Modified Rutherford Equation Research Articles

Effect of electron cyclotron beam width to neoclassical tearing mode stabilization by minimum seeking control in ITER

We report the effect of the electron cyclotron (EC) beam width on the full suppression time of neoclassical tearing mode (NTM) using the finite difference method (FDM) based minimum seeking controller in ITER. An integrated numerical system is setup for time-dependent simulations of the NTM evolution in ITER by solving the modified Rutherford equation together with the plasma equilibrium, transport, and EC heating and current drive. The calculated magnetic island width and growth rate is converted to the Mirnov diagnostic signal as an input to the controller to mimic the real experiment. In addition, 10% of the noise is enforced to this diagnostic signal to evaluate the robustness of the controller. To test the dependency of the NTM stabilization time on the EC beam width, the EC beam width scan is performed for a perfectly aligned case first, then for cases with the feedback control using the minimum seeking controller. When the EC beam is perfectly aligned, the narrower the EC beam width, the smaller the NTM stabilization time is observed. As the beam width increases, the required EC power increases exponentially. On the other hand, when the minimum seeking controller is applied, NTM stabilization sometimes fails as the EC beam width decreases. This is consistently observed in the simulation with various representations of the noise as well as without the noise in the Mirnov signal. The higher relative misalignment, misalignment divided by the beam width, is found to be the reason for the failure with the narrower beam widths. The EC stabilization effect can be lower for the narrower beam widths than the broader ones even at the same misalignment due to the smaller ECCD at the island O-point. On the other hand, if the EC beam is too wide, the NTM stabilization time takes too long. Accordingly, the optimal EC beam width range is revealed to exist in the feedback stabilization of NTM.

Towards self-consistent plasma modelisation in presence of neoclassical tearing mode and sawteeth: effects on transport coefficients

The neoclassical tearing modes (NTM) increase the effective heat and particle radial transport inside the plasma, leading to a flattening of the electron and ion temperature and density profiles at a given location depending on the safety factor q rational surface (Hegna and Callen 1997 Phys. Plasmas 4 2940). In burning plasma such as in ITER, this NTM-induced increased transport could reduce significantly the fusion performance and even lead to a disruption. Validating models describing the NTM-induced transport in present experiment is thus important to help quantifying this effect on future devices. In this work, we apply an NTM model to an integrated simulation of current, heat and particle transport on JET discharges using the European transport simulator. In this model, the heat and particle radial transport coefficients are modified by a Gaussian function locally centered at the NTM position and characterized by a full width proportional to the island size through a constant parameter adapted to obtain the best simulations of experimental profiles. In the simulation, the NTM model is turned on at the same time as the mode is triggered in the experiment. The island evolution is itself determined by the modified Rutherford equation, using self-consistent plasma parameters determined by the transport evolution. The achieved simulation reproduces the experimental measurements within the error bars, before and during the NTM. A small discrepancy is observed on the radial location of the island due to a shift of the position of the computed q = 3/2 surface compared to the experimental one. To explain such small shift (up to about 12% with respect to the position observed from the experimental electron temperature profiles), sensitivity studies of the NTM location as a function of the initialization parameters are presented. First results validate both the transport model and the transport modification calculated by the NTM model.

Electron cyclotron power management for control of neoclassical tearing modes in the ITER baseline scenario

Time-dependent simulations are used to evolve plasma discharges in combination with a modified Rutherford equation for calculation of neoclassical tearing mode (NTM) stability in response to electron cyclotron (EC) feedback control in ITER. The main application of this integrated approach is to support the development of control algorithms by analyzing the plasma response with physics-based models and to assess how uncertainties in the detection of the magnetic island and in the EC alignment affect the ability of the ITER EC system to fulfill its purpose. Simulations indicate that it is critical to detect the island as soon as possible, before its size exceeds the EC deposition width, and that maintaining alignment with the rational surface within half of the EC deposition width is needed for stabilization and suppression of the modes, especially in the case of modes with helicity . A broadening of the deposition profile, for example due to wave scattering by turbulence fluctuations or not well aligned beams, could even be favorable in the case of the NTM, by relaxing an over-focussing of the EC beam and improving the stabilization at the mode onset. Pre-emptive control reduces the power needed for suppression and stabilization in the ITER baseline discharge to a maximum of 5 MW, which should be reserved and available to the upper launcher during the entire flattop phase. Assuming continuous triggering of NTMs, with pre-emptive control ITER would be still able to demonstrate a fusion gain of .

Shrinking of core neoclassical tearing mode magnetic islands due to edge localized modes and the role of ion-scale turbulence in island recovery in DIII-D

Experimental signature of long-wavelength turbulence accelerating the recovery of Neoclassical Tearing Mode (NTM) magnetic islands after they have been transiently reduced in size due to interaction with Edge Localized Modes (ELMs) is reported for the first time. This work shows that perturbations associated with ELMs result in peaking of the electron temperature (Te) in the O-point region of saturated core m/n = 2/1 islands (m/n being the poloidal/toroidal mode numbers). In synchronization with this Te peak, the island width shrinks by as much as 30% suggesting a key role of the Te peak in NTM stability due to modified pressure gradient (∇p) and perturbed bootstrap current (δjBS) at the O-point. Next, this Te peak relaxes via anomalous transport (i.e., the diffusivity is 2 orders of magnitude larger than the neoclassical value) and the island recovers. Long-wavelength turbulent density fluctuations (ñ) are reduced at the O-point of flat islands but these fluctuations are increased when Te is peaked which offers an explanation for the observed anomalous transport that is responsible for the relaxation of the Te peak. Linear gyrokinetic simulations indicate that ñ inside the peaked island is dominantly driven by the Ion Temperature Gradient instability. These measurements suggest that ñ accelerates NTM recovery after an ELM crash via accelerating the relaxation of ∇p at the O-point. These observations are qualitatively replicated by coupled predator-prey equations and modified Rutherford equation. In this simple model, turbulence accelerates NTM recovery via relaxing ∇p and therefore restoring δjBS at the O-point. The key physics of the relationship between the Te peak and NTM stability has potentially far-reaching consequences, such as NTM control via pellet injection in high-β tokamak plasmas.

Stabilization of Neoclassical Tearing Modes by rf Current

Neoclassical tearing mode (NTM) can degrade plasma confinement or even cause disruptions in existing tokamaks. Stabilization of the NTMs by radio frequency (rf) current is investigated by the modified Rutherford equation (MRE) in this paper. In a range of parameters, the required rf current for mode stabilization is obtained, which is linearly proportional to the bootstrap current density for both modulated current drive (MCD) and non-modulated current drive (NMCD), linearly (quadratically) to the radial width of the rf current for MCD (NMCD), and quadratically to the radial deviation of the rf current from the rational surface for both MCD and NMCD.

Nonlinear asymmetric tearing mode evolution in cylindrical geometry

The growth of a tearing mode is described by reduced MHD equations. For a cylindrical equilibrium, tearing mode growth is governed by the modified Rutherford equation, i.e., the nonlinear Δ′(w). For a low beta plasma without external heating, Δ′(w) can be approximately described by two terms, Δ′ql(w), ΔA′(w) [White et al., Phys. Fluids 20, 800 (1977); Phys. Plasmas 22, 022514 (2015)]. In this work, we present a simple method to calculate the quasilinear stability index Δql′ rigorously, for poloidal mode number m≥2. Δql′ is derived by solving the outer equation through the Frobenius method. Δ′ql is composed of four terms proportional to: constant Δ′0, w, wlnw, and w2. ΔA′ is proportional to the asymmetry of island that is roughly proportional to w. The sum of Δql′ and ΔA′ is consistent with the more accurate expression calculated perturbatively [Arcis et al., Phys. Plasmas 13, 052305 (2006)]. The reduced MHD equations are also solved numerically through a 3D MHD code M3D-C1 [Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)]. The analytical expression of the perturbed helical flux and the saturated island width agree with the simulation results. It is also confirmed by the simulation that the ΔA′ has to be considered in calculating island saturation.

Local measurement of error field using naturally rotating tearing mode dynamics in EXTRAP T2R

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 rotates rapidly in the presence of an unknown EF and a known, deliberately applied RMP. The RMP amplitude is fixed while the toroidal phase is varied from one discharge to the other, completing a full toroidal scan. Using three such scans with different RMP amplitudes, the EF amplitude and phase are inferred from the phases at which the TM amplitude maximizes. The estimated EF amplitude is consistent with other estimates (e.g. based on the best EF-cancelling RMP, resulting in the fastest TM rotation). A passive variant of this technique is also presented, where no RMPs are applied, and the EF phase is deduced.

Non-perturbative measurement of cross-field thermal diffusivity reduction at the O-point of 2/1 neoclassical tearing mode islands in the DIII-D tokamak

Neoclassical tearing modes (NTMs) often lead to the decrease of plasma performance and can lead to disruptions, which makes them a major impediment in the development of operating scenarios in present toroidal fusion devices. Recent gyrokinetic simulations predict a decrease of plasma turbulence and cross-field transport at the O-point of the islands, which in turn affects the NTM dynamics. In this paper, a heat transport model of magnetic islands employing spatially non-uniform cross-field thermal diffusivity (χ⊥) is presented. This model is used to derive χ⊥ at the O-point from electron temperature data measured across 2/1 NTM islands in DIII-D. It was found that χ⊥ at the O-point is 1 to 2 orders of magnitude smaller than the background plasma transport, in qualitative agreement with gyrokinetic predictions. As the anomalously large values of χ⊥ are often attributed to turbulence driven transport, the reduction of the O-point χ⊥ is consistent with turbulence reduction found in recent experiments. Finally, the implication of reduced χ⊥ at the O-point on NTM dynamics was investigated using the modified Rutherford equation that predicts a significant effect of reduced χ⊥ at the O-point on NTM saturation.

Modeling of neoclassical tearing mode stabilization by electron cyclotron heating and current drive in tokamak plasmas

An integrated numerical system is established to model time-dependent behavior of the neoclassical tearing mode (NTM) in a tokamak which solves the modified Rutherford equation (MRE) by coupling with plasma transport, equilibrium, heating and current drive. The MRE is formulated in a simple form to be well-suited for time-dependent simulations including a predictive purpose for the feedback controller design by coupling the electron cyclotron effect self-consistently. In particular the electron cyclotron heating (ECH) effect is newly included to the MRE in addition to the electron cyclotron current drive (ECCD) effect to investigate their impact to stabilize the NTM. The integrated numerical system is applied to experiments for benchmarking in which NTMs are stabilized by ECCD and by ECH at ASDEX Upgrade and TCV, respectively. The impact of ECCD and ECH on stabilizing NTMs is identified in the simulations.

First principles fluid modelling of magnetic island stabilization by electron cyclotron current drive (ECCD)

Tearing modes are MagnetoHydroDynamics (MHD) instabilities that reduce the performance of fusion devices. They can however be controlled and suppressed using electron cyclotron current drive (ECCD) as demonstrated in various tokamaks. In this work, simulations of island stabilization by ECCD-driven current have been carried out using the toroidal nonlinear 3D full MHD code xtor-2f, in which a current source term modeling the ECCD has been implemented. The efficiency parameter, , has been computed and its variations with respect to source width and location were also computed. The influence of parameters such as current intensity, source width and position with respect to the island was evaluated and compared to the modified Rutherford equation. We retrieved a good agreement between the simulations and the analytical predictions concerning the variations of control efficiency with source width and position. We also show that the 3D nature of the current source term can lead to the onset of an island if the source term is precisely applied on a rational surface. We report the observation of a flip phenomenon in which the O- and X-points of the island rapidly switch their position in order for the island to take advantage of the current drive to grow.

The tokamak density limit: A thermo-resistive disruption mechanism

The behavior of magnetic islands with 3D electron temperature and the corresponding 3D resistivity effects on growth are examined for islands with near-zero net heating in the island interior. We refer to the resulting class of non-linearities as thermo-resistive effects. In particular, the effects of varying impurity mix on the previously proposed local island onset threshold [Gates and Delgado-Aparicio, Phys. Rev. Lett. 108, 165004 (2012)] are examined and shown to be consistent with the well established experimental scalings for tokamaks at the density limit. A surprisingly simple semi-analytic theory is developed which imposes the effects of heating/cooling in the island interior as well as the effects of island geometry. For the class of current profiles considered, it is found that a new term that accounts for the thermal effects of island asymmetry is required in the modified Rutherford equation. The resultant model is shown to exhibit a robust onset of a rapidly growing tearing mode—consistent with the disruption mechanism observed at the density limit in tokamaks. A fully non-linear 3D cylindrical calculation is performed that simulates the effect of net island heating/cooling by raising/suppressing the temperature in the core of the island. In both the analytic theory and the numerical simulation, the sudden threshold for rapid growth is found to be due to an interaction between three distinct thermal non-linearities which affect the island resistivity, thereby modifying the growth dynamics.

Amplitude based feedback control for NTM stabilisation at ASDEX Upgrade

Neoclassical Tearing Modes (NTMs) degrade the confinement in tokamak plasmas at high beta, placing a major limitation on the projected fusion performance. Furthermore, NTMs can lead to disruptions with even more severe consequences. Therefore methods to stabilise NTMs are being developed with high priority at several research institutes worldwide. The favoured method is to deposit Electron Cyclotron Current Drive (ECCD) precisely at the mode location by controlling a movable mirror in the ECCD launcher. This method requires both the mode location and the deposition location to be known with high accuracy in real time. The required accuracy is given by half of the marginal island width, or approximately 1cm for a m/n=3/2 NTM at ASDEX Upgrade. Despite considerable development on a range of diagnostics, it remains challenging to provide the necessary accuracy reliably and in real time.To relax the accuracy requirements and add robustness, the feedback controller can additionally consider the effect of ECCD on the NTM amplitude directly. Then the optimal deposition location is simply where the NTM amplitude is minimised. The simplest implementation sweeps the ECCD beam across the expected NTM location. After the sweep, the beam can be returned to the optimal location and held there to stabilise the NTM. Unfortunately, waiting for a full sweep takes too long. Therefore a second method assesses the NTM growth every 100ms and hones in on the NTM in a series of steps. The two algorithms employ similar principles to the “Target Lock” and “Search & Suppress” algorithms implemented at DIII-D [1]. This paper presents the development of both algorithms using a simulation of the full feedback loop including NTM dynamics modelled by a Modified Rutherford Equation, and a demonstration of their effectiveness during ASDEX Upgrade experiments.

Self-consistent modeling of the dynamic evolution of magnetic island growth in the presence of stabilizing electron-cyclotron current drive

The most promising technique for the control of neoclassical tearing modes in tokamak experiments is the compensation of the missing bootstrap current with an electron-cyclotron current drive (ECCD). In this frame, the dynamics of magnetic islands has been studied extensively in terms of the modified Rutherford equation (MRE), including the presence of a current drive, either analytically described or computed by numerical methods. In this article, a self-consistent model for the dynamic evolution of the magnetic island and the driven current is derived, which takes into account the island's magnetic topology and its effect on the current drive. The model combines the MRE with a ray-tracing approach to electron-cyclotron wave-propagation and absorption. Numerical results exhibit a decrease in the time required for complete stabilization with respect to the conventional computation (not taking into account the island geometry), which increases by increasing the initial island size and radial misalignment of the deposition.

Physics of radiation-driven islands near the tokamak density limit

In previous work (Gates and Delgado-Aparicio 2012 Phys. Rev. Lett. 108 165004), the onset criterion for radiation-driven islands (Rebut et al 1985 Proc. 10th Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research 1984 (London, UK, 1984) vol 2 (Vienna: IAEA) p 197) in combination with a simple cylindrical model of tokamak current channel behaviour was shown to be consistent with the empirical scaling of the tokamak density limit (Greenwald et al 1988 Nucl. Fusion 28 2199). A number of the unexplained phenomena at the density limit are consistent with this novel physics mechanism. In this work, a more formal theoretical underpinning, consistent with cylindrical tearing mode theory, is developed for the onset criteria of these modes. The appropriate derivation of the radiation-driven addition to the modified Rutherford equation (MRE) is discussed. Additionally, the ordering of the terms in the MRE is examined in a regime near the density limit. It is hoped that, given the apparent success of this simple model in explaining the observed global scalings, it will lead to a more comprehensive analysis of the possibility that radiation-driven islands are the physics mechanism responsible for the density limit. In particular, with modern diagnostic capabilities detailed measurements of current densities, electron densities and impurity concentrations at rational surfaces should be possible, enabling verification of the concepts described above.

On the formation and stability of long-lived impurity-ion snakes in Alcator C-Mod

Long-lived (1, 1) ‘snake’ modes were discovered nearly three decades ago, but basic questions regarding their formation, stability, and superb particle confinement—shown by surviving tens to hundreds of sawtooth cycles—have remained unanswered. High-resolution spectroscopic imaging diagnostics permit studies of heavy-impurity-ion snakes with unprecedented temporal and spatial resolution, making it possible to positively identify the SXR signals with specific ion charge states and to infer, for the first time, the perturbed impurity density, Zeff, and resistivity at the centre of these long-lived helical modes. The results show a new scenario for the formation of heavy-impurity-ion snakes, which can begin as a broad 1/1 kink asymmetry of the central impurity-ion density, that grows and undergoes a seamless transition to a large crescent-shaped helical island-like structure inside q < 1, with a regularly sawtoothing core. This type of formation departs strongly from the nonlinear island model based on a modified Rutherford equation proposed originally to describe the pellet-induced snakes and expanded further to account for the impurity effects (e.g. and ). These new high-resolution observations show details of their evolution and the accompanying sawtooth oscillations that suggest important differences between the density and temperature dynamics, ruling out a purely pressure-driven process. Instead, many features arise naturally from nonlinear interactions in a 3D MHD model that separately evolves the plasma density and temperature.

Electron-cyclotron wave propagation, absorption and current drive in the presence of neoclassical tearing modes

We analyze the propagation of electron-cyclotron waves, their absorption and current drive when neoclassical tearing modes (NTMs), in the form of magnetic islands, are present in a tokamak plasma. So far, the analysis of the wave propagation and power deposition in the presence of NTMs has been performed mainly in the frame of an axisymmetric magnetic field, ignoring any effects from the island topology. Our analysis starts from an axisymmetric magnetic equilibrium, which is perturbed such as to exhibit magnetic islands. In this geometry, we compute the wave evolution with a ray-tracing code, focusing on the effect of the island topology on the efficiency of the absorption and current drive. To increase the precision in the calculation of the power deposition, the standard analytical flux-surface labeling for the island region has been adjusted from the usual cylindrical to toroidal geometry. The propagation up to the O-point is found to be little affected by the island topology, whereas the power absorbed and the driven current are significantly enhanced, because the resonant particles are bound to the small volumes in between the flux surfaces of the island. The consequences of these effects on the NTM evolution are investigated in terms of the modified Rutherford equation.

The influence of finite radial transport on the structure and evolution of m/n = 2/1 neoclassical tearing modes on MAST

Finite radial transport around magnetic islands is believed to play an important role in the threshold, spatial structure and temporal evolution of neoclassical tearing modes (NTMs). We report on novel measurements of NTMs with mode structure m/n = 2/1 on the MAST spherical tokamak (ST), which have allowed a direct evaluation of the effect of transport on island behaviour for the first time on an ST. Temperature profiles obtained with the upgraded Thomson scattering (TS) system on MAST have been used to constrain the solutions of a heat transport equation for the NTM magnetic island (Fitzpatrick 1995 Phys. Plasmas 2 825), allowing the determination of the critical width for temperature flattening across an island wc, an important parameter in the modified Rutherford equation (MRE) for NTM evolution. The measured value of wc = 0.7 ± 0.2 cm obtained for an ensemble of high βN MAST discharges has been used in an analysis of the MRE for 2/1 NTM growth and saturation on MAST. Using a probabilistic method for parameter and error estimation, which takes account of the experimental uncertainty on measured equilibrium parameters, it is found that the temporal evolution of island size is well described by marginally, classically unstable NTMs with strongly destabilizing bootstrap current and stabilizing curvature terms. Finally, further analysis of a β ramp-down discharge is presented, in which the measured wc value explains the observed threshold width well.

Hybrid-like 2/1 flux-pumping and magnetic island evolution due to edge localized mode-neoclassical tearing mode coupling in DIII-D

Direct analysis of internal magnetic field pitch angles measured using the motional Stark effect diagnostic shows m/n=2/1 neoclassical tearing modes exhibit stronger poloidal magnetic flux-pumping than typical hybrids containing m/n=3/2 modes. This flux-pumping causes the avoidance of sawteeth, and is present during partial electron cyclotron current drive suppression of the tearing mode. This finding could lead to hybrid discharges with higher normalized fusion performance at lower q95. The degree of edge localized mode-neoclassical tearing mode (ELM-NTM) coupling and the strength of flux-pumping increase with beta and the proximity of the modes to the ELMing pedestal. Flux-pumping appears independent of magnetic island width. Individual ELM-NTM coupling events show a rapid timescale drop in the island width followed by a resistive recovery that is successfully modeled using the modified Rutherford equation. The fast transient drop in island width increases with ELM size.

Stabilization Analysis of Neoclassical Tearing Mode in Fusion Plasmas

For the stabilization of neoclassical tearing mode (NTM), Electron Cyclotron Current Drive (ECCD) is used. The change of the EC control efficiency depends on EC modulation width and EC injection phase lag from the Opoint of magnetic island. In this work, NTM stabilization by ECCD is analyzed using 1.5-dimensional transport code TOTAL, in which the time variation of magnetic island is described by the modified Rutherford equation. NTM in ITER can be stabilized when the EC phase lag is smaller than 10% and the EC modulation width is around 20%, when the time-averaged EC current and power is fixed.

Effect of magnetic island geometry on ECRH/ECCD and consequences to the NTM stabilization dynamics

In the majority of codes that model ECCD-based NTM stabilization, the analysis of the EC propagation and absorption is performed in terms of the axisymmetric magnetic field, ignoring effects due to the island topology. In this paper, we analyze the wave propagation, absorption and current drive in the presence of NTMs, as well as the ECCD-driven island growth, focusing on the effect of the island geometry on the wave de- position. A primary evaluation of the consequences of these effects on the NTM evolution is also made in terms of the modified Rutherford equation.