Modified Rutherford Equation Research Articles

Application of the Bayesian analysis to the modified Rutherford equation for NTM stabilization

The determination of the free parameters present in the modified Rutherford equation (MRE), which is routinely used for studying the physics of neoclassical tearing mode (NTM) stabilization, is addressed by making use of the Bayesian probability theory. The evaluation of the free coefficients is particularly sensitive to the assumptions used in the modelled equation, to the correlation of various physical parameters and to the uncertainties of the experimental measurements. A probabilistic method was applied for the consistent evaluation of the coefficients and their uncertainties using a large database of discharges and by considering the correlations and the uncertainties of the multiple physical quantities present in the MRE. The estimated values and uncertainties of the coefficients are related to the precise determination of the minimum amount of electron cyclotron current drive power necessary to stabilize NTMs in the International Thermonuclear Experimental Reactor.

ASDEX Upgrade—JT-60U comparison and ECRH power requirements for NTM stabilization in ITER

Neoclassical tearing modes (NTMs) are experimentally controlled with local electron cyclotron current drive (ECCD) and the island width decay during NTM stabilization is modelled using the so-called modified Rutherford equation (MRE). In this paper, a modelling of the MRE is carried out and simulations of the island width decay are compared with the experimentally observed ones in order to fit the two free machine-independent parameters present in the equation. A systematic study on a database of NTM stabilization discharges from ASDEX Upgrade and JT-60U is done for extrapolating the ECCD power requirements for ITER. The extrapolation to ITER of the NTM stabilization results from ASDEX Upgrade and JT-60U shows that 10 MW of ECCD power are enough to stabilize large NTMs. The 10 MW power estimate for ITER is based on the assumption that the free parameters in the MRE are machine independent. Indeed, this assumption is verified in this paper for ASDEX Upgrade and JT-60U. An interesting consequence of the relatively modest power requirement for ITER is that the installed 20 MW will suffice for simultaneous 2/1 and 3/2 NTM stabilization.

On the merits of heating and current drive for tearing mode stabilization

Neoclassical tearing modes (NTMs) are magnetohydrodynamic modes that can limit the performance of high β discharges in a tokamak, leading eventually to a plasma disruption. A NTM is sustained by the perturbation of the ‘bootstrap’ current, which is a consequence of the pressure flattening across a magnetic island. Control and suppression of this mode can be achieved by means of electron cyclotron waves (ECWs) which allow the deposition of highly localized power at the island location. The ECW power replenishes the missing bootstrap current by generating a current perturbation either inductively, through a temperature perturbation (electron cyclotron resonance heating), or non-inductively by direct current drive (electron cyclotron current drive). Although both methods have been applied successfully to experiments showing a predominance of ECRH for medium-sized limiter tokamaks (TEXTOR, T-10) and of ECCD for mid-to-large-sized divertor tokamaks (AUG, DIII-D, JT-60), conditions determining their relative importance are still unclear. We address this problem with a numerical study focused on the contributions of heating and current drive to the temporal evolution of NTMs as described by the modified Rutherford equation. For the effects of both heating as well as current drive, simple analytical expressions have been found in terms of an efficiency fore-factor times a ‘geometrical’ term depending on the power deposition width wdep, location and modulation. When the magnetic island width w equals the width of the deposition profile, w ≈ wdep, both geometric terms are practically identical. Whereas for current drive the geometric term approaches a constant for small island widths and is inversely proportional to (w/wdep)2 for large island widths, the heating term approaches a constant for large island widths and is proportional to (w/wdep) for small island widths. For medium-sized tokamaks (TEXTOR, AUG) the heating and current drive efficiencies are of the same order of magnitude, whereas in a future, large reactor like ITER the current drive efficiency is expected to be significantly larger.

H-mode plasma transport simulation in ITER with effect of neoclassical tearing mode

For the prediction of the ITER plasmas, the effect of the neoclassical tearing mode (NTM) on the plasma confinement has been calculated using the 1.5-dimensional equilibrium and transport simulation code TOTAL. The time evolution of the NTM magnetic island has been analyzed using the modified Rutherford equation for ITER normal shear plasmas. The anomalous transport model used here is GLF23. The saturated magnetic island widths are w/a ∼ 0.048 at 3/2 mode and w/a∼0.21 at 2/1 mode, and the reduction in fusion power output by NTM is 27% at the 3/2 mode, and 82% at the 2/1 mode. The stabilization effect of the electron cyclotron current drive (ECCD) with EC is also clarified. The threshold of ECCD power for the full stabilization at high beta NTM island formation is ∼10 MW against the 3/2 mode, and ∼23 MW against the 2/1 mode.

Stabilization of neoclassical tearing modes by electron cyclotron current drive in JT-60U

Results of active control of neoclassical tearing modes (NTMs) by electron cyclotron current drive (ECCD) in JT-60U are described. Growth of an NTM with poloidal mode number m = 3 and toroidal mode number n = 2 has been suppressed by ECCD inside the sawtooth inversion radius in the co-direction, showing the possibility of the coexistence of sawtooth oscillations and a small-amplitude m/n = 3/2 NTM without large confinement degradation. Stabilization of an m/n = 2/1 NTM by ECCD at the mode rational surface has been demonstrated with a small ratio of the current density driven by the electron cyclotron (EC) wave to the local bootstrap current density (∼ 0.5). In addition, dependence of the stabilization effect on ECCD location has been investigated in detail. It has been found that an m/n = 2/1 NTM can be completely stabilized with the misalignment of the ECCD location less than about half of the full island width, and that the m/n = 2/1 NTM is destabilized with the misalignment comparable to the full island width. Time-dependent, self-consistent simulation of magnetic island evolution using the TOPICS code has shown that the stabilization and destabilization of an m/n = 2/1 NTM are well reproduced with the same set of coefficients of the modified Rutherford equation. The TOPICS simulation has also clarified that EC wave power required for complete stabilization can be significantly reduced by narrowing the ECCD deposition width.

Stabilization and prevention of the 2/1 neoclassical tearing mode for improved performance in DIII-D

The m = 2/n = 1 neoclassical tearing mode (NTM) has been observed to strongly degrade confinement and frequently lead to a disruption in high β discharges in DIII-D if allowed to grow to a large size. Stabilization of grown NTMs by the application of a highly localized electron cyclotron current drive (ECCD) at the island location has led to operation at an increased plasma pressure, up to the no-wall kink limit. After the NTM is stabilized by the ECCD, the correct location for the current drive is maintained using information from real-time equilibrium reconstructions which include measurements from the motional Stark effect diagnostic. This same process is used alternatively to prevent the mode from ever growing, leading to performance at the pressure limit in high performance hybrid discharges with β above 4%. Modelling using the modified Rutherford equation shows that the required power is in close agreement with the experimental threshold for the prevention of the 2/1 NTM.

Integrated Simulation Code for Burning Plasma Analysis

A strategy for integrated modeling of burning plasmas at Japan Atomic Energy Agency is described. In order to simulate the burning plasma, which has the complex feature of widely different timescales and spatial scales, a simulation code cluster based on the TOPICS transport code is being developed by integrating heating and current drive, impurity transport, the edge pedestal model, the divertor model, the magnetohydrodynamics (MHD), and the high-energy behavior model. The developed integration models are validated by fundamental research from JT-60U experiments and the simulation based on the First Principle in our strategy. The integration of MHD stability and the transport progresses for three phenomena with different timescales of neoclassical tearing modes (NTMs) (~τNTM ~ 10-2τR), beta limits (~τAlfvén), and edge-localized modes (ELMs) (intermittent of τE and τAlfvén). Here, τR, τAlfvén, and τE are the resistive skin time, the Alfvén transit time, and the energy confinement time, respectively. The integrated model of the NTM is produced by coupling the modified Rutherford equation with the transport equation. The integrated model of the beta limits is developed by the low-n stability analysis of downstreaming data from the TOPICS code. The integrated model of the ELM is developed by the iterative calculation of the MARG2D ideal MHD stability code and the TOPICS code. These models are being validated by the data from the JT-60 experiments and estimate the plasma performance for burning plasmas.

Suppression of NTM Towards Stationary High-Beta Plasmas in JT-60U

Results from stabilization of neoclassical tearing modes (NTMs) in JT-60U are described. NTM stabilization and confinement improvement have been demonstrated by employing a real-time NTM stabilization system, where the identification of the location of an NTM and the optimization of the injection angle of the electron cyclotron wave are performed in real time. Also, a high-beta plasma with the normalized beta of 3 has been sustained by suppressing NTM by applying the electron cyclotron current drive (ECCD) before the onset (`preemptive ECCD'). In addition, a simulation code for analysis of the NTM evolution has been developed by combining the modified Rutherford equation with the transport code TOPICS. It is found that the simulation well reproduces the NTM behavior in JT-60U. The simulation also shows that the ECCD width is also important for NTM stabilization, and that the EC wave power for complete stabilization can be reduced by narrowing the ECCD profile.

Stabilization of neoclassical tearing mode by ECCD and its evolution simulation on JT-60U tokamak**This manuscript is based on the paper EX/7-4 presented at the 20th IAEA Fusion Energy Conference, Vilamoura, Portugal.

Stabilization of an m = 3/n = 2 neoclassical tearing mode (NTM) by a local electron cyclotron current drive (ECCD) has been studied in high βp ELMy H-mode plasmas in the JT-60U tokamak. In the stabilization experiment, the electron cyclotron (EC) injection timing is scanned from the ‘before mode onset’ phase (preemptive ECCD) to the mode saturation phase (late ECCD). It has been demonstrated that the ECCD is more effective when it is applied before the mode onset and the critical timing for effective ECCD is related to the mode excitation phase. The EC power to prevent the 3/2 NTM excitation by preemptive ECCD is lower by about 20% compared with complete stabilization by late ECCD. Even when the 3/2 NTM is excited, the stabilization effect is stronger than for late ECCD. For a high performance plasma of the normalized beta, βN = 3.0, the H-factor H89PL improves from 1.9 to 2.1 when the 3/2 NTM is suppressed. A numerical study has been made based on the modified Rutherford equation coupled with a 1.5D transport code and an EC code. The simulation reproduces well the time evolution of the magnetic island both at the growing and stabilizing phases. The simulation results show that it is important to adjust not only the EC current position but also its width to stabilize the NTM effectively. The reduction of EC power for NTM prevention in experiments for preemptive ECCD can be explained by the increase of the minimum seed island size for NTM excitation.

Resonant effect of a background turbulence on the rotating magnetic island

Resonant interaction between wave packets of the background, short wavelength ion-temperature-gradient mode turbulence and macroscopic, rotating magnetic island is considered. Corresponding contribution to the toroidal torque balance is calculated. This resonant torque is comparable with nonresonant viscous torque produced by turbulent transport of toroidal momentum. Balance of resonant and viscous torques results in the equilibrium island rotation frequency residing in the gap where contribution of the polarization current to the modified Rutherford equation is stabilizing.

On the contribution of local current density to neoclassical tearing mode stabilization

Neoclassical tearing modes are driven by the reduction of bootstrap current inside the island due to the flattening of the pressure profile. This current perturbation enhances the magnetic perturbation responsible for the island formation. Therefore it is well known that local current drive (CD) can be used to compensate this perturbation and stabilize the mode. Several forms of the current drive contribution to the modified Rutherford equation have been proposed. Analytical fits of these contributions are provided in order to facilitate their comparison with experiments and another contribution is proposed. Since the bounce and transit frequencies are much larger than the collision frequency and the modulation frequency of the CD source in phase with the island, it is argued that the effective current driven density profile is a flux function. Assuming an exponential profile leads to no difference between modulated and continuous application of current drive. The various forms differ mainly at small island width and the possibility to differentiate amongst them experimentally is discussed.

ECCD power necessary for the neoclassical tearing mode stabilization in ITER

The stabilization of the neoclassical tearing mode (NTM) by an electron cyclotron current drive (ECCD) has been numerically studied for the parameters of ITER in order to evaluate the necessary EC power for the stabilization of NTMs in ITER. The time evolution of an island width of NTM is calculated by the modified Rutherford equation. The modification of the background current density profile by the EC current located on the rational surface is calculated by the current diffusion equation and the variation of the tearing stability index Δ′ is taken into account. When the EC power is higher than a threshold value, NTMs with any island width can be fully stabilized. The dependence of the threshold power on the parameters in the modified Rutherford equation is examined. The threshold power depends significantly on the parameters of the bootstrap current term in the modified Rutherford equation. The injected EC current decreases the value of Δ′ through the background current modification, which results in the reduction of the threshold power. The effects of the peaked nature of the EC current density profile and the EC power modulation on the threshold power are investigated. When the width of the EC current density profile becomes half, the threshold power is reduced to half or less. The EC power modulation is found to be inessential for the threshold power reduction if the EC current density profile can be peaked. As regards the maximum value of the threshold power in the range of parameters, an EC power of about 25 MW is found to be sufficient for the simultaneous stabilization of both the m/n = 3/2 and 2/1 NTMs in ITER.

Complete suppression of the m = 2/n = 1 neoclassical tearing mode using electron cyclotron current drive in DIII-D

The first suppression of the important and deleterious m = 2/n = 1 neoclassical tearing mode (NTM) is reported using electron cyclotron current drive (ECCD) to replace the ‘missing’ bootstrap current in the island O-point. Experiments on the DIII-D tokamak verify that the maximum shrinkage of the m = 2/n = 1 island occurs when the ECCD location coincides with the q = 2 surface. The DIII-D plasma control system is put into a ‘search and suppress’ mode to make small changes in the toroidal field to find and lock onto the optimum position, based on real time measurements of dBθ/dt, for complete m = 2/n = 1 NTM suppression by ECCD. The requirements of the ECCD for complete island suppression are well modelled by the modified Rutherford equation for the DIII-D plasma conditions.

Numerical Analysis of Neoclassical Tearing Mode Stabilization by Electron Cyclotron Current Drive

Neoclassical tearing mode (NTM) stabilization by an electron cyclotron drive (ECCD) has been studied by using the numerical model on the basis of the modified Rutherford equation coupled with the 1.5 D transport code and the EC code. The transport code solves the current diffusion equation, including the EC current profile. The background current modification and the resultant movement of rational surface by ECCD are taken into account. The EC code consists of the ray tracing method and the Fokker-Planck calculation. Undetermined parameters in the modified Rutherford equation are estimated from a comparison with the JT-60U experiments. Sensitivity of stabilization to the EC current location is investigated by simulation. The low EC current and peaked EC current profile mitigates the sensitivity, whereas the high EC current and peaked EC current profile moves the rational surface more largely via background current modification by the EC current and intensifies the sensitivity. The high EC current and broad EC current profile mitigates the sensitivity. The EC current necessary for the full stabilization is studied for ITER parameters. The necessary EC current strongly depends on the parameters of bootstrap current and ECCD terms in the modified Rutherford equation. Necessary ECCD power on ITER is evaluated on the basis of parameters estimated from comparisons with JT-60U experiments. (author)

Marginal β-limit for neoclassical tearing modes in JET H-mode discharges

A neoclassical tearing mode (NTM) requires a finite size seed island tobecome unstable. Usually the local pressure gradient is relatively largeat the β-values needed for these seed islands to destabilize theNTMs. Therefore, the island has a large growth rate at mode onset andgrows rapidly to its saturated island width. This width is proportionalto β as long as it is well above the marginal β-limit below whichthe mode is stable. The marginal β-limit is independent of the seedisland trigger mechanism and provides detailed information on thestabilizing terms in the modified Rutherford equation, which are notunambiguously determined theoretically. It is shown that in JET themarginal normalized β-limit for the 3/2 mode, βN,marg, is ofthe order of 0.5-1 for magnetic fields between 3.3 and 1 T, withq95≈3.3, and near the H-L transition. Therefore, all H-modeswith typical q-profiles (q95≈3.3) are metastable in JET to3/2 NTMs. In addition, the marginal island width is of the order of 2-4 cmand the stabilizing terms are such that they influence the saturatedisland width when it is smaller than 4-6 cm in these H-mode discharges.It is also shown that detailed analyses of the time evolution of theisland width with slow β ramp-down suggest that the convective formof the stabilization term due to the `χ⊥ model' is moreappropriate and can explain the island decay between 4 and 6 cm to themarginal island width, while the polarization current model can explainthe rapid stabilization when β<βmarg. The range of valuesof the different stabilizing terms are discussed in detail. Inparticular, it is shown that the mode is stabilized and has a large negative growth rate, when the effect of the stabilizing terms is suchas to reduce the saturated width by a factor of 2.

Interaction of neoclassical tearing modes with trapped fast ions

Resonance between rotating neoclassical tearing mode (NTM) and precessing trapped energetic ions has been considered. Resonant ions experience large radial excursions of their banana guiding centers leading to the loss of a considerable fraction of fast ions deposited in the resonance region. The resulting toroidal torque accelerates the magnetic island rotating in the direction of the ion diamagnetic drift. The torque balance equation has been coupled with the modified Rutherford equation, retaining the contributions from the bootstrap current and the ion polarization current. It has been shown that, due to the resonant interaction with energetic ions, this autonomous system of equations exhibits a Hopf bifurcation for some critical ratio of the momentum damping rate to the resistive diffusion rate. The resulting stable limit cycle resembles the frequency jump events observed during NTM activity in the Axisymmetric Divertor Experiment Upgrade (ASDEX Upgrade) tokamak [S. Sesnic et al., Phys. Plasmas 7, 935 (2000)].

Threshold for metastable tearing modes in DIII-D

At sufficiently high poloidal beta βθ and low collisionality, m/n = 3/2 tearing modes are excited in the DIII-D tokamak by a q = 1 sawtooth crash perturbation. The resulting islands of full radial width w are long lived and reduce global confinement (and βθ) by up to 30%. These are tearing modes produced by the perturbed neoclassical bootstrap current destabilization of a seed island arising from the m/n = 2/2 sawtooth precursor. The tearing modes are metastable. Even if the modes are conventionally stable, i.e. Δ' < 0, for sufficiently high βθ, there is a range of island widths w above some threshold value over which the modified Rutherford equation predicts island growth. This threshold value increases with collisionality. Both `transport' and `polarization' threshold models are compared with experiment. The transport model gives the correct scaling with experiment, but the threshold island wd is 3 to 9 times too small, depending on whether ion or electron parallel heat transport is used. The polarization model threshold island wg gives good agreement with experiment, utilizing an approximately hyperbolic tangent transition from collisionless to collisional regimes.