Tearing Mode Stability Research Articles

Utilizing resonant magnetic perturbations to enhance neoclassical tearing mode stabilization by rf current

A new method to stabilize the rotating neoclassical tearing mode (NTM) by using both the rf current drive and the static resonant magnetic perturbation (RMP) is investigated. When a non-uniform mode rotation is induced by the RMP, the stabilization of NTM by the rf current is found to be enhanced if the RMP phase has a half period difference from that of the rf wave deposition along the helical angle. The required rf current for mode stabilization is reduced by about one third if an appropriate RMP amplitude is applied.

Coupling extended magnetohydrodynamic fluid codes with radiofrequency ray tracing codes for fusion modeling

Neoclassical tearing modes are macroscopic (L∼1 m) instabilities in magnetic fusion experiments; if unchecked, these modes degrade plasma performance and may catastrophically destroy plasma confinement by inducing a disruption. Fortunately, the use of properly tuned and directed radiofrequency waves (λ∼1 mm) can eliminate these modes. Numerical modeling of this difficult multiscale problem requires the integration of separate mathematical models for each length and time scale (Jenkins and Kruger, 2012 [21]); the extended MHD model captures macroscopic plasma evolution while the RF model tracks the flow and deposition of injected RF power through the evolving plasma profiles. The scale separation enables use of the eikonal (ray-tracing) approximation to model the RF wave propagation. In this work we demonstrate a technique, based on methods of computational geometry, for mapping the ensuing RF data (associated with discrete ray trajectories) onto the finite-element/pseudospectral grid that is used to model the extended MHD physics. In the new representation, the RF data can then be used to construct source terms in the equations of the extended MHD model, enabling quantitative modeling of RF-induced tearing mode stabilization. Though our specific implementation uses the NIMROD extended MHD (Sovinec et al., 2004 [22]) and GENRAY RF (Smirnov et al., 1994 [23]) codes, the approach presented can be applied more generally to any code coupling requiring the mapping of ray tracing data onto Eulerian grids.

Modelling and analytic studies of sheared flow effects on tearing modes

The effects of flow shear on the stability of a (2,1) tearing mode are examined using numerical and analytic studies on a number of model systems. For a cylindrical reduced magnetohydrodynamic (MHD) model, linear computations using the CUTIE code show that sheared axial flows have a destabilizing effect, while sheared poloidal flows tend to reduce the growth rate of the mode. These effects are independent of the direction of the flow. For helical flows the sign of the shear in the flow matters. This symmetry breaking is also seen in the nonlinear regime where the island saturation level is found to depend on the sign of the flows. In the absence of flow, the CUTIE simulations show that the linear mode is more stable in a two fluid as compared to a single fluid model. However, in the presence of sheared axial flows a negative sheared flow is more destabilizing while a positive sheared flow is more stabilizing, compared to the single fluid model. In contrast to the cylindrical model, simulations in a toroidal model, using the MHD code NEAR, always show a stabilizing effect in the presence of a sheared toroidal flow. This is understood analytically in terms of a flow induced ‘Shafranov’ like shift in the profiles of the equilibrium current that results in a stabilizing change in Δ′ and the saturated island size.

The role of thermal conduction in tearing mode theory

The role of anisotropic thermal diffusivity in tearing mode stability is analysed in general toroidal geometry following similar techniques to Glasser et al (1975 Phys. Fluids 18 875), although a stronger ordering of the plasma compressibility is required. Resistive layer equations are obtained for a resistive magnetohydrodynamic (MHD) model with anisotropic transport of pressure. A dispersion relation linking the growth rate to the tearing mode stability parameter, Δ′, characterising the external ideal MHD region, is derived. By using a resistive MHD code modified to include such thermal transport to calculate tearing mode growth rates, this dispersion relation is employed to determine Δ′ in situations with finite plasma pressure that are stabilised by favourable average curvature when using a simple resistive MHD code. We also demonstrate that the same code can be used to obtain the basis-functions (Ham et al 2012 Plasma Phys. Control. Fusion 54 105014) needed for an alternative approach to calculating Δ′.

Numerical study on the stabilization of neoclassical tearing modes by electron cyclotron current drive

It is well known that electron cyclotron current drive (ECCD) around the o-point of magnetic island along the plasma current direction can stabilize neoclassical tearing modes (NTMs) in tokamak devices. The effects of the radial misalignment between the island and the driven current, the phase misalignment, and the on-duty ratio for modulated current drive on NTM stabilization are studied numerically in this paper. A small radial misalignment is found to significantly decrease the stabilizing effect. When a sufficiently large phase misalignment occurs for the modulated ECCD, the stabilization effect is also reduced a lot. The optimal on-duty ratio of modulated ECCD to stabilize NTMs is found to be in the range of 60%–70%. A larger on-duty ratio than 50% could also mitigate the effect of phase misalignment if it is not too large. There is no benefit from modulation if the phase misalignment is larger than a threshold.

Numerical study on neoclassical tearing mode stabilization via minimum seeking method for the island width growth rate

We report that the control of the ‘island width growth rate’, which is defined as dW/dt, is more efficient than that of the ‘island width’ for neoclassical tearing mode stabilization using the minimum seeking method. A concept of the minimum island width growth rate seeking method is newly proposed for the real-time feedback control of the neoclassical tearing mode by using electron cyclotron current drive. To evaluate the performance of the proposed concept, predictive feedback control simulations with an integrated numerical system are performed based on two types of minimum seeking controllers; the first is a finite difference method based controller and the second is a sinusoidal perturbation based controller. The results are compared with the minimum ‘island width’ seeking method. It is revealed that the proposed control concept is less limited in minimum seeking, and more robust and efficient in reducing the misalignment in shorter time scales.

Influence of energetic ions on tearing modes in a reversed field pinch

Tearing mode stability analysis in the presence of circulating energetic ions (CEI) is studied in the reversed field pinch (RFP) magnetic configuration. In contrast to the minimal effect of precessional drift of CEI on tearing modes in tokamaks, the effect of precessional drift of CEI on tearing modes is important in the RFP. It is found that the effects of CEI on tearing modes in RFP depend on their toroidal circulating direction, and have a strong relation to the pressure gradient of CEI. For co-CEI, tearing modes can be stabilized if the pressure gradient of energetic ions is sufficiently large, which is qualitatively consistent with experimental results in the Madison Symmetric Torus.

Development of Resonant Diplexers for high-power ECRH – Status, Applications, Plans

The development of diplexers for ECRH has been pursued at a number of institutes because of their attractive variety of applications: Power combination, non-mechanical, electrically controlled switching (of combined beams) between launchers with tens of kHz, and discrimination of low-power ECE signals from high-power ECRH is feasible. In a first part, this paper reports on plasma experiments with a ring resonator (Mk IIa) at ASDEX Upgrade. Commissioning experiments on fast switching between two launchers for synchronous stabilization of neoclassical tearing modes, as well as in-line ECE measurements have been performed, and experimental issues and first results are discussed. A clear influence of the switching phase on the amplitude of the 3/2 NTM mode was measured, complete stabilization could, however, not be demonstrated yet mainly due to imperfect resonator control. Concepts for improved tracking of the diplexers to the gyrotron frequency are presented. In a second part, the design of diplexers with ring resonators matched to HE11 fields is briefly discussed; these devices can be connected to corrugated waveguides without any mode converters. A compact version (MQ IV) is under investigation, which is compatible with the ITER ECRH system (170 GHz, 63.5 mm waveguide, vacuum tight casing), with the final goal of high-power tests at the 170 GHz gyrotron facility at JAEA in Naka, Japan. First low-power test results are presented.

Linear and nonlinear effect of sheared plasma flow on resistive tearing modes

The effect of sheared plasma flow on the m/n = 2/1 tearing mode is studied numerically (m and n are the poloidal and toroidal mode numbers). It is found that in the linear phase the plasma flow with a weak or moderate shear plays a stabilizing effect on tearing mode. However, the mode is driven to be more unstable by sufficiently strong sheared flow when approaching the shear Alfvén resonance (AR). In the nonlinear phase, a moderate (strong) sheared flow leads to a smaller (larger) saturated island width. The stabilization of tearing modes by moderate shear plasma flow is enhanced for a larger plasma viscosity and a lower Alfvén velocity. It is also found that in the nonlinear phase AR accelerates the plasma rotation around the 2/1 rational surface but decelerates it at the AR location, and the radial location satisfying AR spreads inwards towards the magnetic axis.

Instabilities of collisionless current sheets revisited: The role of anisotropic heating

In this work, we investigate the influence of the anisotropic heating on the spontaneous instability and evolution of thin Harris-type collisionless current sheets, embedded in antiparallel magnetic fields. In particular, we explore the influence of the macroparticle shape-function using a 2D version of the PIC code ACRONYM. We also investigate the role of the numerical collisionality due to the finite number of macroparticles in PIC codes. It is shown that it is appropriate to choose higher order shape functions of the macroparticles compared to a larger number of macroparticles per cell. This allows to estimate better the anisotropic electron heating due to the collisions of macroparticles in a PIC code. Temperature anisotropies can stabilize the tearing mode instability and trigger additional current sheet instabilities. We found a good agreement between the analytically derived threshold for the stabilization of the anisotropic tearing mode and other instabilities, either spontaneously developing or initially triggered ones. Numerical effects causing anisotropic heating at electron time scales become especially important for higher mass ratios (above mi/me=180). If numerical effects are carefully taken into account, one can recover the theoretical estimated linear growth rates of the tearing instability of thin isotropic collisionless current sheets, also for higher mass ratios.

Improved accuracy in the estimation of the tearing mode stability parameters (Δ′ and wc) using 2D ECEI data in KSTAR

The accuracy in estimation of two important tearing mode stability parameters (Δ′ and wc) is improved by employing two-dimensional (2D) ECE imaging data which help one to overcome the resolution limit of conventional one-dimensional data. The experimentally measured 2D images are directly compared with synthetic ones from a tearing mode Te model to estimate the parameters and an excellent agreement is achieved. The results imply that the observed tearing mode is classically stable but has non-negligible bootstrap current drive.

Time-dependent simulations of feedback stabilization of neoclassical tearing modes in KSTAR plasmas

A simulation is performed for feedback stabilization of neoclassical tearing mode (NTM) by electron cyclotron current drive (ECCD) for KSTAR in preparation for experiments. An integrated numerical system is constructed by coupling plasma transport, NTM stability, and heating and current drive modules and applied to a KSTAR plasma by assuming similar experimental conditions as ASDEX Upgrade to predict NTM behaviors in KSTAR. System identification is made with database produced by predictive simulations with this integrated numerical system so that three plasma response models are extracted which describe the relation between the EC poloidal launcher angle and the island width in KSTAR. Among them, the P1DI model exhibiting the highest fit accuracy is selected for designing a feedback controller based on the classical Proportional–Integral–Derivative (PID) concept. The controller is coupled with the integrated numerical system and applied to a simulation of NTM stabilization. It is observed that the controller can search and fully stabilize the mode even though the poloidal launch angle is misaligned with the island initially.

Non-linear effects in electron cyclotron current drive applied for the stabilization of neoclassical tearing modes

Due to the smallness of the volumes associated with the flux surfaces around the O-point of a magnetic island, the electron cyclotron power density applied inside the island for the stabilization of neoclassical tearing modes (NTMs) can exceed the threshold for non-linear effects as derived previously by Harvey et al (1989 Phys. Rev. Lett. 62 426). We study the non-linear electron cyclotron current drive (ECCD) efficiency through bounce-averaged, quasi-linear Fokker–Planck calculations in the magnetic geometry as created by the islands. The calculations are performed for the parameters of a typical NTM stabilization experiment on ASDEX Upgrade. A particular feature of these experiments is that the rays of the EC wave beam propagate tangential to the flux surfaces in the power deposition region. The calculations show significant non-linear effects on the ECCD efficiency, when the ECCD power is increased from its experimental value of 1 MW to a larger value of 4 MW. The nonlinear effects are largest in the case of locked islands or when the magnetic island rotation period is longer than the collisional time scale. The non-linear effects result in an overall reduction of the current drive efficiency for this case with absorption of the EC power on the low-field side of the electron cyclotron resonance layer. As a consequence of the non-linear effects, also the stabilizing effect of the ECCD on the island is reduced from linear expectations.

Recent Advances in the Worldwide Fusion Gyrotron Development

Progress in the worldwide development of high-power gyrotrons for magnetic confinement fusion plasma applications is presented. Gyrotron oscillators are used for electron cyclotron heating, electron cyclotron current drive, stability control, and plasma diagnostics. After technology breakthroughs in the research on gyrotron components in the 1990s, significant progress has been achieved in the 2000s, in the field of long-pulse and continuous wave (CW) operation for a wide range of frequencies. Currently, the development of 1-MW-class CW gyrotrons for the tokamak ITER (170 GHz), the stellarator Wendelstein 7-X (140 GHz), and the tokamaks DIII-D and JT-60SA (110 GHz) has been very successful in EU, Japan, Russia, and USA. The Japan 170-GHz ITER gyrotron holds the energy world record of 2.88 GJ (0.8 MW at 1-h pulse duration). For this progress in the field of high-power long-pulse gyrotrons, innovations such as the realization of high-efficiency stable oscillation in very high-order cavity modes with low ohmic losses, the use of single-stage depressed collectors for energy recovery (efficiency enhancement and simpler power supplies), highly efficient internal quasi-optical (q.o.) mode converters (low level of internal stray radiation), and synthetic diamond windows have essentially contributed. The total tube efficiencies are around 50% and the purity of the linearly polarized fundamental Gaussian output mode is 97% and higher. Power modulation technologies for stabilization of neoclassical tearing modes have proceeded. Future prospects of advanced high-power fusion gyrotrons are in the areas of two- and three-frequency gyrotrons, fast step-wise frequency tuneability, higher unit power (coaxial cavities), and higher frequencies for more efficient plasma stabilization and noninductive current drive as well as reliability, availability, maintainability and inspectability for next step fusion power stations. The GYCOM step-tuneable 1-MW gyrotron for ASDEX Upgrade employing a broadband travelling-wave-resonator window (with two diamond disks) operates at 105, 117, 127 and 140 GHz. The EU 170 GHz coaxial-cavity gyrotron prototype achieved in millisecond pulses the power of 2.1 MW at 46% efficiency and 96% Gaussian mode purity. A new power record of second harmonic oscillation has been achieved in the subterahertz band (83 kW/389 GHz/3 ms) in Japan for application to collective Thomson scattering diagnostics. A fundamental frequency 670-GHz gyrotron with pulsed magnet generated 210 kW in 20-30- μs pulses at 20% efficiency (collaboration of institutions in Russia and USA).

Neoclassical effects on the stabilization of tearing mode by current modulation

The neoclassical effects on the stabilization of tearing modes by current modulation have been investigated. Neoclassical effects enhance the resistivity and reduce the resistive diffusion time of the modulation current. Therefore, the oscillating current can penetrate deeper into the plasma. With an oscillating loop voltage, the plasma oscillates radially at the Ware-pinch velocity. These neoclassical effects improve the efficiency of tearing mode stabilization by the current modulation.

Dynamics of magnetic islands in large Δ′ regimes

Saturation of magnetic islands at large values of the tearing mode stability parameter Δ′ is investigated numerically. In such regimes, the island dynamics exhibit a number of transient features such as coalescence instability, X-point collapse, and plasmoid generation. It is shown that while conditions for these transient instabilities to appear depend on the viscosity and resistivity, the final width of the saturated island is independent of such phenomena, as well as viscosity and resistivity values. It is found that the saturated island width is strongly influenced by global properties of the current profile and related to the equilibrium reconnected flux across the magnetic island.

Resistive wall tearing mode generated finite net electromagnetic torque in a static plasma

The MARS-F code [Y. Q. Liu et al., Phys. Plasmas 7, 3681 (2000)] is applied to numerically investigate the effect of the plasma pressure on the tearing mode stability as well as the tearing mode-induced electromagnetic torque, in the presence of a resistive wall. The tearing mode with a complex eigenvalue, resulted from the favorable averaged curvature effect [A. H. Glasser et al., Phys. Fluids 18, 875 (1975)], leads to a re-distribution of the electromagnetic torque with multiple peaking in the immediate vicinity of the resistive layer. The multiple peaking is often caused by the sound wave resonances. In the presence of a resistive wall surrounding the plasma, a rotating tearing mode can generate a finite net electromagnetic torque acting on the static plasma column. Meanwhile, an equal but opposite torque is generated in the resistive wall, thus conserving the total momentum of the whole plasma-wall system. The direction of the net torque on the plasma is always opposite to the real frequency of the mode, agreeing with the analytic result by Pustovitov [Nucl. Fusion 47, 1583 (2007)]. When the wall time is close to the oscillating time of the tearing mode, the finite net torque reaches its maximum. Without wall or with an ideal wall, no net torque on the static plasma is generated by the tearing mode. However, re-distribution of the torque density in the resistive layer still occurs.

The role of pressure flattening in calculating tearing mode stability

Calculations of tearing mode stability in tokamaks split conveniently into one in an external region, where marginally stable ideal magnetohydrodynamics (MHD) is applicable, and one in a resonant layer around the rational surface where sophisticated kinetic physics is needed. These two regions are coupled by the stability parameter Δ′. Axisymmetric pressure and current perturbations localized around the rational surface significantly alter Δ′. Equations governing the changes in the external solution and Δ′ are derived for arbitrary perturbations in axisymmetric toroidal geometry. These equations can be used in two ways: (i) the Δ′ can be calculated for a physically occurring perturbation to the pressure or current; (ii) alternatively we can use these equations to calculate Δ′ for profiles with a pressure gradient at the rational surface in terms of the value when the perturbation removes this gradient. It is the second application we focus on here since resistive magnetohydrodynamics (MHD) codes do not contain the appropriate layer physics and therefore cannot predict stability for realistic hot plasma directly. They can, however, be used to calculate Δ′. Existing methods (Ham et al 2012 Plasma Phys. Control. Fusion 54 025009) for extracting Δ′ from resistive codes are unsatisfactory when there is a finite pressure gradient at the rational surface and favourable average curvature because of the Glasser stabilizing effect (Glasser et al 1975 Phys. Fluids 18 875). To overcome this difficulty we introduce a specific artificial pressure flattening function that allows the earlier approach to be used. The technique is first tested numerically in cylindrical geometry with an artificial favourable curvature. Its application to toroidal geometry is then demonstrated using the toroidal tokamak tearing mode stability code T7 (Fitzpatrick et al 1993 Nucl. Fusion 33 1533) which employs an approximate analytic equilibrium. The prospects for applying this approach to resistive MHD codes such as MARS-F (Liu et al 2000 Phys. Plasmas 7 3681) which utilize a fully toroidal equilibrium are discussed.

A data acquisition system for real-time magnetic equilibrium reconstruction on ASDEX Upgrade and its application to NTM stabilization experiments

The pre-emptive stabilization of a neoclassical tearing mode, NTM, requires the calculation of the tokamak magnetic equilibrium in real-time. A launcher mirror is positioned to deposit electron cyclotron current drive on the rational surface where the NTM should appear. A real-time Grad–Shafranov solver using constraints from magnetic probe, flux loop and Motional Stark Effect measurements has been developed to locate these rational surfaces and deliver this information to the mirror controller in real-time. A novel algorithm significantly reduces the number of operations required in the first and second step of the solver. Contour integrals are carried out to calculate the q profile as a function of normalized radius and the rational surfaces are found by spline interpolation. A cycle time of 0.6ms for calculating two tokamak equilibria in parallel using four current basis functions with magnetic constraints only and using six current basis functions with magnetic and MSE constraints has been achieved. Using these tools, pre-emptive stabilization of a m/n=3/2 NTM mode in ASDEX Upgrade could be demonstrated.

An overview of FTU results

Since the 2010 IAEA-FEC Conference, FTU has exploited improvements in cleaning procedures and in the density control system to complete a systematic exploration of access to high-density conditions in a wide range of plasma currents and magnetic fields. The line-averaged densities at the disruptive limit increased more than linearly with the toroidal field, while no dependence on plasma current was found; in fact, the maximum density of 4.3 × 1020 m−3 was reached at B = 8 T even at the minimum current of 0.5 MA, corresponding to twice the Greenwald limit. The lack of plasma current dependence was due to the increase in density peaking with the safety factor. Experiments with the 140 GHz electron cyclotron resonance heating (ECRH) system were focused on the sawtooth (ST) period control and on the commissioning of a new launcher with real-time steering capability that will act as the front-end actuator of a real-time system for ST period control and tearing mode stabilization. Various ECRH and electron cyclotron current-drive modulation schemes were used; with the fastest one, the ST period synchronized with an 8 ms modulation period. The observed period variations were simulated using the JETTO code with a critical shear model for the crash trigger. The new launcher was of the plug-in type, allowing quick insertion and connection to the transmission line. Both beam characteristics and steering speed were in line with design expectation. Experimental results on the connection between improved coupling of lower hybrid waves in high-density plasmas and reduced wave spectral broadening were interpreted by fully kinetic, non-linear model calculations. A dual-frequency, time-of-flight diagnostic for the measurement of density profiles was developed and successfully tested. Fishbone-like instabilities driven by energetic electrons were simulated by the hybrid MHD-gyrokinetic XHMGC code.