The equilibrium of a finite-beta RFP plasma in the presence of saturated-amplitude tearing modes is investigated. The singularities of the MHD force balance equation JXB=grad(p) at the modes rational surfaces are resolved through a proper regularization of the zeroth-order (equilibrium) profiles, by setting to zero there the gradient of the pressure and parallel current density. An equilibrium model, which satisfies the regularization rule at the various rational surfaces, is developed. The comparison with the experimental data from the Reversed Field eXperiment (RFX) gives encouraging results. The model provides an easy tool for magnetic analysis: many aspects of the perturbations can be analyzed and reconstructed.
The equilibrium of a finite-beta cylindrical RFP plasma in the presence of saturated-amplitude tearing modes is investigated. The singularities arising from the perturbative analysis of the MHD force balance equation JxB=grad(p) are resolved through a proper regularization of the cylindrical symmetric component of the pressure and parallel current density profiles, by setting to zero their gradient at the modes rational surfaces. An equilibrium model, which satisfies the regularization rule at the various rational surfaces, is developed. The comparison with the experimental data from the Rever…
We calculate the stability of a multiple relaxation region MHD (MRXMHD) plasma, or stepped-Beltrami plasma, using both variational and tearing mode treatments. The configuration studied is a periodic cylinder. In the variational treatment, the problem reduces to an eigenvalue problem for the interface displacements. For the tearing mode treatment, analytic expressions for the tearing mode stability parameter $Δ'$, being the jump in the logarithm in the helical flux across the resonant surface, are found. The stability of these treatments is compared for $m=1$ displacements of an illustrative R…
Effects of externally applied resonant magnetic perturbation (RMP) on the locked mode of neo-classical tearing mode (NTM) are numerically investigated by means of a set of reduced magnetohydrodynamic equations. It is found that for a small bootstrap current fraction, three regimes, namely slight suppression regime, small locked island (SLI) regime and big locked island (BLI) regime, are discovered with the increase of RMP strength. For a much higher bootstrap current fraction, however, a new oscillation regime appears instead of the SLI regime, although the other regimes still remain. The phys…
Configurations with nearby multiple resonant surfaces have broad spectra of linearly unstable coupled tearing modes with dominant high poloidal mode numbers m. This was recently shown for the case of multiple q = 1 resonances [Bierwage et al., Phys. Rev. Lett. 94 (6), 65001 (2005)]. In the present work, similar behavior is found for double tearing modes (DTM) on resonant surfaces with q >= 1. A detailed analysis of linear instability characteristics of DTMs with various mode numbers m is performed using numerical simulations. The mode structures and dispersion relations for linearly unstable m…
The linear instability and nonlinear dynamics of collisional (resistive) and collisionless (due to electron inertia) double tearing modes (DTMs) are compared with the use of a reduced cylindrical model of a tokamak plasma. We focus on cases where two q = 2 resonant surfaces are located a small distance apart. It is found that regardless of the magnetic reconnection mechanism, resistivity or electron inertia, the fastest growing linear eigenmodes may have high poloidal mode numbers m ~ 10. The spectrum of unstable modes tends to be broader in the collisionless case. In the nonlinear regime, it …
Double-tearing modes (DTMs) have been proposed as a driver of `off-axis sawtooth' crashes in reverse magnetic shear tokamak configurations. Recently differential rotation provided by equilibrium sheared flows has been shown capable of decoupling the two DTM resonant layers, slowing the growth the instability. In this work we instead supply this differential rotation using an electron diamagnetic drift, which emerges in the presence of an equilibrium pressure gradient and finite Larmor radius physics. Diamagnetic drifts have the additional benefit of stabilizing reconnection local to the two te…
Effect of externally applied resonant magnetic perturbations on resistive tearing modes
State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology（华中科技大学电磁工程与技术国家重点实验室）；Max-Planck-Institut für Plasmaphysik, EURATOM association（德国马普等离子体物理研究所，欧洲原子能共同体联合机构）
We present experimental results of magnetic feedback control on the m=2, n=1 tearing mode in RFX-mod operated as a circular ohmically heated tokamak. The feedback suppression of the non-resonant m=2, n=1 Resistive Wall Mode (RWM) in q(a)<2 plasmas is a well-established result of RFX-mod. The control of the tearing counterpart, which develops in q(a)>2 equilibrium, is instead a more difficult issue. In fact, the disruption induced by a growing amplitude m=2, n=1 tearing mode can be prevented by feedback only when the resonant surface q=2 is close to the plasma edge, namely 2<q(a)<2.5, and the e…
Kinetic treatments of drift-tearing modes that match an inner resonant layer solution to an external MHD region solution, characterised by $Δ^{\prime}$, fail to properly match the ideal MHD boundary condition on the parallel electric field, $E_{\parallel}.$ In this paper we demonstrate how consideration of ion sound and ion Landau damping effects achieves this and place the theory on a firm footing. As a consequence, these effects contribute quite significantly to the critical value of $Δ^{\prime}$ for instability of drift-tearing modes and play a key role in determining the minimum value for …
The nonlinear saturation of the tearing mode is revisited in slab geometry by taking into account higher-order harmonics in the outer solution. The general formalism for tackling this problem in the case of a vanishing current gradient at the resonant surface is derived. It is shown that, although the higher-order harmonics lead to corrections in the final saturation equation, they are of higher order in the perturbation parameter, which provides a formal proof that the standard one-harmonic approach is asymptotically correct.
In JT-60U, simultaneous excitation of n = 1 helical cores (HCs) and m/n = 2/1 Tearing Modes (TMs) was observed [T. Bando et al., Plasma Phys. Control. Fusion 61 115014 (2019)]. In this paper, we have investigated the excitation mechanism of n = 1 HCs with m/n = 2/1 TMs based on the experimental observations and a simple quasi-linear MHD model. In the previous study, it was reported that a "coupling" on the phase of the MHD mode is observed between n = 1 HCs and m/n = 2/1 TMs. In this study, it is found that the coupling is observed with the mode frequency from several Hz to 6 kHz. This indicat…
Starting from expressions in Connor et al. (1988) [1], we derive a one-dimensional tearing equation similar to the approximate equation obtained by Hegna and Callen (1984) [2] and by Nishimura et al (1998) [3], but for more realistic toroidal equilibria. The intention is to use this approximation to explore the role of steep profiles, bootstrap currents and strong shaping in the vicinity of a separatrix, on the stability of tearing modes which are resonant in the H-mode pedestal region of finite aspect ratio, shaped cross-section tokamaks, e.g. JET. We discuss how this one-dimensional model fo…
In magnetic configurations with two or three q=1 (with q being the safety factor) resonant surfaces in a tokamak plasma, resistive magnetohydrodynamic modes with poloidal mode numbers m much larger than 1 are found to be linearly unstable. It is found that these high-m double or triple tearing modes significantly enhance through nonlinear interactions, the growth of the m=1 mode. This may account for the sudden onset of the internal resistive kink, i.e., the fast sawtooth trigger. Based on the subsequent reconnection dynamics that can proceed without formation of the m=1 islands, it is propose…
The dynamics of a global reconnection in the presence of a poloidal shear flow which is located in between magnetic islands is investigated. Different linear regimes are identified according to the value of the resistivity and the distance between the low-order resonant surfaces. It is found that the presence of a small shear flow affects and significantly delays the global reconnection processes. It is shown that this delay is linked to a breaking of symmetry imposed by the existence of the shear flow and the generation of a mean poloidal flow in the resistive layers.
The global ideal kink equation, for cylindrical geometry and zero beta, is simplified in the high poloidal mode number limit and used to determine the tearing stability parameter, $Δ^\prime$. In the presence of a steep monotonic current gradient, $Δ^\prime$ becomes a function of a parameter, $σ_0$, characterising the ratio of the maximum current gradient to magnetic shear, and $x_s$, characterising the separation of the resonant surface from the maximum of the current gradient. In equilibria containing a current "spike", so that there is a non-monotonic current profile, $Δ^\prime$ also depends…
A comprehensive two-fluid nonlinear theory of magnetic reconnection driven at a single, tearing-stable, rational surface embedded in an H-mode tokamak plasma is presented. The surface is assumed to be resonant with one of the dominant helical harmonics of an applied resonant magnetic perturbation (RMP). The theory described in this paper is highly relevant to the problem of understanding the physics of RMP-induced edge localized mode (ELM) suppression in tokamak plasmas.
The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for a Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle ($θ$) with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes ($θ\neq 0$). We find th…
The multi-scale interaction of self-consistently driven magnetic islands with electromagnetic turbulence is studied within the three dimensional, toroidal gyro-kinetic framework. It can be seen that, even in the presence of electromagnetic turbulence the linear structure of the mode is retained. Turbulent fluctuations do not destroy the growing island early in its development, which then maintains a coherent form as it grows. The island is seeded by the electromagnetic turbulence fluctuations, which provide an initial island structure through nonlinear interactions and which grows at a rate si…
Direct evidence of micro-turbulence effect on the onset of neoclassical tearing mode (NTM) is reported for the first time in this letter. A puzzling positive correlation between critical width of seed island of NTM and normalized plasma pressure beta_p is first observed employing a novel method for clearly separating the processes of seed island and the onset of NTM in the EAST tokamak. Different from the methods developed before, the width of the seed island is well controlled by slowly ramping up the current in resonant magnetic perturbation coils. It is revealed that the positive correlatio…
The nonlinear evolution of resistive double tearing modes (DTMs) with safety factor values q=1 and q=3 is studied in a reduced cylindrical model of a tokamak plasma. We focus on cases where the resonant surfaces are a small distance apart. Recent numerical studies have shown that in such configurations high-m modes are strongly unstable. In this paper, it is first demonstrated that linear DTM theory predicts the dominance of high-m DTMs. A semi-empirical formula for estimating the poloidal mode number of the fastest growing mode, m_peak, is obtained from the existing linear theory. Second, usi…
Screening effect of plasma flow on the resonant magnetic perturbation penetration in tokamak based on two-fluid model
Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams (Ministry of Education), School of Physics, Dalian University of Technology（大连理工大学物理学院激光、离子和电子束材料改性教育部重点实验室）；Key Laboratory of Geospace Environment, University of Science and Technology of China（中国科学技术大学空间环境重点实验室）；Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology（大连理工大学电子信息与电气工程学部）
Calculations of tearing mode stability in tokamaks split conveniently into an external region, where marginally stable ideal MHD is applicable, and a resonant layer around the rational surface where sophisticated kinetic physics is needed. These two regions are coupled by the stability parameter. Pressure and current perturbations localized around the rational surface alter the stability of tearing modes. Equations governing the changes in the external solution and - are derived for arbitrary perturbations in axisymmetric toroidal geometry. The relationship of - with and without pressure flatt…
First direct experimental evidence of high frequency, high toroidal mode number (n>20), magnetic fluctuations due to unstable resistive interchange modes (g-modes) resonant in the edge region of a reversed field pinch (RFP) plasma is presented. Experimental characterization of time and space periodicities of the modes is provided by means of highly resolved in-vessel edge and insertable magnetic diagnostics. It is found that the spectral mode properties are in good agreement with the predictions of the theoretical linear resistive magnetohydrodynamic stability analysis. A simple model is propo…
A detailed understanding of island seeding is crucial to avoid (N)TMs 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, 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 e…
The interaction between Edge Localized Modes (ELMs) and Resonant Magnetic Perturbations (RMPs) is modeled with the magnetohydrodynamic code JOREK using experimental parameters from ASDEX Upgrade discharges. The ELM mitigation or suppression is optimal when the amplification of both tearing and peeling-kink responses result in a better RMP penetration. The ELM mitigation or suppression is not only due to the reduction of the pressure gradient, but predominantly arises from the toroidal coupling between the ELMs and the RMP-induced mode at the plasma edge, forcing the edge modes to saturate at a…
The effects that energetic trapped ions have on linear resistive magnetohydrodynamic (MHD) instabilities are studied in a reduced model that captures the essential physics driving or damping the modes through variations in the magnetic shear. The drift-kinetic orbital interaction of a slowing down distribution of trapped energetic ions with a resistive MHD instability is integrated to a scalar contribution to the perturbed pressure, and entered into an asymptotic matching formalism for the resistive MHD dispersion relation. Toroidal magnetic field line curvature is included to model trapping i…
An error field (EF) detection technique using the amplitude modulation of a naturally rotating tearing mode (TM) is developed and validated in the EXTRAP T2R reversed field pinch. The technique was used to identify intrinsic EFs of $m/n = 1/-12$, where $m$ and $n$ are the poloidal and toroidal mode numbers. The effect of the EF and of a resonant magnetic perturbation (RMP) on the TM, in particular on amplitude modulation, is modeled with a first-order solution of the Modified Rutherford Equation. In the experiment, the TM amplitude is measured as a function of the toroidal angle as the TM rota…
Resonant field amplification or error field penetration involves driving a weakly stable tearing perturbation in a rotating toroidal plasma. In this paper it is shown that the locking characteristics for modes with finite real frequencies $ω_{r}$ are quite different from the conventional results. A calculation of the tearing mode amplitude assuming modes with frequencies $\pmω_{r}$ in the plasma frame shows that it is maximized when the frequency of the stable backward propagating mode ($-ω_{r}$) in the lab frame is zero, i.e. when $v=ω_{r}/k$. Even more importantly, the locking torque is exac…
Sustained ELM mitigation has been achieved on MAST and AUG using RMPs with a range of toroidal mode numbers over a wide region of low to medium collisionality discharges. The ELM energy loss and peak heat loads at the divertor targets have been reduced. The ELM mitigation phase is typically associated with a drop in plasma density and overall stored energy. In one particular scenario on MAST, by carefully adjusting the fuelling it has been possible to counteract the drop in density and to produce plasmas with mitigated ELMs, reduced peak divertor heat flux and with minimal degradation in pedes…
Plasma response to resonant magnetic perturbations (RMPs) near the pedestal top is crucial for accessing edge localized modes (ELMs) suppression in tokamaks. Since radial location of rotation zero-crossing plays a key role in determining the threshold for field penetration of RMP, plasma response may be different in low input torque plasmas. In this work, the linear MHD code MARS-F is applied to reveal the dependence of plasma response to RMP on rotation zero-crossing by a scan of rotation profiles based on an EAST equilibrium. It is shown that the plasma response is enhanced when zero-crossin…
JOREK is a massively parallel fully implicit non-linear extended MHD code for realistic tokamak X-point plasmas. It has become a widely used versatile code for studying large-scale plasma instabilities and their control developed in an international community. 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 …
