Magnetohydrodynamic Relaxation Research Articles

A Magnetohydrodynamic Relaxation Method for Non-force-free Magnetic Field in Magnetohydrostatic Equilibrium

A nonlinear force-free field (NLFFF) extrapolation is widely used to reconstruct the three-dimensional magnetic field in the solar corona from the observed photospheric magnetic field. However, the pressure gradient and gravitational forces are ignored in the NLFFF model, even though the photospheric and chromospheric magnetic fields are not in general force-free. Here we develop a magnetohydrodynamic (MHD) relaxation method that reconstructs the solar atmospheric (chromospheric and coronal) magnetic field as a non-force-free magnetic field (NFFF) in magnetohydrostatic equilibrium where the Lorentz, pressure gradient, and gravitational forces are balanced. The system of basic equations for the MHD relaxation method is derived, and mathematical properties of the system are investigated. A robust numerical solver for the system is constructed based on the modern high-order shock capturing scheme. Two-dimensional numerical experiments that include the pressure gradient and gravitational forces are also demonstrated.

A magnetic bald-patch flare in solar active region 11117

With SDO observations and a data-constrained magnetohydrodynamics (MHD) model, we identify a confined multi-ribbon flare that occurred on 2010 October 25 in solar active region 11117 as a magnetic bald patch (BP) flare with strong evidence. From the photospheric magnetic field observed by SDO/HMI, we find there are indeed magnetic BPs on the polarity inversion lines (PILs) which match parts of the flare ribbons. From the 3D coronal magnetic field derived from an MHD relaxation model constrained by the vector magnetograms, we find strikingly good agreement of the BP separatrix surface (BPSS) footpoints with the flare ribbons, and the BPSS itself with the hot flaring loop system. Moreover, the triggering of the BP flare can be attributed to a small flux emergence under the lobe of the BPSS, and the relevant change of coronal magnetic field through the flare is reproduced well by the pre-flare and post-flare MHD solutions, which match the corresponding pre- and post-flare AIA observations, respectively. Our work contributes to the study of non-typical flares that constitute the majority of solar flares but which cannot be explained by the standard flare model.

Ideal relaxation of the Hopf fibration

Ideal magnetohydrodynamics relaxation is the topology-conserving reconfiguration of a magnetic field into a lower energy state where the net force is zero. This is achieved by modeling the plasma as perfectly conducting viscous fluid. It is an important tool for investigating plasma equilibria and is often used to study the magnetic configurations in fusion devices and astrophysical plasmas. We study the equilibrium reached by a localized magnetic field through the topology conserving relaxation of a magnetic field based on the Hopf fibration in which magnetic field lines are closed circles that are all linked with one another. Magnetic fields with this topology have recently been shown to occur in non-ideal numerical simulations. Our results show that any localized field can only attain equilibrium if there is a finite external pressure, and that for such a field a Taylor state is unattainable. We find an equilibrium plasma configuration that is characterized by a lowered pressure in a toroidal region, with field lines lying on surfaces of constant pressure. Therefore, the field is in a Grad-Shafranov equilibrium. Localized helical magnetic fields are found when plasma is ejected from astrophysical bodies and subsequently relaxes against the background plasma, as well as on earth in plasmoids generated by, e.g., a Marshall gun. This work shows under which conditions an equilibrium can be reached and identifies a toroidal depression as the characteristic feature of such a configuration.

Summary of the 6th asia-pacific transport working group (APTWG) meeting

This report summarizes the contributions to, and discussions at, the 6th Asia-Pacific Transport Working Group (APTWG) meeting, held on 21–25 June 2016 at Korea University, Seoul, Republic of Korea. The objective of the meeting was to develop an integrated understanding of transport phenomena in magnetically confined plasmas. To accomplish this objective, four technical working groups were organized under the headings: (1) turbulence suppression and transport bifurcation, (2) effect of magnetic topology on transport and magnetohydrodynamics-turbulence interaction, (3) nonlocality and non-diffusive transport, and (4) Energetic particles and particle/impurity transport. A summary is also given of the three plenary review talks on impurity transport, the magnetohydrodynamics relaxation process in reversed field pinch, and recent experimental and modelling results on the quiescent H-mode operation.

MHD Relaxation with Flow in a Sphere

Relaxation process of magnetohydrodynamics (MHD) inside a sphere is investigated by a newly developed spherical grid system, Yin–Yang–Zhong grid. An MHD fluid with low viscosity and resistivity is confined in a perfectly conducting, stress-free, and thermally insulating spherical boundary. Starting from a simple and symmetric state (a ring-shaped magnetic flux with no flow), a dynamical relaxation process of the magnetic energy is numerically integrated. The system settles down to a quasi-stationary state after an initial phase of turbulence. The relaxed state has a highly symmetric structure of the flow in tandem with the magnetic field in which four swirls with the magnetic field are located on vertices of a regular square.

Yin–Yang–Zhong grid: An overset grid system for a sphere

For numerical simulations inside a sphere, an overset grid system, Yin-Yang-Zhong grid, is proposed. The Yin-Yang-Zhong grid is an extension of the Yin-Yang grid, which is widely used in various simulations in spherical shell geometry. The Yin-Yang grid is itself an overset grid system with two component grids, and a new component grid called Zhong is placed at the center of the Yin-Yang grid. The Zhong grid component is constructed on Cartesian coordinates. Parallelization is intrinsically embedded in the Yin-Yang-Zhong grid system because the Zhong grid points are defined with cuboid blocks that are decomposed domains for parallelization. The computational efficiency approaches the optimum as the process number increases. Quantitative test simulations are performed for a diffusion problem in a sphere with the Yin-Yang-Zhong grid. Correct decay rates are obtained by the simulations. Two other tests in magnetohydrodynamics (MHD) in a sphere are also performed. One is an MHD dynamo simulation, and the other is an MHD relaxation simulation in a sphere.

A METHOD FOR EMBEDDING CIRCULAR FORCE-FREE FLUX ROPES IN POTENTIAL MAGNETIC FIELDS

We propose a method for constructing approximate force-free equilibria in pre-eruptive configurations in which a thin force-free flux rope is embedded into a locally bipolar-type potential magnetic field. The flux rope is assumed to have a circular-arc axis, a circular cross-section, and electric current that is either concentrated in a thin layer at the boundary of the rope or smoothly distributed across it with a maximum of the current density at the center. The entire solution is described in terms of the magnetic vector potential in order to facilitate the implementation of the method in numerical magnetohydrodynamic (MHD) codes that evolve the vector potential rather than the magnetic field itself. The parameters of the flux rope can be chosen so that its subsequent MHD relaxation under photospheric line-tied boundary conditions leads to nearly exact numerical equilibria. To show the capabilities of our method, we apply it to several cases with different ambient magnetic fields and internal flux-rope structures. These examples demonstrate that the proposed method is a useful tool for initializing data-driven simulations of solar eruptions.

NONLINEAR FORCE-FREE EXTRAPOLATION OF THE CORONAL MAGNETIC FIELD BASED ON THE MAGNETOHYDRODYNAMIC RELAXATION METHOD

We develop a nonlinear force-free field (NLFFF) extrapolation code based on the magnetohydrodynamic (MHD) relaxation method. We extend the classical MHD relaxation method in two important ways. First, we introduce an algorithm initially proposed by cite{2002JCoPh.175..645D} to effectively clean the numerical errors associated with $nabla cdot vec{B}$. Second, the multi-grid type method is implemented in our NLFFF to perform direct analysis of the high-resolution magnetogram data. As a result of these two implementations, we successfully extrapolated the high resolution force-free field introduced by cite{1990ApJ...352..343L} with better accuracy in a drastically shorter time. We also applied our extrapolation method to the MHD solution obtained from the flux-emergence simulation by cite{2012ApJ...748...53M}. We found that NLFFF extrapolation may be less effective for reproducing areas higher than a half-domain, where some magnetic loops are found in a state of continuous upward expansion. However, an inverse S shaped structure consisting of the sheared and twisted loops formed in the lower region can be captured well through our NLFFF extrapolation method. We further discuss how well these sheared and twisted fields are reconstructed by estimating the magnetic topology and twist quantitatively.

MAGNETIC STRUCTURE PRODUCING X- AND M-CLASS SOLAR FLARES IN SOLAR ACTIVE REGION 11158

We study the three-dimensional magnetic structure of solar active region 11158, which produced one X-class and several M-class flares on 2011 February 13$-$16. We focus on the magnetic twist in four flare events, M6.6, X2.2, M1.0, and M1.1. The magnetic twist is estimated from the nonlinear force-free field extrapolated from the vector fields obtained from the Helioseismic and Magnetic Imager on board the Solar Dynamic Observatory using magnetohydrodynamic relaxation method developed by \cite{2011ApJ...738..161I}. We found that strongly twisted lines ranging from half-turn to one-turn twist were built up just before the M6.6- and X2.2 flares and disappeared after that. Because most of the twist remaining after these flares was less than half-turn twist, this result suggests that the buildup of magnetic twist over the half-turn twist is a key process in the production of large flares. On the other hand, even though these strong twists were also built up just before the M1.0 and M1.1 flares, most of them remained afterwords. Careful topological analysis before the M1.0 and M1.1 flares shows that the strongly twisted lines were surrounded mostly by the weakly twisted lines formed in accordance with the clockwise motion of the positive sunspot, whose footpoints are rooted in strong magnetic flux regions. These results imply that these weakly twisted lines might suppress the activity of the strongly twisted lines in the last two M-class flares.

ON THE SUPPORT OF SOLAR PROMINENCE MATERIAL BY THE DIPS OF A CORONAL FLUX TUBE

The dense prominence material is believed to be supported against gravity through the magnetic tension of dipped coronal magnetic field. For quiescent prominences, which exhibit many gravity-driven flows, hydrodynamic forces are likely to play an important role in the determination of both the large and small scale magnetic field distributions. In this study, we present the first steps toward creating three-dimensional magneto-hydrostatic prominence model where the prominence is formed in the dips of a coronal flux tube. Here 2.5D equilibria are created by adding mass to an initially force-free magnetic field, then performing a secondary magnetohydrodynamic relaxation. Two inverse polarity magnetic field configurations are studied in detail, a simple o-point configuration with a ratio of the horizontal field (B_x) to the axial field (B_y) of 1:2 and a more complex model that also has an x-point with a ratio of 1:11. The models show that support against gravity is either by total pressure or tension, with only tension support resembling observed quiescent prominences. The o-point of the coronal flux tube was pulled down by the prominence material, leading to compression of the magnetic field at the base of the prominence. Therefore tension support comes from the small curvature of the compressed magnetic field at the bottom and the larger curvature of the stretched magnetic field at the top of the prominence. It was found that this method does not guarantee convergence to a prominence-like equilibrium in the case where an x-point exists below the prominence flux tube. The results imply that a plasma beta of ~ 0.1 is necessary to support prominences through magnetic tension.

Computation of multi-region relaxed magnetohydrodynamic equilibria

We describe the construction of stepped-pressure equilibria as extrema of a multi-region, relaxed magnetohydrodynamic (MHD) energy functional that combines elements of ideal MHD and Taylor relaxation, and which we call MRXMHD. The model is compatible with Hamiltonian chaos theory and allows the three-dimensional MHD equilibrium problem to be formulated in a well-posed manner suitable for computation. The energy-functional is discretized using a mixed finite-element, Fourier representation for the magnetic vector potential and the equilibrium geometry; and numerical solutions are constructed using the stepped-pressure equilibrium code, SPEC. Convergence studies with respect to radial and Fourier resolution are presented.

A NEW CODE FOR NONLINEAR FORCE-FREE FIELD EXTRAPOLATION OF THE GLOBAL CORONA

Reliable measurements of the solar magnetic field are still restricted to the photosphere, and our present knowledge of the three-dimensional coronal magnetic field is largely based on extrapolation from photospheric magnetogram using physical models, e.g., the nonlinear force-free field (NLFFF) model as usually adopted. Most of the currently available NLFFF codes have been developed with computational volume like Cartesian box or spherical wedge while a global full-sphere extrapolation is still under developing. A high-performance global extrapolation code is in particular urgently needed considering that Solar Dynamics Observatory (SDO) can provide full-disk magnetogram with resolution up to $4096\times 4096$. In this work, we present a new parallelized code for global NLFFF extrapolation with the photosphere magnetogram as input. The method is based on magnetohydrodynamics relaxation approach, the CESE-MHD numerical scheme and a Yin-Yang spherical grid that is used to overcome the polar problems of the standard spherical grid. The code is validated by two full-sphere force-free solutions from Low & Lou's semi-analytic force-free field model. The code shows high accuracy and fast convergence, and can be ready for future practical application if combined with an adaptive mesh refinement technique.

Minimum energy states of the cylindrical plasma pinch in single-fluid and Hall magnetohydrodynamics

Relaxed states of a plasma column are found analytically in single-fluid and Hall magnetohydrodynamics (MHD). We perform complete minimization of the energy with constraints imposed by invariants inherent in the corresponding models. It is shown that the relaxed state in Hall MHD is a force-free magnetic field with uniform axial flow and/or rigid azimuthal rotation. In contrast, the relaxed states in single-fluid MHD are more complex due to the coupling between velocity and magnetic field. Cylindrically and helically symmetric relaxed states are considered for both models. Helical states may be time dependent and analogous to helical waves, propagating on a cylindrically symmetric background. Application of our results to reversed-field pinches (RFP) is discussed. The radial profile of the parallel momentum predicted by the single-fluid MHD relaxation theory is shown to be in reasonable agreement with experimental observation from the Madison symmetric torus RFP experiment.

Magnetohydrodynamic simulation on co- and counter-helicity merging of spheromaks and driven magnetic reconnection

A magnetohydrodynamic relaxation process of spheromak merging is studied by means of an axisymmetric numerical simulation. As a result of counter-helicity merging, a field-reversed configuration is obtained in the final state, while a larger spheromak is formed after co-helicity merging. In the counter-helicity case, a clear pressure profile of which iso-surfaces coincide with flux surfaces is generated by thermal transport of a poloidal flow induced by driven reconnection. It is also found that a sharp pressure gradient formed in the vicinity of a reconnection point causes a bouncing motion of spheromaks. According to the bounce motion, the reconnection rate changes repeatedly. As shown by the Tokyo University Spherical Torus No. 3 (TS-3) experiments [M. Yamada, et al., Phys. Rev. Lett. 65, 721 (1990)], furthermore, strong acceleration of a toroidal flow and reversal of a toroidal field in the counter-helicity merging were observed.

Vacuum UV spectroscopy at REPUTE-1

A vacuum UV spectroscopic measurement system has been constructed on the REPUTE-1 device, and electron temperature measurements using the line intensity ratio method have been successfully carried out for ultralow γ plasmas. The electron temperatures measured are consistent with the results obtained by Thomson scattering measurement. It is observed that the electron temperature decreases in the phases of magnetohydrodynamic relaxation.

Three-dimensional simulation study of the magnetohydrodynamic relaxation process in the solar corona. 1: Spontaneous generation of Taylor-Heyvaerts-Priest state

Three-dimensional simulation study of the magnetohydrodynamic relaxation process in the solar corona. 1: Spontaneous generation of Taylor-Heyvaerts-Priest state

Transit-time magnetic pumping induced by spontaneous fluctuations in a magnetohydrodynamic relaxation process

In the magnetohydrodynamic relaxation accompanying fast reconnections, magnetic fluctuations originating from kink-type instabilities yield a finite magnetic compression, and the corresponding transverse electric field is dissipated through transit-time damping. This dissipation mechanism does not change the helicity, while it dissipates the fluctuation energy to result in direct heating of ions. This is in contrast to a slow relaxation process based on the tearing mode turbulence, where the parallel electric field is predominantly dissipated, which accompanies preferential heating of electrons and dissipation of the helicity.

MHD simulation of the toroidal phase locking mechanism in a reversed field pinch plasma

The toroidal phase locking process of kink modes in a reversed field pinch (RFP) plasma is investigated in detail by means of magnetohydrodynamic (MHD) simulation. The physical mechanism of phase locking is clarified. The two most dominant linearly unstable kink modes govern the evolution of other kink modes, whereby phase locking takes place. It is confirmed that the phase locking process is not a special phenomenon of plasmas with a resistive boundary, but a common feature of the MHD relaxation process in RFPs. The relation between the phase locking process and the MHD relaxation process is briefly discussed. It is found that phase locking is reproduced cyclically in the sustainment process.

Three-dimensional numerical simulation of the multi-helicity magnetohydrodynamic relaxation process in low-q spheromaks

Three-dimensional magnetohydrodynamic computer simulations have been made on the dynamic behaviour of the high temperature spheromak plasma whose conductivity profile is peaked at the magnetic axis. On the assumption of stationary spatial profiles of the plasma conductivity, these simulations examine the transient process from the current peaking to the subsequent relaxation. They reveal low-q relaxations caused by multi-helicity (current driven) kink modes. The low-q relaxations are classified into two types: the fast-type relaxation without n = 2 mode saturation and the slow-type relaxation with n = 2 mode saturation. In these simulations, resistive current loss in the outer region of the plasma causes a peaking of the current density profile, resulting in a departure from the initial Taylor state to a low-q state. As the q value at the magnetic axis, q0, decreases to <0.5, the internal kink mode with a toroidal mode number n = 2 is first destabilized. The feature of the subsequent relaxation process depends on the degree of peaking of the conductivity profile at the magnetic axis, including its change during the relaxation phase. When the peaking of the conductivity profile is strong enough to decrease q0 to much less than 0.5, the higher mode (the n = 3 mode) is destabilized, which is found to trigger the subsequent relaxation. During the relaxation phase, the non-linear coupling of these n = 2 and 3 (and sometimes 4) modes leads to flux conversion from poloidal to toroidal and the configuration with the excessive poloidal flux can relax back to a state close to the Taylor state with a balanced ratio of the poloidal flux to the toroidal flux. This is the scenario of the fast-type relaxation. On the other hand, when the conductivity profile is weakly peaked, the slow decrease in q0 causes a slow growth of the n = 3 mode, resulting in the saturation of the n = 2 mode. Even if the coupling of the n = 2 and 3 modes triggers a relaxation, the relaxation event is weak, unclear and incomplete. This is the scenario of the slow-type relaxation. It is also found that if the peaked conductivity proflle is maintained during the relaxation phase, relaxation back to the Taylor state is less complete.

Simulation study of the self-sustainment mechanism in the reversed field pinch configuration

Three-dimensional magnetohydrodynamic (MHD) simulations are carried out in order to reveal the fundamental mechanism of the self-sustainment process in the reversed field pinch (RFP) plasma. It is confirmed that the RFP configuration is sustained in a cyclic process, where the MHD relaxation phase and the resistive diffusion phase appear cyclically and alternatively. In the MHD relaxation process, the RFP plasma approaches a Taylor's minimum energy state, but departs from there in the diffusion process. In other words, since MHD relaxation processes periodically release excess magnetic energy accumulated in the resistive diffusion phase, the RFP plasma can stay in the neighbourhood of the minimum energy state. The mechanism of this cyclic process is disclosed. Specifically, when at least two ideal kink (m = 1) modes become unstable, MHD relaxation can take place. This is because the MHD relaxation progresses through non-linear reconnection of the m = 0 mode, which is driven by non-linear coupling between the unstable kink modes. Therefore, self-sustainment processes can be achieved by the non-linear effects of, essentially, m = 0 and m = 1 modes. The quantitative dependence of the relaxation-diffusion cycle on the aspect ratio of the device is considered, along with its dependence on the magnetic Reynolds number. These results are consistent with recent experiments and indicate that a coherent oscillation, which is often observed in experiments, is necessary for self-sustainment. The influence of self-sustainment processes on particle confinement is briefly discussed.