Toroidal Effects Research Articles

Systematic simulation studies on the penetration of resonant magnetic perturbations in an Experimental Advanced Superconducting Tokamak

The penetration properties of the n = 1 resonant magnetic perturbations (RMPs) with toroidal rotation are systematically studied by the upgraded three-dimensional toroidal magnetohydrodynamic code CLTx, through both linear and nonlinear simulations. It is found that in the presence of toroidal plasma rotation, the saturation state for high resonant harmonics is obtained in linear simulations due to the mode becoming unlocked from the internal magnetic islands. In nonlinear simulations, nonlinear effects become important when the toroidal plasma rotation is not included. The zonal component resulting from the nonlinear mode coupling is necessary for the saturation of the whole system including the internal kink mode and the m /n = 2/1 tearing mode. The simulations of RMP penetration demonstrate that the mode coupling is associated with the toroidal effect rather than nonlinear effects. With a low resistivity close to the experimental value, the single-harmonic-RMP is hard to penetrate the mode-rational surface because of the plasma screening effects, resulting in a truncation of the radial mode structure. On the other hand, the non-resonant components in the multiple-harmonic-RMP could largely reduce the effect of the plasma shielding, which leads to that the RMP is able to penetrate deeply into the central plasma region through the poloidal harmonic coupling.

Locking effects of error fields on a tearing mode in tokamak

Locking effects of error fields on a tearing mode in Tokamak are studied numerically using the three-dimensional toroidal code based on a full set of magnetohydrodynamic equations. It is found that a threshold of the error field for mode locking exists and depends on the plasma rotation and the ramp-up time of the error field. The mode locking threshold increases with increasing the rotation frequency and the ramp-up time of the error field. Moreover, the results from the multiple helical error field suggest that the m/n = 3/1 and 4/2 error field along with the m/n = 2/1 error field can increase both the m/n = 2/1 perturbation and its higher-harmonics through the mode coupling due to both the toroidal and nonlinear effects, but it becomes more effective if the 4/2 error field is imposed directly. The 3/1 error field in-phase (anti-phase) with the 2/1 error field leads to a positive (negative) contribution to intensification of the 2/1 tearing mode and mode locking.

Toroidicity effects on the absorption of IC waves by electrons

Poloidal modulation of the parallel velocity of charged particles in a tokamak magnetic field modifies the strength of electron absorption of the compressional wave in the ion cyclotron range of frequencies. We have developed a model of this effect which is sufficiently simple to be numerically tractable, and we have implemented it in TORIC full-wave code, which solves the wave equation in toroidal axisymmetric geometry. We find that the effect is normally negligible in present-day devices, but its importance increases with the dimensions and the temperature of the plasma, and should be taken into account when estimating the competition between electron and ion heating and the efficiency of the current drive in future reactor-grade plasmas.

Temperature Gradient, Toroidal and Ion FLR Effects on Drift-Tearing Modes**Supported by the National MCF Energy R&D Program (Grant Nos. 2018YFE0304100, 2018YFE0311300, and 2017YFE0301300), the National Natural Science Foundation of China (Grant Nos. 11675256, 11675257, 11835016, and 11705275), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB16010300), the Key Research Program of Frontier Science of Chinese Academy

The influences of the temperature gradient and toroidal effects on drift-tearing modes have been studied using the Gyrokinetic Toroidal code. After the thermal force term is introduced into the parallel electron force balance equation, the equilibrium temperature gradient can cause a significant increase in the growth rate of the drift-tearing mode and a broadening of the mode structure. The simulation results show that the toroidal effects increase the growth rate of the drift-tearing mode, and the contours of the perturbation field “squeeze” toward the stronger field side in the poloidal section. Finally, the hybrid model for fluid electrons and kinetic ions has been studied briefly, and the dispersion relation of the drift-tearing mode under the influence of ion finite Larmor radius effects is obtained. Compared with the dispersion relation under the fluid model, a stabilizing effect of the ion finite Larmor radius is observed.

The effects of toroidal equilibrium flow on the stabilization of ballooning modes in tokamaks

Effects of toroidal flows on ballooning mode instabilities have been studied extensively in recent years. In the published works, toroidal flow effects are taken into account only in the perturbation equation while the tokamak equilibrium is assumed to be unchanged. In this work, the influence of a toroidal flow on ballooning mode instabilities is numerically investigated by using the BOUT++ code while the modification of the tokamak equilibrium due to the flow has been taken into account in fixing the profiles of the quasi-toroidal magnetic field, quasi-density, and quasi-pressure and only changing the toroidal Mach number. A toroidal flow can significantly modify the safety factor q profile and the pressure gradient ∇P, which then affects the ballooning instability. The simulation results show that in the presence of shear flows, the growth rate of the ballooning mode will be affected if the modified equilibrium is considered. For a shear flow, the growth rates of high-n modes are significantly reduced.

On the multiplicity of axisymmetric Alfvén continuous spectra in tokamaks

Axisymmetric Alfvén continuous spectra are derived using the ideal MHD equations. The usual toroidal coupling effect due to the variation of the magnetic field strength is neglected, but the curvature coupling of the pressure fluctuation and the geodesic compressional effect are included. The cylindrical continuous spectrum of Alfvén waves splits into two branches due to these effects. The frequency separation of the two m = 2 branches is much smaller than that of the two m = 1 branches. It is likely that the experimentally observed modes with smaller frequency separation are related to the two m = 1 continuum branches, while the modes with large frequency separation have different poloidal numbers.

Theory of gyrokinetic velocity moment and its application for zonal flows in a tokamak plasma

The calculation of the velocity moment in the gyrokinetic theory and its application in the study of zonal flows in a tokamak plasma are investigated. Based on the scalar invariance property, the expression for the particle velocity in the gyrocenter coordinates is obtained, and a method to systematically calculate the velocity moment in the gyrokinetic theory is proposed, especially for the calculation of the first-order perpendicular velocity moment. The kinetic equation which describes the evolution of the perturbed distribution on the meso-scale of the zonal flow is derived. The effects of the turbulent particle flux, the turbulent energy flux, the turbulent toroidal Reynolds stress and the turbulent poloidal Reynolds stress (PRS) are explicitly included in the phase-space particle flux. In the cylindrical geometry, the zonal radial electric field is driven by the turbulent PRS and the turbulent energy flux, and the result agrees with the radial force balance equation. In the toroidal geometry, the zonal radial electric field is driven by the turbulent PRS, the turbulent energy flux and the turbulent toroidal Reynolds stress; in contrast to the case in the cylindrical geometry, the effect of the turbulent PRS is shielded by the toroidal effect. By combining the radial force balance equation, the poloidal momentum equation in the toroidal geometry is obtained.

Local profiles of line emission of impurity ions in rotating fusion plasmas

We analyze the poloidal distribution of impurity ions, such as C V, C VI and O VIII, in fast rotating KSTAR plasmas using the impurity emission lines in the 1–7 nm wavelength range obtained from the space-resolved extreme ultra violet spectrometer. This is aided particularly by a new tomographic reconstruction technique with limited viewing angles. The vacuum and extreme ultraviolet spectrometer system at ITER and its prototypes installed at KSTAR are one of the main diagnostic systems to monitor the impurity species in fusion plasmas; however, they have an insufficient field of view to utilize the conventional reconstruction method. The new algorithm developed in this work considers a curvilinear grid of pixels mapped onto the magnetic equilibrium and a weight factor considering toroidal rotational effect on impurity species. Validation tests with poloidally asymmetric images of synthetic plasma demonstrate that this method is highly useful for emission diagnostics that have a narrow field of view. The experimental results of the spatially resolved spectra from KSTAR plasma clearly show that even the light impurities such as carbon and oxygen are significantly affected by the toroidal rotation. Accordingly, the method is expected to be utilized for estimating the poloidal density profile of each charge state of an impurity species, which will play a key role in impurity transport studies.

Toroidal dipolar bound state in the continuum and antiferromagnetic in asymmetric metasurface

The toroidal dipole (TD) with weak coupling to the electromagnetic fields promises tremendous capability in sensing, light absorption and optical nonlinearities. Here, we analyze the near-field coupling effects of an asymmetric all-dielectric metasurface with an array of high-index Mie-resonant dielectric tetramer cluster. Two distinct optical toroidal dipolar bound states in the continuum are identified from the asymmetric metasurface. One occurs at the internal gap of the tetramer cluster array (intra-cluster toroidal modes), and the other arises from two neighboring clusters (inter-cluster toroidal modes). Via control of the asymmetry of the cluster, the TD could transform from bound states in the continuum into leaky resonances with high-Q factors. The optical antiferromagnetic response could also be identified from the tetramer cluster arrays. The toroidal and antiferromagnetic effects come from the trapped modes with symmetry breaking. These robust responses of the asymmetric metasurface remain stable as the asymmetry degree increases and the polarization of the incident light changes. The proposed metasurface with efficient light–matter interaction serves as a platform for controlling and exploiting optical toroidal and antiferromagnetic excitations.

Actively tunable toroidal excitations in graphene based terahertz metamaterials

Toroidal dipoles, unlike electric and magnetic dipoles, are formed due to currents flowing on the surface of a torus along it’s meridians. We investigate the toroidal dipole response in a graphene based infinite 2D array of terahertz (THz) metamaterial (MM) and report the active tuning of the toroidal resonances. Each metamolecule in our proposed MM configuration consists of two joint split ring resonators (SRRs) made up of graphene material with two gaps in each resonator. We examine actively tunable response of the toroidal excitations with respect to a change in the Fermi energy of graphene material. The toroidal effect is elaborated using surface current and magnetic field profiles at the resonance frequency of the metamolecule structure. The modulation of the metamaterial’s toroidal response and effect on Q factor is further examined by changing the relaxation time of graphene. The excitation of toroidal resonances at terahertz frequencies and their dynamic control could be significant in designing efficient tunable modulators, filters and sensors.

Real frequency tearing layers with parallel dynamics and the effect on error field locking and resistive wall modes

The response to tearing perturbations of a resonant layer within a toroidal plasma often includes real frequencies. These real frequencies are of importance not only for the stability of the tearing mode but also for determining the response to an error field, including locking torques, and the stability properties when interacting with a resistive wall. Unfortunately, including the physics which drives these frequencies can make the model highly complicated and mask the root physics mechanisms driving the response, making it difficult to draw connections between the physics mechanisms occurring in different regimes. In this paper, we present a simple and intuitive method to derive the tearing mode layer properties in various regimes of resistive MHD with the key physics effects of pressure gradient, toroidal field line curvature and parallel dynamics, focusing on two important regimes: The resistive-inertial (RI) and the viscoresistive (VR) regimes. The usual Glasser effect, a toroidal effect which involves real frequencies, has been discussed only in the context of the RI regime. We find that it occurs in the VR regime as well. Based on these results and the presence of real frequencies in many other tearing regimes, we reach the conclusion that real frequencies in tearing modes are the rule rather than the exception. We also find that the existence of tearing modes with real frequencies in the VR and RI regimes is related to nearby electrostatic resistive interchange modes, themselves with real frequencies. Another important effect we find is that the threshold for the destabilization of resistive wall tearing modes is significantly lowered by plasma rotation in the presence of real frequency layers.

Toroidal effect on runaway vortex and avalanche growth rate

The momentum-space dynamics of runaway electrons in a slab geometry, in terms of both the geometry and topological transition of the runaway vortex when synchrotron radiative damping is taken into account, has recently been shown to play a crucial role in runaway mitigation and avoidance. In a tokamak geometry, magnetic trapping arises from parallel motion along the magnetic field that scales as 1/R in strength with R the major radius. Since the transit time for a runaway electron moving along the field is of order 10−8 s while the collisional time is of ∼0.01 s in ITER-like plasmas, a bounce-averaged formulation can drastically reduce computational cost. Here, the Los Alamos Plasma Simulation – Relativistic Fokker-Planck Solver code's implementation of a bounce-averaged relativistic Fokker-Planck model, along with the essential physics of synchrotron radiation damping and knock-on collisions, is described. It is found that the magnetic trapping can reduce the volume of the runaway vortex as the momentum-space fluxes are strongly modified inside the trapped-region. As a result, the avalanche growth rate is reduced at off-axis locations. In addition to benchmarking with previous calculations that did not take into account radiation damping, we also clarify how synchrotron radiation damping modifies the avalanche growth rate in a tokamak.

Unified modeling of both resonant and non-resonant neoclassical transport under non-axisymmetric magnetic perturbations in tokamaks

A numerical model for neoclassical transport under nonaxisymmetric magnetic perturbations in low collisionality plasmas in tokamaks is developed. To take into account bounce-drift resonances and magnetic drift effects, a Fourier decomposition of the drift kinetic equation in new coordinates, rather than bounce average of it, is employed. A pitch angle scattering collisional operator is used to keep accuracy in the nonresonant regimes or resonant plateau regimes with resonant pitch near pitch space boundaries. Full toroidal geometry effects are also included to increase the accuracy in the modeling of neoclassical transport in the resonant regimes. Neoclassical transport in the most important collisionless regimes, including resonant super-banana plateau and bounce-drift resonances regimes, nonresonant 1/ν and ν−ν regimes, and the transitions between them, can be modeled simultaneously in this model by numerically solving the drift kinetic equation. By application to the neoclassical toroidal viscosity modeling in one discharge in the EAST tokamak, it is found that the bounce-drift resonances dominate the contributions near the plasma core where the plasma E→×B→ drift frequency is close to the bounce frequency, while the precessional resonance dominates the contribution near the edge pedestal top where the E→×B→ drift frequency is close to zero.

Rotation braking with n = 1 nonaxisymmetric magnetic perturbation in the EAST tokamak

The toroidal plasma rotation braking effect during the application of n = 1 static resonant magnetic perturbation is studied by momentum transport analysis in the EAST tokamak. The braking torque shows a global profile and two peaks located near the plasma core and the edge, respectively. The effect of momentum diffusion contributes significantly to the calculated torque. Simulation results with the obtained torque and momentum diffusion coefficients well reproduce the observed plasma rotation evolution. Neoclassical toroidal viscosity (NTV) torque is modeled for comparison with the experimental torque. The total integrated NTV torque is around −0.12 Nm, which is comparable to the observed braking torque (around −0.33 Nm). In the plasma edge, there is a peak in the NTV torque profile, which agrees well in amplitude with the obvious peak in the observed torque density profile. An additional peak in the NTV torque profile due to the ion bounce resonance is also located in the core region. However, the magnitude of this peak is much smaller than the observed one near the plasma core.

Spontaneous magnetic field multipolar structure in toroidal plasmas based on 2D equilibrium

The multipolar magnetic field structure is investigated by the momentum conservation equation with self-consistent 3D sheared flows during transition of plasma properties from local paramagnetic to diamagnetic fields. Numerical results show that the traditional poloidal magnetic field () is one part of equilibrium magnetic fields. The non-zero-order quantities are originated from the higher-order terms of 2D equilibrium treatment based on a Fourier expansion of ψ (r, θ). The distributions of magnetic field vectors of the order of 1, 2, and 3 terms are presented respectively in two, four, and six polar fields with the local vortex structures (spontaneous magnetic connection). The excitation mechanisms of the magnetic vortices are the coupling effects of the magnetic fluid structure pattern and the toroidal effects. These results can help us understand the physical mechanism of the interaction between the external perturbation fields and control tearing modes, as well as the radial plasma flow and magnetic vortices.

Multipolar Structure of Equilibrium Shear Flow Field in Toroidal Plasmas**Supported by National Natural Science Foundation of China under Grant No. 11575066 and Domestic ITER under Grant No. 2009GB105003

The multipolar velocity field structures are investigated by 2D momentum conservation equation with 3D equilibrium sheared flows in the full toroidal system. Numerical results show that the non-existence of radial velocity field in equilibrium surfaces is suitable only for the zero-order term of our 2D simulation. The non-zero-order radial velocity field is still preserved, even when converted to conventional magnetic surface coordinates. The distribution of velocity field vectors of the order of 1, 2, and 3 are presented respectively in 2, 4, and 6 polar fields with the local vortex structure. The excitation mechanisms of these velocity vortices are the coupling effects of the magneto-fluid structure patterns and the toroidal effects. These results can help us understand the complexity of core physics, the transverse transport across magnetic field by the radial plasma flow and the formation of velocity vortices.

Linear and nonlinear simulations of the visco-resistive internal kink mode using the M3D code

Numerical simulations of visco-resistive internal kink modes with dominant poloidal and toroidal mode numbers m/n = 1 in tokamak plasma configurations have been carried out using the M3D code. The numerical scaling of the growth rate of linear visco-resistive internal kink is found to agree reasonably well with the analytical theory. The impact of toroidal effects is assessed. Nonlinear simulations show that when viscosity increases, the nonlinear evolution of resistive internal kinks transits from cyclic, Kadomtsev-like sawtooth oscillations at relatively low values of the viscosity parameter, to pulsating m/n = 1 magnetic islands at intermediate viscosity, to steady state islands at high viscosity. For the pulsating and steady state scenarios, the safety factor profile is nearly flat and almost equals to unity in the core region. Regimes with pulsating or steady state islands may be relevant to the interpretation of partial sawteeth and of the so-called snake phenomenon sometimes observed in tokamak experiments.

Studies of various physics effects on an on-axis LHCD experiment using numerical tools

The lower hybrid current drive (LHCD) plays an important role in the long pulse and high performance experiments on the EAST Tokamak, using LHW systems with frequencies of 2.45 GHz and 4.6 GHz. With the initial parallel refractive index mostly peaking at n∥∼1.8−2.1, it is difficult to have on-axis LHCD due to the accessibility conditions which prevents the LHW from propagating into the plasma core. However, according to the investigation of HXR diagnostics, LHW did provide an on-axis current drive in an EAST experiment. To interpret this on-axis LHCD phenomenon, various physics effects that give rise to variations in parallel wave numbers, thus influencing the propagation and the absorption of the LHW, have been taken into account, such as toroidal effects, full wave effects with focusing and diffraction, and scattering effects due to density fluctuations. Numerical tools including ray-tracing and full wave codes that contain one or two of the above physics effects have been used to model the heating and current drive of the LHW on EAST. Linear Landau damping and quasi-linear Landau damping have been compared by using the coupling between ray-tracing and the Fokker-Planck codes. Convergence studies have been performed for the full wave modeling, the maximum poloidal modes in the full wave have been scanned, and the results are compared with each other. The factors that characterize the amplitude, the radial thickness, and the location of the density fluctuations have been studied and scanned. The measurement of HXR diagnostics has been compared with synthetic diagnostics by reconstructing the HXR data from both the standard GENRAY/CQL3D LHCD modelling and the modeling taking the scattering effects into account.

Plasma equilibrium with fast ion orbit width, pressure anisotropy, and toroidal flow effects

We formulate the problem of tokamak plasma equilibrium including the toroidal flow and fast ion (or energetic particle, EP) pressure anisotropy and the finite drift orbit width (FOW) effects. The problem is formulated via the standard Grad–Shafranov equation (GShE) amended by the solvability condition which imposes physical constraints on allowed spacial dependencies of the anisotropic pressure.The GShE problem employs the pressure coupling scheme and includes the dominant diagonal terms and non-diagonal corrections to the standard pressure tensor. The anisotropic tensor elements are obtained via the distribution function represented in the factorized form via the constants of motion. Considered effects on the plasma equilibrium are estimated analytically, if possible, to understand their importance for GShE tokamak plasma problem. The novelty of the proposed approach is in the way FOW is included into the GShE via the non-diagonal pressure tensor, factorized distribution function of the fast ions representation and in the prescription of the spacial dependence of given the spacial dependence of .

Relaxation models for single helical reversed field pinch plasmas at low aspect ratio

A recent study [R. Paccagnella, Phys. Plasmas 23, 092512 (2016)] about the scaling with the aspect ratio (i.e., the ratio between the major and minor radius of the torus) of the dominant mode in single helical reverse field pinch plasmas has shown, at intermediate aspect ratio, that the dominant toroidal mode number in the helical states can be interpreted as the result of a relaxation process. In this work, the same theoretical model is compared and validated with the experimental data obtained in the low aspect ratio RELAX device [S. Masamune et al., J. Phys. Soc. Jpn. 76, 123501 (2007).]. Some difficulties have emerged in this comparison that may be interpreted either as related to a residual level of magnetic chaos in the RELAX plasmas or possibly to genuine toroidal effects not analyzed in this paper, where a cylindrical approximation is considered.