Resistive MHD Equations Research Articles

Effect of fast electrons on the stability of resistive interchange modes in the TJ-II stellarator

In this paper, we report on electromagnetic phenomena in low-β plasmas at the TJ-II stellarator, controlled by external heating. To understand the observations qualitatively, we introduce a simple modification of the standard resistive MHD equations, to include the potential impact of fast electrons on instabilities. The dominant instabilities of the modeling regime are resistive interchange modes, and calculations are performed in a configuration with similar characteristics as the TJ-II stellarator. The main effect of the trapping of fast electrons by magnetic islands induced by MHD instabilities is to increase the magnetic component of the fluctuations, changing the character of the instability to tearing-like and modifying the frequency of the modes. These effects seem to be consistent with some of the experimental observations.

Application of MFS–MPS to the current-hole simulation in a Tokamak

A meshless method based on fundamental and particular solutions (MFS–MPS) has been implemented for the current-hole simulation in cylindrical tokamaks. We first benchmark the method by solving the Grad–Shafranov (GS) equation for zero as well as nonzero inverse aspect ratios and noncircular cross-sections. Thereafter, the method is implemented to solve the time-dependent reduced resistive MHD equations for the current-hole simulation. The initial current density profile with a negative current density near the center is chosen and the corresponding equilibrium magnetic flux profile is first computed using the GS equation. These profiles are then used to solve the coupled nonlinear system of MHD equations using MFS–MPS and a semi-implicit time differencing scheme. After an initial linear phase extending up to a few thousand Alfvén times, the nonlinear oscillations are observed which continue for several Alfvén times. This oscillating behavior is in agreement with the previous simulations using other numerical methods. It is also found that these oscillations are damped out at higher resistivity.

Bifurcation of magnetic island saturation controlled by plasma viscosity

Two nonlinear regimes, depending on the magnetic Prandtl number Prm, are identified for the magnetic islands described by resistive MHD equations. The frontier between these two regimes is sharp, and has the characteristics of a phase transition controlled by plasma viscosity. In the low Prm regime, a new form of the so-called flip instability, consisting of a sudden change in the island phase, is identified. Already known in the context of the forcing by external magnetic perturbations and localized current drive, it occurs spontaneously at low Prm. The main characteristics of this new structural instability are described. The low Prm regime is well described by the slab visco-resistive model in the linear phase, and is characterized by both a large saturation of the island and strong nonlinearly driven zonal flows (that do not significantly impact the island dynamics, however), while curvature physics strongly impacts the viscous regime.

Space–time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations

In this paper, we address the space–time decay properties for strong solutions to the incompressible viscous resistive Hall-MHD equations. We obtained the same space–time decay rates as those of the heat equation. Based on the temporal decay results in [10] , we find that one can obtain weighted estimates of the magnetic field B by direct weighted energy estimate, and then by regarding the magnetic convection term as a forcing term in the velocity equations, we can obtain the weighted estimates for the vorticity, which yields the corresponding estimates for the velocity field. The higher order derivative estimates will be obtained by using a parabolic interpolation inequality proved in [22] . It should be emphasized that the magnetic field has stronger decay properties than the velocity field in the sense that there is no restriction on the exponent of the weight. The same arguments also yield the sharp space–time decay rates for strong solutions to the usual MHD equations.

On Arbitrary-Lagrangian-Eulerian One-Step WENO Schemes for Stiff Hyperbolic Balance Laws

AbstractIn this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws. High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkin method recently proposed in [20]. In the Lagrangian framework considered here, the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points. The moving space-time elements are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE. We show some computational examples in one space-dimension for non-stiff and for stiff balance laws, in particular for the Euler equations of compressible gas dynamics, for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.

MAGNETOSPHERIC ACCRETION AND EJECTION OF MATTER IN RESISTIVE MAGNETOHYDRODYNAMIC SIMULATIONS

The ejection of matter in the close vicinity of a young stellar object is investigated, treating the accretion disk as a gravitationally bound reservoir of matter. By solving the resistive MHD equations in 2D axisymmetry using our version of the Zeus-3D code with newly implemented resistivity, we study the effect of magnetic diffusivity in the magnetospheric accretion-ejection mechanism. Physical resistivity was included in the whole computational domain so that reconnection is enabled by the physical as well as the numerical resistivity. We show, for the first time, that quasi-stationary fast ejecta of matter, which we call {\em micro-ejections}, of small mass and angular momentum fluxes, can be launched from a purely resistive magnetosphere. They are produced by a combination of pressure gradient and magnetic forces, in presence of ongoing magnetic reconnection along the boundary layer between the star and the disk, where a current sheet is formed. Mass flux of micro-ejection increases with increasing magnetic field strength and stellar rotation rate, and is not dependent on the disk to corona density ratio and amount of resistivity.

Formation of magnetic flux tubes in cylindrical wedge geometry

Three-dimensional (3D) MHD numerical simulations have not been able to demonstrate convincingly the spontaneous formation of large vertical flux tubes. Two-dimensional (2D) magnetoconvection in axisymmetric cylinders forms a central magnetic flux tube surrounded by annular convection rings. To study the robustness of this type of solution in three dimensions, the nonlinear resistive MHD equations are solved numerically in a 3D cylindrical wedge from an initially uniform vertical magnetic field. It is shown that the 2D result is retrieved for small domain radii. However, for larger radii the central axis loses its importance and in this case many convection cells form in the numerical domain. Magnetic flux is captured between cells where flow converges and the reduced amount of flux that congregates at the central axis is eroded by the surrounding convection.

On fan-shaped cold MHD winds from Keplerian accretion discs

We investigate under which conditions cold, fan-shaped winds can be steadily launched from thin (Keplerian) accretion discs. Such winds are magneto-centrifugal winds launched from a thin annulus in the disc, along open magnetic field lines that fan out above the disc. In principle, such winds could be found in two situations: (1) at the interface between an inner Jet Emitting Disc, which is itself powering magneto-centrifugally driven winds, and an outer standard accretion disc; (2) at the interface between an inner closed stellar magnetosphere and the outer standard accretion disc. We refer to Terminal or T-winds to the former kind and to Magnetospheric or M-winds to the latter. The full set of resistive and viscous steady state MHD equations are analyzed for the disc (the annulus), which allow us to derive general expressions valid for both configurations. We find that, under the framework of our analysis, the only source of energy able to power any kind of fan-shaped winds is the viscous transport of rotational energy coming below the inner radii. Using standard local $\alpha$ prescriptions for the anomalous (turbulent) transport of angular momentum and magnetic fields in the disc, we derive the strength of the transport coefficients that are needed to steadily sustain the global configuration. It turns out that, in order for these winds to be dynamically relevant and explain observed jets, the disc coefficients must be far much larger than values expected from current knowledge of turbulence occurring inside proto-stellar discs. Either the current view on MHD turbulence must be deeply reconsidered or steady-state fan-shaped winds are never realized in Nature. The latter hypothesis seems to be consistent with current numerical simulations.

BIPOLAR JETS LAUNCHED FROM MAGNETICALLY DIFFUSIVE ACCRETION DISKS. I. EJECTION EFFICIENCY VERSUS FIELD STRENGTH AND DIFFUSIVITY

We investigate the launching of jets and outflows from magnetically diffusive accretion disks. Using the PLUTO code we solve the time-dependent resistive MHD equations taking into account the disk and jet evolution simultaneously. The main question we address is which kind of disks do launch jets and which kind of disks do not? In particular, we study how the magnitude and distribution of the (turbulent) magnetic diffusivity affect mass loading and jet acceleration. We have applied a turbulent magnetic diffusivity based on \alpha-prescription, but have also investigate examples where the scale height of diffusivity is larger than that of the disk gas pressure. We further investigate how the ejection efficiency is governed by the magnetic field strength. Our simulations last for up to 5000 dynamical time scales corresponding to 900 orbital periods of the inner disk. As a general result we observe a continuous and robust outflow launched from the inner part of the disk, expanding into a collimated jet of super fast magneto-sonic speed. For long time scales the disk internal dynamics changes, as due to outflow ejection and disk accretion the disk mass decreases. For magneto-centrifugally driven jets we find that for i) less diffusive disks, ii) a stronger magnetic field, iii) a low poloidal diffusivity, or a iv) lower numerical diffusivity (resolution), the mass loading of the outflow is increased - resulting in more powerful jets with high mass flux. For weak magnetization the (weak) outflow is driven by the magnetic pressure gradient. We further investigate the jet asymptotic velocity and the jet rotational velocity in respect of the different launching scenarios. We find a lower degree of jet collimation than previous studies, most probably due to our revised outflow boundary condition.

Group velocity and causality in standard relativistic resistive magnetohydrodynamics

Group velocity of electromagnetic waves in plasmas derived by standard relativistic resistive MHD (resistive RMHD) equations is superluminal. If we assume that the group velocity represents the propagation velocity of a signal, we have to worry about the causality problem. That is, some acausal phenomena may be induced, such that information transportation to the absolute past and spontaneous decrease in the entropy. Here, we tried to find the acausal phenomena using standard resistive RMHD numerical simulations in the suggested situation of the acausal phenomena. The calculation results showed that even in such situations no acausal effect happens. The numerical result with respect to the velocity limit of the information transportation is consistent with a linear theory of wave train propagation. Our results assure that we can use these equations without problems of acausal phenomena.

Intuitive approach to magnetic reconnection

Two reconnection problems are considered. The first problem concerns global physics. The plasma in the global reconnection region is in magnetostatic equilibrium. It is shown that this equilibrium can be uniquely characterized by a set of constraints. During reconnection and independently of the local reconnection physics, these constraints can be uniquely evolved from any initial state. The second problem concerns Petschek reconnection. Petschek’s model for fast reconnection, which is governed by resistive MHD equations with constant resistivity is not validated by numerical simulations. Malyshkin et al. [Phys. Plasmas 12, 102920 (2005)], showed that the reason for the discrepancy is that Petschek did not employ Ohm’s law throughout the local diffusion region, but only at the X-point. A derivation of Petschek reconnection, including Ohm’s law throughout the entire diffusion region, removes the discrepancy. This derivation is based largely on Petschek’s original 1964 calculation [in AAS-NASA Symposium on Solar Flares (National Aeronautics and Space Administration, Washington, D.C., 1964), NASA SP50, p. 425]. A useful physical interpretation of the role which Ohm’s law plays in the diffusion region is presented.

Resistive MHD reconstruction of two-dimensional coherent structures in space

Abstract. We present a reconstruction technique to solve the steady resistive MHD equations in two dimensions with initial inputs of field and plasma data from a single spacecraft as it passes through a coherent structure in space. At least two components of directly measured electric fields (the spacecraft spin-plane components) are required for the reconstruction, to produce two-dimensional (2-D) field and plasma maps of the cross section of the structure. For convenience, the resistivity tensor η is assumed diagonal in the reconstruction coordinates, which allows its values to be estimated from Ohm's law, E+v×B=η·j. In the present paper, all three components of the electric field are used. We benchmark our numerical code by use of an exact, axi-symmetric solution of the resistive MHD equations and then apply it to synthetic data from a 3-D, resistive, MHD numerical simulation of reconnection in the geomagnetic tail, in a phase of the event where time dependence and deviations from 2-D are both weak. The resistivity used in the simulation is time-independent and localized around the reconnection site in an ellipsoidal region. For the magnetic field, plasma density, and pressure, we find very good agreement between the reconstruction results and the simulation, but the electric field and plasma velocity are not predicted with the same high accuracy.

Influence of poloidal equilibrium rotation in MHD simulations of edge-localized modes

In nonlinear MHD simulations of edge localized modes (ELMs) with the code JOREK, a strong m = n = 0 poloidal flow localized in the H-mode pedestal region shears density filaments off the separatrix. This flow is nonlinearly induced by the ballooning instability (Huysmans and Czarny 2007 Nucl. Fusion 47 659, Huysmans et al 2009 Plasma Phys. Control. Fusion 51 124012). An approach of the reduced resistive MHD equations used in JOREK leads to an analytical interpretation of the equilibrium flow in both circular and X-point plasmas. Transitions between two different flow structures have been found. Similar transitions between equilibrium flow states had been predicted by Strauss (1995 Phys. Plasmas 2 1229). Finally, results concerning the influence of the equilibrium flow on the linear and nonlinear stabilities of the ELM are presented. ELM-like multiple crashes were simulated.

Flux emergence and coronal eruption

Our aim is to study the photospheric flux distribution of a twisted flux tube that emerges from the solar interior. We also report on the eruption of a new flux rope when the emerging tube rises into a pre-existing magnetic field in the corona. To study the evolution, we use 3D numerical simulations by solving the time-dependent and resistive MHD equations. We qualitatively compare our numerical results with MDI magnetograms of emerging flux at the solar surface. We find that the photospheric magnetic flux distribution consists of two regions of opposite polarities and elongated magnetic tails on the two sides of the polarity inversion line (PIL), depending on the azimuthal nature of the emerging field lines and the initial field strength of the rising tube. Their shape is progressively deformed due to plasma motions towards the PIL. Our results are in qualitative agreement with observational studies of magnetic flux emergence in active regions (ARs). Moreover, if the initial twist of the emerging tube is small, the photospheric magnetic field develops an undulating shape and does not possess tails. In all cases, we find that a new flux rope is formed above the original axis of the emerging tube that may erupt into the corona, depending on the strength of the ambient field.

Recurrent solar jets in active regions

We study the emergence of a toroidal flux tube into the solar atmosphere and its interaction with a pre-existing field of an active region. We investigate the emission of jets as a result of repeated reconnection events between colliding magnetic fields. We perform 3D simulations by solving the time-dependent, resistive MHD equations in a highly stratified atmosphere. A small active region field is constructed by the emergence of a toroidal magnetic flux tube. A current structure is build up and reconnection sets in when new emerging flux comes into contact with the ambient field of the active region. The topology of the magnetic field around the current structure is drastically modified during reconnection. The modification results in a formation of new magnetic systems that eventually collide and reconnect. We find that reconnection jets are taking place in successive recurrent phases in directions perpendicular to each other, while in each phase they release magnetic energy and hot plasma into the solar atmosphere. After a series of recurrent appearance of jets, the system approaches an equilibrium where the efficiency of the reconnection is substantially reduced. We deduce that the emergence of new magnetic flux introduces a perturbation to the active region field, which in turn causes reconnection between neighboring magnetic fields and the release of the trapped energy in the form of jet-like emissions. This is the first time that self-consistent recurrency of jets in active regions is shown in a three-dimensional experiment of magnetic flux emergence.

Formation of Ellerman bombs due to 3D flux emergence

Aims. We investigate the emergence of a “sea-serpent” magnetic field into the outer solar atmosphere and the connection between undulating fieldlines and formation of Ellerman bombs. Methods. We perform 3D numerical experiments solving the time-dependent and resistive MHD equations. Results. A sub-photospheric magnetic flux sheet develops undulations due to the Parker instability. It rises from the convectively unstable sub-photospheric layer and emerges into the highly stratified atmosphere through successive reconnection events along the undulating system. Brightenings with the characteristics of Ellerman bombs are produced due to reconnection, which occurs during the emergence of the field. At an advanced stage of the evolution of the system, the resistive emergence leads to the formation of long, arch-like magnetic fields that expand into the corona. The enhancement of the magnetic field at the low atmosphere and episodes of emergence of new magnetic flux are also discussed.

Observations and 3D MHD simulations of a solar active region jet

Aims. We study an active region jet originating from NOAA 8531 on May 15 1999. We perform 3D MHD numerical simulations of magnetic flux emergence and its subsequent reconnection with preexisting magnetic flux. Then, we compare the physical properties of the observed jet with the reconnecting outflow produced in the numerical model. Methods. We report observations of this jet using a series of TRACE 171 A filtergrams, simultaneous observations from SUMER in Ne viii 770A and C iv 1548 A as well as MDI magnetograms. In the numerical simulation, the full compressible and resistive MHD equations are solved, including viscous and Ohmic heating. Results. A high-velocity upflow ($\simeq$100 km s -1 ) is observed after the emergence of new magnetic flux at the edge of the active region. The jet is recorded over a range of temperatures between 10 5 K and $1.5 \times 10^6$ K. In our numerical experiments, we find that the jet is the result of magnetic reconnection between newly emerging flux and the preexisting magnetic field of the active region. Conclusions. The hot and high-velocity bidirectional flows occur as a result of the interaction between oppositely directed magnetic fields. Observations and numerical results are strongly suggestive of effective reconnection process being responsible for producing jets when emerging flux appears in solar active regions.

MAGNETOHYDRODYNAMIC WAVES IN A PARTIALLY IONIZED FILAMENT THREAD

Oscillations and propagating waves are commonly seen in high-resolution observations of filament threads, i.e., the fine-structures of solar filaments/prominences. Since the temperature of prominences is typically of the order of 10^4 K, the prominence plasma is only partially ionized. In this paper, we study the effect of neutrals on the wave propagation in a filament thread modeled as a partially ionized homogeneous magnetic flux tube embedded in an homogeneous and fully ionized coronal plasma. Ohmic and ambipolar magnetic diffusion are considered in the basic resistive MHD equations. We numerically compute the eigenfrequencies of kink, slow, and Alfven linear MHD modes, and obtain analytical approximations in some cases. We find that the existence of propagating modes is constrained by the presence of critical values of the longitudinal wavenumber. In particular, the lower and upper frequency cut-offs of kink and Alfven waves owe their existence to magnetic diffusion parallel and perpendicular to magnetic field lines, respectively. The slow mode only has a lower frequency cut-off, which is caused by perpendicular magnetic diffusion and is significantly affected by the ionization degree. In addition, ion-neutral collisions is the most efficient damping mechanism for short wavelengths while ohmic diffusion dominates in the long-wavelength regime.

Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations

In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperbolic and contains a source term, the one for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization we propose to use high order P N P M schemes as introduced in Dumbser et al. [M. Dumbser, D. Balsara, E.F. Toro, C.D. Munz, A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes, Journal of Computational Physics 227 (2008) 8209–8253] for hyperbolic conservation laws and a high order accurate unsplit time-discretization is achieved using the element-local space–time discontinuous Galerkin approach proposed in Dumbser et al. [M. Dumbser, C. Enaux, E.F. Toro, Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, Journal of Computational Physics 227 (2008) 3971–4001] for one-dimensional balance laws with stiff source terms. The divergence-free character of the magnetic field is accounted for through the divergence cleaning procedure of Dedner et al. [A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, M. Wesenberg, Hyperbolic divergence cleaning for the MHD equations, Journal of Computational Physics 175 (2002) 645–673]. To validate our high order method we first solve some numerical test cases for which exact analytical reference solutions are known and we also show numerical convergence studies in the stiff limit of the RRMHD equations using P N P M schemes from third to fifth order of accuracy in space and time. We also present some applications with shock waves such as a classical shock tube problem with different values for the conductivity as well as a relativistic MHD rotor problem and the relativistic equivalent of the Orszag–Tang vortex problem. We have verified that the proposed method can handle equally well the resistive regime and the stiff limit of ideal relativistic MHD. For these reasons it provides a powerful tool for relativistic astrophysical simulations involving the appearance of magnetic reconnection.

Can magnetic breakout be achieved from multiple flux emergence?

Aims. We study the breakout model using multiple flux emergence to produce the magnetic configuration and the trigger. We do not impose any artificial motions on the boundaries. Once the original flux tube configuration is chosen the system is left to evolve itself. Methods. We perform non-linear simulations in 2.5D by solving the compressible and resistive MHD equations using a Lagrangian remap, shock capturing code (Lare2D). To produce a quadrupolar configuration from flux emergence we build on previous work where the interaction of two flux tubes forms the required quadrupole. Instead of imposing a shearing flow, a third flux tube is then allowed to emerge up through the central arcade. Results. Breakout is not achieved in any of the experiments. This is due to the interaction of the third tube with the quadrupole and the effect of the plasma β being O(1) at the photosphere and β > O(1) in the solar interior. When β is of these orders, flows generated in the plasma can influence the magnetic field and so photospheric footpoints do not remain fixed.