Lundquist Number Research Articles

RECONNECTION PROPERTIES OF LARGE-SCALE CURRENT SHEETS DURING CORONAL MASS EJECTION ERUPTIONS

ABSTRACT We present a detailed analysis of the properties of magnetic reconnection at large-scale current sheets (CSs) in a high cadence version of the Lynch & Edmondson 2.5D MHD simulation of sympathetic magnetic breakout eruptions from a pseudostreamer source region. We examine the resistive tearing and break-up of the three main CSs into chains of X- and O-type null points and follow the dynamics of magnetic island growth, their merging, transit, and ejection with the reconnection exhaust. For each CS, we quantify the evolution of the length-to-width aspect ratio (up to ∼100:1), Lundquist number (∼103), and reconnection rate (inflow-to-outflow ratios reaching ∼0.40). We examine the statistical and spectral properties of the fluctuations in the CSs resulting from the plasmoid instability, including the distribution of magnetic island area, mass, and flux content. We show that the temporal evolution of the spectral index of the reconnection-generated magnetic energy density fluctuations appear to reflect global properties of the CS evolution. Our results are in excellent agreement with recent, high-resolution reconnection-in-a-box simulations even though our CSs’ formation, growth, and dynamics are intrinsically coupled to the global evolution of sequential sympathetic coronal mass ejection eruptions.

NONAXISYMMETRIC MHD INSTABILITIES OF CHANDRASEKHAR STATES IN TAYLOR-COUETTE GEOMETRY

ABSTRACT We consider axially periodic Taylor–Couette geometry with insulating boundary conditions. The imposed basic states are so-called Chandrasekhar states, where the azimuthal flow U ϕ and magnetic field B ϕ have the same radial profiles. Mainly three particular profiles are considered: the Rayleigh limit, quasi-Keplerian, and solid-body rotation. In each case we begin by computing linear instability curves and their dependence on the magnetic Prandtl number . For the azimuthal wavenumber m = 1 modes, the instability curves always scale with the Reynolds number and the Hartmann number. For sufficiently small these modes therefore only become unstable for magnetic Mach numbers less than unity, and are thus not relevant for most astrophysical applications. However, modes with can behave very differently. For sufficiently flat profiles, they scale with the magnetic Reynolds number and the Lundquist number, thereby allowing instability also for the large magnetic Mach numbers of astrophysical objects. We further compute fully nonlinear, three-dimensional equilibration of these instabilities, and investigate how the energy is distributed among the azimuthal (m) and axial (k) wavenumbers. In comparison spectra become steeper for large m, reflecting the smoothing action of shear. On the other hand kinetic and magnetic energy spectra exhibit similar behavior: if several azimuthal modes are already linearly unstable they are relatively flat, but for the rigidly rotating case where m = 1 is the only unstable mode they are so steep that neither Kolmogorov nor Iroshnikov–Kraichnan spectra fit the results. The total magnetic energy exceeds the kinetic energy only for large magnetic Reynolds numbers .

Fast reconnection in relativistic plasmas: the magnetohydrodynamics tearing instability revisited

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 light. It is found that the linear phase of the instability closely matches the analysis in classical MHD, where the growth rate scales with the Lundquist number S as S^-1/2, with the only exception of an enhanced inertial term due to the thermal and magnetic energy contributions. In addition, when thin current sheets of inverse aspect ratio scaling as S^-1/3 are considered, the so-called "ideal" tearing regime is retrieved, with modes growing independently on S and extremely fast, on only a few light crossing times of the sheet length. The overall growth of fluctuations is seen to solely depend on the value of the background Alfven velocity. In the fully nonlinear stage we observe an inverse cascade towards the fundamental mode, with Petschek-type supersonic jets propagating at the external Alfven speed from the X-point, and a fast reconnection rate at the predicted value R~(ln S)^-1.

Internal Transport Barrier Broadening through Subdominant Mode Stabilization in Reversed Field Pinch Plasmas.

The reversed field pinch (RFP) device RFX-mod features strong internal transport barriers when the plasma accesses states with a single dominant helicity. Such transport barriers enclose a hot helical region with high confinement whose amplitude may vary from a tiny one to an amplitude encompassing an appreciable fraction of the available volume. The transition from narrow to wide thermal structures has been ascribed so far to the transport reduction that occurs when the dominant mode separatrix, which is a preferred location for the onset of stochastic field lines, disappears. In this Letter we show instead that the contribution from the separatrix disappearance, by itself, is marginal and the main role is instead played by the progressive stabilization of secondary modes. The position and the width of the stochastic boundary encompassing the thermal structures have been estimated by applying the concept of a 3D quasiseparatrix layer, developed in solar physics to treat reconnection phenomena without true separatrices and novel to toroidal laboratory plasmas. Considering the favorable scaling of secondary modes with the Lundquist number, these results open promising scenarios for RFP plasmas at temperatures higher than the presently achieved ones, where lower secondary modes and, consequently, larger thermal structures are expected. Furthermore, this first application of the quasiseparatrix layer to a toroidal plasma indicates that such a concept is ubiquitous in magnetic reconnection, independent of the system geometry under investigation.

Turbulent reconnection of magnetic bipoles in stratified turbulence

We consider strongly stratified forced turbulence in a plane-parallel layer with helicity and corresponding large-scale dynamo action in the lower part and non-helical turbulence in the upper. The magnetic field is found to develop strongly concentrated bipolar structures near the surface. They form elongated bands with a sharp interface between opposite polarities. Unlike earlier experiments with imposed magnetic field, the inclusion of rotation does not strongly suppress the formation of these structures. We perform a systematic numerical study of this phenomenon by varying magnetic Reynolds number, scale separation ratio, and Coriolis number. We focus on the formation of a current sheet between bipolar regions where reconnection of oppositely oriented field lines occurs. We determine the reconnection rate by measuring either the inflow velocity in the vicinity of the current sheet or by measuring the electric field in the reconnection region. We demonstrate that for large Lundquist numbers, S>10^3, the reconnection rate is nearly independent of S in agreement with results of recent numerical simulations performed by other groups in simpler settings.

“Ideal” tearing and the transition to fast reconnection in the weakly collisional MHD and EMHD regimes

AbstractThis paper discusses the transition to fast growth of the tearing instability in thin current sheets in the collisionless limit where electron inertia drives the reconnection process. It has been previously suggested that in resistive MHD there is a natural maximum aspect ratio (ratio of sheet length and breadth to thickness) which may be reached for current sheets with a macroscopic length L, the limit being provided by the fact that the tearing mode growth time becomes of the same order as the Alfvén time calculated on the macroscopic scale. For current sheets with a smaller aspect ratio than critical the normalized growth rate tends to zero with increasing Lundquist number S, while for current sheets with an aspect ratio greater than critical the growth rate diverges with S. Here we carry out a similar analysis but with electron inertia as the term violating magnetic flux conservation: previously found scalings of critical current sheet aspect ratios with the Lundquist number are generalized to include the dependence on the ratio , where de is the electron skin depth, and it is shown that there are limiting scalings which, as in the resistive case, result in reconnecting modes growing on ideal time scales. Finite Larmor radius effects are then included, and the rescaling argument at the basis of “ideal” reconnection is proposed to explain secondary fast reconnection regimes naturally appearing in numerical simulations of current sheet evolution.

Visco-resistive plasmoid instability

The plasmoid instability in visco-resistive current sheets is analyzed in both the linear and nonlinear regimes. The linear growth rate and the wavenumber are found to scale as S1/4(1+Pm)−5/8 and S3/8(1+Pm)−3/16 with respect to the Lundquist number S and the magnetic Prandtl number Pm. Furthermore, the linear layer width is shown to scale as S−1/8(1+Pm)1/16. The growth of the plasmoids slows down from an exponential growth to an algebraic growth when they enter into the nonlinear regime. In particular, the time-scale of the nonlinear growth of the plasmoids is found to be τNL∼S−3/16(1+Pm)19/32τA,L. The nonlinear growth of the plasmoids is radically different from the linear one, and it is shown to be essential to understand the global current sheet disruption. It is also discussed how the plasmoid instability enables fast magnetic reconnection in visco-resistive plasmas. In particular, it is shown that the recursive plasmoid formation can trigger a collisionless reconnection regime if S≳Lcs(ϵclk)−1(1+Pm)1/2, where Lcs is the half-length of the global current sheet and lk is the relevant kinetic length scale. On the other hand, if the current sheet remains in the collisional regime, the global (time-averaged) reconnection rate is shown to be 〈dψ/dt|X〉≈ϵcvA,uBu(1+Pm)−1/2, where ϵc is the critical inverse aspect ratio of the current sheet, while vA,u and Bu are the Alfvén speed and the magnetic field upstream of the global reconnection layer.

TURBULENT MAGNETOHYDRODYNAMIC RECONNECTION MEDIATED BY THE PLASMOID INSTABILITY

ABSTRACT It has been established that the Sweet–Parker current layer in high Lundquist number reconnection is unstable to the super-Alfvénic plasmoid instability. Past two-dimensional magnetohydrodynamic simulations have demonstrated that the plasmoid instability leads to a new regime where the Sweet–Parker current layer changes into a chain of plasmoids connected by secondary current sheets, and the averaged reconnection rate becomes nearly independent of the Lundquist number. In this work, a three-dimensional simulation with a guide field shows that the additional degree of freedom allows plasmoid instabilities to grow at oblique angles, which interact and lead to self-generated turbulent reconnection. The averaged reconnection rate in the self-generated turbulent state is of the order of a hundredth of the characteristic Alfvén speed, which is similar to the two-dimensional result but is an order of magnitude lower than the fastest reconnection rate reported in recent studies of externally driven three-dimensional turbulent reconnection. Kinematic and magnetic energy fluctuations both form elongated eddies along the direction of the local magnetic field, which is a signature of anisotropic magnetohydrodynamic turbulence. Both energy fluctuations satisfy power-law spectra in the inertial range, where the magnetic energy spectral index is in the range from −2.3 to −2.1, while the kinetic energy spectral index is slightly steeper, in the range from −2.5 to −2.3. The anisotropy of turbulence eddies is found to be nearly scale-independent, in contrast with the prediction of the Goldreich–Sridhar theory for anisotropic turbulence in a homogeneous plasma permeated by a uniform magnetic field.

Magnetic reconnection: from the Sweet–Parker model to stochastic plasmoid chains

Magnetic reconnection is the topological reconfiguration of the magnetic field in a plasma, accompanied by the violent release of energy and particle acceleration. Reconnection is as ubiquitous as plasmas themselves, with solar flares perhaps the most popular example. Other fascinating processes where reconnection plays a key role include the magnetic dynamo, geomagnetic storms and the sawtooth crash in tokamaks.Over the last few years, the theoretical understanding of magnetic reconnection in large-scale fluid systems has undergone a major paradigm shift. The steady-state model of reconnection described by the famous Sweet–Parker (SP) theory, which dominated the field for ∼50 years, has been replaced with an essentially time-dependent, bursty picture of the reconnection layer, dominated by the continuous formation and ejection of multiple secondary islands (plasmoids). Whereas in the SP model reconnection was predicted to be slow, a major implication of this new paradigm is that reconnection in fluid systems is fast (i.e. independent of the Lundquist number), provided that the system is large enough. This conceptual shift hinges on the realization that SP-like current layers are violently unstable to the plasmoid (tearing) instability—implying, therefore, that such current sheets are super-critically unstable and thus can never form in the first place. This suggests that the formation of a current sheet and the subsequent reconnection process cannot be decoupled, as is commonly assumed.This paper provides an introductory-level overview of the recent developments in reconnection theory and simulations that led to this essentially new framework. We briefly discuss the role played by the plasmoid instability in selected applications, and describe some of the outstanding challenges that remain at the frontier of this subject. Amongst these are the analytical and numerical extension of the plasmoid instability to (i) 3D and (ii) non-magnetohydrodynamics (MHD) regimes. New results are reported in both cases.

MAGNETIC RECONNECTION: RECURSIVE CURRENT SHEET COLLAPSE TRIGGERED BY “IDEAL” TEARING

We study, by means of MHD simulations, the onset and evolution of fast reconnection via the "ideal" tearing mode within a collapsing current sheet at high Lundquist numbers ($S\gg10^4$). We first confirm that as the collapse proceeds, fast reconnection is triggered well before a Sweet-Parker type configuration can form: during the linear stage plasmoids rapidly grow in a few Alfv\'en times when the predicted "ideal" tearing threshold $S^{-1/3}$ is approached from above; after the linear phase of the initial instability, X-points collapse and reform nonlinearly. We show that these give rise to a hierarchy of tearing events repeating faster and faster on current sheets at ever smaller scales, corresponding to the triggering of "ideal" tearing at the renormalized Lundquist number. In resistive MHD this process should end with the formation of sub-critical ($S \leq10^4$) Sweet Parker sheets at microscopic scales. We present a simple model describing the nonlinear recursive evolution which explains the timescale of the disruption of the initial sheet.

IRISSi iv LINE PROFILES: AN INDICATION FOR THE PLASMOID INSTABILITY DURING SMALL-SCALE MAGNETIC RECONNECTION ON THE SUN

Our understanding of the process of fast reconnection has undergone a dramatic change in the last 10 years driven, in part, by the availability of high-resolution numerical simulations that have consistently demonstrated the break-up of current sheets into magnetic islands, with reconnection rates that become independent of Lundquist number, challenging the belief that fast magnetic reconnection in flares proceeds via the Petschek mechanism that invokes pairs of slow-mode shocks connected to a compact diffusion region. The reconnection sites are too small to be resolved with images but these reconnection mechanisms, Petschek and the plasmoid instability, have reconnection sites with very different density and velocity structures and so can be distinguished by high-resolution line-profiles observations. Using IRIS spectroscopic observations we obtain a survey of typical line profiles produced by small-scale events thought to be reconnection sites on the Sun. Slit-jaw images are used to investigate the plasma heating and re-configuration at the sites. A sample of 15 events from two active regions is presented. The line profiles are complex with bright cores and broad wings extending to over 300 km/s. The profiles can be reproduced with the multiple magnetic islands and acceleration sites that characterise the plasmoid instability but not by bi-directional jets that characterise the Petschek mechanism. This result suggests that if these small-scale events are reconnection sites, then fast reconnection proceeds via the plasmoid instability, rather than the Petschek mechanism during small-scale reconnection on the Sun.

The Tayler instability at low magnetic Prandtl numbers: between chiral symmetry breaking and helicity oscillations

The Tayler instability is a kink-type, current driven instability that plays an important role in plasma physics but might also be relevant in liquid metal applications with high electrical currents. In the framework of the Tayler–Spruit dynamo model of stellar magnetic field generation (Spruit 2002 Astron. Astrophys. 381 923–32), the question of spontaneous helical (chiral) symmetry breaking during the saturation of the Tayler instability has received considerable interest (Zahn et al 2007 Astron. Astrophys. 474 145–54; Gellert et al 2011 Mon. Not. R. Astron. Soc. 414 2696–701; Bonanno et al 2012 Phys. Rev. E 86 016313). Focusing on fluids with low magnetic Prandtl numbers, for which the quasistatic approximation can be applied, we utilize an integro-differential equation approach (Weber et al 2013 New J. Phys.15 043034) in order to investigate the saturation mechanism of the Tayler instability. Both the exponential growth phase and the saturated phase are analysed in terms of the action of the α and β effects of mean-field magnetohydrodynamics. In the exponential growth phase we always find a spontaneous chiral symmetry breaking which, however, disappears in the saturated phase. For higher degrees of supercriticality, we observe helicity oscillations in the saturated regime. For Lundquist numbers in the order of one we also obtain chiral symmetry breaking of the saturated magnetic field.

Control of 3D equilibria with resonant magnetic perturbations in MST

To aid in diagnosis of 3D equilibria in the Madison Symmetric Torus, it has become necessary to control the orientation of the equilibria. In reversed field pinch experiments a transition to a 3D equilibrium is common with sufficiently large plasma current (and Lundquist number). Diagnosis of this state is hampered by the fact that the helical structure is stationary but with an orientation that varies shot-to-shot. A resonant magnetic perturbation (RMP) technique has been developed to vary controllably the orientation of the 3D equilibria and optimized to minimize the plasma wall interaction due to its use. Application of an RMP now allows alignment of the structure with key diagnostics, including Thomson scattering and an interferometer-polarimeter.

Phase diagrams of forced magnetic reconnection in Taylor’s model

Recent progress in the understanding of how externally driven magnetic reconnection evolves is organized in terms of parameter space diagrams. These diagrams are constructed using four pivotal dimensionless parameters: the Lundquist number $S$, the magnetic Prandtl number $P_{m}$, the amplitude of the boundary perturbation $\hat{{\it\Psi}}_{0}$, and the perturbation wave number $\hat{k}$. This new representation highlights the parameter regions of a given system in which the magnetic reconnection process is expected to be distinguished by a specific evolution. Contrary to previously proposed phase diagrams, the diagrams introduced here take into account the dynamical evolution of the reconnection process and are able to predict slow or fast reconnection regimes for the same values of $S$ and $P_{m}$, depending on the parameters that characterize the external drive, which have not been considered until now. These features are crucial to understanding the onset and evolution of magnetic reconnection in diverse physical systems.

RESISTIVE MAGNETOHYDRODYNAMICS SIMULATIONS OF THE IDEAL TEARING MODE

We study the linear and nonlinear evolution of the tearing instability on thin current sheets by means of two-dimensional numerical simulations, within the framework of compressible, resistive magnetohydrodynamics. In particular we analyze the behavior of current sheets whose inverse aspect ratio scales with the Lundquist number S as S 1=3 . This scaling has been recently recognized to yield the threshold separating fast, ideal reconnection, with an evolution and growth which are independent of S provided this is high enough, as it should be natural having the ideal case as a limit for S! 1. Our simulations conrm that the tearing instability growth rate can be as fast as 0:6 A 1 , where A is the ideal Alfv enic time set by the macroscopic scales, for our least diusive case with S = 10 7 . The expected instability dispersion relation and eigenmodes are also retrieved in the linear regime, for the values of S explored here. Moreover, in the nonlinear stage of the simulations we observe secondary events obeying the same critical scaling with S, here calculated on the local, much smaller lengths, leading to increasingly faster reconnection. These ndings strongly support the idea that in a fully dynamic regime, as soon as current sheets develop, thin and reach this critical threshold in their aspect ratio, the tearing mode is able to trigger plasmoid formation and reconnection on the local (ideal) Alfv enic timescales, as required to explain the explosive aring activity often observed in solar and astrophysical plasmas. Subject headings: plasmas { MHD { methods: numerical.

SELF-GENERATED TURBULENCE IN MAGNETIC RECONNECTION

Classical Sweet–Parker models of reconnection predict that reconnection rates depend inversely on the resistivity, usually parameterized using the dimensionless Lundquist number (S). We describe magnetohydrodynamic (MHD) simulations using a static, nested grid that show the development of a three-dimensional (3D) instability in the plane of a current sheet between reversing field lines without a guide field. The instability leads to rapid reconnection of magnetic field lines at a rate independent of S over at least the range resolved by the simulations. We find that this instability occurs even for cases with that in our models appear stable to the recently described, two-dimensional, plasmoid instability. Our results suggest that 3D, MHD processes alone produce fast (resistivity independent) reconnection without recourse to kinetic effects or external turbulence. The unstable reconnection layers provide a self-consistent environment in which the extensively studied turbulent reconnection process can occur.

Quantifying three dimensional reconnection in fragmented current layers

There is growing evidence that when magnetic reconnection occurs in high Lundquist number plasmas such as in the Solar Corona or the Earth's Magnetosphere it does so within a fragmented, rather than a smooth current layer. Within the extent of these fragmented current regions, the associated magnetic flux transfer and energy release occur simultaneously in many different places. This investigation focusses on how best to quantify the rate at which reconnection occurs in such layers. An analytical theory is developed which describes the manner in which new connections form within fragmented current layers in the absence of magnetic nulls. It is shown that the collective rate at which new connections form can be characterized by two measures; a total rate which measures the true rate at which new connections are formed and a net rate which measures the net change of connection associated with the largest value of the integral of E|| through all of the non-ideal regions. Two simple analytical models are presented which demonstrate how each should be applied and what they quantify.

THE TEARING MODE INSTABILITY OF THIN CURRENT SHEETS: THE TRANSITION TO FAST RECONNECTION IN THE PRESENCE OF VISCOSITY

This paper studies the growth rate of reconnection instabilities in thin current sheets in the presence of both resistivity and viscosity. In a previous paper, Pucci & Velli, it was argued that at sufficiently high Lundquist number S it is impossible to form current sheets with aspect ratios that scale as with because the growth rate of the tearing mode would then diverge in the ideal limit . Here we extend their analysis to include the effects of viscosity, always present in numerical simulations along with resistivity, and which may play a role in the solar corona and other astrophysical environments. A finite Prandtl number allows current sheets to reach larger aspect ratios before becoming rapidly unstable in pileup-type regimes. Scalings with Lundquist and Prandtl numbers are discussed, as well as the transition to kinetic reconnection.

Plasmoid instability in double current sheets

The linear behavior of plasmoid instability in double current sheet configurations, namely, double plasmoid mode (DPM), is analytically and numerically investigated within the framework of a reduced magnetohydrodynamic model. Analytical analysis shows that if the separation of double current sheets is sufficiently small [κxs≪κ2/9SL1/3], the growth rate of DPMs scales as κ2/3SL0 in the non-constant-ψ regime, where κ=kLCS/2 is the wave vector measured by the half length of the system LCS/2, 2xs is the separation between two resonant surfaces, and SL=LCSVA/2η is Lundquist number with VA and η being Alfven velocity and resistivity, respectively. If the separation is very large [κxs≫κ2/9SL1/3], the growth rate scales as κ−2/5SL2/5 in the constant-ψ regime. Furthermore, it is also analytically found that the maximum wave number scales as xs−9/7SL3/7 at the transition position between these two regimes, and the corresponding maximum growth rate scales as xs−6/7SL2/7 there. The analytically predicted scalings are verified in some limits through direct numerical calculations.

Repetitive formation and decay of current sheets in magnetic loops: An origin of diverse magnetic structures

In this work, evolution of an incompressible, thermally homogeneous, infinitely conducting, viscous magnetofluid is numerically explored as the fluid undergoes repeated events of magnetic reconnection. The initial magnetic field is constructed by a superposition of two linear force-free fields and has similar morphology as the magnetic loops observed in the solar corona. The results are presented for computations with three distinct sets of footpoint geometries. To onset reconnection, we rely on numerical model magnetic diffusivity, in the spirit of implicit large eddy simulation. It is generally expected that in a high Lundquist number fluid, repeated magnetic reconnections are ubiquitous and hence can lead to a host of magnetic structures with considerable observational importance. In particular, the simulations presented here illustrate formations of magnetic islands, rotating magnetic helices and rising flux ropes—depending on the initial footpoint geometry but through the common process of repeated magnetic reconnections. Further, we observe the development of extended current sheets in two case studies, where the footpoint reconnections generate favorable dynamics.