Resistive effects on line-tied magnetohydrodynamic modes in cylindrical geometry

The tearing instability in a current sheet, which has a sub-Alfvénic or super-Alfvénic plasma flow in the current layer, is investigated based on the linearized compressible magnetohydrodynamic (MHD) equations. An initial-value method is employed to obtain the linear growth rate and eigenmode profiles of the fastest growing mode. The results show that for a sub-Alfvénic plasma flow parallel to the neutral sheet, the growth rate of the tearing instability is only slightly larger than that of the pure tearing mode without the flow. On the other hand, a large increase in the growth rate of the most unstable mode is observed, when the streaming speed V0m in the central region of the current sheet increases above a critical speed VC≂1.2VA∞. Here VA∞ is the Alfvén speed far away from the current layer. This study shows that when the electric resistivity η is zero, the sausage mode is excited because of a super-Alfvénic plasma flow parallel to the current sheet. This flow-induced sausage mode is called the streaming sausage mode. In the presence of a finite resistivity, the streaming sausage mode becomes a mixed sausage–tearing mode, because of the presence of magnetic field line reconnections in the current sheet. This mixed sausage–tearing mode, or simply the streaming tearing mode, has a high growth rate, γ≂0.1τ−1A, where τA is the Alfvén transit time across the current layer. This growth rate is larger than the growth rate of the pure tearing mode, which is approximately the inverse of the geometric mean of the Alfvén time τA and the diffusion time τD across the current layer.

Magnetic field topology frozen in ideal magnetohydrodynamics (MHD) and its breakage in near-ideal MHD are reviewed in two parts, clarifying and expanding basic concepts. The first part gives a physically complete description of the frozen field topology derived from magnetic flux conservation as the fundamental property, treating four conceptually related topics: Eulerian and Lagrangian descriptions of three dimensional (3D) MHD, Chandrasekhar-Kendall and Euler-potential field representations, magnetic helicity, and inviscid vortex dynamics as a fluid system in physical contrast to ideal MHD. A corollary of these developments clarifies the challenge of achieving a high degree of the frozen-in condition in numerical MHD. The second part treats field-topology breakage centered around the Parker Magnetostatic Theorem on a general incompatibility of a continuous magnetic field with the dual demand of force-free equilibrium and an arbitrarily prescribed, 3D field topology. Preserving field topology as a global constraint readily results in formation of tangential magnetic discontinuities, or, equivalently, electric current-sheets of zero thickness. A similar incompatibility is present in the steady force-thermal balance of a heated radiating fluid subject to an anisotropic thermal flux conducted strictly along its frozen-in magnetic field in the low-β limit. In a weakly resistive fluid the thinning of current sheets by these general incompatibilities inevitably results in sheet dissipation, resistive heating and topological changes in the field notwithstanding the small resistivity. Strong Faraday induction drives but also macroscopically limits this mode of energy dissipation, trapping or storing free energy in self-organized ideal-MHD structures. This property of MHD turbulence captured by the Taylor hypothesis is reviewed in relation to the Sun’s corona, calling for a basic quantitative description of the breakdown of flux conservation in the low-resistivity limit. A cylindrical initial-boundary value problem provides specificity in the general MHD ideas presented.

Interest in real-time control of magnetic field errors and magnetohydrodynamic (MHD) instabilities has been growing in the last decades due to the demanding stability requirements of high-performance scenarios in fusion devices. In this framework, the RFX-mod experiment (Sonato et al 2003 Fusion Eng. Des. 66 161) plays an important role. One of the main goals of RFX-mod is to explore high-plasma current regimes up to 2 MA for the first time in a reversed-field pinch. To this aim, RFX-mod is equipped with an advanced active coil system for the control of error fields and MHD modes, such as tearing and resistive-wall modes. As far as tearing modes are concerned, both controlling their edge radial magnetic field and maintaining them into slow (∼10–100 Hz) rotation are crucial to reduce both the plasma–wall interaction and the core magnetic stochasticity. In this paper, a model-based optimization of the RFX-mod feedback control is presented. The aim is to find an optimal gain set for a spectrum of multiple tearing modes, which produces the lowest possible value of the edge radial magnetic field, maintaining at the same time the modes into slow rotation and avoiding coil current saturations. These optimal gains have first been calculated offline by simulating the non-linear dynamics of a spectrum of tearing modes interacting through viscous and electromagnetic torques, using an adaptation to the RFX-mod multiple-shell layout of the model described in Zanca (2009 Plasma Phys. Control. Fusion 51 015006). This gain set has been obtained by scanning the proportional and derivative gains and has been tested in an extensive experimental campaign, showing good agreement with the model. With this approach, a reduction in the edge radial magnetic field up to 15%, with respect to discharges in which an empirical optimization was used, has been obtained. The above model proved to be a powerful tool to tune a multi-mode controller offline, which allowed us to save a large amount of experimental time.

Inertial confinement fusion (ICF) is a promising candidate for renewable energy generation. It involves the compression of a spherical capsule of deuterium-tritium fuel by the action of an imploding shock wave. In concept, the shock wave provides thermonuclear ignition of the fuel on convergence to the capsule centre, and the inward inertial motion of the surrounding fuel provides confinement long enough for the fusion burn to be sustained. The process, however, suffers from the presence of hydrodynamic instabilities in the confinement process. As the fluids involved in such processes are plasmas, they may be affected by magnetic fields. There is some evidence that application of a magnetic field to a planar flow in ideal magnetohydrodynamics (MHD) suppresses the Richtmyer-Meshkov instability, and the possibility exists of it suppressing this instability in converging flows as well. There is also experimental evidence suggesting increased performance of ICF due to electron and alpha particle confinement by the magnetic field. However, the application of a magnetic field to a converging plasma may disrupt the ability of the imploding shock to produce thermonuclear ignition. Here, the dynamic effects of the external application of magnetic fields (referred to as seed magnetic fields) to converging plasma flows such as those seen in ICF are investigated numerically and analytically under the framework of ideal magnetohydrodynamics (MHD). In the first study, cylindrical and spherical implosions in MHD are examined. We formulate Riemann problems to generate a full set of imploding waves and determine the symmetry features in the flow. Three configurations of seed magnetic fields are applied: a uniform, unidirectional field, an azimuthal, axisymmetric field which is zero in part of the domain, and a field with a stagnation point at the centre of the domain (referred to as a “saddle”-field). The application of a given field results in the breaking of the axi- or spherisymmetry (for cylindrical or spherical geometry respectively) of the implosion into a symmetry type which matches that of the applied field. The cylindrical geometry with the azimuthal, axisymmetric field therefore shows an axisymmetric implosion, unlike its spherical counterpart which shows a profound disruption of spherisymmetry. The least distorted shock wave implosions are provided by “saddle”-fields, in both cylindrical and spherical problems. Second, we consider the cylindrical collapse of a shock wave onto a constant current which runs along the cylinder axis. This current produces an axisymmetric magnetic field which varies with radius r as 1/r and which, unlike the similar field considered in the first study, is singular on the axis. The shock Mach number and pressure ratio are both found to weaken as M(r)-1∼r and p(r)-1∼r respectively, making the use of a constant current such as this unsuitable since it would inhibit thermonuclear fusion. By varying the current to zero precisely as the shock collapses onto it in a power-law decrease with decay exponent μ, as is done in the third study, the picture becomes more complicated. A strong dependency of shock behaviour on the choice of m shows up to five distinct behaviours. However, a strong-shock singular pressure of, at weakest, p(r)∼r^((4-13μ)/(4(μ+1))) could be assured by choosing μ>4/13. A choice of μ>0.816 could further ensure strong-shock collapse comparable to that of gas-dynamic shocks, with M(r)∼r^(-1/n) and p(r)∼r^(-2/n), where n≃0.225425 for the plasma of interest. Finally, we investigate the effect of the magnetic field on the Richtmyer-Meshkov instability in cylindrical and spherical numerical formulations similarly to the first study. The uniform, unidirectional and “saddle”-field configurations are considered only. A perturbed density interface is initialized on the interior of the initial Riemann interface, and the Richtmyer-Meshkov and Rayleigh-Taylor instabilities are provoked by the action of the incoming MHD shocks. Both instabilities are suppressed by the seed magnetic field. Suppression extent is largely insensitive to the choice of field configuration, but does show different behaviour depending on the local orientation of the magnetic field to the interface, as the vorticity which causes perturbation growth is transported along field lines. Three-dimensional magnetic fields show weaker suppression of the instabilities for a given field strength. The studies suggest that a seed magnetic field could be used to increase performance of ICF by suppression of hydrodynamic instabilities such as, in particular, the Richtmyer-Meshkov instability. However, the field should be judiciously chosen in both its configuration and strength in order to avoid the effects of distortion or weakening of imploding shock waves.

We model an imaged Kelvin-Helmholtz (KH) instability on a coronal mass ejecta (CME) in the lower corona by investigating conditions under which kink ($m = 1$) and $m = -3$ magnetohydrodynamic (MHD) modes in an uniformly twisted flux tube moving along its axis become unstable. We employ the dispersion relations of MHD modes derived from the linearised magnetohydrodynamic equations. We assume real wave numbers and complex angular wave frequencies, namely complex wave phase velocities. The dispersion relations are solved numerically at fixed input parameters (taken from observational data) and various mass flow velocities. It is shown that the stability of the modes depends upon four parameters, the density contrast between the flux tube and its environment, the ratio of the background magnetic fields in the two media, the twist of the magnetic field lines inside the tube, and the value of the Alfv\'en Mach number (the ratio of the tube velocity to Alfv\'en speed inside the flux tube). For a twisted magnetic flux tube at a density contrast of $0.88$, background magnetic field ratio of $1.58$, and a normalised magnetic field twist of $0.2$, the critical speed for the kink ($m = -3$) mode (where $m$ is the azimuthal mode number) is $678$ km\,s$^{-1}$ just as it is observed. The growth rate for this harmonic at KH wavelength of $18.5$ Mm and ejecta width of $4.1$ Mm is equal to $0.037$ s$^{-1}$, in agreement with observations. KH instability of the $m = -3$ mode may also explain why the KH vortices are seen only at the one side of arising CME. The good agreement between observational and computational data shows that the imaged KH instability on CME can be explained in terms of emerging KH instability of the $m = -3$ MHD mode in twisted magnetic flux tube moving along its axis.

Kelvin–Helmholtz instability in a twisting solar polar coronal hole jet observed by SDO/AIA

JOREK is a massively parallel fully implicit non-linear extended magneto-hydrodynamic (MHD) code for realistic tokamak X-point plasmas. It has become a widely used versatile simulation code for studying large-scale plasma instabilities and their control and is continuously developed in an international community with strong involvements in the European fusion research programme and ITER organization. This article gives a comprehensive overview of the physics models implemented, numerical methods applied for solving the equations and physics studies performed with the code. A dedicated section highlights some of the verification work done for the code. A hierarchy of different physics models is available including a free boundary and resistive wall extension and hybrid kinetic-fluid models. The code allows for flux-surface aligned iso-parametric finite element grids in single and double X-point plasmas which can be extended to the true physical walls and uses a robust fully implicit time stepping. Particular focus is laid on plasma edge and scrape-off layer (SOL) physics as well as disruption related phenomena. Among the key results obtained with JOREK regarding plasma edge and SOL, are deep insights into the dynamics of edge localized modes (ELMs), ELM cycles, and ELM control by resonant magnetic perturbations, pellet injection, as well as by vertical magnetic kicks. Also ELM free regimes, detachment physics, the generation and transport of impurities during an ELM, and electrostatic turbulence in the pedestal region are investigated. Regarding disruptions, the focus is on the dynamics of the thermal quench (TQ) and current quench triggered by massive gas injection and shattered pellet injection, runaway electron (RE) dynamics as well as the RE interaction with MHD modes, and vertical displacement events. Also the seeding and suppression of tearing modes (TMs), the dynamics of naturally occurring TQs triggered by locked modes, and radiative collapses are being studied.

The Kelvin-Helmholtz instability (KHI) is a standard test of hydrodynamic and magnetohydrodynamic (MHD) simulation codes and finds many applications in astrophysics. The classic linear theory considers a discontinuity in density and velocity at the interface of two fluids. However, for numerical simulations of the KHI such initial conditions do not yield converged results even at the linear stage of the instability. Instead, smooth profiles of velocity and density are required for convergence. This renders analytical theory to be only approximately valid and hinders quantitative comparisons between the classical theory and simulations. In this paper we derive a linear theory for the KHI with smooth profiles and illustrate code testing with the MHD code Athena. We provide the linear solution for the KHI with smooth initial conditions in three different limits: inviscid hydrodynamics, ideal MHD and Braginskii-MHD. These linear solutions are obtained numerically with the framework Psecas (Pseudo-Spectral Eigenvalue Calculator with an Automated Solver), which generates and solves numerical eigenvalue problems using an equation-parser and pseudo-spectral methods. The Athena simulations are carried out on a periodic, Cartesian domain which is useful for code testing purposes. Using Psecas and analytic theory, we outline the differences between this artificial numerical setup and the KHI on an infinite Cartesian domain and the KHI in cylindrical geometry. We discuss several astrophysical applications, such as cold flows in galaxy formation and cold fronts in galaxy cluster mergers. Psecas, and the linear solutions used for code testing, are publicly available and can be downloaded from the web.

Soft x-ray (SX) tomography is a useful diagnostic in fusion research, and a multi-channel SX diagnostic will be installed in JT-60SA, the largest elongated tokamak in the world. However, in the SX diagnostic of JT-60SA, plasmas will be only viewed from the low field side and the upper side of plasmas; the sight lines are limited, which would be common in future devices as well as JT-60SA. This kind of limited sight lines is not preferred for SX tomography to investigate the spatial structure of magnetohydrodynamics (MHD) modes because inadequate information of plasmas makes artifacts in the reconstructed SX profiles. One of the solutions to reduce the artifacts is to employ L1 regularization, which gives the essential and sparse contributions [Kaptanoglu etal., Phys. Plasmas 30, 033906 (2023)]. In this study, as a first topic, the applicability of L1 regularization to reduce the artifacts in SX tomography with limited sight lines is investigated with traditional L2 regularization for a high beta scenario of JT-60SA where MHD modes would occur. Here, as a series of basis functions, the Fourier-Bessel series (FBS) is employed because FBS has the poloidal Fourier modes explicitly. A disadvantage of FBS is that the accurate equilibrium inside the last closed flux surface (LCFS) is needed; interior measurement such as the motional Stark effect measurement is required, which is not always available during a whole discharge. The second topic of this study is to investigate other appropriate basis functions to study the spatial structure of MHD modes in elongated tokamak plasmas. Here, we introduce Saito's Laplacian eigenfunction (LEF). Saito's LEF can be calculated if LCFS is given and the LEF is expected to show the explicit poloidal Fourier mode. Because the calculation of LCFS with magnetic measurements is a basic task of plasma operations, Saito's LEF may be used anytime. Our investigation showed that L1 regularization can strongly improve the SX tomography with the traditional L2 regularization having FBS/LEF and would be effective against other tomographic problems in fusion devices.

The detailed poloidal structure of internal magnetohydrodynamic (MHD) modes is imaged using broadband visible bremsstrahlung emission from the core of the DIII-D tokamak. Spectral analysis of individual pixel time series recorded by a fast framing camera (up to 26,000 frames/s at 256x256 spatial resolution) is used to reconstruct two-dimensional images of mode amplitude. Application of this spectrally filtered fast imaging (SFFI) technique reveals spatially extended coherent structures that correspond to a (m,n)=(1,1) kink mode and a (2,1) neoclassical tearing mode rotating in the laboratory frame. The SFFI technique produces images with significantly less noise than images produced with the commonly used background subtraction method. Extension of SFFI to other core MHD events and coherent fluctuations in general is straightforward and could lead to further understanding of core MHD activity in fusion devices.

Forced reconnexion due to tearing modes driven by fast magnetosonic waves in a current sheet with stagnation-point flows is discussed. The current sheet with stagnation-point flows which is weakly unstable against tearing modes can be strongly destabilized by vortex motions due to the ponderomotive force of the fast magnetosonic wave. This forced tearing instability can be driven when the incident fast magnetosonic wave intensity, I, exceeds a critical value given by where is the Alfvén velocity, vg the group velocity of the fast wave, vo the background inflow velocity, l the thickness of the current sheet and k the wavenumber of the forced tearing mode. The growth rate is estimated. Applications to solar flares and magnetopause reconnexion processes are briefly discussed.

Tearing modes of finite amplitude are studied numerically using the “almost ideal magnetohydrodynamics (MHD)” constraint, which is a modification of the ideal MHD constraints that allows reconnection of the magnetic field lines. For the one-dimensional initial equilibria studied here, the stability criterion is found to be the same as that of the linear tearing mode theory, namely Δ′<0. The nonlinear saturation level of the mode can also be determined; it is found to be smaller than that estimated from Δ′(Wsat)=0 [R. B. White et al., Phys. Fluids 20, 800 (1977)].

Progress in the understanding of magnetohydrodynamic (MHD) stability is summarized on JT-60U tokamak discharges with improved confinement such as the (hot-ion) H-mode, high-βp mode, high-βp H-mode, and reversed shear discharges. Transport barriers, which are essential for the improved confinement, play key roles in the local and global MHD stability owing to the local large pressure gradient and the related bootstrap current. Disruptive β limits of these discharges are consistent with theoretical ideal kink-ballooning stability limits with low toroidal mode numbers n. Achievable β limit is improved by broadening of the pressure profile with high plasma internal inductance, plasma shaping, and wall stabilization. Edge localized modes (ELMs) and barrier localized modes (BLMs), which are associated with edge and internal transport barriers, respectively, are analyzed carefully. Resistive interchange modes with n ≤ 3 are excited in the negative shear region in reversed shear discharges with the internal transport barrier and lead to major collapse occasionally through nonlinear coupling with a tearing mode in the positive shear region. MHD characteristics of low m/n (m: poloidal mode number) tearing modes, which are attributed to the neoclassical tearing mode, are investigated. Stabilization of tearing modes and control of sawtooth activity are demonstrated using the fundamental O-mode electron cyclotron wave injection. Resistive wall modes associated with current-driven and pressure-driven low n external kink modes are identified.

Magnetic reconnection

Dissipation processes responsible for fast magnetic reconnection in collisionless plasmas are investigated using 3D electromagnetic particle-in-cell simulations. The present study revisits the two simulation runs performed in the previous study (Fujimoto, Phys. Plasmas 16, 042103 (2009)); one with small system size in the current density direction, and the other with larger system size. In the case with small system size, the reconnection processes are almost the same as those in 2D reconnection, while in the other case a kink mode evolves along the current density and deforms the current sheet structure drastically. Although fast reconnection is achieved in both the cases, the dissipation mechanism is very different between them. In the case without kink mode, the electrons transit the electron diffusion region without thermalization, so that the magnetic dissipation is supported by the inertia resistivity alone. On the other hand, in the kinked current sheet, the electrons are not only accelerated in bulk, but they are also partly scattered and thermalized by the kink mode, which results in the anomalous resistivity in addition to the inertia resistivity. It is demonstrated that in 3D reconnection the thickness of the electron current sheet becomes larger than the local electron inertia length, consistent with the theoretical prediction in Fujimoto and Sydora (Phys. Plasmas 16, 112309 (2009)).