Lundquist Number Research Articles

Joint action of phase mixing and nonlinear effects in MHD waves propagating in coronal loops

Context. The evolution of Alfvén waves in cylindrical magnetic flux tubes, which represent a basic model for loops observed in the solar corona, can be affected by phase mixing and turbulent cascade. Phase mixing results from transverse inhomogeneities in the Alfvén speed, causing different shells of the flux tube to oscillate at different frequencies, thus forming increasingly smaller spatial scales in the direction perpendicular to the guide field. Turbulent cascade also contributes to the dissipation of the bulk energy of the waves through the generation of smaller spatial scales. Both processes present characteristic timescales. Different regimes can be envisaged according to how those timescales are related and to the typical timescale at which dissipation is at work. Aims. We investigate the interplay of phase mixing and the nonlinear turbulent cascade in the evolution and dissipation of Alfvén waves using compressible magnetohydrodynamic numerical simulations. We consider perturbations in the form of torsional waves, both propagating and standing, or turbulent fluctuations, or a combination of the two. The main purpose is to study how phase mixing and nonlinear couplings jointly work to produce small scales in different regimes. Methods. We conducted a numerical campaign to explore the typical parameters, such as the loop length, the amplitude and spatial profile of the perturbations, and the dissipative coefficients. A pseudo-spectral code was employed to solve the three-dimensional compressible magnetohydrodynamic equations, modeling the evolution of perturbations propagating in a flux tube corresponding to an equilibrium configuration with cylindrical symmetry. Results. We find that phase mixing takes place for moderate amplitudes of the turbulent component even in a distorted, nonaxisymmetric configuration, building small scales that are locally transverse to the density gradient. The dissipative time decreases with increasing the percentage of the turbulent component. This behavior is verified both for propagating and standing waves. Even in the fully turbulent case, a mechanism qualitatively similar to phase mixing occurs: it actively generates small scales together with the nonlinear cascade, thus providing the shortest dissipative time. General considerations are given to identify this regime in the parameter space. The turbulent perturbation also distorts the background density, locally increasing the Alfvén velocity gradient and further contributing to accelerating the formation of small scales. Conclusions. Our campaign of simulations is relevant for the coronal plasma where Reynolds and Lundquist numbers are extremely high. For sufficiently low perturbation amplitudes, phase mixing and turbulence work synergically, speeding up the dissipation of the perturbation energy: phase mixing dominates at early times and nonlinear effects at later times. We find that the dissipative time is shorter than those of phase mixing and the nonlinear cascade when individually considered.

Resistively controlled primordial magnetic turbulence decay

Context. Magnetic fields generated in the early Universe undergo turbulent decay during the radiation-dominated era. The decay is governed by a decay exponent and a decay time. It has been argued that the latter is prolonged by magnetic reconnection, which depends on the microphysical resistivity and viscosity. Turbulence, on the other hand, is not usually expected to be sensitive to microphysical dissipation, which affects only very small scales. Aims. We want to test and quantify the reconnection hypothesis in decaying hydromagnetic turbulence. Methods. We performed high-resolution numerical simulations with zero net magnetic helicity using the PENCIL CODE with up to 20483 mesh points and relate the decay time to the Alfvén time for different resistivities and viscosities. Results. The decay time is found to be longer than the Alfvén time by a factor that increases with increasing Lundquist number to the 1/4 power. The decay exponent is as expected from the conservation of the Hosking integral, but a timescale dependence on resistivity is unusual for developed turbulence and not found for hydrodynamic turbulence. In two dimensions, the Lundquist number dependence is shown to be leveling off above values of ≈25 000, independently of the value of the viscosity. Conclusions. Our numerical results suggest that resistivity effects have been overestimated in earlier work. Instead of reconnection, it may be the magnetic helicity density in smaller patches that is responsible for the resistively slow decay. The leveling off at large Lundquist number cannot currently be confirmed in three dimensions.

Simulations of radiatively cooled magnetic reconnection driven by pulsed power

Magnetic reconnection is an important process in astrophysical environments, as it reconfigures magnetic field topology and converts magnetic energy into thermal and kinetic energy. In extreme astrophysical systems, such as black hole coronae and pulsar magnetospheres, radiative cooling modifies the energy partition by radiating away internal energy, which can lead to the radiative collapse of the reconnection layer. In this paper, we perform two- and three-dimensional simulations to model the MARZ (Magnetic Reconnection on Z) experiments, which are designed to access cooling rates in the laboratory necessary to investigate reconnection in a previously unexplored radiatively cooled regime. These simulations are performed in GORGON, an Eulerian two-temperature resistive magnetohydrodynamic code, which models the experimental geometry comprising two exploding wire arrays driven by 20 MA of current on the Z machine (Sandia National Laboratories). Radiative losses are implemented using non-local thermodynamic equilibrium tables computed using the atomic code Spk, and we probe the effects of radiation transport by implementing both a local radiation loss model and $P_{1/3}$ multi-group radiation transport. The load produces highly collisional, super-Alfvénic (Alfvén Mach number $M_A \approx 1.5$ ), supersonic (Sonic Mach number $M_S \approx 4-5$ ) strongly driven plasma flows which generate an elongated reconnection layer (Aspect Ratio $L/\delta \approx 100$ , Lundquist number $S_L \approx 400$ ). The reconnection layer undergoes radiative collapse when the radiative losses exceed the rates of ohmic and compressional heating (cooling rate/hydrodynamic transit rate = $\tau _{\text {cool}}^{-1}/\tau _{H}^{-1}\approx 100$ ); this generates a cold strongly compressed current sheet, leading to an accelerated reconnection rate, consistent with theoretical predictions. Finally, the current sheet is also unstable to the plasmoid instability, but the magnetic islands are extinguished by strong radiative cooling before ejection from the layer.

A comparative study of resistivity models for simulations of magnetic reconnection in the solar atmosphere

Context. Plasmoid-mediated reconnection plays a fundamental role in different solar atmospheric phenomena. Numerical reproduction of this process is therefore essential for developing robust solar models. Aims. Our goal is to assess plasmoid-mediated reconnection across various numerical resistivity models in order to investigate how plasmoid numbers and reconnection rates depend on the Lundquist number. Methods. We used the Bifrost code to drive magnetic reconnection in a 2D coronal fan-spine topology, carrying out a parametric study of several experiments with different numerical resolution and resistivity models. We employed three anomalous resistivity models: (1) the original hyper-diffusion from Bifrost, (2) a resistivity proportional to current density, and (3) a resistivity quadratically proportional to electron drift velocity. For comparisons, experiments with uniform resistivity were also run. Results. Plasmoid-mediated reconnection is obtained in most of the experiments. With uniform resistivity, increasing the resolution reveals higher plasmoid frequency with weaker scaling to the Lundquist number, obtaining 7.9–12 plasmoids per minute for SL ∈ [1.8 × 104, 2.6 × 105] with a scaling of SL0.210 in the highest-resolution resistivity cases, transcending into Petschek reconnection in the high-SL limit (where the diffusive effects of the resistivity become small compared to the non-uniform viscosity) and Sweet-Parker reconnection in the low-SL limit. Anomalous resistivity leads to similar results even with lower resolution. The drift-velocity-dependent resistivity excellently reproduces Petschek reconnection for any Lundquist number, and similar results are seen with resistivity proportional to current-density though with slightly lower reconnection rates and plasmoid numbers. Among the different resistivity models applied on the given numerical resolution, the hyper-diffusion model reproduced plasmoid characteristics in closest resemblance to those obtained with uniform resistivity at a significantly higher resolution.

The gap-size influence on the excitation of magnetorotational instability in cylindricTaylor–Couette flows

The excitation conditions of the magnetorotational instability (MRI) are studied for axially unbounded Taylor–Couette (TC) flows of various gap widths between the cylinders. The cylinders are considered as made from both perfect-conducting or insulating material and the conducting fluid with a finite but small magnetic Prandtl number rotates with a quasi-Keplerian velocity profile. The solutions are optimized with respect to the wavenumber and the Reynolds number of the rotation of the inner cylinder. For the axisymmetric modes, we find the critical Lundquist number of the applied axial magnetic field: the lower, the wider the gap between the cylinders. A similar result is obtained for the induced cell structure: the wider the gap, the more spherical the cells are. The marginal rotation rate of the inner cylinder – for a fixed size of the outer cylinder – always possesses a minimum for not too wide and not too narrow gap widths. For perfect-conducting walls the minimum lies at $r_{{\rm in}}\simeq 0.4$ , where $r_{{\rm in}}$ is the ratio of the radii of the two rotating cylinders. The lowest magnetic field amplitudes to excite the instability are required for TC flows between perfect-conducting cylinders with gaps corresponding to $r_{{\rm in}}\simeq ~0.2$ . For even wider and also for very thin gaps the needed magnetic fields and rotation frequencies are shown to become rather huge. Also the non-axisymmetric modes with $|m|=1$ have been considered. Their excitation generally requires stronger magnetic fields and higher magnetic Reynolds numbers in comparison with those for the axisymmetric modes. If TC experiments with too slow rotation for the applied magnetic fields yield unstable modes of any azimuthal symmetry, such as the currently reported Princeton experiment (Wang et al., Phys. Rev. Lett., vol. 129, 115001), then also other players, including axial boundary effects, than the MRI-typical linear combination of current-free fields and differential rotation should be in the game.

Magnetic reconnection during sawteeth crashes

Sawteeth oscillations are periodic relaxations of the core plasma density and temperature in tokamaks. The rise of the temperature due to external heating is terminated by the crash phase, which involves magnetic reconnection. This is the case of fast magnetic reconnection in collisionless plasmas (Lundquist number S≥108) with a strong guide field. (The toroidal magnetic field in a tokamak is a few orders larger compared to the reconnected helical field.) Experimental measurements show non-linear behavior before and during the crash phase. Simplified single-fluid models are not able to explain the reconnection dynamics during the crash, and two-fluid effects have to be considered. In this case, numerical simulations give good agreement with the observations for the crash duration. At the same time, the present simulations explain experimentally observed phenomena only partially, and several questions remain an area of active research: evolution during the crash, the onset of the fast phase, the existence of the post-cursors, the degree of stochasticity, and others. This overview paper summarizes the current understanding of the crash process, highlights remaining problems, and shows connections to magnetic reconnection research in other plasmas.

Modeling the Formation and Evolution of Solar Wind Microstreams: From Coronal Plumes to Propagating Alfvénic Velocity Spikes

We investigate the origin of mesoscale structures in the solar wind called microstreams, defined as enhancements in the solar wind speed and temperature that last several hours. They were first clearly detected in Helios and Ulysses solar wind data and are now omnipresent in the “young” solar wind measured by the Parker Solar Probe and Solar Orbiter. These recent data reveal that microstreams transport a profusion of Alfvénic perturbations in the form of velocity spikes and magnetic switchbacks. In this study, we use a very-high-resolution 2.5D MHD model of the corona and the solar wind to simulate the emergence of magnetic bipoles interacting with the preexisting ambient corona and the creation of jets that become microstreams propagating in the solar wind. Our high-resolution simulations reach sufficiently high Lundquist numbers that capture the tearing mode instability that develops in the reconnection region and produces plasmoids released with the jet into the solar wind. Our domain runs from the lower corona to 20 R ⊙, which allows us to track the formation process of plasmoids and their evolution into Alfvénic velocity spikes. We obtain perturbed solar wind flows lasting several hours with velocity spikes occurring at characteristic periodicities of about 19 minutes. We retrieve several properties of the microstreams measured in the pristine solar wind by the Parker Solar Probe, namely an increase in wind velocity of about 100 km s−1 during a stream's passage together with superposed velocity spikes of also about 100 km s−1 released into the solar wind.

Statistical Signatures of Nanoflare Activity. III. Evidence of Enhanced Nanoflaring Rates in Fully Convective stars as Observed by the NGTS

Previous examinations of fully convective M-dwarf stars have highlighted enhanced rates of nanoflare activity on these distant stellar sources. However, the specific role the convective boundary, which is believed to be present for spectral types earlier than M2.5V, plays on the observed nanoflare rates is not yet known. Here, we utilize a combination of statistical and Fourier techniques to examine M-dwarf stellar lightcurves that lie on either side of the convective boundary. We find that fully convective M2.5V (and later subtypes) stars have greatly enhanced nanoflare rates compared with their pre-dynamo mode-transition counterparts. Specifically, we derive a flaring power-law index in the region of 3.00 ± 0.20, alongside a decay timescale of 200 ± 100 s for M2.5V and M3V stars, matching those seen in prior observations of similar stellar subtypes. Interestingly, M4V stars exhibit longer decay timescales of 450 ± 50 s, along with an increased power-law index of 3.10 ± 0.18, suggesting an interplay between the rate of nanoflare occurrence and the intrinsic plasma parameters, e.g., the underlying Lundquist number. In contrast, partially convective (i.e., earlier subtypes from M0V to M2V) M-dwarf stars exhibit very weak nanoflare activity, which is not easily identifiable using statistical or Fourier techniques. This suggests that fully convective stellar atmospheres favor small-scale magnetic reconnection, leading to implications for the flare-energy budgets of these stars. Understanding why small-scale reconnection is enhanced in fully convective atmospheres may help solve questions relating to the dynamo behavior of these stellar sources.

Plasmoid-mediated reconnection during nonlinear peeling–ballooning edge-localized modes

Plasmoid-mediated reconnection is investigated for nonlinear peeling–ballooning edge-localized modes (P-B ELMs). The formation of current sheets and the transition to 3D current sheet instability is demonstrated through fully nonlinear resistive MHD simulations of P-B ELMs in DIII-D discharges. Large-scale axisymmetric current sheets, as well as small-scale poloidally extended current sheets, are formed as the coherent P-B ELM filaments nonlinearly evolve. It is observed that, at high Lundquist numbers, these current sheets break during a reconnection burst, i.e. a secondary exponential growth of intermediate modes followed by relaxation due to the suppression of P-B drive.

The Evolution of Ion Charge States in Coronal Mass Ejections

We model the observed charge states of the elements C, O, Mg, Si, and Fe in the ejecta of coronal mass ejections (CMEs). We concentrate on “halo” CMEs observed in situ by the Advanced Composition Explorer/Solar Wind Ion Composition Spectrometer to measure ion charge states, and also remotely by the Solar Terrestrial Relations Observatory when in near quadrature with the Earth, so that the CME expansion can be accurately specified. Within this observed expansion, we integrate equations for the CME ejecta ionization balance, including electron heating parameterized as a fraction of the kinetic and gravitational energy gain of the CME. We also include the effects of non-Maxwellian electron distributions, characterized as a κ function. Focusing first on the 2010 April 3 CME, we find a somewhat better match to the observed charge states with κ close to the theoretical minimum value of κ = 3/2, implying a hard spectrum of nonthermal electrons. Similar but more significant results come from the 2011 February 15 event, although it is quite different in terms of its evolution. We discuss the implications of these values, and of the heating required, in terms of the magnetic reconnection Lundquist number and anomalous resistivity associated with CME evolution close to the Sun.

On a fully-implicit VMS-stabilized FE formulation for low Mach number compressible resistive MHD with application to MCF

This study presents the development and evaluation of a fully-implicit variational multiscale (VMS) stabilized unstructured finite element (FE) formulation for compressible magnetohydrodynamics (MHD) model, at low Mach number regime. The model describes the dynamics of a compressible conducting fluid in the low Mach number limit in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics phenomena. The specific applications that motivate this study are macroscopic simulations of the longer time-scale stability and disruptions of magnetic confinement fusion (MCF) devices, specifically the ITER tokamak. The discussion considers the development of the VMS FE representation, the structure of the stabilizing terms that deal with significant convective flows, the stabilization of the nearly incompressible response of the fluid flow, and the stabilization of the constraint that enforces the solenoidal involution on the magnetic field. The nonlinear discretized system is solved with scalable preconditioned Newton–Krylov iterative methods, which employs a multiphysics block preconditioning method based on approximate block factorizations and Schur complements. The study presents an evaluation of the VMS method on a 2D cartesian tearing mode instability, and illustrates the scalability of the solvers on MCF relevant problems. A set of results are also presented for longer time-scale stability and disruptions for the ITER tokamak. These include a vertical displacement event (VDE), and a (1,1) internal kink mode. The formulation is demonstrated to be scalable and also reasonably robust with respect to the Lundquist number scaling.

Dynamic Formation of a Transient Jet from Arched Magnetized Laboratory Plasma

A laboratory plasma experiment was built to explore the eruptive behavior of arched magnetized plasmas with dimensionless parameters relevant to the Sun’s photosphere (β ≈ 10−3, Lundquist number ≈104, plasma radius/ion gyroradius ≈20, ion–neutral collision frequency ≫ion cyclotron frequency). Dynamic formation of a transient plasma jet was observed in the presence of the strapping magnetic field. The eruption leading to the jet is unintuitive because the arched plasma is both kink- and torus-stable. The jet structure erupts within a few Alfvén transit times from the formation of the arched plasma. Extensive measurements of plasma temperature, density, magnetic field, and flows are presented. In its early stages, the jet plasma flows away from the arch with supersonic speeds (Mach 1.5). This high-speed flow persists up to the resistive diffusion time in the arched plasma and is driven by large gradients in the magnetic and thermal pressures near the birthplace of jets. There are two distinct electric current channels within the jet, one consisting of outgoing electrons and another composed of electrons returning to the anode footpoint. Significant current density around the jet is a consequence of the diamagnetic current produced by a large thermal pressure gradient in the jet. Ion–neutral charge-exchange collisions provide an efficient mechanism to produce the cross-field current and control the dynamics of the complex current channels of the jet.

High-frequency magnetohydrodynamics

We consider the case that the displacement current is not neglected in the classical MHD equations, as it is usually done. This amounts to cast them in the relativistic framework of a finite speed of light. We show some consequences in describing magnetic reconnection phenomena and for hydromagnetic waves. In the first case, the equation for the magnetic induction is changed from (formally) parabolic to (formally) hyperbolic, in the second case both, the perturbed magnetic field and the particle velocity, obey to a certain third-order in time partial differential equation, rather than to the classical wave equation. We stress the role of two typically small but nonzero parameters, the magnetic diffusivity, η (corresponding to large values of the Lundquist number), and ε:=c−2.

Multifluid Simulations of Upper-chromospheric Magnetic Reconnection with Helium–Hydrogen Mixture

Our understanding of magnetic reconnection (MR) under chromospheric conditions remains limited. Recent observations have demonstrated the important role of ion–neutral interactions in the dynamics of the chromosphere. Furthermore, the comparison between the spectral profiles and synthetic observations of reconnection events suggests that current MHD approaches appear to be inconsistent with observations. First, collisions and multithermal aspects of the plasma play a role in these regions. Second, hydrogen and helium ionization effects are relevant to the energy balance of the chromosphere. This work investigates the multifluid multispecies (MFMS) effects on MR in conditions representative of the upper chromosphere using the multifluid Ebysus code. We compare an MFMS approach based on a helium–hydrogen mixture with a two-fluid MHD model based on hydrogen only. The simulations of MR are performed in a Lundquist number regime high enough to develop plasmoids and instabilities. We study the evolution of the MR and compare the two approaches including the structure of the current sheet and plasmoids, the decoupling of the particles, the evolution of the heating mechanisms, and the composition. The presence of helium species leads to more efficient heating mechanisms than the two-fluid case. This scenario, which is out of reach of the two-fluid or single-fluid models, can reach transition region temperatures starting from upper-chromospheric thermodynamic conditions, representative of a quiet Sun scenario. The different dynamics between helium and hydrogen species could lead to chemical fractionation and, under certain conditions, enrichment of helium in the strongest outflows. This could be of significance for recent observations of helium enrichment in the solar wind in switchbacks and coronal mass ejections.

A multilevel block preconditioner for the HDG trace system applied to incompressible resistive MHD

We present a scalable block preconditioning strategy for the trace system arising from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block for which a system algebraic multigrid (AMG) is used to approximate the inverse. The complexity of the MHD equations together with the algebraic nature of the statically condensed HDG trace system makes the choice of smoother in the system AMG part critical for the convergence and performance of the block preconditioner. Our numerical experiments show GMRES preconditioned by ILU(0) of overlap zero as a smoother inside system AMG performs best in terms of robustness, time per nonlinear iteration and memory requirements. With several transient test cases in 2D and 3D including the island coalescence problem at high Lundquist number we demonstrate the robustness and parallel scalability of the block preconditioner. Additionally for the upper block a preliminary study of an alternate nodal block system solver based on a multilevel approximate nested dissection is presented. On a 2D island coalescence problem the multilevel approximate nested dissection preconditioner shows better scalability with respect to mesh refinement than the system AMG, but is relatively less robust with respect to Lundquist number scaling. In the Appendix B, we rigorously show: (1) the uniqueness of the solution of the nonlinear MHD system for small time, (2) the convergence of the Picard iterations for each backward Euler step, and (3) the convergence of the entire time stepping procedure.

Do chaotic field lines cause fast reconnection in coronal loops?

Over the past decade, Boozer has argued that three-dimensional (3D) magnetic reconnection fundamentally differs from two-dimensional reconnection due to the fact that the separation between any pair of neighboring field lines almost always increases exponentially over distance in a 3D magnetic field. According to Boozer, this feature makes 3D field-line mapping chaotic and exponentially sensitive to small non-ideal effects; consequently, 3D reconnection can occur without intense current sheets. We test Boozer's theory via ideal and resistive reduced magnetohydrodynamic simulations of the Boozer–Elder coronal loop model driven by sub-Alfvénic footpoint motions [A. H. Boozer and T. Elder, Phys. Plasmas 28, 062303 (2021)]. Our simulation results significantly differ from their predictions. The ideal simulation shows that Boozer and Elder under-predict the intensity of current density due to missing terms in their reduced model equations. Furthermore, resistive simulations of varying Lundquist numbers show that the maximal current density scales linearly rather than logarithmically with the Lundquist number.

Scaling of the Hosking integral in decaying magnetically dominated turbulence

The Saffman helicity invariant of Hosking & Schekochihin (Phys. Rev. X, vol. 11, issue 4, 2021, 041005), which we here call the Hosking integral, has emerged as an important quantity that may govern the decay properties of magnetically dominated non-helical turbulence. Using a range of different computational methods, we confirm that this quantity is indeed gauge invariant and nearly perfectly conserved in the limit of large Lundquist numbers. For direct numerical simulations with ordinary viscosity and magnetic diffusivity operators, we find that the solution develops in a nearly self-similar fashion. In a diagram quantifying the instantaneous decay coefficients of magnetic energy and integral scale, we find that the solution evolves along a line that is indeed suggestive of the governing role of the Hosking integral. The solution settles near a line in this diagram that is expected for a self-similar evolution of the magnetic energy spectrum. The solution will settle in a slightly different position when the magnetic diffusivity decreases with time, which would be compatible with the decay being governed by the reconnection time scale rather than the Alfvén time.

Thermally enhanced tearing in solar current sheets: Explosive reconnection with plasmoid-trapped condensations

Context. Thermal instability plays a major role in condensation phenomena in the solar corona (e.g., for coronal rain and prominence formation). In flare-relevant current sheets, tearing instability may trigger explosive reconnection and plasmoid formation. However, how the two instabilities influence the disruption of current concentrations in the solar corona has received less attention to date. Aims. We explore how the thermal and tearing modes reinforce each other in the fragmentation of a current sheet in the solar corona through an explosive reconnection process, characterized by the formation of plasmoids which interact and trap condensing plasma. Methods. We used a resistive magnetohydrodynamic simulation of a 2D current layer, incorporating the non-adiabatic effects of optically thin radiative energy loss and background heating using the open-source code MPI-AMRVAC. Multiple levels of adaptive mesh refined grids are used to achieve a high resolution to resolve the fine structures during the evolution of the system. Results. Our parametric survey explores different resistivities and plasma-β to quantify the instability growth rate in the linear and nonlinear regimes. We note that for dimensionless resistivity values within 10−4 − 5 × 10−3, we obtain explosive behavior where thermal instability and tearing behavior reinforce each other. This is clearly below the usual critical Lundquist number range of pure resistive explosive plasmoid formation. We calculate the mean growth rate for the linear phase and different nonlinear phases of the evolution. The nonlinear growth rates follow a weak power-law dependency with resistivity. The fragmentation of the current sheet and the formation of the plasmoids in the nonlinear phase of the evolution due to the thermal and tearing instabilities are obtained. The formation of plasmoids is found for the Lundquist number (SL) range 4.6 × 103 − 2.34 × 105. We quantify the temporal variation in the plasmoid numbers and the density filling factor of the plasmoids for different physical conditions. We also find that the maximum plasmoid numbers scale as SL0.223.Within the nonlinearly coalescing plasmoid chains, localized cool condensations gather, realizing density and temperature contrasts similar to coronal rain or prominences.

Two-dimensional Modeling of the Tearing-mode-governed Magnetic Reconnection in the Large-scale Current Sheet above the Two-ribbon Flare

We attempt to model magnetic reconnection during the two-ribbon flare in a gravitationally stratified solar atmosphere with the Lundquist number of S = 106 using 2D simulations. We found that the tearing mode instability leads to inhomogeneous turbulence inside the reconnecting current sheet (CS) and invokes the fast phase of reconnection. Fast reconnection brings an extra dissipation of magnetic field which enhances the reconnection rate in an apparent way. The energy spectrum in the CS shows a power law pattern and the dynamics of plasmoids govern the associated spectral index. We noticed that the energy dissipation occurs at a scale l ko of 100–200 km, and the associated CS thickness ranges from 1500 to 2500 km, which follows the Taylor scale l T = l ko S 1/6. The termination shock (TS) appears in the turbulent region above flare loops, which is an important contributor to heating flare loops. Substantial magnetic energy is converted into both kinetic and thermal energies via TS, and the cumulative heating rate is greater than the rate of the kinetic energy transfer. In addition, the turbulence is somehow amplified by TS, in which the amplitude is related to the local geometry of the TS.

Instabilities in a current sheet with plasma jet

We study the stability problem of a magnetohydrodynamic current sheet with the presence of a plasma jet. The flow direction is perpendicular to the normal of the current sheet and we analyse two cases: (1) the flow is along the antiparallel component of the magnetic field; (2) the flow is perpendicular to the antiparallel component of the magnetic field. A generalized equation set with the condition of incompressibility is derived and solved as a boundary value problem. For the first case we show that the streaming kink mode is stabilized by the magnetic field at $V_0/B_0 \lesssim 2$ , where $V_0$ and $B_0$ are the jet speed and upstream Alfvén speed, and it is not affected by resistivity significantly. The streaming sausage mode is stabilized at $V_0/B_0 \lesssim 1$ , and it can transit to the streaming tearing mode with a finite resistivity. The streaming tearing mode has larger growth rate than the pure tearing mode, though the scaling relation between the maximum growth rate and the Lundquist number remains unchanged. When the jet is perpendicular to the antiparallel component of the magnetic field, the most unstable sausage mode is usually perpendicular (wavevector along the jet) without a guide field. But with a finite guide field, the most unstable sausage mode can be oblique, depending on the jet speed and guide field strength.