Collisionless Tearing Research Articles

Impact of electron temperature anisotropy on the collisionless tearing mode instability in the presence of a strong guide field

We derive and analyze a dispersion relation for the growth rate of collisionless tearing modes, driven by electron inertia and accounting for equilibrium electron temperature anisotropy in a strong guide field regime. For this purpose, a new gyrofluid model is derived and subsequently simplified to make the derivation of the dispersion relation treatable analytically. The main simplifying assumptions consist in assuming cold ions, neglecting electron finite Larmor radius effects, decoupling ion gyrocenter fluctuations, and considering β⊥e≪1, with β⊥e indicating the ratio between the perpendicular electron thermal pressure and the magnetic pressure exerted by the guide field. This simplified version of the gyrofluid model is shown to possess a noncanonical Hamiltonian structure. The dispersion relation is obtained by applying the theory of asymptotic matching and does not predict an enhancement of the growth rate as the ratio Θe between perpendicular and parallel equilibrium electron temperatures increases. This indicates a significant difference with respect to the case of absent or moderate guide field. For an equilibrium magnetic shear length of the order of the perpendicular sonic Larmor radius and at a fixed β⊥e, we obtain that the tearing mode in the strong guide field regime gets actually weakly damped, as Θe increases. In the isotropic limit Θe=1, the dispersion relation reduces to a previously known formula. The analytical predictions are tested against numerical simulations, showing a very good quantitative agreement. We also provide a detailed discussion of the range of validity of the derived dispersion relation and of the compatibility among the different adopted assumptions.

Tearing mode analysis in electron magnetohydrodynamics with pressure gradient

A two-dimensional three-component (2D/3C) electron magnetohydrodynamic (EMHD) model is implemented to investigate the linear behavior of collisionless tearing modes in slab geometry. Owing to nonuniformity of thermal pressure and plasma density, the electron diamagnetic drift and Biermann battery effects are involved. The linear structures, growth rate, and real frequency are analyzed with a thin current sheet in the electron inertia scale. The ratio of the electron current to the total current in equilibrium can notably promote the growth of the tearing mode in EMHD. More numerical results then show that the effect of the pressure gradient on the tearing mode is dependent on the plasma beta, stabilizing the mode in a low enough beta limit but destabilizing it with the higher beta. The frequency of the mode caused by the pressure gradient is found to be increasing with it. The Biermann battery effect slightly stabilizes the tearing mode in low beta plasma but is indicated to be significant in much higher beta conditions.

The physics of a small-scale tearing mode in collisionless slab plasmas

Microtearing modes have been widely reported as a tearing parity electron temperature gradient driven plasma instability, which leads to fine scale tearing of the magnetic flux surfaces thereby resulting in reconnection of magnetic field lines and formation of magnetic islands. In slab geometry it has previously been shown that the drive mechanism requires a finite collision frequency. However, we find in linear gyrokinetic simulations that a collisionless fine-scale tearing parity instability exists even at low and zero collision frequency. Detailed studies reveal that these slab modes are also driven by electron temperature gradient but are sensitive to electron finite Larmor radius effects, and have a radial wavenumber much smaller than the binormal wavenumber, which is comparable to the ion Larmor radius. Furthermore, they exist even in the electrostatic limit and electromagnetic effects actually have a stabilising influence on this collisionless tearing mode. An analytic model shows that this collisionless small scale tearing mode is consistent with a tearing parity slab electron temperature gradient (ETG) mode, which can be more unstable than the twisting parity ETG mode that is often studied. This small-scale tearing parity mode can lead to magnetic islands, which, in turn, can influence turbulent transport in magnetised plasmas.

Spontaneous Onset of Collisionless Magnetic Reconnection on an Electron Scale

Abstract Using particle-in-cell simulations, we investigate the onset of magnetic reconnection from a quiescent Harris current sheet in collisionless plasmas. After the current sheet is destabilized by the collisionless tearing mode instability, it proceeds to onset of reconnection, which manifests spontaneous thinning of current sheet and pileup of upstream magnetic flux. Once the current sheet thins to a critical thickness, about two electron inertial lengths, reconnection begins to grow explosively in this electron current sheet. This study shows that the spontaneous onset of collisionless magnetic reconnection is controlled by electron kinetics.

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.

Interchange destabilization of collisionless tearing modes by temperature gradient

Using a fluid theory, the stability of collisionless tearing modes in plasmas is analyzed in the presence of an inhomogeneous magnetic field, electron temperature and density gradients. It is shown that small scale modes, characterized by a negative stability parameter (Δ′<0), can be driven unstable due to a combination of the magnetic field and electron temperature gradients. The destabilization mechanism is identified as of the interchange type similar to that for toroidal electron temperature gradient modes.

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 conservative scheme of drift kinetic electrons for gyrokinetic simulation of kinetic-MHD processes in toroidal plasmas

A conservative scheme of drift kinetic electrons for gyrokinetic simulations of kinetic-magnetohydrodynamic processes in toroidal plasmas has been formulated and verified. Both vector potential and electron perturbed distribution function are decomposed into adiabatic part with analytic solution and non-adiabatic part solved numerically. The adiabatic parallel electric field is solved directly from the electron adiabatic response, resulting in a high degree of accuracy. The consistency between electrostatic potential and parallel vector potential is enforced by using the electron continuity equation. Since particles are only used to calculate the non-adiabatic response, which is used to calculate the non-adiabatic vector potential through Ohm's law, the conservative scheme minimizes the electron particle noise and mitigates the cancellation problem. Linear dispersion relations of the kinetic Alfvén wave and the collisionless tearing mode in cylindrical geometry have been verified in gyrokinetic toroidal code simulations, which show that the perpendicular grid size can be larger than the electron collisionless skin depth when the mode wavelength is longer than the electron skin depth.

Structure and computation of two-dimensional incompressible extended MHD

A comprehensive study of the extended magnetohydrodynamic model obtained from the two-fluid theory for electrons and ions with the enforcement of quasineutrality is given. Starting from the Hamiltonian structure of the fully three-dimensional theory, a Hamiltonian two-dimensional incompressible four-field model is derived. In this way, the energy conservation along with four families of Casimir invariants is naturally obtained. The construction facilitates various limits leading to the Hamiltonian forms of Hall, inertial, and ideal MHD, with their conserved energies and Casimir invariants. Basic linear theory of the four-field model is treated, and the growth rate for collisionless reconnection is obtained. Results from nonlinear simulations of collisionless tearing are presented and interpreted using, in particular, normal fields, a product of the Hamiltonian theory that gives rise to simplified equations of motion.

Verification of gyrokinetic particle simulation of current-driven instability in fusion plasmas. III. Collisionless tearing mode

A finite-mass electron fluid model for low frequency electromagnetic fluctuations, particularly the collisionless tearing mode, has been implemented in the gyrokinetic toroidal code. Using this fluid model, linear properties of the collisionless tearing mode have been verified. Simulations verify that the linear growth rate of the single collisionless tearing mode is proportional to De2, where De is the electron skin depth. On the other hand, the growth rate of a double tearing mode is proportional to De in the parameter regime of fusion plasmas.

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.

Explosive magnetic reconnection caused by an X-shaped current-vortex layer in a collisionless plasma

A mechanism for explosive magnetic reconnection is investigated by analyzing the nonlinear evolution of a collisionless tearing mode in a two-fluid model that includes the effects of electron inertia and temperature. These effects cooperatively enable a fast reconnection by forming an X-shaped current-vortex layer centered at the reconnection point. A high-resolution simulation of this model for an unprecedentedly small electron skin depth de and ion-sound gyroradius ρs, satisfying de=ρs, shows an explosive tendency for nonlinear growth of the tearing mode, where it is newly found that the explosive widening of the X-shaped layer occurs locally around the reconnection point with the length of the X shape being shorter than the domain length and the wavelength of the linear tearing mode. The reason for the onset of this locally enhanced reconnection is explained theoretically by developing a novel nonlinear and nonequilibrium inner solution that models the local X-shaped layer, and then matching it to an outer solution that is approximated by a linear tearing eigenmode with a shorter wavelength than the domain length. This theoretical model proves that the local reconnection can release the magnetic energy more efficiently than the global one and the estimated scaling of the explosive growth rate agrees well with the simulation results.

The linear tearing instability in three dimensional, toroidal gyro-kinetic simulations

Linear gyro-kinetic simulations of the classical tearing mode in three-dimensional toroidal geometry were performed using the global gyro-kinetic turbulence code, GKW. The results were benchmarked against a cylindrical ideal MHD and analytical theory calculations. The stability, growth rate, and frequency of the mode were investigated by varying the current profile, collisionality, and the pressure gradients. Both collisionless and semi-collisional tearing modes were found with a smooth transition between the two. A residual, finite, rotation frequency of the mode even in the absence of a pressure gradient is observed, which is attributed to toroidal finite Larmor-radius effects. When a pressure gradient is present at low collisionality, the mode rotates at the expected electron diamagnetic frequency. However, the island rotation reverses direction at high collisionality. The growth rate is found to follow a η1∕7 scaling with collisional resistivity in the semi-collisional regime, closely following the semi-collisional scaling found by Fitzpatrick. The stability of the mode closely follows the stability analysis as performed by Hastie et al. using the same current and safety factor profiles but for cylindrical geometry, however, here a modification due to toroidal coupling and pressure effects is seen.

Numerical comparison between a gyrofluid and gyrokinetic model investigating collisionless magnetic reconnection

The first detailed comparison between gyrokinetic and gyrofluid simulations of collisionless magnetic reconnection has been carried out. Both the linear and nonlinear evolution of the collisionless tearing mode have been analyzed. In the linear regime, we have found a good agreement between the two approaches over the whole spectrum of linearly unstable wave numbers, both in the drift kinetic limit and for finite ion temperature. Nonlinearly, focusing on the small-Δ′ regime, with Δ′ indicating the standard tearing stability parameter, we have compared relevant observables such as the evolution and saturation of the island width, as well as the island oscillation frequency in the saturated phase. The results are basically the same, with small discrepancies only in the value of the saturated island width for moderately high values of Δ′. Therefore, in the regimes investigated here, the gyrofluid approach can describe the collisionless reconnection process as well as the more complete gyrokinetic model.

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.

PROTON TEMPERATURE ANISOTROPY AND MAGNETIC RECONNECTION IN THE SOLAR WIND: EFFECTS OF KINETIC INSTABILITIES ON CURRENT SHEET STABILITY

We investigate the role of kinetic instabilities driven by a proton anisotropy on the onset of magnetic reconnection by means of 2-D hybrid simulations. The collisionless tearing of a current sheet is studied in the presence of a proton temperature anisotropy in the surrounding plasma. Our results confirm that anisotropic protons within the current sheet region can significantly enhance/stabilize the tearing instability of the current. Moreover, fluctuations associated to linear instabilities excited by large proton temperature anisotropies can significantly influence the stability of the plasma and perturb the current sheets, triggering the tearing instability. We find that such a complex coupling leads to a faster tearing evolution in a regime with larger perpendicular temperature when an ion-cyclotron instability is generated by the anisotropic proton distribution functions. On the contrary, in the presence of the opposite anisotropy, fire hose fluctuations excited by the unstable background protons with larger parallel temperature are not able to efficiently destabilize the current sheets, which remain stable for a long time after fire hose saturation. We discuss possible influences of this novel coupling on the solar wind and heliospheric plasma dynamics.

Gyrokinetic simulations of collisionless tearing modes

We investigate the effects of temperature and density gradients on collisionless magnetic reconnection by means of PIC-δf simulations with standard gyrokinetic equations using gyrokinetic ions and drift kinetic electrons. In a slab model ion and electron temperature gradients act stabilising on the tearing mode instability in the cases that have been considered. An additional ion and electron density gradient completely stabilises the mode below a critical η (ratio of temperature gradient to density gradient). The critical η becomes larger with decreasing equilibrium current.Nonlinear single-mode simulations show that the island width saturates at the level predicted by [1], if the linear instability growth rate is sufficiently small. Including ion and electron temperature and density gradients in nonlinear calculations the saturated island width vanishes below a critical η.

Investigation of tearing instability using GeFi particle simulation model

The gyrokinetic (GK) electron and fully kinetic ion (GeFi) simulation model of Lin et al. [Plasmas Phys. Controlled Fusion 53, 054013 (2011)] has been thoroughly benchmarked and validated for a two-dimensional (2D) Harris current sheet with a finite guide field. First, a gyrokinetic eigenmode theory for the collisionless tearing mode in the small Larmor radius limit is presented. The linear eigenmode structure and growth rate of the tearing mode obtained from the GeFi simulation are benchmarked against those from the GK eigenmode analysis in the limit of L≫ρi>ρe, where L is the current sheet half-width, ρi is ion Larmor radius, and ρe is electron Larmor radius. Second, to valid the GeFi model, both the linear and nonlinear tearing instabilities obtained from the GeFi simulations are compared with the Darwin particle-in-cell (PIC) simulation. The validation of the GeFi model for laboratory and space plasmas is also discussed. Meanwhile, the GeFi simulation is carried out to investigate both the linear and nonlinear tearing instabilities for cases with a broad range of L and guide magnetic field BG. It is found that in a wide current sheet with L > 4.5ρeK, the nonlinear saturation level of the island half-width is ws ≃ 3ρeK, where ρeK = ρeB0/Bx0, B0 is the strength of the asymptotic magnetic field, and Bx0 is the antiparallel field. On the other hand, in a thin current sheet with L < 2.5ρeK, ws ≃ 2.2 L. In addition, a high frequency electrostatic drift mode is found to coexist with the tearing mode.

An improved gyrokinetic electron and fully kinetic ion particle simulation scheme: benchmark with a linear tearing mode

An improved gyrokinetic electron and fully kinetic ion (GeFi) particle simulation scheme is presented for the investigation of linear collisionless tearing mode instability in a two-dimensional Harris current sheet under a finite guide field BG and a realistic ion-to-electron mass ratio mi/me. Due to the removal of the rapid electron cyclotron motion while retaining the finite electron Larmor radii, wave–particle interaction, and off-diagonal components of the electron pressure tensor, the GeFi model can be used to investigate the physics of magnetic reconnection with a realistic mi/me in a large-scale current sheet, which in general possesses wave modes ranging from Alfvén waves to lower hybrid/whistler waves, with wave frequency ω < Ωe, where Ωe is the electron gyrofrequency. As a necessary step of utilizing the code for magnetic reconnection, the linearized GeFi scheme is benchmarked by comparing the simulation results using a δf method against direct numerical solutions of the tearing-instability eigenmode equations, as well as those obtained analytically via asymptotic matching.