Resistive Magnetohydrodynamics Research Articles

Hall magnetohydrodynamic reconnection in the plasmoid unstable regime

A set of reduced Hall magnetohydrodynamic (MHD) equations are used to evaluate the stability of large aspect ratio current sheets to the formation of plasmoids (secondary islands). Reconnection is driven by resistivity in this analysis, which occurs at the resistive skin depth dη≡SL-1/2LνA/γ, where SL is the Lundquist number, L, the length of the current sheet, νA, the Alfvén speed, and γ, the growth rate. Modifications to a recent resistive MHD analysis [N. F. Loureiro et al., Phys. Plasmas 14, 100703 (2007)] arise when collisions are sufficiently weak that dη is shorter than the ion skin depth di ≡ c/ωpi. Secondary islands grow faster in this Hall MHD regime: the maximum growth rate scales as (di/L)6/13SL7/13νA/L and the number of plasmoids as (di/L)1/13SL11/26, compared to SL1/4νA/L and S3/8, respectively, in resistive MHD.

Zero-β modeling of coaxial helicity injection in the HIT-II spherical torus

Coaxial helicity injection in the HIT-II [A. J. Redd et al., Phys. Plasmas 9, 2006 (2002)] spherical torus is modeled with time-dependent resistive magnetohydrodynamics computations run to steady state for conditions without strong relaxation. Laboratory and computed results on injector current and plasma current agree reasonably well as toroidal magnetic field and injector flux are scaled. The scalings are consistent with a dimensional estimate from the Grad-Shafranov equation that provides a new perspective on a previously published model based on a current-sheet equilibrium and the magnetic pressure required for the “bubble-burst” criterion. Numerical solutions of the Grad-Shafranov equation with an assumed current profile also indicate large qualitative changes as the predicted criterion is crossed.

Magnetic Reconnection Under Solar Coronal Conditions with the 2.5D AMR Resistive MHD Model

The evolutionary process of magnetic reconnection under solar coronal conditions is investigated with our recently developed 2.5D adaptive mesh refinement (AMR) resistive magneto hydrodynamics (MHD) model. We reveal the successive fragmentation and merging of plasmoids in a long-thin current sheet with Lundquist number Rm = 5.0 × 104. It is found that several big magnetic islands are formed eventually, with many slow-mode shocks bounding around the outflow regions. The multi-scale hierarchical-like structures of the magnetic reconnection are well resolved by the model and the AMR technique of the model can capture many fine pictures (e.g., the near-singular diffusion regions) of the development and simultaneously it can save a great deal of computing resources.

The role of the electron convection term for the parallel electric field and electron acceleration in MHD simulations

There has been a great concern about the origin of the parallel electric field in the frame of fluid equations in the auroral acceleration region. This paper proposes a new method to simulate magnetohydrodynamic (MHD) equations that include the electron convection term and shows its efficiency with simulation results in one dimension. We apply a third-order semi-discrete central scheme to investigate the characteristics of the electron convection term including its nonlinearity. At a steady state discontinuity, the sum of the ion and electron convection terms balances with the ion pressure gradient. We find that the electron convection term works like the gradient of the negative pressure and reduces the ion sound speed or amplifies the sound mode when parallel current flows. The electron convection term enables us to describe a situation in which a parallel electric field and parallel electron acceleration coexist, which is impossible for ideal or resistive MHD.

A Modified Third-Order Semi-Discrete Central-Upwind Scheme for MHD Simulation

The Kurganov scheme is a third-order semi-discrete central numerical algorithm. The high solution of the scheme is ensured by a piecewise quadratic non-oscillatory reconstruction which consists of the cell-average data. We employ a modification of the smooth limiter of reconstruction in a simple way. The modified limiter possesses rigorous positivity and the reformulation does not change the non-oscillatory property of reconstruction. In order to explore the potential capability of application of the modified Kurganov scheme to magnetohydrodynamics (MHD) and resistive magnetohydrodynamics (RMHD) equations, two numerical problems are simulated in two dimensions (2D). These numerical simulations demonstrate that the modified Kurganov scheme keeps high precision and has stable reliable results for MHD and RMHD applications.

Effect of Magnetic Braking on Circumstellar Disk Formation in a Strongly Magnetized Cloud

Abstract Using resistive magnetohydrodynamics simulation, we consider circumstellar disk formation in a strongly magnetized cloud. As the initial state, an isolated cloud core embedded in a low-density interstellar medium with a uniform magnetic field was adopted. The cloud evolution was calculated until almost all gas inside the initial cloud fell onto either the circumstellar disk or a protostar, and a part of the gas was ejected into the interstellar medium by the protostellar outflow driven by the circumstellar disk. In the early main accretion phase, the disk size is limited to $\sim\ $10 AU because the angular momentum of the circumstellar disk is effectively transferred by both magnetic braking and the protostellar outflow. In the later main accretion phase, however, the circumstellar disk grows rapidly and exceeds $\gtrsim\ $100 AU by the end of the main accretion phase. This rapid growth of the circumstellar disk is caused by depletion of the infalling envelope, while magnetic braking is effective when the infalling envelope is more massive than the circumstellar disk. The infalling envelope cannot brake the circumstellar disk when the latter is more massive than the former. In addition, the protostellar outflow weakens and disappears in the later main accretion phase, because the outflow is powered by gas accretion onto the circumstellar disk. Although the circumstellar disk formed in a magnetized cloud is considerably smaller than that in an unmagnetized cloud, a circumstellar disk exceeding 100 AU can form even in a strongly magnetized cloud.

A NEW NUMERICAL SCHEME FOR RESISTIVE RELATIVISTIC MAGNETOHYDRODYNAMICS USING METHOD OF CHARACTERISTICS

We present a new numerical method of special relativistic resistive magnetohydrodynamics with scalar resistivity that can treat a range of phenomena, from nonrelativistic to relativistic (shock, contact discontinuity, and Alfv\'en wave). The present scheme calculates the numerical flux of fluid by using an approximate Riemann solver, and electromagnetic field by using the method of characteristics. Since this scheme uses appropriate characteristic velocities, it is capable of accurately solving problems that cannot be approximated as ideal magnetohydrodynamics and whose characteristic velocity is much lower than light velocity. The numerical results show that our scheme can solve the above problems as well as nearly ideal MHD problems. Our new scheme is particularly well suited to systems with initially weak magnetic field, and mixed phenomena of relativistic and non-relativistic velocity; for example, MRI in accretion disk, and super Alfv\'enic turbulence.

A new matching method for linear stability analysis of MHD modes close to the ideal MHD marginal stability

A new matching method has been invented for linear stability analysis of magnetohydrodynamics (MHD) modes for plasmas marginally stable against ideal MHD. An inner region with a finite width is utilized as in our previous study [M. Furukawa, S. Tokuda, and L.-J. Zheng, Phys. Plasmas 17, 052502 (2010)]. An ordering scheme for the outer region has been newly developed, thereby it succeeds to include effects of small plasma inertia and resistivity perturbatively in the outer region. The corresponding boundary condition requires direct, not asymptotic, matching of the outer and inner solutions, which assumes nothing special for the behavior of parallel electric field across the matching points. The union of the ordering scheme and the boundary condition enables us to apply our matching method even for plasmas marginally stable against ideal MHD. Because our matching method is not asymptotic, it is easy to implement numerically. The stability analysis of resistive MHD modes, such as internal kink and tearing modes, is satisfactory.

Predictor-based multivariable closed-loop system identification of the EXTRAP T2R reversed field pinch external plasma response

The usage of computationally feasible overparametrized and nonregularized system identification signal processing methods is assessed for automated determination of the full reversed-field pinch external plasma response spectrum for the experiment EXTRAP T2R. No assumptions on the geometry of eigenmodes are imposed. The attempted approach consists of high-order autoregressive exogenous estimation followed by Markov block coefficient construction and Hankel matrix singular value decomposition. It is seen that the obtained ‘black-box’ state-space models indeed can be compared with the commonplace ideal magnetohydrodynamics (MHD) resistive thin-shell model in cylindrical geometry. It is possible to directly map the most unstable autodetected empirical system pole to the corresponding theoretical resistive shell MHD eigenmode.

SPATIAL DAMPING OF PROPAGATING KINK WAVES DUE TO RESONANT ABSORPTION: EFFECT OF BACKGROUND FLOW

Observations show the ubiquitous presence of propagating magnetohydrodynamic (MHD) kink waves in the solar atmosphere. Waves and flows are often observed simultaneously. Due to plasma inhomogeneity in the perpendicular direction to the magnetic field, kink waves are spatially damped by resonant absorption. The presence of flow may affect the wave spatial damping. Here, we investigate the effect of longitudinal background flow on the propagation and spatial damping of resonant kink waves in transversely nonuniform magnetic flux tubes. We combine approximate analytical theory with numerical investigation. The analytical theory uses the thin tube (TT) and thin boundary (TB) approximations to obtain expressions for the wavelength and the damping length. Numerically, we verify the previously obtained analytical expressions by means of the full solution of the resistive MHD eigenvalue problem beyond the TT and TB approximations. We find that the backward and forward propagating waves have different wavelengths and are damped on length scales that are inversely proportional to the frequency as in the static case. However, the factor of proportionality depends on the characteristics of the flow, so that the damping length differs from its static analogue. For slow, sub-Alfvenic flows the backward propagating wave gets damped on a shorter length scale than in the absence of flow, while for the forward propagating wave the damping length is longer. The different properties of the waves depending on their direction of propagation with respect to the background flow may be detected by the observations and may be relevant for seismological applications.

Necessary criterion for magnetic field reversal in the reversed-field pinch

We derive analytically a necessary criterion for the reversal of the axial magnetic field of ohmic single-helicity states of the reversed-field pinch. This is done in the frame of resistive magnetohydrodynamics (MHD) in cylindrical geometry using perturbation theory for a paramagnetic pinch with low edge conductivity and axial magnetic field. The criterion involves the radial profile of the logarithmic derivative of the Newcomb eigenfunction of the pinch. It is suggestive that a finite edge radial magnetic field Br(a) might be favourable for field reversal. In accordance with this, visco-resistive MHD simulations show that helical equilibria with smaller maximum radial magnetic fields achieve reversal when a finite Br(a) is applied. Numerical simulations also show that the criterion works for large perturbations of the pinch too, in particular those leading to states with a single helical axis. The necessary criterion is found to be satisfied for reversed states where a finite Br(a) is driven in the RFX-mod machine.

Quasi-separatrix layers and three-dimensional reconnection diagnostics for line-tied tearing modes

In three-dimensional magnetic configurations for a plasma in which no closed field line or magnetic null exists, no magnetic reconnection can occur, by the strictest definition of reconnection. A finitely long pinch with line-tied boundary conditions, in which all the magnetic field lines start at one end of the system and proceed to the opposite end, is an example of such a system. Nevertheless, for a long system of this type, the physical behavior in resistive magnetohydrodynamics (MHD) essentially involves reconnection. This has been explained in terms comparing the geometric and tearing widths [1, 2]. The concept of a quasi-separatrix layer[3, 4] was developed for such systems. In this paper we study a model for a line-tied system in which the corresponding periodic system has an unstable tearing mode. We analyze this system in terms of two magnetic field line diagnostics, the squashing factor[3-5] and the electrostatic potential difference used in kinematic reconnection studies[6, 7]. We discuss the physical and geometric significance of these two diagnostics and compare them in the context of discerning tearing-like behavior in line-tied modes. [1] G. L. Delzanno and J. M. Finn. Physics of Plasmas, 15(3):032904, 2008. [2] Y.-M. Huang and E. G. Zweibel. Physics of Plasmas, 16(4):042102, 2009. [3] E. R. Priest and P. D\'emoulin. J. Geophys. Res., 100(A12):23443-23463, 1995. [4] P. D\'emoulin, J. C. Henoux, E. R. Priest, and C. H. Mandrini. Astron. Astrophys., 308:643-655, Apr. 1996. [5] V. S. Titov and G. Hornig. Advances in Space Research, 29(7):1087-1092, 2002. [6] Y. Lau and J. M. Finn. The Astrophysical Journal, 350:672-691, Feb. 1990. [7] Y. Lau and J. M. Finn. The Astrophysical Journal, 366:577-591, 1991.

Profile effect of shear flows on double tearing modes

The profile effect of shear flows on double tearing modes is investigated in a two-dimensional resistive magnetohydrodynamics model. It is found that the flow shear between the resonant surfaces rather than the flow shear on an individual resonant surface is the dominant effect on island suppression.

First-order finite-Larmor-radius effects on magnetic tearing in pinch configurations

The linear and nonlinear evolution of a single-helicity tearing mode in a cylindrical, force-free pinch are investigated using a fluid model with first-order finite-Larmor-radius corrections. Linear results computed with the nimrod [nonideal magnetohydrodynamics (MHD) with rotation, open discussion] code [Sovinec et al., J. Comput. Phys. 195, 355 (2004)] produce a regime at small ρs where the growth rate is reduced relative to resistive MHD, though the Hall term is not significant. The leading order contributions from ion gyroviscosity may be expressed as a drift associated with ∇B0 and poloidal curvature for experimentally relevant β=0.1, S~105-106 force-free equilibria. The heuristic analytical dispersion relation, γ4(γ-iω*gv)=γMHD5 where ω*gv is the gyroviscous drift frequency, confirms numerical results. The behavior of our cylindrical computations at large ρs corroborates previous analytic slab studies where an enhanced growth rate and radially localized Hall dynamo are predicted. Similar to previous drift-tearing results, nonlinear computations with cold ions demonstrate that the Hall dynamo is small when the island width is large in comparison with the scale for electron–ion coupling. The saturation is then determined by the resistive MHD physics. However, with warm ions the gyroviscous stress supplements the nonlinear Lorentz force, and the saturated island width is reduced.

Relaxation model for extended magnetohydrodynamics: Comparison to magnetohydrodynamics for dense Z-pinches

It is shown that the two-fluid model under a generalized Ohm’s law formulation and the resistive magnetohydrodynamics (MHD) can both be described as relaxation systems. In the relaxation model, the under-resolved stiff source terms constrain the dynamics of a set of hyperbolic equations to give the correct asymptotic solution. When applied to the collisional two-fluid model, the relaxation of fast time scales associated with displacement current and finite electron mass allows for a natural transition from a system where Ohm’s law determines the current density to a system where Ohm’s law determines the electric field. This result is used to derive novel algorithms, which allow for multiscale simulation of low and high frequency extended-MHD physics. This relaxation formulation offers an efficient way to implicitly advance the Hall term and naturally simulate a plasma-vacuum interface without invoking phenomenological models. The relaxation model is implemented as an extended-MHD code, which is used to analyze pulsed power loads such as wire arrays and ablating foils. Two-dimensional simulations of pulsed power loads are compared for extended-MHD and MHD. For these simulations, it is also shown that the relaxation model properly recovers the resistive-MHD limit.

Lattice Boltzmann formulation for Braginskii magnetohydrodynamics

Braginskii magnetohydrodynamics is a single-fluid description of large-scale motions in a strongly magnetised plasma, one in which the ion gyroradius is much smaller than the mean free path between collisions in an unmagnetised plasma of the same density and temperature. The relation between stress and strain rate then becomes highly anisotropic, with the viscous stress being predominantly aligned parallel to the magnetic field lines. We modify an existing lattice Boltzmann formulation of isotropic resistive magnetohydrodynamics to simulate Braginskii magnetohydrodynamics by introducing an anisotropic collision operator for the hydrodynamic distribution functions. This collision operator applies a different relaxation time to the component of stress directed parallel to the magnetic field. The method is illustrated by comparison with independent numerical solutions for steady planar channel flows.

Oscillation regimes of the internal kink mode in tokamak plasmas

The long-term dynamics of the internal kink mode are studied using the non-linear, two-fluid magnetohydrodynamics (MHD) code XTOR-2F (Lütjens et al 2010 J. Comput. Phys. 229 8130). The dynamics of the internal kink are studied in resistive MHD plus transport, and, in addition, also including some two-fluid effects. The simulation results yield a pattern of kink cycles or stationary states with m/n = 1/1 helicity. The threshold for sustained (non-decaying) kink cycles, which characterize sawtooth oscillations, is shown to depend on the plasma pressure, the current and energy diffusion times, and the ion and electron diamagnetic drifts. In particular, the diamagnetic flow stabilization extends the cyclic kink regime into a parameter space that is adequate to carry out studies of sawtooth oscillations in ohmic tokamak discharges.

Control of ideal and resistive magnetohydrodynamic modes in reversed field pinches with a resistive wall

Numerical studies of magnetohydrodynamic (MHD) instabilities with feedback control in reversed field pinches (RFPs) are presented. Specifically, investigations are performed of the stability of m=1 modes in RFPs with control based on sensing the normal and tangential magnetic fields at the resistive wall and applying two-parameter feedback proportional to these fields. The control scheme is based on that of [J. M. Finn, Phys. Plasmas 13, 082504 (2006)], which is here modified to use a more realistic plasma model. The plasma model now uses full resistive MHD rather than reduced MHD, and it uses three realistic classes of equilibrium parallel current density profiles appropriate to RFPs. Results with these modifications are in qualitative agreement with [J. M. Finn, Phys. Plasmas 13, 082504 (2006)]: the feedback can stabilize tearing modes (with resistive or ideal-wall) and resistive wall ideal modes. The limit for stabilization is again found to be near the threshold for ideal modes with an ideal-wall. In addition to confirming these predictions, the nature of the instabilities limiting the range of feedback stabilization near the ideal-wall ideal-plasma threshold are studied, and the effects of viscosity, resistive wall time, and plasma resistivity are reported.

AMR Simulations of Magnetohydrodynamic Problems by the CESE Method in Curvilinear Coordinates

The objective of this paper is to present new extensions of the space – time conservation element and solution element (CESE) method for simulations of magnetohydrodynamic (MHD) problems in general curvilinear coordinates by using an adaptive mesh refinement (AMR) grid system. By transforming the governing MHD equations from the physical space (x,y,z) to the computational space (ξ,η,ζ) while retaining the form of conservation, the CESE method is established for MHD in the curvilinear coordinates. Utilizing the parallel AMR package PARAMESH, we present the first implementation of applying the AMR CESE method for MHD (AMR-CESE-MHD) in both Cartesian and curvilinear coordinates. To show the validity and capabilities of the AMR-CESE-MHD code, a suite of numerical tests in two and three dimensions including ideal MHD and resistive MHD are carried out, with two of them in both Cartesian and curvilinear coordinates. Numerical tests show that our results are highly consistent with those obtained previously by other authors, and the results under both coordinate systems confirm each other very well.

2D Magnetohydrodynamics simulations of induced plasma dynamics in the near-core region of a galaxy cluster

We present results from numerical simulations of the cooling-core cluster A2199 produced by the two-dimensional (2-D) resistive magnetohydrodynamics (MHD) code MACH2. In our simulations we explore the effect of anisotropic thermal conduction on the energy balance of the system. The results from idealized cases in 2-D axisymmetric geometry underscore the importance of the initial plasma density in ICM simulations, especially the near-core values since the radiation cooling rate is proportional to ${n_e}^2$. Heat conduction is found to be non-effective in preventing catastrophic cooling in this cluster. In addition we performed 2-D planar MHD simulations starting from initial conditions deliberately violating both thermal balance and hydrostatic equilibrium in the ICM, to assess contributions of the convective terms in the energy balance of the system against anisotropic thermal conduction. We find that in this case work done by the pressure on the plasma can dominate the early evolution of the internal energy over anisotropic thermal conduction in the presence of subsonic flows, thereby reducing the impact of the magnetic field. Deviations from hydrostatic equilibrium near the cluster core may be associated with transient activity of a central active galactic nucleus and/or remnant dynamical activity in the ICM and warrant further study in three dimensions.