid-dominated regime. Visco-resistive magnetohydrodynamic simulations complement the analytical calculations. The plasmoid formation plays a crucial role in allowing fast reconnection in a magnetohydrodynamical plasma, and the presented results suggest that it may occur and have profound consequences even if the plasma is tearing-stable.

Gyro-induced acceleration of magnetic reconnection

The linear and nonlinear evolution of magnetic reconnection in collisionless high-temperature plasmas with a strong guide field is analyzed on the basis of a two-dimensional gyrofluid model. The linear growth rate of the reconnecting instability is compared to analytical calculations over the whole spectrum of linearly unstable wave numbers. In the strongly unstable regime (large Δ′), the nonlinear evolution of the reconnecting instability is found to undergo two distinctive acceleration phases separated by a stall phase in which the instantaneous growth rate decreases. The first acceleration phase is caused by the formation of strong electric fields close to the X-point due to ion gyration, while the second acceleration phase is driven by the development of an open Petschek-like configuration due to both ion and electron temperature effects. Furthermore, the maximum instantaneous growth rate is found to increase dramatically over its linear value for decreasing diffusion layers. This is a consequence of the fact that the peak instantaneous growth rate becomes weakly dependent on the microscopic plasma parameters if the diffusion region thickness is sufficiently smaller than the equilibrium magnetic field scale length. When this condition is satisfied, the peak reconnection rate asymptotes to a constant value.

Secondary instabilities in two-and three-dimensional magnetic reconnection in fusion relevant plasmas

The fast collisionless reconnection process typical of fusion relevant plasma regimes is analyzed with both two-dimensional and three-dimensional models. The vorticity and current density layers, which typically form in these regimes, are followed during all the phases of their dynamical evolution. Here, these structures are shown to be unstable in the cold electron case to secondary Kelvin-Helmholtz-like instabilities not only in the two-dimensional approximation but also in the full three-dimensional setting.

Ion Larmor radius effects in collisionless reconnection

The nonlinear evolution of collisionless magnetic field line reconnection is investigated numerically in plasma regimes where the effects of the electron and ion temperatures are important. These effects modify the structure of the current and vorticity layers that are formed during the onset of the reconnection instability. The results of investigations in a two-dimensional periodic configuration including ion Larmorradius effects to all orders are presented and compared with the results obtained in regimes with a large sound Larmor radius. It is found that, while the roles of the sound Larmor radius and the ion Larmor radius are inter-changeable as far as the nonlinear reconnection rate is concerned, the structure of the vorticity and current density layers is different in the two cases.

Nonlinear gyrofluid simulations of collisionless reconnection

The Hamiltonian gyrofluid model recently derived by Waelbroeck et al. [Phys. Plasmas 16, 032109 (2009)] is used to investigate nonlinear collisionless reconnection with a strong guide field by means of numerical simulations. Finite ion Larmor radius gives rise to a cascade of the electrostatic potential to scales below both the ion gyroradius and the electron skin depth. This cascade is similar to that observed previously for the density and current in models with cold ions. In addition to density cavities, the cascades create electron beams at scales below the ion gyroradius. The presence of finite ion temperature is seen to modify, inside the magnetic island, the distribution of the velocity fields that advect two Lagrangian invariants of the system. As a consequence, the fine structure in the electron density is confined to a layer surrounding the separatrix. Finite ion Larmor radius effects produce also a different partition between the electron thermal, potential, and kinetic energy, with respect to the cold-ion case. Other aspects of the dynamics such as the reconnection rate and the stability against Kelvin-Helmholtz modes are similar to simulations with finite electron compressibility but cold ions.

Magnetic islands and spontaneous generation of zonal flows

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