Harris-type Current Sheet Research Articles

Current Sheet as an Optimal Synchrotron Maser on a Radio Pulsar

Using a relativistic plasma with an isotropic monoenergetic distribution of electrons andpositrons as an example, we show that in the maser regime the maximum possible amplification ofsynchrotron radiation at a distance of one wavelength is achieved in a medium where the magnetic energydensity is of the order of the particle energy density. This ratio of the energy densities corresponds to a(Harris-type) current sheet. We have obtained an electron Lorentz factor of 350 and a magnetic fieldstrength of 10 kG in the maser radio emission region for the Crab pulsar. Our estimate suggests thatthe optical and coherent radio emissions of the object originate from one synchrotron source in the form ofa current sheet. The diameter of the source must exceed the light-cylinder radius approximately by a factorof 6 for the maser wave field to interact with particles in the linear regime, in particular, to keep its phasevelocity higher than the speed of light in a vacuum—a necessary condition for the synchrotron instability.

Magnetic Reconnection and Hot Spot Formation in Black Hole Accretion Disks

Abstract Hot spots, or plasmoids, which form due to magnetic reconnection in current sheets, are conjectured to power frequent X-ray and near-infrared flares from Sgr A*, the black hole in the center of our Galaxy. It is unclear how, where, and when current sheets form in black hole accretion disks. We perform axisymmetric general-relativistic resistive magnetohydrodynamics simulations to model reconnection and plasmoid formation in a range of accretion flows. Current sheets and plasmoids are ubiquitous features that form regardless of the initial magnetic field in the disk, the magnetization in the quasisteady-state phase of accretion, and the spin of the black hole. Within 10 Schwarzschild radii from the event horizon, we observe plasmoids forming, after which they can merge, grow to macroscopic scales of the order of a few Schwarzschild radii, and are ultimately advected along the jet’s sheath or into the disk. Large plasmoids are energized to relativistic temperatures via reconnection and contribute to the jet’s limb brightening. We find that only hot spots forming in magnetically arrested disks can potentially explain the energetics of Sgr A* flares. The flare period is determined by the reconnection rate, which we find to be between and in all cases, consistent with studies of reconnection in isolated Harris-type current sheets. We quantify magnetic dissipation and nonideal electric fields, which can efficiently inject nonthermal particles. We show that explicit resistivity allows for converged numerical solutions, such that the electromagnetic energy evolution and dissipation become independent of the grid scale for the extreme resolutions considered here.

Effect of Ionization and Recombination on the Evolution of the Harris-type Current Sheet in Partially Ionized Plasmas

Abstract Two-dimensional magnetohydrodynamics (MHD) simulations, treating plasma and neutral populations (hereafter, neutrals) as two separate components of the magneto-fluid, are performed in order to investigate the effect of ionization and recombination (or I/R) on the time evolution of the Harris-type current sheet in partially ionized plasmas. Our MHD simulations, including the effect of ambipolar diffusion (arising due to ion-neutral interactions) along with the I/R, show that the current sheet thinning occurs due to the diffusion of neutral particles from the current sheet. In addition to ambipolar diffusion, frictional heating also appears and affects the evolution of the current sheet. In a current sheet that is formed in a partially ionized plasma, the neutral population tries to spread outward and the plasma population tries to converge toward the center of the current sheet, and the overall process is influenced by the I/R. One of the important feature that is captured in our 2D simulations is that the escape of neutrals from the current sheet is sometimes suppressed due to the increase in ionization rate at the center of the current sheet, for the case of collisional I/R. As long as the ionization degree is kept low inside the current sheet, the current sheet thinning and elongation takes place and the current sheet becomes unstable due to the tearing-mode and plasmoid formation. The ion-neutral interactions coupled with I/R and the dynamics of the magnetic reconnection play an important role in plasmoid-mediated reconnection, therefore, the present study on the current sheet thinning and plasmoid formation could serve as a key for understanding bursty and intermittent plasma ejections observed in the solar chromosphere.

Analysis of fast turbulent reconnection with self-consistent determination of turbulence timescale

We present results of a Reynolds-averaged turbulence model simulation on the problem of magnetic reconnection. In the model, in addition to the mean density, momentum, magnetic field, and energy equations, the evolution equations of the turbulent cross-helicity W, turbulent energy K, and its dissipation rate ε are simultaneously solved to calculate the rate of magnetic reconnection for a Harris-type current sheet. In contrast to previous studies based on algebraic modeling, the turbulence time scale is self-determined by the nonlinear evolutions of K and ε, their ratio being a time scale. We compare the reconnection rate produced by our mean-field model to the resistive nonturbulent MHD rate. To test whether different regimes of reconnection are produced, we vary the initial strength of turbulent energy and study the effect on the amount of magnetic flux reconnected in time. We obtain that the time scale of turbulence self-adjusts to always produce fast reconnection.

Why does Steady-State Magnetic Reconnection have a Maximum Local Rate of Order 0.1?

Simulations suggest collisionless steady-state magnetic reconnection of Harris-type current sheets proceeds with a rate of order 0.1, independent of dissipation mechanism. We argue this long-standing puzzle is a result of constraints at the magnetohydrodynamic (MHD) scale. We predict the reconnection rate as a function of the opening angle made by the upstream magnetic fields, finding a maximum reconnection rate close to 0.2. The predictions compare favorably to particle-in-cell simulations of relativistic electron-positron and nonrelativistic electron-proton reconnection. The fact that simulated reconnection rates are close to the predicted maximum suggests reconnection proceeds near the most efficient state allowed at the MHD scale. The rate near the maximum is relatively insensitive to the opening angle, potentially explaining why reconnection has a similar fast rate in differing models.

Slow shock and rotational discontinuity in MHD and Hall MHD models with anisotropic pressure

AbstractPressure anisotropy may modify the characteristics of magnetohydrodynamic (MHD) waves, in particular, the slow mode wave and the corresponding shocks and discontinuities. In this study the formation of slow shocks (SSs) in anisotropic plasmas is examined by solving the gyrotropic MHD and Hall MHD equations numerically for one‐dimensional Riemann problem. The MHD shocks and discontinuities are generated by imposing a finite normal magnetic field on the Harris type current sheet with a guide magnetic By component. It is shown that anomalous SSs moving faster than the intermediate wave or with positive density‐magnetic field correlation may be generated in gyrotropic MHD and Hall MHD models. Moreover, for some parameter values SSs may exhibit upstream wave trains with right‐handed polarization in contrast with the earlier prediction that SSs shall possess downstream left‐hand polarized wave trains based on the isotropic Hall MHD theory. For the cases of By ≠ 0, SSs with increased density and decreased magnetic field followed by noncoplanar intermediate mode or rotational discontinuity (RD)‐like structures similar to the compound SS‐RD structures observed in space plasma environments may possibly form in symmetric and asymmetric current layers. The Walén relation of these anomalous RDs without the correction of pressure anisotropy may significantly be violated.

Sub-grid-scale description of turbulent magnetic reconnection in magnetohydrodynamics

Magnetic reconnection requires, at least locally, a non-ideal plasma response. In collisionless space and astrophysical plasmas, turbulence could transport energy from large to small scales where binary particle collisions are rare. We have investigated the influence of small scale magnetohydrodynamics (MHD) turbulence on the reconnection rate in the framework of a compressible MHD approach including sub-grid-scale (SGS) turbulence. For this sake, we considered Harris-type and force-free current sheets with finite guide magnetic fields directed out of the reconnection plane. The goal is to find out whether unresolved by conventional simulations MHD turbulence can enhance the reconnection process in high-Reynolds-number astrophysical plasmas. Together with the MHD equations, we solve evolution equations for the SGS energy and cross-helicity due to turbulence according to a Reynolds-averaged turbulence model. The SGS turbulence is self-generated and -sustained through the inhomogeneities of the mean fields. By this way, the feedback of the unresolved turbulence into the MHD reconnection process is taken into account. It is shown that the turbulence controls the regimes of reconnection by its characteristic timescale τt. The dependence on resistivity was investigated for large-Reynolds-number plasmas for Harris-type as well as force-free current sheets with guide field. We found that magnetic reconnection depends on the relation between the molecular and apparent effective turbulent resistivity. We found that the turbulence timescale τt decides whether fast reconnection takes place or whether the stored energy is just diffused away to small scale turbulence. If the amount of energy transferred from large to small scales is enhanced, fast reconnection can take place. Energy spectra allowed us to characterize the different regimes of reconnection. It was found that reconnection is even faster for larger Reynolds numbers controlled by the molecular resistivity η, as long as the initial level of turbulence is not too large. This implies that turbulence plays an important role to reach the limit of fast reconnection in large Reynolds number plasmas even for smaller amounts of turbulence.

Instabilities of collisionless current sheets revisited: The role of anisotropic heating

In this work, we investigate the influence of the anisotropic heating on the spontaneous instability and evolution of thin Harris-type collisionless current sheets, embedded in antiparallel magnetic fields. In particular, we explore the influence of the macroparticle shape-function using a 2D version of the PIC code ACRONYM. We also investigate the role of the numerical collisionality due to the finite number of macroparticles in PIC codes. It is shown that it is appropriate to choose higher order shape functions of the macroparticles compared to a larger number of macroparticles per cell. This allows to estimate better the anisotropic electron heating due to the collisions of macroparticles in a PIC code. Temperature anisotropies can stabilize the tearing mode instability and trigger additional current sheet instabilities. We found a good agreement between the analytically derived threshold for the stabilization of the anisotropic tearing mode and other instabilities, either spontaneously developing or initially triggered ones. Numerical effects causing anisotropic heating at electron time scales become especially important for higher mass ratios (above mi/me=180). If numerical effects are carefully taken into account, one can recover the theoretical estimated linear growth rates of the tearing instability of thin isotropic collisionless current sheets, also for higher mass ratios.

COMBINING PARTICLE ACCELERATION AND CORONAL HEATING VIA DATA-CONSTRAINED CALCULATIONS OF NANOFLARES IN CORONAL LOOPS

We model nanoflare heating of extrapolated active-region coronal loops via the acceleration of electrons and protons in Harris-type current sheets. The kinetic energy of the accelerated particles is estimated using semi-analytical and test-particle-tracing approaches. Vector magnetograms and photospheric Doppler velocity maps of NOAA active region 09114, recorded by the Imaging Vector Magnetograph (IVM), were used for this analysis. A current-free field extrapolation of the active-region corona was first constructed. The corresponding Poynting fluxes at the footpoints of 5000 extrapolated coronal loops were then calculated. Assuming that reconnecting current sheets develop along these loops, we utilized previous results to estimate the kinetic-energy gain of the accelerated particles and we related this energy to nanoflare heating and macroscopic loop characteristics. Kinetic energies of 0.1 to 8 keV (for electrons) and 0.3 to 470 keV (for protons) were found to cause heating rates ranging from $10^{-6}$ to 1 $\mathrm{erg\, s^{-1} cm^{-3}}$. Hydrodynamic simulations show that such heating rates can sustain plasma in coronal conditions inside the loops and generate plasma thermal distributions which are consistent with active region observations. We concluded the analysis by computing the form of X-ray spectra generated by the accelerated electrons using the thick target approach that were found to be in agreement with observed X-ray spectra, thus supporting the plausibility of our nanoflare-heating scenario.

General formulation for magnetohydrodynamic wave propagation, fire-hose, and mirror instabilities in Harris-type current sheets

Harris-type current sheets with the magnetic field model of B⇀=Bx(z)x̂+By(z)ŷ have many important applications to space, astrophysical, and laboratory plasmas for which the temperature or pressure usually exhibits the gyrotropic form of p↔=p∥b̂b̂+p⊥(I↔−b̂b̂). Here, p∥ and p⊥ are, respectively, to be the pressure component along and perpendicular to the local magnetic field, b̂=B→/B. This study presents the general formulation for magnetohydrodynamic (MHD) wave propagation, fire-hose, and mirror instabilities in general Harris-type current sheets. The wave equations are expressed in terms of the four MHD characteristic speeds of fast, intermediate, slow, and cusp waves, and in the local (k∥,k⊥,z) coordinates. Here, k∥ and k⊥ are, respectively, to be the wave vector along and perpendicular to the local magnetic field. The parameter regimes for the existence of discrete and resonant modes are identified, which may become unstable at the local fire-hose and mirror instability thresholds. Numerical solutions for discrete eigenmodes are shown for stable and unstable cases. The results have important implications for the anomalous heating and stability of thin current sheets.

Magnetic reconnection in an anisotropic plasma: Observation and theory

[1] Space plasma usually exhibits temperature or pressure anisotropy with two distinct components, p∥ and p⊥, denoting the pressure parallel and perpendicular to the magnetic field, respectively. This paper presents the first result of two-dimensional (∂/∂y = 0) magnetic reconnection with single X-line in anisotropic plasma based on gyrotropic magnetohydrodynamic (MHD) simulations. The field-line reconnection is initiated by imposing a localized resistivity on the Harris-type current sheet with a guide magnetic field By. It is shown that the observational feature of a slow shock (SS) followed by a contact discontinuity (CD) near the central sheet, across which the magnetic field is decreased and the pressure anisotropy is changed from p∥ > p⊥ to p∥ ≈ p⊥, as identified by Hoshino et al. (2000) in the Geotail data, may be reproduced in our simulations. For the initial equilibrium with relatively large pressure anisotropy of p∥ ≫ p⊥, the steady reconnection layer may consist of SS followed by rotational discontinuity (RD) with both upstream and downstream states of SS being super-Alfvenic which is similar to the anomalous SS-RD compound structure found in the solar wind and magnetosphere. While for the cases of p⊥ > p∥, the reconnection layer may consist of RD followed by SS and the reconnection rate is larger as compared to the cases of p∥ ≥ p⊥.

Statistical visualization of the Earth's magnetotail based on Geotail data and the implied substorm model

Abstract. We investigated the temporal and spatial development of the near-Earth magnetotail during substorms based on multi-dimensional superposed-epoch analysis of Geotail data. The start time of the auroral break-up (t=0) of each substorm was determined from auroral data obtained by the Polar and IMAGE spacecraft. The key parameters derived from the plasma, magnetic-field, and electric-field data from Geotail were sorted by their meridional X(GSM)–Z(proxy) coordinates. The results show that the Poynting flux toward the plasma-sheet center starts at least 10 min before the substorm onset, and is further enhanced at X~−12 RE (Earth radii) around 4 min before the onset. Simultaneously, large-amplitude fluctuations occurred, and earthward flows in the central plasma sheet between X~−11 RE and X~−19 RE and a duskward flow around X=−10 RE were enhanced. The total pressure starts to decrease around X=−16 RE about 4 min before the onset of the substorm. After the substorm onset, a notable dipolarization is observed and tailward flows commence, characterised by southward magnetic fields in the form of a plasmoid. We confirm various observable-parameter variations based on or predicted by the relevant substorm models; however, none of these can explain our results perfectly. Therefore, we propose a catapult (slingshot) current-sheet relaxation model, in which an earthward convective flow produced by catapult current-sheet relaxation and a converted duskward flow near the Earth are enhanced through flow braking around 4 min before the substorm onset. These flows induce a ballooning instability or other instabilities, causing the observed current disruption. The formation of the magnetic neutral line is a natural consequence of the present model, because the relaxation of a highly stretched catapult current-sheet produces a very thin current at its tailward edge being surrounded by intense earthward and tailward magnetic fields which were formerly the off-equatorial lobe magnetic fields. This location is the boundary between a highly stressed catapult current sheet and a Harris-type current sheet characterized by little stress. In addition, the flows induced around the boundary toward the current-sheet center may enhance the formation of the magnetic neutral line and the efficiency of magnetic reconnection. After magnetic reconnection is induced, it plays a significant role in driving the substorm.

Characteristics of magnetohydrodynamic waves in Harris-type current sheet with guide magnetic fieldBy

[1] An overview is presented of the characteristics of hydromagnetic waves propagating in Harris-type current sheet with magnetic field of Bx(z) and an embedded guide By component based on the ideal magnetohydrodynamic (MHD) model. The wave equations are expressed in the (k∥, k⊥, z) coordinates, and singularities are identified for general cases of ky ≠ 0 and By ≠ 0, where k∥ and k⊥ are the components of the wave vector parallel and perpendicular to the local magnetic field, respectively, and the inhomogeneity is in the z direction. The presence of By may lead to the rotation of the magnetic field relative to the wave vector so that a singular layer with k⊥ = 0 surrounded by a neighboring region of k⊥ ≠ 0 may exist. This result is in contrast with the case of By = 0 for which Alfven or field-line resonance associated with the mathematical singularity tends to occur for k⊥ ≠ 0 and ω = k∥CA, where CA is the Alfven speed. The possibility for MHD waves with frequency of ω = k∥CA to propagate through the inhomogeneous layer with By ≠ 0 is analyzed, and numerical solutions for discrete eigenmodes with various free parameter values are presented.

Particle Interactions with Single or Multiple 3D Solar Reconnecting Current Sheets

The acceleration of charged particles (electrons and protons) in flaring solar active regions is analyzed using numerical experiments. The acceleration is modeled as a stochastic process taking place by the interaction of the particles with local magnetic reconnection sites via multiple steps. Two types of local reconnecting topologies are studied: the Harris-type and the X-point. A formula for the maximum kinetic energy gain in a Harris-type current sheet, found in a previous work of ours, fits well the numerical data for a single step of the process. A generalization is then given approximating the kinetic energy gain through an X-point. In the case of the multiple step process, in both topologies the kinetic energy distribution of the particles is found to acquire a practically invariant form after a small number of steps. This tendency is interpreted theoretically. Other characteristics of the acceleration process are given, such as the mean acceleration time and the pitch angle distributions of the particles.

Reconnection rates in driven magnetic reconnection

Using resistive magnetohydrodynamic simulations, we investigate the influence of various parameters on the reconnection rate in two scenarios of magnetic reconnection. The first scenario consists of the “Newton Challenge” problem [Birn et al., Geophys. Res. Lett. 32, L06105 (2005)]. In this scenario, reconnection is initiated in a plane Harris-type current sheet by temporally limited, spatially varying, inflow of magnetic flux. The second scenario consists of the well-studied island coalescence problem. This scenario starts from an equilibrium containing periodic magnetic islands with parallel current filaments. Due to the attraction between parallel currents, pairs of islands may move toward each other, forming a current sheet in between. This leads to reconnection and ultimately the merging of islands. In either scenario, magnetic reconnection may be considered as being driven by external or internal forcing. Consistent with that interpretation we find that in either case the maximum reconnection rate (electric field) depends approximately linearly on the maximum driving electric field, when other parameters remain unchanged. However, this can be understood mostly from the change of characteristic background parameters; particularly, the increase of the magnetic field strength in the inflow region due to the added magnetic flux. This interpretation is consistent with the result that the maximum of the reconnection electric field is assumed significantly later (tens of Alfvén times) than the maximum driving and typically does not match the instantaneous driving electric field. Furthermore, the reconnection rate also depends on the resistivity and the time scale of the driving.

Entropy conservation in simulations of magnetic reconnection

Entropy and mass conservation are investigated for the dynamic field evolution associated with fast magnetic reconnection, based on the “Newton Challenge” problem [Birn et al., Geophys. Res. Lett. 32, L06105 (2005)]. In this problem, the formation of a thin current sheet and magnetic reconnection are initiated in a plane Harris-type current sheet by temporally limited, spatially varying, inflow of magnetic flux. Using resistive magnetohydrodynamic (MHD) and particle-in-cell (PIC) simulations, specifically the entropy and mass integrated along the magnetic flux tubes are compared between the simulations. In the MHD simulation these should be exactly conserved quantities, when slippage and Ohmic dissipation are negligible. It is shown that there is very good agreement between the conservation of these quantities in the two simulation approaches, despite the effects of dissipation, provided that the resistivity in the MHD simulation is strongly localized. This demonstrates that dissipation is highly localized in the PIC simulation also, and that heat flux across magnetic flux tubes has negligible effect as well, so that the entropy increase on a full flux tube remains small even during reconnection. The mass conservation also implies that the frozen-in flux condition of ideal MHD is a good integral approximation outside the reconnection site. This result lends support for using the entropy-conserving MHD approach not only before and after reconnection but even as a constraint connecting the two phases.

Electromagnetic full particle code with adaptive mesh refinement technique: Application to the current sheet evolution

We have developed a new two and a half dimensional electromagnetic particle code with adaptive mesh refinement (AMR) technique in an effort to give a self-consistent description of the dynamic change of the plasma sheet in association with magnetic reconnection, which includes multi-scale processes from the electron scale to the magnetohydrodynamic scale. The AMR technique subdivides and removes cells dynamically in accordance with a refinement criterion and it is quite effective to achieve high-resolution simulations of phenomena that locally include micro-scale processes. Since the number of particles per cell decreases in the subdivided cells and a numerical noise increases, we subdivide not only cells but also particles therein and control the number of particles per cell. Our code is checked against several well-known processes such as the Landau damping of the Langmuir waves and we show that the AMR technique and particle splitting algorithm are successfully applied to the conventional particle codes. We have also examined the nonlinear evolution of the Harris-type current sheet and realized basically the same properties as those in other full particle simulations. We show that the numbers of cells and particles are greatly reduced so that the time to complete the simulation is significantly shortened in our AMR code, which enables us to conduct large-scale simulations on the current sheet evolution.

Full particle simulation of the plasma sheet using adaptive mesh refinement technique

We have developed a new 2-1/2 dimensional electromagnetic particle code with adaptive mesh refinement (AMR) technique in an effort to give a self-consistent description of the dynamic change of the plasma sheet in association with magnetic reconnection, which includes multi-scale processes from the electron scale to the magnetohydrodynamic scale. The AMR technique dynamically subdivides the cells that satisfy a certain refinement criterion and it is quite effective to achieve high-resolution simulations such as those around the X-type neutral line. Our code is checked against several well known processes such as the Landau damping of the Langmuir waves and we show that the AMR technique is successfully applied to the conventional particle code. A preliminary result on the time evolution of the Harris-type current sheet using our code is also presented. The number of cells and particles are dramatically reduced by using the AMR.

Sausage mode instability of thin current sheets as a cause of magnetospheric substorms

Abstract. Observations have shown that, prior to substorm explosions, thin current sheets are formed in the plasma sheet of the Earth's magnetotail. This provokes the question, to what extent current-sheet thinning and substorm onsets are physically, maybe even causally, related. To answer this question, one has to understand the plasma stability of thin current sheets. Kinetic effects must be taken into account since particle scales are reached in the course of tail current-sheet thinning. We present the results of theoretical investigations of the stability of thin current sheets and about the most unstable mode of their decay. Our conclusions are based upon a non-local linear dispersion analysis of a cross-magnetic field instability of Harris-type current sheets. We found that a sausage-mode bulk current instability starts after a sheet has thinned down to the ion inertial length. We also present the results of three-dimensional electromagnetic PIC-code simulations carried out for mass ratios up to Mi / me=64. They verify the linearly predicted properties of the sausage mode decay of thin current sheets in the parameter range of interest.Key words. Magnetospheric physics (plasma waves and instabilities; storms and substorms) · Space plasma physics (magnetic reconnection)

Sausage mode instability of thin current sheets as a cause of magnetospheric substorms

Observations have shown that, prior to substorm explosions, thin current sheets are formed in the plasma sheet of the Earth's magnetotail. This provokes the question, to what extent current-sheet thinning and substorm onsets are physically, maybe even causally, related. To answer this question, one has to understand the plasma stability of thin current sheets. Kinetic effects must be taken into account since particle scales are reached in the course of tail current-sheet thinning. We present the results of theoretical investigations of the stability of thin current sheets and about the most unstable mode of their decay. Our conclusions are based upon a non-local linear dispersion analysis of a cross-magnetic field instability of Harris-type current sheets. We found that a sausage-mode bulk current instability starts after a sheet has thinned down to the ion inertial length. We also present the results of three-dimensional electromagnetic PIC-code simulations carried out for mass ratios up to Mi / me=64. They verify the linearly predicted properties of the sausage mode decay of thin current sheets in the parameter range of interest.Key words. Magnetospheric physics (plasma waves and instabilities; storms and substorms) · Space plasma physics (magnetic reconnection)