Double Tearing Modes Research Articles

Radio Emission of Pulsars. I. Slow Tearing of a Quantizing Magnetic Field

The pulsed radio emission of rotating neutron stars is connected to slow tearing instabilities feeding off an inhomogeneous twist profile within the open circuit. This paper considers the stability of a weakly sheared, quantizing magnetic field in which the current is supported by a relativistic particle flow. The electromagnetic field is almost perfectly force free, and particles are confined to the lowest Landau state, experiencing no appreciable curvature drift. In a charge-neutral plasma, we find multiple branches of slowly growing tearing modes, relativistic analogs of the double-tearing mode, with peak growth rate . Here, B z is the strong (nearly potential) guide magnetic field, J z the field-aligned current density, and is the mode wavenumber normalized by the current gradient scale. These modes are overstable when the plasma carries a net charge, with the real frequency proportional to the imbalance in the densities of positive and negative charges. An isolated current sheet thinner than the skin depth supports localized tearing modes with growth rate scaling as (sheet thickness/skin depth)−1/2. In a pulsar, the peak growth rate is comparable to the angular frequency of rotation, , slow compared with the longitudinal oscillations of particles and fields in a polar gap. The tearing modes experience azimuthal drift reminiscent of subpulse drift and are a promising driver of pulse-to-pulse flux variations. A companion paper demonstrates a Cerenkov-like instability of current-carrying Alfvén waves in thin current sheets with relativistic particle flow and proposes coherent curvature emission by these waves as a source of pulsar radio emission.

Linear analysis of plasma pressure-driven mode in reversed shear cylindrical tokamak plasmas

The linear behavior of the dominant unstable mode (m = 2, n = 1) and its high order harmonics (m = 2n, n ≥ 2) are numerically investigated in a reversed magnetic shear cylindrical plasma with two q = 2 rational surfaces on the basis of the non-reduced magnetohydrodynamics (MHD) equations. The results show that with low beta (beta is defined as the ratio of plasma pressure to magnetic field pressure), the dominant mode is a classical double tearing mode (DTM). However, when the beta is sufficiently large, the mode is driven mainly by plasma pressure. In such a case, both the linear growth rate and mode structures are strongly affected by pressure, while almost independent of the resistivity. This means that the dominant mode undergoes a transition from DTM to pressure-driven mode with the increase of pressure, which is consistent with the experimental result in ASDEX Upgrade. The simulations also show that the distance between two rational surfaces has an important influence on the pressure needed in mode transition. The larger the distance between two rational surfaces, the larger the pressure for driving the mode transition is. Motivated by the phenomena that the high-m modes may dominate over low-m modes at small inter-resonance distance, the high-m modes with different pressures and q profiles are studied too.

Sawtooth-like oscillations and steady states caused by the m/n = 2/1 double tearing mode

The sawtooth-like oscillations resulting from the double tearing mode (DTM) are numerically investigated through the three-dimensional, toroidal, nonlinear resistive-MHD code (CLT). We find that the nonlinear evolution of the DTM can lead to sawtooth-like oscillations, which are similar to those driven by the kink mode. The perpendicular thermal conductivity and the external heating rate can significantly alter the behaviors of the DTM driven sawtooth-like oscillations. With a high perpendicular thermal conductivity, the system quickly evolves into a steady state with magnetic islands and helical flow. However, with a low perpendicular thermal conductivity, the system tends to exhibit sawtooth-like oscillations. With a sufficiently high or low heating rate, the system exhibits sawtooth-like oscillations, while with an intermediate heating rate, the system quickly evolves into a steady state. At the steady state, there exist the non-axisymmetric magnetic field and strong radial flow, and both are with helicity of Like the steady state with radial flow, which is beneficial for preventing the helium ash accumulation in the core, the steady state with radial flow might also be a good candidate for the advanced steady state operations in future fusion reactors. We also find that the behaviors of the sawtooth-like oscillations are almost independent of tokamak geometry, which implies that the steady state with saturated islands might exist in different tokamaks.

Numerical study on nonlinear double tearing mode in ITER

The nonlinear dynamics of the m/n = 2/1 double tearing mode (DTM) in ITER are systematically studied using the three-dimensional toroidal magnetohydrodynamic code, CLT. We carefully investigate the effects of the radial locations and magnetic shear strengths of the inner and outer rational surfaces r 1, r 2, s 1 and s 2, as well as the safety factor at the magnetic axis q 0 on DTM. It is found that the explosive burst takes place only with the moderate separation of the two rational surfaces or the stronger magnetic shear strength in which the strong interaction of magnetic islands in the two rational surfaces happens in the early nonlinear phase of the island development. The explosive burst can result from either the direct mutual driving associated with the fast growth island in the two rational surfaces or a strong nonlinear mode–mode coupling. For a large separation and a weak shear strength of the two rational surfaces, the magnetic islands saturate without strong interaction with each other, and (w in + w out)/2 is always below the separation Δr s. For a small separation, the kinetic evolution of DTM only exhibits an oscillation with a very low level and then decreases.

Influence of circulating energetic ions on MHD instabilities in negative magnetic shear plasma

The influence of circulating energetic ions (CEIs) on double tearing mode (DTM) is investigated by a global kinetic-MHD hybrid simulation. It is shown that the co-CEIs with a small orbit width have a destabilizing effect on the DTM. As the orbit width increases, this destabilizing effect is reduced. If the orbit width is large enough, the DTM is transformed to single tearing modes due to the co-CEIs, so the mode becomes more stable than the DTM without CEIs. The effect of the counter-CEIs with small orbit width is similar to that of the co-CEIs. For a large orbit width, the counter-CEIs also have a stabilizing effect. If the energetic ion beta is larger than a threshold, an energetic particle driven mode (EPM) is excited due to the resonance between counter-CEIs and the bulk plasma. The effects of energetic ion beta, orbit width and resistivity on this EPM are studied systematically. The onset threshold of EPM depends on the orbit width and resistivity. It decreases with orbit width increasing, while it increases with resistivity increasing. The mode transition from DTM to EPM with energetic ion beta increasing is also found for large orbit width.

Gyrokinetic simulations of double tearing modes in toroidal plasma

An investigation of the characteristics of double-tearing modes (DTMs) and the influence of the kinetic effects of ions has been carried out with the gyrokinetic code GTC. A simplified linear fluid and a gyro-kinetic model has been used for the electron and ion dynamics, respectively. It was found that as the separation of the rational surfaces was increased, the growth rates of DTMs were enhanced and the DTM system tended to decouple into a system of two single-tearing modes. When the width between the rational surfaces is larger, the corresponding amplitude of the inner-tearing mode is smaller. We also demonstrate that the existence of thermal ions destabilizes the DTM. The toroidal effect also has a mild destabilizing effect (∼8%) on the growth of DTMs, which is consistent with the theoretical analysis.

Investigation of annular/central collapse events triggered by the double tearing modes in EAST

Annular/central collapses of electron temperature have been investigated in the EAST tokamak. The precursor modes are excited before the annular/central collapse events, and the coexistence of multi magnetic island chains can be observed synchronously for the core concentration of high-Z impurities. The primary magnetic islands for the two cases are located in the same region of 0.25 ⩽ ρ(r/a) ⩽ 0.35, while the secondary magnetic islands for central and annular cases are located inside (ρ ≈ 0.1) and outside (ρ ≈ 0.4) respectively. The similar simulation has been performed by the toroidal resistive MHD code CLT (Zhang et al 2020 Phys. Plasmas 27 122509), and the annular/central collapse events can be reproduced by the magnetic reconnection of m/n = 2/1 double tearing modes, m and n are the poloidal and toroidal mode numbers respectively. The combination of lower hybrid current driven and off-axis electron cyclotron resonance heating is an essential condition for the establishment of reversed magnetic shear. The different penetration depth of high-Z impurity density plays an important role for excitation of the annular/central collapse events, e.g. the steeper gradient of T e-profile can be formed accordingly in the annular region of 0.1 ⩽ ρ ⩽ 0.4, together with excitation of the precursor modes.

Influence of resonance magnetic perturbation on the asymmetric magnetic perturbation induced double tearing modes

In this paper, we have studied the influence of resonance magnetic perturbation (RMP) on the double tearing modes induced by asymmetric magnetic perturbation in a 2D geometry based on the MHD model. After studying the four cases, we have found that with RMP, no matter which rational surface the magnetic perturbation is in: (1) the width of the outer magnetic island will be greater than that of the inner magnetic island at the end of the mode evolution; (2) The influence of the RMP on the outer magnetic island is far greater than the inner magnetic island; (3) RMP will shorten the time required for the magnetic island to reach saturation.

Nonlinear evolution and secondary island formation of the double tearing mode in a hybrid simulation

Double tearing modes (DTMs), induced by double current sheet configurations or two neighboring rational surfaces with the same safety factor in tokamaks, are widely observed in solar, space, and fusion plasmas. In this paper, the evolution of DTMs without a guide field is investigated numerically using a hybrid model (electron fluid + ion PIC). The overall evolution processes of DTMs are qualitatively consistent with previous works using other models. The particle dynamics during the evolution of DTMs is analyzed in detail. Behaviors of ions and electrons present different characteristics around the reconnection region which gives rise to Hall effects producing the out-of-plane quadrupole magnetic field. In the explosive reconnection process with interactions between two DTMs islands, the asymmetric drive and the thin current layer feature lead to the emergence of secondary magnetic islands which develop with the late evolution of the DTMs.

Magnetohydrodynamic simulations of runaway electron beam termination in JET

A runaway electron (RE) fluid model is used to perform non-linear magnetohydrodynamic simulations of a relativistic electron beam termination event in JET. The case considered is that of a post-disruption low density cold plasma in the runaway plateau phase, wherein high-Z impurities have been largely flushed out via deuterium second injection (Shot:95135). Details of the experiment are found in separate publications. Our studies reveal that a combination of low plasma density and a hollow current profile which is confirmed by experimental studies causes fast growth of a double-tearing mode, which in turn leads to stochastization of the magnetic field and a prompt loss of REs. The phenomenology of events leading to the crash and the timescales of the dynamics are in excellent agreement with the experiment. Simulations also indicate significant toroidal variation in RE deposition but without localized hotspots. The strong stochastization setting in first from the edge leads to a poloidally broad deposition footprint that partly explains the benign nature of the termination event. This work further supports the potential possibility to engineer a benign RE beam termination scenario via deuterium second injection in ITER, as proposed by Reux et al ‘Runaway electron beam suppression using impurity flushing and large magnetohydrodynamic instabilities’ (submitted to Physical Review Letters).

Influence of trapped energetic ions on low-frequency magnetohydrodynamic instabilities with reversed shear profile

The influence of trapped energetic ions (TEIs) on m/n=2/1 low-frequency magnetohydrodynamic (MHD) instabilities with reversed shear profile is investigated by a global kinetic–MHD hybrid simulation. For low energetic-ion beta, TEIs have a stabilizing effect on the double tearing mode (DTM). When the energetic-ion beta is larger than a threshold, an energetic-particle-driven mode, the fishbone-like mode (FLM), is excited; that is, a mode transition from the DTM to the FLM occurs. The threshold increases with increasing resistivity, and the growth rate of the FLM is reduced by the presence of resistivity. In addition, the dependences on energetic-ion beta, gyroradius, and injection velocity of the effects of TEIs are studied systematically.

The off-axis pressure crash associated with the nonlinear evolution of the m/n = 2/1 double tearing mode

The nonlinear evolution of the m/n = 2/1 double tearing mode (DTM) is investigated by the toroidal resistive magnetohydrodynamic code CLT. It is found that the m/n = 2/1 DTM can lead to either a core pressure crash or an off-axis pressure crash. Unlike the core pressure crash, the plasma pressure at the magnetic axis remains almost unchanged during the off-axis pressure crash. The pressure crash only occurs in the annular region during the off-axis crash, and the on-axis plasma pressure slowly reduces after the crash, which is consistent with TFTR observations. A series of simulations are carried out to investigate the influence of the radial position of the inner resonant surface r1, the magnetic shear at the inner resonance surface, and the spatial separation between the two resonant surfaces on nonlinear behaviors of DTMs. We find that r1 plays a dominant role in the nonlinear DTM behaviors. It is more likely for the DTM to lead to the core pressure crash with a smaller r1. It is also found that the magnetic shear at the inner resonant surface and the spatial separation between the two resonant surfaces can also largely influence the nonlinear evolution of the DTM. A simple theoretical formula of the transition criterion between the two pressure crashes is proposed, which agrees well with the simulation results.

Numerical Studies of Fast Pressure Crash Associated with Double Tearing Modes

The fast pressure crash associated with the nonlinear evolution of DTMs is investigated using a three-dimensional toroidal and nonlinear MHD code CLT. Our simulation results indicate that the fast pressure crash is not directly related to the explosive growth of DTM because the explosive growth of the DTM takes place in the earlier nonlinear phase and the fast pressure crash happens in the much later nonlinear stage. It is also found that the explosive growth lasts much longer than the fast pressure crash. At almost the end of the explosive growth phase, the magnetic fields around the magnetic axis become stochastic, and a strong radial flow is generated. The radial flow can effectively transfer the hot plasma around the magnetic axis into the outer cold region and consequently leads to the fast crash of the pressure in the core. As a result, the dependence of the crash time on the resistivity is very weak. It is also found that the geometry of Tokamaks has very weak influence on the fast pressure crash.

Effect of Specific Pressure of Fast Particles on Double-Tearing Mode of Resistance Magnetic Fluid

In the advanced Tokamak magnetic field configuration, the instability of the double tear mode excited by the diamagnetic shear configuration plays an important role. At the same time, due to the fusion reaction and auxiliary heating means, a large number of fast particles will be generated. The existence of fast particles will affect the growth of double tear mode. In the context of resistive magnetic fluids, the role of fast particle specific pressure in the instability of the double tear mode was analyzed, and the CLT-K program was used to simulate the instability of the fast particle specific pressure to the double-tearing mode to explore the physical mechanism of its impact.

Destabilization and nonlinear interaction of tearing modes in tokamak plasmas with locally reversed shear

Linear destabilization and nonlinear dynamics of m/n = 2 tearing modes in tokamak plasmas with locally reversed shear have been investigated in a reduced magnetohydrodynamic model. It is found that a transformation from the single tearing mode structure to the global double tearing mode structure takes place depending on the local shear parameter. The resistivity scaling law of the mode growth rate also changes from ∼SHp−3/5 to ∼ SHp−1/3 with qmin decreasing, where SHp is the magnetic Reynolds number, and qmin is the local minimum q value at position outsider q = 2 surfaces. Nonlinear simulation shows that once qmin < 2, the fast growing islands on the outer rational surfaces interact each other and trigger a plasmoid instability, which can couple with the inner tearing mode and generate a turbulent structure finally. This might provide an implication for understanding of relative nonlinear physics in ITER or astrophysical plasmas.

Influence of shear flows on dynamic evolutions of double tearing modes

In this paper, we investigate the influences of shear flows on dynamic evolutions of the m/n = 2/1 double tearing mode (DTM), using three-dimensional, toroidal, and nonlinear resistive magnetohydrodynamic code CLT. It is found that weak shear flows can lead to the decoupling of tearing modes on the two resonant surfaces, reducing the linear growth rate of DTMs. When the two tearing modes grow to large amplitudes at the nonlinear stage, they start to lock with each other. Consequently, DTMs with weak shear flows exhibit almost the same behavior as those without shear flows, i.e. weak shear flow can significantly reduce the linear growth rate of the DTM and delay a pressure crash, but has almost no influence on the time and the amplitude of the pressure crash. It is also found that the linear growth rates of the modes can become even larger than those without shear flows when the shear flow exceeds a critical value, due to Kelvin–Helmholtz (KH)-like instability. This KH-like instability can broaden the spectrum of the modes and then form a broad region with stochastic magnetic fields. Therefore, strong shear flows might be even more destructive in relation to plasma confinement.

Linear, resistive stability studies for Wendelstein 7-X-type equilibria with external current drive

Comprehensive linear stability studies of resistive modes are presented for Wendelstein 7-X-type stellarator equilibria with electron cyclotron current drive (ECCD). The external co-current drive leads to an increase of the rotational transform and the formation of one or two ι = 1 rational flux surfaces. Using the 3D linear stability CASTOR3D code, low n*-type resistive modes (n* = 1,2...,8 with n* being the dominant toroidal Fourier harmonic contributing to the mode) are investigated. We studied the dependence of their growth rates on plasma resistivity, parallel viscosity, and shape of the rotational transform profile (especially various distances between two ι = 1 flux surfaces). Similarly to tokamak configurations, single and double tearing modes, and internal resistive kink modes are found. In addition, modes oscillating between two n*-types of the same mode family are observed. The frequencies of those modes are in the range of ≈ 10–260 Hz. Equilibria with either a large distance between the ι = 1 flux surfaces, or a single ι = 1 surface are most unstable with respect to n* = 1 resistive kink modes. The latter finding fits to the experimental observation of sawtooth-like oscillations followed by a thermal quench in W7-X discharges with ECCD.

Nonlinear study of tearing modes and plasma flows around magnetic islands in tokamak plasmas

The behavior of non-linear tearing modes and the plasma flows around magnetic islands is studied numerically in a cylindrical geometry using the method of reduced magneto-hydro-dynamic simulation. The results show that the single tearing mode (STM) becomes stable in a short time. The magnetic islands of double tearing modes (DTMs) first develop to a sufficiently large scale and then couple together when entering the non-linear phase. It is also shown that, before the non-linear growth stage, the perturbed kinetic energy and magnetic energy of both the STM and DTM have a process from different signs to constant positive values. For the STM island and plasma flow, it is demonstrated that the shear flow is formed on both sides of the islands and the poloidal flow velocity around the O-point is larger than that around the X-point. For the DTM island and plasma flow, we found that the peak of the poloidal flow velocity is mainly distributed at the O-point edge and the X-point of the magnetic island and the plasma flow velocities at the magnetic island edges are opposite. In addition, the process of plasma flow accompanying the nonlinear evolution of tearing modes is also presented.

Numerical modeling of the electron temperature crashes observed in Wendelstein 7-X stellarator experiments

Numerical calculations based on nonlinear two-fluid equations have been carried out to understand the sawtooth-like crashes of electron temperature observed in Wendelstein 7-X stellarator experiments with electron cyclotron current drive (ECCD). The application of ECCD leads to non-monotonic radial profiles of the safety factor q with two q = 1 surfaces in the plasma core region. Using input parameters similar to the experiment, numerical calculations show two types of crashes of the central electron temperature depending on the growth of the internal kink mode coupled to double tearing modes. When the distance between two equilibrium q = 1 surfaces is sufficiently large, the internal kink mode leads to a fast radial displacement of the hot core from the center to the region of the q = 1 surfaces. In agreement with experimental observation, this causes a full crash of central electron temperature in about 20–30 microseconds. When the distance between the two q = 1 surfaces is not large enough, partial crashes of the central electron temperature are found. During these crashes the hot core only temporally moves outwards. Then it moves back, because the double tearing mode perturbation leads to an increase of the minimum q value about one.

Effect of local toroidal flow on double‐tearing modes in cylindrical geometry

AbstractThe effect of local toroidal flow (LTF) on double‐tearing modes (DTMs) is investigated in cylindrical geometry using the reduced magnetohydrodynamics (RMHD) model. The results indicate that the LTF between the two rational surfaces is the dominant suppression effect on DTMs. The suppression effect is enhanced with the flow width increasing, and the DTMs become more stable with the increase of the shear of the LTF between the two rational surfaces. So, in the reversed shear magnetic field configuration, the local flow is driven between the two rational surfaces, which can effectively suppress the development of DTMs and maintain the high‐performance state.