Collisionless Tearing Mode 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.

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.

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.

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.

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.

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.

Fast linear growth of collisionless double tearing modes in a cylindrical plasma

Scaling laws of fast growth of collisionless double tearing modes in the linear phase are studied numerically by a reduced magnetohydrodynamics model in a cylindrical plasma. It is found that in the case ρs ≪ de, with de the electron inertial skin depth and ρs the ion sound gyroradius, the scaling of the linear growth rate changes gradually from to as the rational surface separation Δrs increases. In the case de ≪ ρs, on the other hand, the scaling shifts gradually from or to or as Δrs increases. In the small Δrs regime, furthermore, it is shown that for short wavelength unstable modes, the scaling on de and ρs as their poloidal mode number varies is similar to that as Δrs does. In addition, ρs is found to play an important role in reducing the scaling dependence on de. These numerical scaling laws are testified to be reasonable in comparison with previous analytical theories, based on the similarity of physical characteristics of the same Δ′ categories in tearing modes, where Δ′ is the linear instability parameter for tearing modes. Finally, the characteristics of the second unstable eigenmode with different mode numbers, under the influence of de and ρs, are obtained and analysed.

Is the Near-Earth Current Sheet Prior to Reconnection Unstable to Tearing Mode?

The tearing mode instability plays a key role in the triggering process of reconnection. The triggering collisionless tearing mode instability has been theoretically and numerically analyzed by many researchers. However, due to the difficulty in obtaining the observational wave number, it is still unknown whether the tearing mode instability can be excited in an actual plasma sheet prior to reconnection onset. Using the data from four Cluster satellites prior to a magnetospheric reconnection event on 13 September 2002, we utilized the wave telescope technique to obtain the wave number which corresponds to the peak of power spectral density. The wavelength is about 18RE and is consistent with previous theoretic and numerical results. After substituting the wave vector and other necessary parameters of the observed current sheet into the triggering condition of tearing mode instability, we find that the near-Earth current sheet prior to reconnection is unstable to tearing mode.

Collisionless instability of thin current sheets in the presence of sheared parallel flows

The first results of an ongoing investigation of the influence of sheared parallel flows on the onset of collisionless magnetic reconnection in thin current sheets are reported. In particular, an exact kinetic equilibrium that incorporates flow with a symmetric profile into the well-known Harris model is proposed and its linear stability is described. The complete linearized Vlasov–Maxwell system is solved using a numerical approach that introduces no approximations regarding the shape of the particle orbits or relative magnitude of the different components of the electromagnetic potentials. Thus accurate results are obtained in a difficult, but practically important limit where the characteristic length scales for the variation of the equilibrium magnetic field and flow are comparable to the ion kinetic length scales. In particular, the dispersion relation for an instability that produces magnetic reconnection is traced as a function of the flow speed V0 at the center of the sheet, starting with the collisionless tearing mode at V0=0. The effects of the sheared flow are shown to qualitatively depend on the thickness of the current sheet. In relatively thick sheets the characteristic features of the dispersion relation depend mostly on the value of the flow shear. In this regime there exist regions of the parameter space where the flow is destabilizing. However, the growth rate of the instability never significantly exceeds that of the collisionless tearing mode and the mode is always stabilized at high enough values of V0. When the sheet is thinner than the ion gyroradius in the asymptotic magnetic field, the flow shear introduces strong non-Maxwellian features into the equilibrium distribution function. In this regime, the flow is purely stabilizing for the equilibrium considered. The instability produces magnetic reconnection in all the parameter regimes considered. Finally, the results of the linear analysis are verified using large-scale particle-in-cell simulations.

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 thresholds sufficient for energy-Casimir stability are obtained. The Hamiltonian formulation is also 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.

Collisionless tearing modes in cylindrical geometry

Magnetic reconnection due to electron inertia is responsible for fast and violent phenomena in magnetized plasmas in which magnetic energy is converted into electron kinetic energy on very short timescales. To study the nonlinear evolution of a reconnecting mode, the collisionless two-fluid drift-Alfvén equations are applied to an equilibrium that consists of a z-independent, cylindrically symmetric current region, with jumps in the radial direction. The Lagrangian advective formulation of the two-fluid equations makes it possible to use the powerful contour dynamics (CD) method to study linear stability and calculate numerically the nonlinear evolution of a tearing unstable equilibrium using a two-fluid model. The analytical expressions obtained for the linear dispersion relation are in good agreement with the CD simulations.

Kinetic theory and simulation of collisionless tearing in bifurcated current sheets

Observations from the Earth’s geomagnetic tail have established that the current sheet is often bifurcated with two peaks in the current density. These so-called bifurcated current sheets have also been reported in a variety of simulations and often occur in conjunction with significant temperature anisotropy. In this work, a new self-consistent Vlasov equilibrium is developed that permits both the current profile and temperature anisotropy to be independently adjusted. The stability of these layers with respect to the collisionless tearing mode is examined using both standard analytic techniques and a formally exact treatment involving a numerical evaluation of the full orbit integral. The resulting linear growth rate and mode structure are verified with fully kinetic particle-in-cell simulations. These results demonstrate that a bifurcated current profile has a strong stabilizing influence on the tearing mode in comparison to centrally peaked layers with a similar thickness. In contrast, electron temperature anisotropy is strongly destabilizing in the limit Te⊥&amp;gt;Te∥ and strongly stabilizing when Te⊥&amp;lt;Te∥. The simplified analytic theory is reasonably accurate in capturing these dependencies for long-wavelength modes, but the short-wavelength regime generally requires the full numerical treatment to accurately compute the growth rate.

Comparison between resistive and collisionless double tearing modes for nearby resonant surfaces

The linear instability and nonlinear dynamics of collisional (resistive) and collisionless (due to electron inertia) double tearing modes (DTMs) are compared with the use of a reduced cylindrical model of a tokamak plasma. We focus on cases where two q = 2 resonant surfaces are located a small distance apart. It is found that regardless of the magnetic reconnection mechanism, resistivity or electron inertia, the fastest growing linear eigenmodes may have high poloidal mode numbers m ∼ 10. The spectrum of unstable modes tends to be broader in the collisionless case. In the nonlinear regime, it is shown that in both cases fast growing high-m DTMs lead to an annular collapse involving small magnetic island structures. In addition, collisionless DTMs exhibit multiple reconnection cycles due to reversibility of collisionless reconnection and strong E × B flows. Collisionless reconnection leads to a saturated stable state, while in the collisional case resistive decay keeps the system weakly dynamic by driving it back towards the unstable equilibrium maintained by a source term.