Collisionless Tearing Instability Research Articles

Three‐Dimensional X‐line Spreading in Asymmetric Magnetic Reconnection

AbstractThree‐dimensional X‐line spreading is important for the coupling between global dynamics and local kinetic physics of magnetic reconnection. Using large‐scale 3‐D particle‐in‐cell simulations, we investigate the spreading of the X‐line out of the reconnection plane under a strong guide field in asymmetric reconnection. The X‐line spreading speed depends strongly on the equilibrium current sheet thickness. In a simulation with a thick, ion‐scale equilibrium current sheet (CS), the X‐line spreads at the ambient species drift speeds, which are significantly lower than the Alfvén speed based on the guide field (sub‐Alfvénic spreading). In simulations with a sub‐ion‐scale CS, the X‐line spreads at instead (Alfvénic spreading). An Alfvénic signal consistent with kinetic Alfvén waves develops and propagates, leading to CS thinning and extending, which ultimately causes reconnection onset. The continuous onset of reconnection along the propagation direction of the signal manifests as Alfvénic X‐line spreading. The strong dependence on the CS thickness of the spreading speeds and the orientation of the X‐line are consistent with the collisionless tearing instability. Our simulations indicate that when the collisionless tearing growth is sufficiently strong in a thinner CS such that , Alfvénic X‐line spreading can effectively take place. Our results compare favorably with a number of numerical simulations and recent magnetopause observations. An important implication of this work is that the magnetopause CS is typically too thick for the X‐line to spread at the Alfvén speed.

Gyro-kinetic theory and global simulations of the collisionless tearing instability: The impact of trapped particles through the magnetic field curvature

The linear instability of the tearing mode is analyzed using a gyrokinetic approach within a Hamiltonian formalism, where the interaction between particles and the tearing mode through the wave-particle resonance is retained. On the one hand, the curvature of the magnetic field is shown to play no role in the linear instability when only passing particles are present in the plasma. On the other hand, the presence of trapped particles leads to an overall increase in the growth rate. Gyrokinetic simulations using the state-of-the-art Gkw code confirm these findings and are further used to investigate the impact of the magnetic field curvature and the temperature gradient on tearing modes including the effect of trapped particles. Without the temperature gradient, wave-particle resonance with the trapped electrons tends to stabilize the tearing mode, while with the finite temperature gradient, the magnetic curvature tends to destabilize the tearing mode, suggesting an interchange mechanism. The balance of these two stabilizing/destabilizing effects leads to a threshold in the temperature gradient beyond which the magnetic curvature plays a destabilizing role. This opens the way for a deeper understanding and control of the tearing instability in fusion plasmas.

Collisionless kinetic theory of oblique tearing instabilities

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. We find that this stabilization is associated with the density-gradient-driven diamagnetic drift. The analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. A simple analytic estimate for the stability criterion is provided.

A two-fluid study of oblique tearing modes in a force-free current sheet

Kinetic simulations have demonstrated that three-dimensional reconnection in collisionless regimes proceeds through the formation and interaction of magnetic flux ropes, which are generated due to the growth of tearing instabilities at multiple resonance surfaces. Since kinetic simulations are intrinsically expensive, it is desirable to explore the feasibility of reduced two-fluid models to capture this complex evolution, particularly, in the strong guide field regime, where two-fluid models are better justified. With this goal in mind, this paper compares the evolution of the collisionless tearing instability in a force-free current sheet with a two-fluid model and fully kinetic simulations. Our results indicate that the most unstable modes are oblique for guide fields larger than the reconnecting field, in agreement with the kinetic results. The standard two-fluid tearing theory is extended to address the tearing instability at oblique angles. The resulting theory yields a flat oblique spectrum and underestimates the growth of oblique modes in a similar manner to kinetic theory relative to kinetic simulations.

On the structure of guide magnetic field in the inertia-driven magnetic reconnection with the presence of shear flow

The effect of equilibrium shear flow on the structure of out-of-plane magnetic field is analytically investigated in the two-fluid magnetohydrodynamic (MHD) regimes of the collisionless tearing instability, where the electron inertia breaks the frozen-in condition. Our scaling analysis reveals that the Alfvénic and sub-Alfvénic shear flows cannot significantly modify the linear regimes of applicability. In addition, we show that the structure of out-of-plane magnetic field can either be quadrupolar or non-quadrupolar in Hall-MHD regimes. In particular, both types of structures can dominate at β < 1 (β is the ratio of plasma kinetic pressure to the pressure in the magnetic field) depending on the value of the normalized ion inertial skin depth. This conclusion, however, is in contradiction to the claim presented by Rogers et al. [J. Geophys. Res. 108, A3 (2003)], which states that the quadrupolar structure cannot appear at β < 1. The reasons of this disagreement are discussed in our study.

On the role of a nonscalar electron pressure in the collisionless magnetic reconnection

Collisionless tearing instability in a sheared force-free magnetic field is considered in the framework of electron magnetohydrodynamics. A rigorous analytical analysis demonstrates that the bulk inertia of electrons is the dominant reconnection mechanism in a sufficiently low-β plasma, when the reconnection current sheet width exceeds the electron gyroradius. Otherwise a fluidlike approach fails, and fully kinetic treatment of the problem is required. A recently raised issue of the role of the electron gyroviscous cancellation in collisionless reconnection is also addressed.

Two-fluid regimes of the resistive and collisionless tearing instability

The two-fluid (electrons and ions) plasma description is used to investigate spontaneous magnetic reconnection (tearing instability) in a sheared force-free magnetic configuration. The study includes both collisional (resistive) and collisionless (electron inertia) reconnection mechanisms. Together with the Hall effect, this yields various instability regimes, which can be identified by the magnitude of the four nondimensional parameters: Lundquist number S⪢1, electron/ion mass ratio μ≡me∕mi⪡1, plasma β, and the scaled ion inertial skin-depth di=c∕(ωpil), with l being the shear length of the magnetic field. The role of the electron gyroviscosity in the collisionless case is also clarified.

Magnetic dissipation in a force-free plasma with a sheet-pinch configuration

This paper describes the study of a force-free plasma with an initial sheet-pinch configuration ∇×B=−αB, where α is a scalar constant, using both linear theory and 2-1/2 dimensional particle-in-cell simulations with doubly periodic boundary conditions. Previous studies have shown that this configuration is unstable to the collisionless tearing instability. In this work, the linear growth rate in the long wavelength limit is found to have an upper bound at 1/τA, where τA is the Alfvén traversal time through the region with sheared magnetic fields. The simulations show that the initial force-free state evolves by seeking a more “relaxed,” lower energy state within the constraints of the simulation geometry. Under the periodic boundary conditions, the amount of magnetic energy available for dissipation is determined mostly by the geometry. The reconnection region shows a multiscale structure, separated by ion and electron inertial lengths. Particle and flow dynamics at the reconnection regions have been analyzed. The reconnection rate is shown to be high, with an ion inflow speed of ∼0.2 Alfvén speed, and the timescale for reconnection is several Ωi−1. Most of the dissipated magnetic energy goes into the thermal energy of the particles.

Magnetic Field Energy Dissipation Driven by Relativistic Flows in Force-Free Collisionless Pair Plasmas

It is shown by using a 2-dimensional fully relativistic electromagnetic particle-in-cell (PIC) code that a force-free collisionless pair plasma which is stable against collisionless tearing instability becomes unstable when driven by relativistic plasma flows and the magnetic field energy can be strongly dissipated and converted to heat plasmas as well as high-energy particle production. The force-free current sheet becomes unstable against the firehose instability and eventually breaks up into current filaments which are associated with the magnetic reconnection. The energy spectrum in the high-energy region shows an exponential-type law. The interaction process between the force-free collisionless plasmas and the relativistic plasma flows is an important mechanism for understanding the effective magnetic energy conversion, filament structures and high-energy particle production in astrophysical plasmas.

The Role of the Drift Kink Mode in Destabilizing Thin Current Sheets

The drift kink mode in current sheets of the Harris type is driven by the relative cross-field streaming of the electrons and ions. The properties of this mode are investigated by means of electromagnetic particle simulations and a two-fluid stability analysis, first in a 2-D (y, z) geometry and then in a full 3D. For thin current sheets (ρi0/L ≤ 1, where ρi0 is the ion gyroradius in the lobe field and L is the current sheet half thickness) the fluid theory indicates that the most unstable mode has kyL ∼ (Mi/me)1/2/2 and that the associated real frequency extends up to several times the ion gyrofrequency Ωi0. The simulations indicate, however, that finite gyroradius effects limit the dominant growing modes to longer wavelengths kyL ∼ 1 and to frequencies ∼Ωi0. In 2D the current sheet kinks due to a bulk displacement of the plasma in z, and this displacement appears to be limited only by the simulation boundary. This transport of plasma across flux tubes reduces the electron compressibility associated with the finite Bz field, and, in 3D, allows the collisionless tearing instability to grow at a rate comparable to that for the simple 1-D neutral sheet. Implications of these results for the triggering of substorms are discussed.

Three-dimensional current sheet tearing in the Earth's magnetotail

Evidently, reconnection takes place in the Earth's magnetospheric tail. The question is, however, what disrupts the tail current sheet and how. A major hypothesis is that of a spontaneous collisionless tearing instability. Past theoretical investigations have revealed, however that the small residual magnetic field at the geomagnetic equator stabilizes the current sheet pretty much against collisionless tearing. We have now revised this problem focusing on the main stumbling block — the effect of electron compressional stabilization. In contrary to past considerations of two-dimensional sheet tearing, we have recalculated the main terms of the variational principle for a sheared tail current sheet against shorter wavelength perturbations (WKB approach). In contrast to previous investigations, however, we have investigated truly three-dimensional perturbations. As a result we have found conditions, for which the stabilizing influence of the electron compression becomes reduced. Our results imply that a localized in all three spatial dimension tearing of the current sheet in the Earth's magnetotail is much more probable rather than the quasi- two-dimensional ones, considered in the past.

Effect of electron dynamics on collisionless reconnection in two‐dimensional magnetotail equilibria

Analytic theory has yielded contradictory predictions regarding the stabilizing role of electron dynamics on the collisionless tearing instability in a field‐reversal region with finite normal magnetic field component Bz. Two‐dimensional (x, z) simulations in which both electrons and ions are treated as particles demonstrate that, for the case of thin current sheets (Ri/ λ ∼ 1, where Ri is the ion Larmor radius based on the lobe field and λ is the current sheet half thickness) and with weakly chaotic electron orbits, the ion tearing mode is indeed stabilized under the mild condition that kρe ≲ 1, where ρe is the electron Larmor radius in the Bz field.

Tearing instability of reconnecting current sheets in space plasmas

Magnetic reconnection provides an efficient conversion of the so-called ‘free magnetic energy’ to kinetic and thermal energies of cosmic plasmas, hard electromagnetic radiation, and accelerated particles. This phenomenon was found in laboratory and space, but it is especially well studied in the solar atmosphere where it manifests itself as flares and flare-like events. We review the works devoted to the tearing instability — the inalienable part of the reconnection process — in current sheets which have, inside of them, a transverse (perpendicular to the sheet ‘plain’) component of the magnetic field and a longitudinal (parallel to the electric current) component of the field. Such ‘non-neutral’ current sheets are well known as the energy sources for flare-like processes in the solar corona. In particular, quasi-steady high-temperature turbulent current sheets are the energy sources during the ‘main’ or ‘hot’ phase of solar flares. These sheets are stabilized with respect to the collisionless tearing instability by a small transverse component of magnetic fiel, normally existing in the reconnecting and reconnected magnetic fluxes. The collision tearing mode plays, however, an important and perhaps dominant role for non-neutral current sheets in solar flares. In the MHD approximation, the theory shows that the tearing instability can be completely stabilized by the transverse fieldB n if its value satisfies the conditionB n /B≫S−3/4B is the reconnecting component of the magnetic field just near the current sheet,S is the magnetic Reynolds number for the sheet. In this case, stable current sheets become sources of temporal spatial oscillations and usual MHD waves. The application of the theory to the solar atmosphere shows that the effect of the transverse field explains high stability of high-temperature turbulent current sheets in the solar corona. The stable current sheets can be sources of radiation in the radio band. If the sheet is destabilized (atB n /B≪S−3/4) the compressibility of plasma leads to the arizing of the tearing instability in a long wave region, in which for an incompressible plasma the instability is absent. When a longitudinal magnetic field exists in the current sheet, the compressibility-induces instability can be dumped by the longitudinal field. These effects are significant in destabilization of reconnecting current sheets in solar flares: in particular, the instability with respect to disturbances comparable with the width of the sheet is determined by the effect of compressibility.

The coalescence instability in collisionless plasmas

The process whereby magnetic island structures tend to merge into larger units is known as magnetic island coalescence. This coalescence process is investigated for the case of a collisionless plasma by means of two-dimensional particle simulations. The pairwise merging of a preformed island chain leads to a gentle redistribution of the density and current density profiles. The linear growth rates substantially exceed the magnetohydrodynamic (MHD) values for ρ/λ≥0.1, where ρ is the asymptotic ion gyroradius and λ is the current sheet half-thickness. The nonlinear evolution of the collisionless tearing instability in a long system is found to lead to a much more violent coalescence process resulting in the formation of a vacuum X region. This stage is characterized by enhanced growth rates for the long-wavelength modes that exceed the maximum linear tearing growth rate, super-Alfvénic flows away from the reconnection X line, substantial bulk heating in the parallel direction, and the formation of energetic particle tails in the direction of the inductive electric field.

The beta dependence of the collisionless tearing instability at the dayside magnetopause

The β dependence of the collisionless tearing instability at the dayside magnetopause current sheet is investigated analytically for various current sheet models, where β is the ratio of plasma pressure to magnetic pressure. In the present models the magnetosheath field (Bs) is separated from the magnetospheric field (Bm) by a current sheet with an antiparallel magnetic field component in the z direction and a common guiding magnetic field component (By) in the y direction. Four different current sheet models are considered. For a symmetric current sheet with equal magnetic field strength (|Bm| = |Bs|) and plasma density on its two sides and By = 0 (case A), it is found that the collisionless tearing instability has a strong β dependence. An increase in β leads to a significant decrease in the tearing growth rate. For an asymmetric current sheet with fixed |Bm| and βm on the magnetospheric side and By = 0 (case B) or By = const ≠ 0 (case C), it is found that the collisionless tearing growth rate is not very sensitive to the values of βs where βs is the ratio of plasma pressure to magnetic pressure in the magnetosheath. An increase of βs only slightly changes the tearing growth rate. For an asymmetric current sheet with fixed |Bm| and βm and a large, nonconstant By (case D), it is found that the collisionless tearing growth rate displays a strong βs dependence. The present study shows that the β dependence of the collisionless tearing instability at the dayside magnetopause is controlled by the magnetic field profile across the magnetopause current sheet.

Implicit particle simulation of electromagnetic plasma phenomena

A direct method for the implicit particle simulation of electromagnetic phenomena in magnetized, multi-dimensional plasmas is developed. The method is second-order accurate for ω Δt < 1, with ω a characteristic frequency and time step Δt. Direct time integration of the implicit equations with simplified space differencing allows the consistent inclusion of finite particle size. Decentered time differencing of the Lorentz force permits the simulation of strongly magnetized plasmas in the limit of zero perpendicular temperature. A Fourier-space iterative technique for solving the implicit field corrector equation, based on the separation of plasma responses perpendicular and parallel to the magnetic field and longitudinal and transverse to the wavevector, is described. Wave propagation properties in a uniform plasma are in excellent agreement with theoretical expectations. Applications to collisionless tearing and coalescence instabilities further demonstrate the usefulness of the algorithm.

Collisionless tearing instability of a bi-Maxwellian neutral sheet: An integrodifferential treatment with exact particle orbits

The integrodifferential equation describing the linear tearing instability in the bi-Maxwellian neutral sheet is solved without approximating the particle orbits or the eigenfunction ψ. Results of this calculation are presented. Comparison between the exact solution and the three-region approximation [Phys. Fluids 27, 1198 (1984)] motivates the piecewise-straight-line approximation, a simplification that allows faster solution of the integrodifferential equation, yet retains the important features of the exact solution.

Nonlinear 3‐D evolution of bounded kinetic Alfven waves due to shear flow and collisionless tearing instability

The three‐dimensional nonlinear simulation of the evolution of a kinetic Alfven wave is presented for simple boundary conditions that resemble those of a discrete auroral breakup arc. The model used is a variant of the reduced magnetohydrodynamic description that includes, in addition, the important dispersive effect of electron inertia. It is found that with a "generator" boundary condition corresponding to a current generator and a fully conducting "load" boundary, the wave evolves three dimensionally through a combination of shear flow and collisionless tearing mode instability.

Turbulent stabilization of collisionless tearing instabilities

Using oscillation centre formalism the electron kinetic response to high frequency fields is derived. Only the non-resonant interaction is renormalized. It is shown that this can lead to a reduction of the growth rates of m ≥ 2 collisionless tearing modes.

Simulation of the collisionless tearing instability in an anisotropic neutral sheet

A two‐dimensional magnetoinductive particle‐in‐cell code is used to compare the stability of isotropic and anisotropic current layers in a collisionless plasma. We find, in agreement with recent theories, that temperature anisotropy with T⊥ &gt; T∥ significantly enhances the instability. The simulation growth rates agree well with analytical predictions of growth rates in the linear phase. Nonlinear growth rates are significantly larger than linear growth rates and are apparently the result of coalescence instability. For a fixed system length the influence of coalescence on the growth rates increases with anisotropy. This is evidently because of a shift in the most unstable mode to shorter wavelength with increasing anisotropy. Thus anisotropic current layers tend to produce large amplitude small wavelength islands which rapidly merge.