\large{\bf Abstract-} Unsteady Hall Magnetohydrodynamics (MHD) near a hyperbolic magnetic neutral line is investigated. An exact analytical solution describing a self-similar evolution is given. This solution shows a negligible impact on the current-sheet formation process near the hyperbolic magnetic neutral line at small times by the Hall effect but, subsequently, a quenching by the Hall effect of the finite-time singularity exhibited in ideal MHD and, hence a prevention of the current density blow-up at large times. The asymptotic result given by this time-dependent solution is in full quantitative agreement with the formulation of \textit{steady} Hall MHD near a $X$-type magnetic neutral line (Shivamoggi [23]). The latter formulation showed that this asymptotic result indeed corresponds to a hyperbolic configuration of the magnetic field lines in the \textit{steady} case.
The nonlinear dynamics of axisymmetric, as well as helical, frozen-in vortex structures is investigated by the Hamiltonian method in the framework of ideal incompressible electron magnetohydrodynamics. For description of current-sheet formation from a smooth initial magnetic field, local and nonlocal nonlinear approximations are introduced and partially analyzed that are generalizations of the previously known exactly solvable local model neglecting electron inertia. Finally, estimations are made that predict finite-time singularity formation for a class of hydrodynamic models intermediate bet…
Three dimensional electron-magnetohydrodynamic (EMHD) simulations of electron current sheet dynamics in a background of stationary and unmagnetized ions and the subsequent generation of electromagnetic fluctuations are carried out. The physical parameters and initial magnetic configuration in the simulations are chosen to be similar to those in the \textsc{Vineta}.II magnetic reconnection experiment. Consistent with the experimental results, our 3D EMHD simulations show the formation of an elongated electron scale current sheet together with the excitation of electromagnetic fluctuations withi…
The formalism of frozen-in vortex lines for two-dimensional (2D) flows in ideal incompressible electron magnetohydrodynamics (EMHD) is formulated. A localized approximation for nonlinear dynamics of two close sheets of the generalized vorticity is suggested and its integrability by the hodograph method is demonstrated.
Inertial effects in nonlinear magnetic reconnection are studied within the context of 2D electron magnetohydrodynamics (EMHD) with resistive and viscous dissipation. Families of nonlinear solutions for relevant current sheet parameters are predicted and confirmed numerically in all regimes of interest. Electron inertia becomes important for current sheet thicknesses $δ$ below the inertial length $d_{e}$. In this case, in the absence of electron viscosity, the sheet thickness experiences a nonlinear collapse. Viscosity regularizes solutions at small scales. Transition from resistive to viscous …
An overview of some recent progress on magnetohydrodynamic stability and current sheet formation in a line-tied system is given. Key results on the linear stability of the ideal internal kink mode and resistive tearing mode are summarized. For nonlinear problems, a counterexample to the recent demonstration of current sheet formation by Low \emph{et al}. [B. C. Low and Å. M. Janse, Astrophys. J. \textbf{696}, 821 (2009)] is presented, and the governing equations for quasi-static evolution of a boundary driven, line-tied magnetic field are derived. Some open questions and possible strategies to…
The adiabatic limit of a recently proposed dynamical extension of Taylor relaxation, \emph{multi-region relaxed magnetohydrodynamics} (MRxMHD) is summarized, with special attention to the appropriate definition of relative magnetic helicity. The formalism is illustrated using a simple two-region, sheared-magnetic-field model similar to the Hahm--Kulsrud--Taylor (HKT) rippled-boundary slab model. In MRxMHD a linear Grad--Shafranov equation applies, even at finite ripple amplitude. The adiabatic switching on of boundary ripple excites a shielding current sheet opposing reconnection at a resonant…
Rigorous theories of the tearing instability are mathematically quite involving. Therefore, the present note aims to demonstrate how their main results can be reproduced by a simple qualitative analysis of the respective magnetohydrodynamic (MHD) equations.
We develop a framework for studying the statistical properties of current sheets in numerical simulations of 3D magnetohydrodynamic (MHD) turbulence. We describe an algorithm that identifies current sheets in a simulation snapshot and then determines their geometrical properties (including length, width, and thickness) and intensities (peak current density and total energy dissipation rate). We then apply this procedure to simulations of reduced MHD turbulence and perform a statistical analysis on the obtained population of current sheets. We evaluate the role of reconnection by separately stu…
Introducing a Clebsch-like parameterization, we have formulated a canonical Hamiltonian system on a symplectic leaf of reduced magnetohydrodynamics. An interesting structure of the equations is in that the Lorentz-force, which is a quadratic nonlinear term in the conventional formulation, appears as a linear term -ΔQ, just representing the current density (Q is a Clebsch variable, and Δ is the two-dimensional Laplacian); omitting this term reduces the system into the two-dimensional Euler vorticity equation of a neutral fluid. A heuristic estimate shows that current sheets grow exponentially (…
The present paper provides comprehensive description of various regimes involved in the two-fluid model of the resistive tearing instability. These include two novel regimes of this instability, which correspond to the long-wave modes that can develop in a highly-elongated current sheet. This issue is relevant to the study of fast magnetic reconnection and magnetic turbulence in magnetohydrodynamic (MHD) objects with a large value of the Lundquist number.
We propose two sets of initial conditions for magnetohydrodynamics (MHD) in which both the velocity and the magnetic fields have spatial symmetries that are preserved by the dynamical equations as the system evolves. When implemented numerically they allow for substantial savings in CPU time and memory storage requirements for a given resolved scale separation. Basic properties of these Taylor-Green flows generalized to MHD are given, and the ideal non-dissipative case is studied up to the equivalent of 2048^3 grid points for one of these flows. The temporal evolution of the logarithmic decrem…
We analyze the effect of the Hall term in the magnetohydrodynamic turbulence under a strong externally supported magnetic field, seeing how this changes the energy cascade, the characteristic scales of the flow and the dynamics of global magnitudes, with particular interest in the dissipation. Numerical simulations of freely evolving three-dimensional reduced magnetohydrodynamics (RHMHD) are performed, for different values of the Hall parameter (the ratio of the ion skin depth to the macroscopic scale of the turbulence) controlling the impact of the Hall term. The Hall effect modifies the tran…
A set of reduced Hall magnetohydrodynamic (MHD) equations are used to evaluate the stability of large aspect ratio current sheets to the formation of plasmoids (secondary islands). Reconnection is driven by resistivity in this analysis, which occurs at the resistive skin depth $d_η\equiv S_L^{-1/2} \sqrt{L v_A/γ}$, where $S_L$ is the Lundquist number, $L$ the length of the current sheet, $v_A$ the Alfvén speed, and $γ$ the growth rate. Modifications to a recent resistive MHD analysis [N.\ F.\ Loureiro, A.\ A.\ Schekochihin, and S.\ C. Cowley, Phys.\ Plasmas {\bf 14}, 100703 (2007)] arise when …
We present high resolution 2D and 3D simulations of magnetized decaying turbulence in relativistic resistive magneto-hydrodynamics. The simulations show dynamic formation of large scale intermittent long-lived current sheets being disrupted by the tearing instability into plasmoid chains. These current sheets are locations of enhanced magnetic field dissipation and heating of the plasma. We find magnetic energy spectra $\propto k^{-3/2}$, together with strongly pronounced dynamic alignment of Elsasser fields and of velocity and magnetic fields, for strong guide-field turbulence, whereas we ret…
Ideal magnetohydrodynamics (IMHD) is strongly constrained by an infinite number of microscopic constraints expressing mass, entropy and magnetic flux conservation in each infinitesimal fluid element, the latter preventing magnetic reconnection. By contrast, in the Taylor relaxation model for formation of macroscopically self-organized plasma equilibrium states, all these constraints are relaxed save for global magnetic fluxes and helicity. A Lagrangian variational principle is presented that leads to a new, fully dynamical, \emph{relaxed magnetohydrodynamics} (RxMHD), such that all static solu…
Newcomb's Lagrangian for ideal magnetohydrodynamics (MHD) in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum-preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instabil…
We investigate the origins of $k^{-2}$ spectrum in a decaying Taylor-Green magnetohydrodynamic flow with zero large scale magnetic flux that was reported in Lee et al. (2010). A possible candidate for this scaling exponent has been the weak turbulence phenomenology. From our numerical simulations, we observe that current sheets in the magnetic Taylor-Green flow are formed in regions of magnetic discontinuities. Based on this observation and by studying the influence of the current sheets on the energy spectrum, using a filtering technique, we argue that the discontinuities are responsible for …
The plasmoid instability in evolving current sheets has been widely studied due to its effects on the disruption of current sheets, the formation of plasmoids, and the resultant fast magnetic reconnection. In this Letter, we study the role of the plasmoid instability in two-dimensional magnetohydrodynamic (MHD) turbulence by means of high-resolution direct numerical simulations. At sufficiently large magnetic Reynolds number ($R_m=10^6$), the combined effects of dynamic alignment and turbulent intermittency lead to a copious formation of plasmoids in a multitude of intense current sheets. The …
Simulations of decaying magnetohydrodynamic (MHD) turbulence are performed with a fluid and a kinetic code. The initial condition is an ensemble of long-wavelength, counter-propagating, shear-Alfvén waves, which interact and rapidly generate strong MHD turbulence. The total energy is conserved and the rate of turbulent energy decay is very similar in both codes, although the fluid code has numerical dissipation whereas the kinetic code has kinetic dissipation. The inertial range power spectrum index is similar in both the codes. The fluid code shows a perpendicular wavenumber spectral slope of…
We examine the effect of an external guide field and current sheet thickness on the growth rates and nature of three dimensional unstable modes of an electron current sheet driven by electron shear flow. The growth rate of the fastest growing mode drops rapidly with current sheet thickness but increases slowly with the strength of the guide field. The fastest growing mode is tearing type only for thin current sheets (half thickness $\approx d_e$, where $d_e=c/ω_{pe}$ is electron inertial length) and zero guide field. For finite guide field or thicker current sheets, fastest growing mode is non…
Reconnection physics at micro-scales is investigated in an electron magnetohydrodynamics frame. A new process of collapse of the neutral current sheet is demonstrated by means of analytical and numerical solutions. It shows how at scales smaller than ion inertia length a compression of the sheet triggers an explosive evolution of current perturbation. Collapse results in the formation of a intense sub-sheet and then an X-point structure embedded into the equilibrium sheet. Hall currents associated with this structure support high reconnection rates. Nonlinear static solution at scales of the e…
The cat-eyes steady state solution in the framework of hydrodynamics describing an infinite row of identical vortices is extended to the magnetohydrodynamic equilibrium equation with incompressible flow of arbitrary direction. The extended solution covers a variety of equilibria including one- and two-dimensional generalized force-free and Harris-sheet configurations which are preferable from those usually employed as initial states in reconnection studies. Although the vortex shape is not affected by the magnetic field, the flow in conjunction with the equilibrium nonlinearity has a strong im…
Magnetic field fluctuations in MHD turbulence can be viewed as current sheets that are progressively more anisotropic at smaller scales. As suggested by Loureiro & Boldyrev (2017) and Mallet et al (2017), below a certain critical thickness $λ_c$ such current sheets become tearing-unstable. We propose that the tearing instability changes the effective alignment of the magnetic field lines in such a way as to balance the eddy turnover rate at all scales smaller than $λ_c$. As a result, turbulent fluctuations become progressively less anisotropic at smaller scales, with the alignment angle increa…
To investigate the impact of current sheet motion on the reconnection process, we perform resistive magnetohydrodynamic (MHD) simulations of two closely located reconnection sites which move apart from each other as reconnection develops. This simulation develops less quickly than an otherwise equivalent single perturbation simulation but eventually exhibits a higher reconnection rate. The unobstructed outflow jets are faster and longer than the outflow jets directed towards the magnetic island that forms between the two current sheets. The X-line and flow stagnation point are located near the…
The effect of compressibility in charged particle energization by magnetohydrodynamic (MHD) fields is studiedin the context of test particle simulations. This problem is relevant to the solar wind and the solar corona due to the compressible nature of the flow in those astrophysical scenarios. We consider turbulent electromagnetic fields obtained from direct numerical simulations of the MHD equations with a strong background magnetic field. In order to explore the flow compressibilty effect over the particle dynamics we performed different numerical experiments: an incompressible case, and two…
The Sweet-Parker layer in a system that exceeds a critical value of the Lundquist number ($S$) is unstable to the plasmoid instability. In this paper, a numerical scaling study has been done with an island coalescing system driven by a low level of random noise. In the early stage, a primary Sweet-Parker layer forms between the two coalescing islands. The primary Sweet-Parker layer breaks into multiple plasmoids and even thinner current sheets through multiple levels of cascading if the Lundquist number is greater than a critical value $S_{c}\simeq4\times10^{4}$. As a result of the plasmoid in…
Hall magnetohydrodynamics (MHD) properties near a two-dimensional (2D) X-type magnetic neutral line in the steady state are considered via heuristic and rigorous developments. Upon considering the steady-state as the asymptotic limit of the corresponding \textit{time-dependent} problem and using a rigorous development, Hall effects are shown to be able to sustain the hyperbolicity of the magnetic field (and hence a more open X-point configuration) near the neutral line in the steady state. The heuristic development misses this subtle connection of the steady state with the corresponding \texti…
Recent results of numerical magnetohydrodynamic simulations suggest that in collisionless space plasmas turbulence can spontaneously generate thin current sheets. These coherent structures can partially explain intermittency and the non-homogenous distribution of localized plasma heating in turbulence. In this Letter Cluster multi-point observations are used to investigate the distribution of magnetic field discontinuities and the associated small-scale current sheets in the terrestrial magnetosheath downstream of a quasi-parallel bow shock. It is shown experimentally, for the first time, that…
Many linear stability aspects in plasmas are heavily influenced by non-ideal effects beyond the basic ideal magnetohydrodynamics (MHD) description. Here, the extension of the modern open-source MHD spectroscopy code Legolas with viscosity and the Hall current is highlighted and benchmarked on a stringent set of historic and recent findings. The viscosity extension is demonstrated in a cylindrical setup featuring Taylor-Couette flow and in a viscoresistive plasma slab with a tearing mode. For the Hall extension, we show how the full eigenmode spectrum relates to the analytic dispersion relation…
Models of magnetohydrodynamic (MHD) equilibia that for computational convenience assume the existence of a system of nested magnetic flux surfaces tend to exhibit singular current sheets. These sheets are located on resonant flux surfaces that are associated with rational values of the rotational transform. We study the possibility of eliminating these singularities by suitable modifications of the plasma boundary, which we prescribe in a fixed boundary setting. We find that relatively straightforward iterative procedures can be used to eliminate weak current sheets that are generated at reson…
In this paper, we have carried out linear and nonlinear analysis of tearing and surface preserving modes of two dimensional (2D) Electron Magnetohydrodynamics (EMHD). A linear analysis shows that the perturbations parallel to equilibrium magnetic field $B_0$ (characteristic tangent hyperbolic spatial profile), driven by the current gradients, lead to two different modes. The first mode is the tearing mode having a non-local behavior which requires the null-line in the magnetic field profile. Whereas, the second mode is a surface preserving local mode which does not require the null-line in the…
A new kind of magnetohydrodynamic instability and waves are analyzed for a current sheet in the presence of a small normal magnetic field component varying along the sheet. These waves and instability are related to existence of two gradients of the tangential and normal magnetic field components along the normal and tangential directions with respect to the current sheet. The current sheet can be stable or unstable if the multiplication of two magnetic gradients is positive or negative. In the stable region, the kink-like wave mode is interpreted as so called flapping waves observed in the Ea…
We present a set of three-dimensional (3D) direct numerical simulations of incompressible decaying magnetohydrodynamic turbulence in which we investigate the influence of an external uniform magnetic field B_0. A parametric study in terms of B_0 intensity is made where, in particular, we distinguish the shear- from the pseudo-Alfven waves dynamics. The initial kinetic and magnetic energies are equal with a negligible cross-correlation. Both the temporal and spectral effects of B_0 are discussed. A sub-critical balance is found between the Alfven and nonlinear times with both a global and a spe…
Magnetic reconnection with a 0.1 rate: Effective resistivity in general relativistic magnetohydrodynamics
Canadian Institute for Theoretical Astrophysics（加拿大理论天体物理研究所）；Department of Physics, University of Toronto（多伦多大学物理系）；David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto（多伦多大学大卫·邓拉普天文与天体物理系）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；Department of Astronomy, Columbia University（哥伦比亚大学天文学系）；Center for Computational Astrophysics, Flatiron Institute（平顿研究所计算天体物理学中心）；Department of Physics, University of Maryland（马里兰大学物理系）；Princeton Center for Theoretical Science and Department of Astrophysical Sciences, Princeton University（普林斯顿大学理论科学中心和天体物理科学系）；Anton Pannekoek Institute（安东·潘内克研究所）
We investigate using direct numerical simulations with grids up to 1536^3 points, the rate at which small scales develop in a decaying three-dimensional MHD flow both for deterministic and random initial conditions. Parallel current and vorticity sheets form at the same spatial locations, and further destabilize and fold or roll-up after an initial exponential phase. At high Reynolds numbers, a self-similar evolution of the current and vorticity maxima is found, in which they grow as a cubic power of time; the flow then reaches a finite dissipation rate independent of Reynolds number.
In an electrically conducting fluid, vortices stretch out a weak, large-scale magnetic field to form strong current sheets on their edges. Associated with these current sheets are magnetic stresses, which are subsequently released through reconnection, leading to vortex disruption, and possibly even destruction. This disruption phenomenon is investigated here in the context of two-dimensional, homogeneous, incompressible magnetohydrodynamics. We derive a simple order of magnitude estimate for the magnetic stresses --- and thus the degree of disruption --- that depends on the strength of the ba…
A general formulation of the problem of calculating the spectrum of stable and unstable eigenmodes of linearized perturbations about a magnetically confined toroidal plasma is presented. The analysis is based on a new hydromagnetic dynamical model, Multi-region Relaxed Magnetohydrodynamics (MRxMHD), which models the plasma-magnetic field system as consisting of multiple regions, containing compressible Euler fluid and Taylor-relaxed magnetic field, separated by flexible ideal-MHD current sheets. This is illustrated using a first-principles analysis of a two-region slab geometry, with periodic …
In space and astrophysical plasmas, turbulence is responsible for transferring energy from large scales driven by violent events or instabilities, to smaller scales where turbulent energy is ultimately converted into plasma heat by dissipative mechanisms. The nonlinear interaction between counterpropagating Alfven waves, denoted Alfven wave collisions, drives this turbulent energy cascade, as recognized by early work with incompressible magnetohydrodynamic (MHD) equations. Recent work employing analytical calculations and nonlinear gyrokinetic simulations of Alfven wave collisions in an ideali…
Fast reconnection operating in magnetically dominated plasmas is often invoked in models for magnetar giant flares, for magnetic dissipation in pulsar winds, or to explain the gamma-ray flares observed in the Crab nebula, hence its investigation is of paramount importance in high-energy astrophysics. Here we study, by means of two dimensional numerical simulations, the linear phase and the subsequent nonlinear evolution of the tearing instability within the framework of relativistic resistive magnetohydrodynamics, as appropriate in situations where the Alfven velocity approaches the speed of l…
We investigated the dynamic evolution of a 3-dimensional (3D) flux rope eruption and magnetic reconnection process in a solar flare, by simply extending 2-dimensional (2D) resistive magnetohydrodynamic simulation model of solar flares with low $β$ plasma to 3D model. We succeeded in reproducing a current sheet and bi-directional reconnection outflows just below the flux rope during the eruption in our 3D simulations. We calculated four cases of a strongly twisted flux rope and a weakly twisted flux rope in 2D and 3D simulations. The time evolution of a weakly twisted flux rope in 3D simulation…
The magnetic topology and field line random walk properties of a nanoflare-heated and magnetically confined corona are investigated in the reduced magnetohydrodynamic regime. Field lines originating from current sheets form coherent structures, called Current Sheet Connected (CSC) regions, extended around them. CSC field line random walk is strongly anisotropic, with preferential diffusion along the current sheets' in-plane length. CSC field line random walk properties remain similar to those of the entire ensemble but exhibit enhanced mean square displacements and separations due to the stron…
Recent studies provide evidence for the multi-scale nature of magnetic turbulence in the plasma sheet. Wavelet methods represent modern time series analysis techniques suitable for the description of statistical characteristics of multi-scale turbulence. Cluster FGM (fluxgate magnetometer) magnetic field high-resolution (~67 Hz) measurements are studied during an interval in which the spacecraft are in the plasma sheet. As Cluster passes through different plasma regions, physical processes exhibit non-steady properties on magnetohydrodynamic (MHD) and small, possibly kinetic scales. As a conse…
We clarify how magnetic reconnection can be derived from magnetohydrodynamics (MHD) equations in a way that is easily understandable to university students. The essential mechanism governing the time evolution of the magnetic field is diffusion dynamics. The magnetic field is represented by two components. It is clarified that the diffusion of a component causes a generation of another component that is initially zero and, accordingly, that the magnetic force lines are reconnected. For this reconnection to occur correctly, the initial magnetic field must be directed oppositely in the two regio…
The role of a super-Alfvénic plasmoid instability in the onset of fast reconnection is studied by means of the largest Hall magnetohydrodynamics simulations to date, with system sizes up to $10^{4}$ ion skin depths ($d_{i}$). It is demonstrated that the plasmoid instability can facilitate the onset of rapid Hall reconnection, in a regime where the onset would otherwise be inaccessible because the Sweet-Parker width is significantly above $d_{i}$. However, the topology of Hall reconnection is not inevitably a single stable X-point. There exists an intermediate regime where the single X-point to…
In situ spacecraft data on the solar wind show events identified as magnetic reconnection with outflows and apparent "`$X$-lines" $10^{3-4}$ times ion scales. To understand the role of turbulence at these scales, we make a case study of an inertial-range reconnection event in a magnetohydrodynamic (MHD) simulation. We observe stochastic wandering of field-lines in space, breakdown of standard magnetic flux-freezing due to Richardson dispersion, and a broadened reconnection zone containing many current sheets. The coarse-grain magnetic geometry is like large-scale reconnection in the solar wind…
In this paper, we investigate the possibility of transient growth in the linear perturbation of current sheets. The resistive magnetohydrodynamic (MHD) operator for a background field consisting of a current sheet is non-normal, meaning that associated eigenvalues and eigenmodes can be very sensitive to perturbation. In a linear stability analysis of a tearing current sheet, we show that modes that are damped as $t\rightarrow\infty$ can produce transient energy growth, contributing faster growth rates and higher energy attainment (within a fixed finite time) than the unstable tearing mode foun…
Numerical evidence for a finite-time singularity in ideal 3D magnetohydrodynamics (MHD) is presented. The simulations start from two interlocking magnetic flux rings with no initial velocity. The magnetic curvature force causes the flux rings to shrink until they come into contact. This produces a current sheet between them. In the ideal compressible calculations, the evidence for a singularity in a finite time $t_c$ is that the peak current density behaves like $|J|_\infty \sim 1/(t_c-t)$ for a range of sound speeds (or plasma betas). For the incompressible calculations consistency with the c…
The Parker model for coronal heating is investigated through a high resolution simulation. An inertial range is resolved where fluctuating magnetic energy E_M (k_perp) \propto k_\perp^{-2.7} exceeds kinetic energy E_K (k_\perp) \propto k_\perp^{-0.6}. Increments scale as δb_\ell \simeq \ell^{-0.85} and δu_\ell \simeq \ell^{+0.2} with velocity increasing at small scales, indicating that magnetic reconnection plays a prime role in this turbulent system. We show that spectral energy transport is akin to standard magnetohydrodynamic (MHD) turbulence even for a system of reconnecting current sheets…
