Two-dimensional MHD Simulations Research Articles

Physical Regimes of Two-dimensional MHD Turbulent Reconnection in Different Lundquist Numbers

Abstract Using two-dimensional MHD simulations in different Lundquist numbers S, we investigate physical regimes of turbulent reconnection and the role of turbulence in enhancing the reconnection rate. Turbulence is externally injected into the system with varying strength. Externally driven turbulence contributes to the conversion of magnetic energy to kinetic energy flowing out of the reconnection site and thus enhances the reconnection rate. The plasmoids formed in high Lundquist numbers contribute to the fast reconnection rate, as well. Moreover, an analysis of the power of turbulence implies its possible association with the generation of plasmoids. Additionally, the presence of turbulence has great impact on the magnetic energy conversion and may be impactful also for the Kelvin–Helmholtz instability in the magnetic reconnection process.

Experimental observations of detached bow shock formation in the interaction of a laser-produced plasma with a magnetized obstacle

The magnetic field produced by planets with active dynamos, like the Earth, can exert sufficient pressure to oppose supersonic stellar wind plasmas, leading to the formation of a standing bow shock upstream of the magnetopause, or pressure-balance surface. Scaled laboratory experiments studying the interaction of an inflowing solar wind analog with a strong, external magnetic field are a promising new way to study magnetospheric physics and to complement existing models, although reaching regimes favorable for magnetized shock formation is experimentally challenging. This paper presents experimental evidence of the formation of a magnetized bow shock in the interaction of a supersonic, super-Alfvénic plasma with a strongly magnetized obstacle at the OMEGA laser facility. The solar wind analog is generated by the collision and subsequent expansion of two counter-propagating, laser-driven plasma plumes. The magnetized obstacle is a thin wire, driven with strong electrical currents. Hydrodynamic simulations using the FLASH code predict that the colliding plasma source meets the criteria for bow shock formation. Spatially resolved, optical Thomson scattering measures the electron number density, and optical emission lines provide a measurement of the plasma temperature, from which we infer the presence of a fast magnetosonic shock far upstream of the obstacle. Proton images provide a measure of large-scale features in the magnetic field topology, and reconstructed path-integrated magnetic field maps from these images suggest the formation of a bow shock upstream of the wire and as a transient magnetopause. We compare features in the reconstructed fields to two-dimensional MHD simulations of the system.

Fast-shear-flow generated multiple magnetic islands and Alfvénic resonance layers in magnetic reconnection

Abstract Magnetic reconnection in the presence of fast sheared plasma flows is investigated using two-dimensional incompressible resistive MHD simulation. It is found that if the initial shear-flow velocity is sufficiently large, multiple Alfvén resonance layers can be formed in the inflow region away from the reconnection separatrices. In particular, two Alfvén layers are formed when the initial asymptotic flow velocity is twice the Alfvén velocity. The Alfvén layers are located in the narrow regions where the flow speed equals or twice the local Alfvén speed. The formation and evolution of the Alfvén layers and magnetic islands are analyzed. It is suggested that the geometry of the magnetic field lines during the islands complex inosculated process is related to the local distribution of velocities. The results may be applied to where the magnetic reconnection occurs with large plasma shear flows in laboratory and space plasma.

Fast Magnetic Wave Could Heat the Solar Low-beta Chromosphere

Abstract Magnetohydrodynamic (MHD) waves are candidates for heating the solar chromosphere, although it is still unclear which mode of the wave is dominant in heating. We perform two-dimensional radiative MHD simulation to investigate the propagation of MHD waves in the quiet region of the solar chromosphere. We identify the mode of the shock waves by using the relationship between gas pressure and magnetic pressure across the shock front and calculate their corresponding heating rate through the entropy jump to obtain a quantitative understanding of the wave-heating process in the chromosphere. Our result shows that the fast magnetic wave is significant in heating the low-beta chromosphere. The low-beta fast magnetic waves are generated from high-beta fast acoustic waves via mode conversion crossing the equipartition layer. Efficient mode conversion is achieved by large attacking angles between the propagation direction of the shock waves and the chromospheric magnetic field.

Two-dimensional resistive MHD simulation of the optimized plasma formation in the spherical tokamaks

We present 2D global MHD simulation results of the optimized start-up scenarios for the spherical tokamak to maximize the ion heating energy determined by the value of the poloidal private flux during the merging start-ups. The series of simulations has successfully revealed the dependence of the poloidal private flux on the in-vessel poloidal coils’ separation length under the fixed amount of coils’ magnetic energy. The radial and azimuthal locations of the internal coils need to be optimized to maximize high private flux and consequently high ion heating energy. In the case of a two poloidal field (PF) coils system, the private flux and ion heating energy increase with the coils’ separation length. Installing two additional PF coils improves the private flux significantly, even if the coils’ separation length is short.

Fundamental Properties of Sheared/Guide Field MHD Magnetic Reconnection

Abstract The properties of the sheared/guide field magnetic reconnection (MRX) are investigated with two-dimensional MHD simulation. We simulate the spontaneous evolution from the isothermal current sheet (CS) equilibrium in which distribution of the thermodynamical quantities is symmetric about the CS. The magnetic shear is characterized by two parameters: the shear parameter and the asymmetry parameter. The asymmetry of the Alfvén speed (V A0x) perpendicular to the X-line along the CS is essential. We focus on the asymptotic self-similarly expanding phase of the evolution. This research is unique for the discussion based on the consistency across the entire MRX system, although the sheared MRX has been studied since the early 1980s. In addition to reconfirmation of the previously reported properties of the sheared MRX, the following new properties are found. (1) The reconnection jet changes to the “core–envelope structure” (a high-density core with a low-density envelope) for the sheared symmetric V A0x case but the “two-layered structure” (the high-speed, low-density layer and the medium-speed, high-density layer) for the asymmetric V A0x case. (2) The parameter dependence of the reconnection rate is clarified. The MRX is fastest for the symmetric case and slows as the asymmetry increases for any fixed shear angle. For the symmetric case, the reconnection rate has a monotonically decreasing dependence on the shear angle. (3) In the asymmetric case, the plasmas from both sides of the CS coexist on the same magnetic field lines in the larger V A0x side plasmoid. This characteristic structure suggests an efficient plasma mixing when the plasmoid breaks.

A Cancellation Nanoflare Model for Solar Chromospheric and Coronal Heating. II. 2D Theory and Simulations

Abstract Recent observations at high spatial resolution have shown that magnetic flux cancellation occurs on the solar surface much more frequently than previously thought, and so this led Priest et al. (2018) to propose magnetic reconnection driven by photospheric flux cancellation as a mechanism for chromospheric and coronal heating. In particular, they estimated analytically the amount of energy released as heat and the height of the energy release during flux cancellation. In the present work, we take the next step in the theory by setting up a two-dimensional resistive MHD simulation of two canceling polarities in the presence of a horizontal external field and a stratified atmosphere in order to check and improve upon the analytical estimates. Computational evaluation of the energy release during reconnection is found to be in good qualitative agreement with the analytical estimates. In addition, we go further and undertake an initial study of the atmospheric response to reconnection. We find that, during the cancellation, either hot ejections or cool ones or a combination of both hot and cool ejections can be formed, depending on the height of the reconnection location. The hot structures can have the density and temperature of coronal loops, while the cooler structures are suggestive of surges and large spicules.

On the effect of parallel shear flow on the plasmoid instability

The effect of symmetric sheared plasma flow parallel to the equilibrium magnetic field on the evolution of plasmoid instability is investigated by using two-dimensional MHD simulations. It is shown that as the velocity amplitude of a shear flow increases from sub-Alfvénic to the level of super-Alfvénic, the current sheet is stabilized against the plasmoid instability. Suppression of plasmoid instability also takes place for a sub-Alfvénic shear flow, provided that the shear length is sufficiently small compared to the magnetic shear. Therefore, an initially elongated thin current sheet with a Lundquist number greater than the critical value does not necessarily guarantee the development of the plasmoid instability. As a result, the Kelvin-Helmholtz (KH) instability is triggered. Although the rate of reconnection is quite fast in the nonlinear regime of the plasmoid instability, the reconnection of twisted field lines during the KH instability is very slow.

Quasi-periodic Oscillations in Flares and Coronal Mass Ejections Associated with Magnetic Reconnection

Abstract We propose a mechanism for quasi-periodic oscillations of both coronal mass ejections (CMEs) and flare loops as related to magnetic reconnection in eruptive solar flares. We perform two-dimensional numerical MHD simulations of magnetic flux rope eruption, with three different values of the global Lundquist number. In the low Lundquist number run, no oscillatory behavior is found. In the moderate Lundquist number run, on the other hand, quasi-periodic oscillations are excited both at the bottom of the flux rope and at the flare loop top. In the high Lundquist number run, quasi-periodic oscillations are also excited; in the meanwhile, the dynamics become turbulent owing to the formation of multiple plasmoids in the reconnection current sheet. In high and moderate Lundquist number runs, thin reconnection jets collide with the flux rope bottom or flare loop top and dig them deeply. Steep oblique shocks are formed as termination shocks where reconnection jets are bent (rather than decelerated) in the horizontal direction, resulting in supersonic backflows. The structure becomes unstable, and quasi-periodic oscillations of supersonic backflows appear at locally confined high-beta regions at both the flux rope bottom and flare loop top. We compare the observational characteristics of quasi-periodic oscillations in erupting flux ropes, post-CME current sheets, flare ribbons, and light curves with corresponding dynamical structures found in our simulation.

Numerical study of Hall effects on counter-helicity spheromak merging by two-dimensional Hall-MHD simulations

Hall effects on counter-helicity spheromak merging were investigated by two-dimensional MHD and Hall-MHD simulations of merging two axisymmetric toroidal flux tubes. In Hall-MHD cases, the structure of the reconnection current sheet and reconnection outflow are modified from the MHD case due to the Hall effect. We compared two cases (called “case-O” and “case-I”) of counter-helicity merging, which are distinguished by the polarity of toroidal magnetic fluxes. Radial motion of the reconnection X-point is controlled by poloidal electron flow accompanying the toroidal flux of the merging two spheromaks, and this creates a large difference in the current sheet and flow structure between the two cases of the Hall-MHD regime. The radial shift of the reconnection X-point depending on the polarity of toroidal magnetic flux of the spheromaks breaks the symmetry between the two cases. It was also found that there widely exists separation of ion and electron flow which are affected by the modification of the current sheet structure due to the radial shift of the X-point in the downstream side of the merging, and its spatial scale of the distribution of the Hall electric field is larger than the ion skin depth.

Nonlinear Dynamics of Non-uniform Current-Vortex Sheets in Magnetohydrodynamic Flows

A theoretical model is proposed to describe fully nonlinear dynamics of interfaces in two-dimensional MHD flows based on an idea of non-uniform current-vortex sheet. Application of vortex sheet model to MHD flows has a crucial difficulty because of non-conservative nature of magnetic tension. However, it is shown that when a magnetic field is initially parallel to an interface, the concept of vortex sheet can be extended to MHD flows (current-vortex sheet). Two-dimensional MHD flows are then described only by a one-dimensional Lagrange parameter on the sheet. It is also shown that bulk magnetic field and velocity can be calculated from their values on the sheet. The model is tested by MHD Richtmyer–Meshkov instability with sinusoidal vortex sheet strength. Two-dimensional ideal MHD simulations show that the nonlinear dynamics of a shocked interface with density stratification agrees fairly well with that for its corresponding potential flow. Numerical solutions of the model reproduce properly the results of the ideal MHD simulations, such as the roll-up of spike, exponential growth of magnetic field, and its saturation and oscillation. Nonlinear evolution of the interface is found to be determined by the Alfven and Atwood numbers. Some of their dependence on the sheet dynamics and magnetic field amplification are discussed. It is shown by the model that the magnetic field amplification occurs locally associated with the nonlinear dynamics of the current-vortex sheet. We expect that our model can be applicable to a wide variety of MHD shear flows.

MHD simulations of three-dimensional resistive reconnection in a cylindrical plasma column

Magnetic reconnection is a plasma phenomenon where a topological rearrangement of magnetic field lines with opposite polarity results in dissipation of magnetic energy into heat, kinetic energy and particle acceleration. Such a phenomenon is considered as an efficient mechanism for energy release in laboratory and astrophysical plasmas. An important question is how to make the process fast enough to account for observed explosive energy releases. The classical model for steady state magnetic reconnection predicts reconnection times scaling as $S^{1/2}$ (where $S$ is the Lundquist number) and yields times scales several order of magnitude larger than the observed ones. Earlier two-dimensional MHD simulations showed that for large Lundquist number the reconnection time becomes independent of $S$ ("fast reconnection" regime) due to the presence of the secondary tearing instability that takes place for $S \gtrsim 1 \times 10^4$. We report on our 3D MHD simulations of magnetic reconnection in a magnetically confined cylindrical plasma column under either a pressure balanced or a force-free equilibrium and compare the results with 2D simulations of a circular current sheet. We find that the 3D instabilities acting on these configurations result in a fragmentation of the initial current sheet in small filaments, leading to enhanced dissipation rate that becomes independent of the Lundquist number already at $S \simeq 1\times 10^3$.

Energy of Alfvén waves generated during magnetic reconnection

A new method for the determination of the Alfvén wave energy generated during magnetic reconnection is introduced and used to analyze the results from two-dimensional MHD simulations. It is found that the regions with strong Alfvén wave perturbations almost coincide with that where both magnetic-field lines and flow-stream lines are bent, suggesting that this method is reliable for identifying Alfvén waves. The magnetic energy during magnetic reconnection is mainly transformed into the thermal energy. The conversion rate to Alfvén wave energy from the magnetic energy is strongly correlated to the magnetic reconnection rate. The maximum conversion rate at the time with the peak reconnection rate is found to be only about 4% for the cases with the plasma beta β=0.01,0.1,and1.0.

Evolution of Kelvin-Helmholtz instability at Venus in the presence of the parallel magnetic field

Two-dimensional MHD simulations were performed to study the evolution of the Kelvin-Helmholtz (KH) instability at the Venusian ionopause in response to the strong flow shear in presence of the in-plane magnetic field parallel to the flow direction. The physical behavior of the KH instability as well as the triggering and occurrence conditions for highly rolled-up vortices are characterized through several physical parameters, including Alfvén Mach number on the upper side of the layer, the density ratio, and the ratio of parallel magnetic fields between two sides of the layer. Using these parameters, the simulations show that both the high density ratio and the parallel magnetic field component across the boundary layer play a role of stabilizing the instability. In the high density ratio case, the amount of total magnetic energy in the final quasi-steady status is much more than that in the initial status, which is clearly different from the case with low density ratio. We particularly investigate the nonlinear development of the case that has a high density ratio and uniform magnetic field. Before the instability saturation, a single magnetic island is formed and evolves into two quasi-steady islands in the non-linear phase. A quasi-steady pattern eventually forms and is embedded within a uniform magnetic field and a broadened boundary layer. The estimation of loss rates of ions from Venus indicates that the stabilizing effect of the parallel magnetic field component on the KH instability becomes strong in the case of high density ratio.

The structure of TeV-bright shell-type supernova remnants

Aims. Two-dimensional MHD simulations are used to model the emission properties of TeV-bright shell-type supernova remnants (SNRs) and to explore their nature. Methods. In the leptonic scenario for the TeV emission, the $\gamma$-ray emission is produced via Inverse Compton scattering of background soft photons by high-energy electrons accelerated by the shocks of the SNRs. The TeV emissivity is proportional to the magnetic field energy density and MHD simulations can be used to model the TeV structure of such remnants directly. 2D MHD simulations for SNRs are then performed under the assumption that the ambient interstellar medium is turbulent with the magnetic field and density fluctuations following a Kolmogorov-like power-law spectrum. Results. (1) As expected, these simulations confirm early 1D and 2D modelings of these sources, namely the hydrodynamical evolution of the shock waves and amplification of magnetic field by Rayleigh-Taylor convective flows and by shocks propagating in a turbulent medium; (2) We reproduce rather complex morphological structure for $\gamma$-rays, suggesting intrinsic variations of the source morphology not related to the structure of the progenitor and environment; (3)The observed radial profile of several remnants are well reproduced with an ambient medium density of $0.1-1$ cm$^{-3}$. An even lower ambient density leads to a sharper drop of the TeV brightness with radius than what is observed near the outer edge of these remnants. Conclusions. In a turbulent background medium, we can reproduce the observed characteristics of several shell-type TeV SNRs with reasonable parameters except for a higher ambient density than that inferred from X-ray observations.

SIMULATION STUDY OF SOLAR PLASMA ERUPTIONS CAUSED BY INTERACTIONS BETWEEN EMERGING FLUX AND CORONAL ARCADE FIELDS

We investigate the triggering mechanisms of plasma eruptions in the solar atmosphere due to interactions between emerging flux and coronal arcade fields by using two-dimensional MHD simulations. We perform parameter surveys with respect to arcade field height, magnetic field strength, and emerging flux location. Our results show that two possible mechanisms exist, and which mechanism is dominant depends mostly on emerging flux location. One mechanism appears when the location of emerging flux is close to the polarity inversion line (PIL) of an arcade field. This mechanism requires reconnection between the emerging flux and the arcade field, as pointed out by previous studies. The other mechanism appears when the location of emerging flux is around the edge of an arcade field. This mechanism does not require reconnection between the emerging flux and the arcade field but does demand reconnection in the arcade field above the PIL. Furthermore, we found that the eruptive condition for this mechanism can be represented by a simple formula.

Parametric decay of parallel and oblique Alfvén waves in the expanding solar wind

The long-term evolution of large-amplitude Alfvén waves propagating in the solar wind is investigated by performing two-dimensional MHD simulations within the expanding box model. The linear and nonlinear phases of the parametric decay instability are studied for both circularly polarized waves in parallel propagation and for arc-polarized waves in oblique propagation. The non-monochromatic case is also considered. In the oblique case, the direct excitation of daughter modes transverse to the local background field is found for the first time in an expanding environment, and this transverse cascade seems to be favored for monochromatic mother waves. The expansion effect reduces the instability growth rate, and it can even suppress its onset for the lowest frequency modes considered here, possibly explaining the persistence of these outgoing waves in the solar wind.

Magnetic field amplification and saturation in turbulence behind a relativistic shock

We have investigated via two-dimensional relativistic MHD simulations the long-term evolution of turbulence created by a relativistic shock propagating through an inhomogeneous medium. In the postshock region, magnetic field is strongly amplified by turbulent motions triggered by preshock density inhomogeneities. Using a long-simulation box we have followed the magnetic-field amplification until it is fully developed and saturated. The turbulent velocity is sub-relativistic even for a strong shock. Magnetic-field amplification is controled by the turbulent motion and saturation occurs when the magnetic energy is comparable to the turbulent kinetic energy. Magnetic-field amplification and saturation depend on the initial strength and direction of the magnetic field in the preshock medium, and on the shock strength. If the initial magnetic field is perpendicular to the shock normal, the magnetic field is first compressed at the shock and then can be amplified by turbulent motion in the postshock region. Saturation occurs when the magnetic energy becomes comparable to the turbulent kinetic energy in the postshock region. If the initial magnetic field in the preshock medium is strong, the postshock region becomes turbulent but significant field amplification does not occur. If the magnetic energy after shock compression is larger than the turbulent kinetic energy in the postshock region, significant field amplification does not occur. We discuss possible applications of our results to gamma-ray bursts and active galactic nuclei.

Active feedback stabilization of the flute instability in a mirror machine using field-aligned coils

A plasma confined in linear mirror machines is unstable even at low β, mainly because of the flute instability. One possible way to stabilize the plasma is to use active feedback to correct the plasma shape in real time. The theoretically investigated apparatus consists of feedback coils aligned with the magnetic field, immersed in a cold plasma around the hot core. When the current through the feedback coils changes, the plasma moves to conserve the magnetic flux via compressional Alfvén waves. An analytical model is used to find a robust feedback algorithm with zero residual currents. It is shown that due to the plasma's rotation, maximal stability is obtained with a large phase angle between the perturbations' modes and the feedback integral-like term. Finally, a two-dimensional MHD simulation implementing the above algorithm in fact shows stabilization of the plasma with zero residual currents.

MAGNETIC FIELD INTENSIFICATION BY THE THREE-DIMENSIONAL “EXPLOSION” PROCESS

We investigate an intensification mechanism for the magnetic field near the base of the solar convection zone that does not rely on differential rotation. Such mechanism in addition to differential rotation has been suggested by studies of flux emergence, which typically require field strength in excess of those provided by differential rotation alone. We study here a process in which potential energy of the superadiabatically stratified convection zone is converted into magnetic energy. This mechanism, know as explosion of magnetic flux tubes, has been previously studied in the thin flux tube approximation as well as two-dimensional MHD simulations, we expand the investigation to three-dimensional MHD simulations. Our main result is that enough intensification can be achieved in a three-dimensional magnetic flux sheet as long as the spatial scale of the imposed perturbation normal to the magnetic field is sufficiently large. When this spatial scale is small, the flux sheet tends to rise toward the surface, resulting in a significant decrease of the magnetic field amplification.