Tearing Parity Research Articles

Bifurcation of coherent vortex flow in a magnetic island through nonlinear parity instability

The topology of the vortex flow associated with the magnetic island plays a significant role in modulating the turbulent transport near the magnetic island. In this paper, self-consistent nonlinear simulations of multi-scale interactions among large scale tearing mode, vortex flow, and small scale ion temperature-gradient (ITG) mode are numerically investigated based on the five-field Landau-fluid model. We found that the coherent vortex flow in a magnetic island has different parities in the nonlinear quasi-steady state, and this can be described by a theoretical framework—nonlinear parity instability. In the ITG stable case, the structure of the vortex flow bifurcates from tearing parity to twisting parity, which is characterized by modulational parity instability, modeled by a four-wave nonlinear coupling process. In the ITG unstable case, the vortex flow stays in tearing parity without parity bifurcation, and the energy is transferred from the twisting parity modes to the tearing parity modes. The impact of the parity instability on the magnetic island width is discussed as well.

Pressure-driven tearing and thermal transport in finite-beta reversed field pinch computations

Nonlinear resistive-magnetohydrodynamics (MHD) computation with heating and anisotropic transport is applied to examine the interaction between thermal energy and magnetic fluctuations in inductively driven reversed-field pinches (RFPs). The magnetic fluctuations underlie magnetic field reversal through dynamo-like correlations, and they enhance thermal energy transport through fluctuations of parallel heat flux density. With the unfavorable magnetic curvature that exists across the RFP profile, thermal energy also affects the magnetic fluctuations. Computations with the NIMROD code [Sovinec et al., J. Comput. Phys. 195, 355–386 (2004)] integrate nonlinear MHD dynamics with energy transport and reproduce an RFP state with experimentally relevant values of plasma-β. Equilibria constructed from results of the 3D computations are analyzed to assess the sources of free energy in the saturated nonlinear state. Linear computations for these profiles show unstable modes of tearing parity. Their eigenfunctions are used to evaluate and compare stabilizing and destabilizing contributions to the kinetic energy integral. An assessment of the drives in the integral reveals that the pressure gradient drive is of comparable magnitude to the parallel current drive, and only the sum of the two surpasses the stabilizing contributions. Correlation of magnetic and parallel heat flux density fluctuations in the nonlinear computations shows that fluctuation-induced thermal conduction is the dominant mode of energy loss, as expected from experimental evidence. Decomposition of the fluctuating heat flux density shows that second-order correlations, alone, do not explain the total energy transport. Higher-order correlations are also important.

Identifying the microtearing modes in the pedestal of DIII-D H-modes using gyrokinetic simulations

Recent evidence points toward the microtearing mode (MTM) as an important fluctuation in the H-mode pedestal for anomalous electron heat transport. A study of the instabilities in the pedestal region carried out using gyrokinetic simulations to model an ELMy H-mode DIII-D discharge (USN configuration, 1.4 MA plasma current, and 3 MW heating power) is presented. The simulations produce MTMs, identified by predominantly electromagnetic heat flux, small particle flux, and a substantial degree of tearing parity. The magnetic spectrogram from Mirnov coils exhibits three distinct frequency bands—two narrow bands at lower frequency (∼35–55 kHz and ∼70–105 kHz) and a broader band at higher frequency (∼300–500 kHz). Global linear GENE simulations produce MTMs that are centered at the peak of the ω * profile and correspond closely with the bands in the spectrogram. The three distinctive frequency bands can be understood from the basic physical mechanisms underlying the instabilities. For example (i) instability of certain toroidal mode numbers (n) is controlled by the alignment of their rational surfaces with the peak in the ω* profile, and (ii) MTM instabilities in the lower n bands are the conventional collisional slab MTM, whereas the higher n band depends on curvature drive. While many features of the modes can be captured with the local approximation, a global treatment is necessary to quantitatively reproduce the detailed band gaps of the low-n fluctuations. Notably, the transport signatures of the MTM are consistent with careful edge modeling by SOLPS.

The physics of a small-scale tearing mode in collisionless slab plasmas

Microtearing modes have been widely reported as a tearing parity electron temperature gradient driven plasma instability, which leads to fine scale tearing of the magnetic flux surfaces thereby resulting in reconnection of magnetic field lines and formation of magnetic islands. In slab geometry it has previously been shown that the drive mechanism requires a finite collision frequency. However, we find in linear gyrokinetic simulations that a collisionless fine-scale tearing parity instability exists even at low and zero collision frequency. Detailed studies reveal that these slab modes are also driven by electron temperature gradient but are sensitive to electron finite Larmor radius effects, and have a radial wavenumber much smaller than the binormal wavenumber, which is comparable to the ion Larmor radius. Furthermore, they exist even in the electrostatic limit and electromagnetic effects actually have a stabilising influence on this collisionless tearing mode. An analytic model shows that this collisionless small scale tearing mode is consistent with a tearing parity slab electron temperature gradient (ETG) mode, which can be more unstable than the twisting parity ETG mode that is often studied. This small-scale tearing parity mode can lead to magnetic islands, which, in turn, can influence turbulent transport in magnetised plasmas.

Experimental observations of an n = 1 helical core accompanied by a saturated m/n = 2/1 tearing mode with low mode frequencies in JT-60U

We report the experimental finding of n = 1 helical cores (HCs) accompanied by saturated m/n = 2/1 tearing modes (TMs) with low mode frequencies in JT-60U. The HCs accompanied by TMs were observed after an increase in the mode amplitude and a decrease in the mode frequency of m/n = 2/1 precursors with tearing parity. The decreased mode frequency is typically lower than 20 Hz. With various diagnostics, the coupling of n = 1 HCs and m/n = 2/1 TMs has been clearly observed. Because the coherent oscillations in the ion temperature are observed in both the core region and the edge region, the flux surfaces including the m/n = 2/1 magnetic island appear to have m = 1 helical deformation. It has also been suggested that the m/n = 2/1 TM and the HC rotate in the electron diamagnetic direction keeping fm/n=1/1(HC) = 2fm/n=2/1(TM) in several plasmas. Here, fm/n=1/1(HC) is the mode frequency of HCs and fm/n=2/1(TM) is the mode frequency of TMs. In addition, the core seems to be shifted to the high-field side when the O-points of the m/n = 2/1 magnetic island line up in the midplane, which is confirmed by reconstructions of magnetohydrodynamic equilibria with motional Stark effect measurement and the MEUDAS code. Our observation of m/n = 2/1 TMs having HCs contributes to the understanding of the excitation mechanism of HCs in tokamak plasmas.

Numerical study of tearing mode seeding in tokamak X-point plasma

A detailed understanding of island seeding is crucial to avoid neoclassical tearing modes and their negative consequences like confinement degradation and disruptions. In the present work, we investigate the growth of 2/1 islands in response to magnetic perturbations. Although we use externally applied perturbations produced by resonant magnetic perturbation (RMP) coils for this study, the results are directly transferable to island seeding by other MHD instabilities creating a resonant magnetic field component at the rational surface. Experimental results for 2/1 island penetration from ASDEX Upgrade are presented extending previous studies. Simulations are based on an ASDEX Upgrade L-mode discharge with low collisionality and active RMP coils. Our numerical studies are performed with the 3D, two-fluid, nonlinear MHD code JOREK. All three phases of mode seeding observed in the experiment are also seen in the simulations: first, a weak response phase characterized by large perpendicular electron flow velocities followed by a fast growth of the magnetic island size accompanied by a reduction of the perpendicular electron velocity and finally the saturation to a fully formed island state with perpendicular electron velocity close to zero. Thresholds for mode penetration are observed in the plasma rotation as well as in the RMP coil current. A hysteresis of the island size and electron perpendicular velocity is observed between the ramping up and down of the RMP amplitude consistent with an analytically predicted bifurcation. The transition from dominant kink/bending to tearing parity during the penetration is investigated.

Multi-scale interactions between turbulence and magnetic islands and parity mixture—a review

This paper presents a review of multi-scale interactions between small-scale turbulence and large scale magnetic islands. In finite beta plasmas, zonal flows are relatively weak, and thus another electromagnetic coherent structure formation such as magnetic islands becomes important for regulating turbulence. In multi-scale interactions, large-scale modes dominate turbulent fluctuations even when the growth rate of the large-scale mode is much smaller than small-scale modes. On the other hand, small-scale modes influence large-scale modes when the large-scale modes are stable/marginally stable. Thus, the multi-scale interactions are categorized according to the stability of tearing mode (TM), which drives large-scale magnetic islands. When the TM is unstable, wide magnetic islands are produced, and as a result of the multi-scale interactions, the turbulent transport is significantly enhanced inside the separatrix of the island, because large-scale stable modes are excited by mutual interactions between turbulence and the island. On the other hand, a steep temperature gradient is formed around the separatrix of the island, which is consistent with zonal flow shear appearing at the separatrix. When the TM is stable/marginally stable, turbulence drives and sustains magnetic islands of width equal to multiples of the Larmor radius. This excitation of islands by turbulence can be related to the seed island formation of neo-classical TMs. The parity of fluctuations plays crucial role in the multi-scale nonlinear interactions, because pure twisting parity mode does not satisfy the nonlinear fluid/gyrokinetic equations. Magnetic islands belongs to the tearing parity mode and drift-wave instabilities normally belong to the twisting parity mode, and each parity is conserved in the linear growth of the instability. However, when the amplitude of the twisting parity mode becomes finite, the nonlinear energy transfer takes place from the twisting parity to tearing parity modes. Through this nonlinear parity mixture, the magnetic islands are produced by the turbulence. The influence of anomalous current drive and polarization current on the multi-scale interactions is discussed as well.

Trigger mechanism for the abrupt loss of energetic ions in magnetically confined plasmas

Interaction between a quasi-stable stationary MHD mode and a tongue-shaped deformation is observed in the toroidal plasma with energetic particle driven MHD bursts. The quasi-stable stationary 1/1 MHD mode with interchange parity appears near the resonant rational surface of q = 1 between MHD bursts. The tongue-shaped deformation rapidly appears at the non-resonant non-rational surface as a localized large plasma displacement and then collapses (tongue event). It curbs the stationary 1/1 MHD mode and then triggers the collapse of energetic particle and magnetic field reconnection. The rotating 1/1 MHD mode with tearing parity at the q = 1 resonant surface, namely, the MHD burst, is excited after the tongue event.

Nonlinear parity mixtures controlling the propagation of interchange modes

The propagation velocity of a resistive interchange mode is numerically investigated based on a two-fluid model. It is newly found that the nonlinearity mixes the interchange parity and the tearing parity to produce magnetic islands and controls the propagation velocity of the instability in the poloidal direction. The parity of the interchange mode is conserved during the linear growing evolution. However, when the amplitude of the mode becomes large and nonlinear effects are dominant, the pure interchange mode does not satisfy the nonlinear two-fluid equation. Thus, the nonlinear energy transfer occurs from the interchange parity mode to the tearing parity mode, which is called the nonlinear parity mixtures, and the magnetic islands are produced by the interchange mode. The nonlinear magnetic island formation by the interchange mode plays a central role in controlling the interchange mode's propagation velocity, which is equal to the electron fluid velocity. This nonlinear process is essential in quantitatively reproducing the propagation velocity of the interchange mode, which is the same as the electron fluid velocity observed in the large helical device experiment. It is also found that one of the mechanisms of parity mixtures is a modulational instability.

Dynamo-driven plasmoid formation from a current-sheet instability

Axisymmetric current-carrying plasmoids are formed in the presence of nonaxisymmetric fluctuations during nonlinear three-dimensional resistive MHD simulations in a global toroidal geometry. We utilize the helicity injection technique to form an initial poloidal flux in the presence of a toroidal guide field. As helicity is injected, two types of current sheets are formed from (1) the oppositely directed field lines in the injector region (primary reconnecting current sheet), and (2) the poloidal flux compression near the plasma edge (edge current sheet). We first find that nonaxisymmetric fluctuations arising from the current-sheet instability isolated near the plasma edge have tearing parity but can nevertheless grow fast (on the poloidal Alfven time scale). These modes saturate by breaking up the current sheet. Second, for the first time, a dynamo poloidal flux amplification is observed at the reconnection site (in the region of the oppositely directed magnetic field). This fluctuation-induced flux amplification increases the local Lundquist number, which then triggers a plasmoid instability and breaks the primary current sheet at the reconnection site. The plasmoids formation driven by large-scale flux amplification, i.e., a large-scale dynamo, observed here has strong implications for astrophysical reconnection as well as fast reconnection events in laboratory plasmas.

Electromagnetic gyrokinetic simulation of turbulence in torus plasmas

Gyrokinetic simulations of electromagnetic turbulence in magnetically confined torus plasmas including tokamak and heliotron/stellarator are reviewed. Numerical simulation of turbulence in finite beta plasmas is an important task for predicting the performance of fusion reactors and a great challenge in computational science due to multiple spatio-temporal scales related to electromagnetic ion and electron dynamics. The simulation becomes further challenging in non-axisymmetric plasmas. In finite beta plasmas, magnetic perturbation appears and influences some key mechanisms of turbulent transport, which include linear instability and zonal flow production. Linear analysis shows that the ion-temperature gradient (ITG) instability, which is essentially an electrostatic instability, is unstable at low beta and its growth rate is reduced by magnetic field line bending at finite beta. On the other hand, the kinetic ballooning mode (KBM), which is an electromagnetic instability, is destabilized at high beta. In addition, trapped electron modes (TEMs), electron temperature gradient (ETG) modes, and micro-tearing modes (MTMs) can be destabilized. These instabilities are classified into two categories: ballooning parity and tearing parity modes. These parities are mixed by nonlinear interactions, so that, for instance, the ITG mode excites tearing parity modes. In the nonlinear evolution, the zonal flow shear acts to regulate the ITG driven turbulence at low beta. On the other hand, at finite beta, interplay between the turbulence and zonal flows becomes complicated because the production of zonal flow is influenced by the finite beta effects. When the zonal flows are too weak, turbulence continues to grow beyond a physically relevant level of saturation in finite-beta tokamaks. Nonlinear mode coupling to stable modes can play a role in the saturation of finite beta ITG mode and KBM. Since there is a quadratic conserved quantity, evaluating nonlinear transfer of the conserved quantity from unstable modes to stable modes is useful for understanding the saturation mechanism of turbulence.

The stochastic field transport associated with the slab ITG modes

Many models for anomalous transport consider the turbulent E × B transport arising from electrostatic micro-instabilities. In this paper we investigate whether the perturbed magnetic field that is associated with such instabilities at small but finite values of β can lead to significant stochastic magnetic field transport. Using the tearing parity, long wavelength ion temperature gradient (ITG) modes in a plasma slab with magnetic shear as an example, we calculate the amplitude of the perturbed magnetic field at the resonant surface that results. The plasma model consists of a Braginskii description for the electrons with the dissipation at the resonant surface evaluated for the semi-collisional regime, while a finite ion Larmor radius kinetic model is invoked for the ions. The resulting stochastic field transport is estimated and also compared with an estimate for the E × B transport due to the ITG mode.

Magnetic stochasticity and transport due to nonlinearly excited subdominant microtearing modes

Subdominant, linearly stable microtearing modes are identified as the main mechanism for the development of magnetic stochasticity and transport in gyrokinetic simulations of electromagnetic ion temperature gradient driven plasma microturbulence. The linear eigenmode spectrum is examined in order to identify and characterize modes with tearing parity. Connections are demonstrated between microtearing modes and the nonlinear fluctuations that are responsible for the magnetic stochasticity and electromagnetic transport, and nonlinear coupling with zonal modes is identified as the salient nonlinear excitation mechanism. A simple model is presented, which relates the electromagnetic transport to the electrostatic transport. These results may provide a paradigm for the mechanisms responsible for electromagnetic stochasticity and transport, which can be examined in a broader range of scenarios and parameter regimes.

Pressure driven tearing and interchange modes in the reversed field pinch

In this work, the magneto-hydro-dynamic stability of pressure driven modes in the reversed field pinch has been analyzed. It is shown that at low and intermediate β's, i.e., typically for values below 20-25%, the tearing parity is dominant, while only at very high β, well above the achieved experimental values, at least part of the modes are converted to ideal interchange instabilities. Before their transition to ideal instabilities, according to their Lundquist number scaling, they can be classified as resistive-g modes.

Towards the construction of a model to describe the inter-ELM evolution of the pedestal on MAST

Pedestal profiles that span the ELM cycle have been obtained and used to test the idea that the pedestal pressure gradient in MAST is limited by the onset of kinetic ballooning modes (KBMs). During the inter-ELM period of a regularly type I ELM-ing discharge on MAST, the pressure pedestal height and width increase together while the pressure gradient increases by only 15% during the ELM cycle. Stability analyses show that the pedestal region over which infinite-n ballooning modes are unstable also broadens during the ELM cycle. To test the relationship between the width of the region that is unstable to n = ∞ ideal magnetohydrodynamic ballooning modes and KBMs the gyrokinetic code, GS2, has been used for microstability analysis of the edge plasma region in MAST. The gyrokinetic simulations find that KBM modes with twisting parity are the dominant microinstabilities in the steep pedestal region, with a transition to tearing parity modes in the shallower pressure gradient core region immediately inside the pedestal top. The region over which KBMs are unstable increases during the ELM cycle, and a good correlation is found between the region where KBMs dominate and the region that is unstable to infinite-n ideal ballooning modes.

Role of subdominant stable modes in plasma microturbulence

In gyrokinetic simulations, thousands of degrees of freedom are available to contribute to the fluctuation spectrum. For wavevectors with a single linear instability, the unstable eigenmode accounts for only one of these degrees of freedom. Little has been known about the role of the remaining fluctuations in the turbulent dynamics. In this paper, these fluctuations are characterized as modes in mode decompositions of gyrokinetic distribution functions from nonlinear simulations. This analysis reveals the excitation of a hierarchy of damped modes at the same perpendicular scales as the driving instabilities. Two effects of these subdominant modes are described: First, these damped modes define a potent energy sink, creating a situation where energy drive and energy dissipation peak at the same perpendicular scales. Second, damped modes with tearing parity (even parity about the outboard midplane for A|| fluctuations) are driven to significant amplitudes and facilitate the development of magnetic stochasticity in electromagnetic gyrokinetic simulations.

Fast ion stabilization of the ion temperature gradient driven modes in the Joint European Torus hybrid-scenario plasmas: a trigger mechanism for internal transport barrier formation

Understanding and modelling turbulent transport in thermonuclear fusion plasmas are crucial for designing and optimizing the operational scenarios of future fusion reactors. In this context, plasmas exhibiting state transitions, such as the formation of an internal transport barrier (ITB), are particularly interesting since they can shed light on transport physics and offer the opportunity to test different turbulence suppression models. In this paper, we focus on the modelling of ITB formation in the Joint European Torus (JET) [1] hybrid-scenario plasmas, where, due to the monotonic safety factor profile, magnetic shear stabilization cannot be invoked to explain the transition. The turbulence suppression mechanism investigated here relies on the increase in the plasma pressure gradient in the presence of a minority of energetic ions. Microstability analysis of the ion temperature gradient driven modes (ITG) in the presence of a fast-hydrogen minority shows that energetic ions accelerated by the ion cyclotron resonance heating (ICRH) system (hydrogen, nH,fast/nD,thermal up to 10%, TH,fast/TD,thermal up to 30) can increase the pressure gradient enough to stabilize the ITG modes driven by the gradient of the thermal ions (deuterium). Numerical analysis shows that, by increasing the temperature of the energetic ions, electrostatic ITG modes are gradually replaced by nearly electrostatic modes with tearing parity at progressively longer wavelengths. The growth rate of the microtearing modes is found to be lower than that of the ITG modes and comparable to the local E × B-velocity shearing rate. The above mechanism is proposed as a possible trigger for the formation of ITBs in this type of discharges.

Micro-tearing modes in the mega ampere spherical tokamak

Recent gyrokinetic stability calculations have revealed that the spherical tokamak is susceptible to tearing parity instabilities with length scales of a few ion Larmor radii perpendicular to the magnetic field lines. Here we investigate this ‘micro-tearing’ mode in greater detail to uncover its key characteristics and compare it with existing theoretical models of the phenomenon. This has been accomplished using a full numerical solution of the linear gyrokinetic–Maxwell equations. Importantly, the instability is found to be driven by the free energy in the electron temperature gradient as described in the literature. However, our calculations suggest it is not substantially affected by either of the destabilizing mechanisms proposed in previous theoretical models. Instead the instability is destabilized by interactions with magnetic drifts and the electrostatic potential. Further calculations reveal that the mode is not significantly destabilized by the flux surface shaping or the large trapped particle fraction present in the spherical tokamak. Its prevalence in spherical tokamak plasmas is primarily due to the higher value of plasma β and the enhanced magnetic drifts due to the smaller radius of curvature.

Microstability physics as illuminated in the spherical tokamak

Spherical tokamaks (STs) have attractive features for fusion, and there is considerable interest in understanding their transport properties which depend on the underlying microinstabilities. STs are capable of operation with low magnetic fields and exhibit large inhomogeneity in the toroidal magnetic field. These factors strongly affect particle dynamics and the potency of magnetic perturbations, which correspondingly impact on the microstability properties of STs. This paper reviews previous microstability studies in ST plasma configurations and presents gyrokinetic microstability calculations for a range of ST equilibria, using the gyrokinetic code GS2. Microstability properties of L-mode and H-mode equilibria, from the MAST experiment at Culham, are compared. In MAST the shearing rates of equilibrium E × B flows usually exceed the growth rates of microinstabilities with k⊥ρi < 1 (including ion temperature gradient, ITG, driven drift waves) and are generally smaller than the growth rates of shorter wavelength modes with k⊥ρi > 1 (electron temperature gradient, ETG, driven drift waves). Electromagnetic effects are significant at mid-radius in these MAST equilibria, where the local β ⩾ 0.1. At k⊥ρi < 1, strongly electromagnetic modes dominate over ITG instabilities, and these modes are found to have tearing parity in the H-mode plasma and twisting parity in the L-mode plasma. Numerical experiments have been carried out to assess the properties of the tearing parity modes and to probe the underlying physical drive mechanism. At shorter wavelengths the electromagnetic effects can significantly stabilize the ETG instabilities. Nonlinear electron scale microturbulence calculations for two surfaces of a MAST H-mode plasma suggest that significant electron heat transport can be carried via this mechanism. In an extremely high β ST equilibrium, which has been proposed as the basis of a conceptual ST power plant, electrostatic instabilities are fully stabilized, but tearing parity modes are predicted to be unstable over-wide range of length scales.

/spl delta/f Simulation of the collisionless tearing mode instability with a gyrokinetic ion response

The evolution of the collisionless tearing mode instability is studied using an electromagnetic gyrokinetic /spl delta/f particle-in-cell simulation model. Drift-kinetic electrons are used. High resolution, small box (less than 10 ion gyroradii) simulations have been well benchmarked with eigenmode analysis and their nonlinear evolution agrees well with theory. In this regime, the ion response is not important and can be either fixed, adiabatic or fully gyrokinetic. Here, results are presented with larger box sizes (64 ion gyroradii radially) where the ion gyrokinetic response is important and cannot be neglected. In these larger box simulations, the instability exhibits an odd parity, different than the even tearing parity.