Magnetic Flux Surfaces Research Articles

Roles of non-axisymmetric perturbations in free drift vertical displacement events on EAST

The safe operation of most tokamaks, especially the large ones, relies on the feedback control of vertical displacement events (VDEs). However, most of these feedback control systems are based on axisymmetric VDE models. In this study, we use NIMROD simulations to study the role of non-axisymmetric perturbations in free drift vertical displacement events on EAST. The high-n modes in the non-axisymmetric VDE grow first, which drives the formation of high-n magnetic island chains. Subsequently, the magnetic island chains grow and overlap with each other, leading to the destruction of the magnetic flux surface, which induces a minor disruption and accelerates the start of the following major disruption. The magnetic island and the stochastic magnetic field allow the toroidally non-axisymmetric poloidal plasma current to jet towards the hoop force direction, forming finger-like and filamentary structures. Such a plasma current non-axisymmetry strongly depends on the anisotropy in the thermal transport coefficients.

Experiments and modelling of negative triangularity ASDEX Upgrade plasmas in view of DTT scenarios

The paper presents experimental and modelling results of a comparison of negative (NT) and positive (PT) triangularity ASDEX Upgrade (AUG) discharges using the plasma shapes presently foreseen in the DTT tokamak, under construction in Italy. This work is part of a broader effort of investigation to understand whether the good properties observed in NT scenarios in DIII-D and TCV may be extrapolated to the DTT device and more generally to DEMO future operations. The experimental results have shown a practical gain of running these AUG plasmas with only ECRH and mixed NBI+ECRH phases in negative triangularity, even if they access the H-mode. Indeed, the NT electron kinetic profiles recover in all cases the PT electron pressures inside mid-radius due to reduced transport in the region ρtor=0.7−0.9 , while exhibiting lower individual ELM (Edge Localised Mode) energy losses. The ion pressure and expected fusion performance are comparable in the case of similar densities. Integrated modelling has been performed using the transport solver ASTRA and the quasi-linear turbulent model TGLF, investigating the transport properties of these discharges. The modelling reproduces the experiments qualitatively with reasonable accuracy. Nonetheless, the heat transport in NT cases is partially overestimated. This may be because TGLF uses the Miller equilibrium, which approximates the magnetic flux surfaces as up-down symmetric. In the caseof these asymmetric NT shapes, the simulated outer surfaces lose part of the tilt with respect to the z-axis, reducing the upper δ < 0 effect. A numerical test to discern the impact of the geometry by symmetrically flipping the shape has shown a beneficial effect of the negative triangularity on heat transport.

Numerical eigenanalysis of continuum geodesic acoustic mode by ideal magnetohydrodynamic model

In this study, we have developed a new eigenvalue code to solve the three-variables ideal magnetohydrodynamic (MHD) equation in flux coordinate system with the help of a symbolic vector analysis module and an automatic numerical discretization module, which were developed with the use of symbolic computation technique and were greatly facilitated the development of toroidal full MHD eigenvalue code SCELT. It has been used to solve for unstable modes and the present work is dedicated to the more difficult challenge of MHD continua, which ordinarily come with spatial singularities. With this new code, we were able to perform numerical calculations to determine the continuum spectrum of geodesic acoustic modes (GAMs), as well as obtain eigenfunctions for both the perturbed plasma displacement and perturbed magnetic field, incorporating a sufficient number of components. Through our global calculations, we have provided numerical evidence of singularities in the eigenfunctions, characterized by the type s−s0−1 and Ins−s0 . Moreover, we have examined the spatial variations of GAM fluctuations in regions both near and far from the singular surface. Here s represents a flux function that characterizes the magnetic flux surfaces and s=s0 defines the singular surface. Additionally, we have investigated the effects of finite aspect ratio, non-circular cross sections as well as the asymmetry on the frequency and eigenfunction of continuum GAMs through numerical analysis.

Geometric properties of lightlike magnetic curves and magnetic flux surfaces in Minkowski 3-space

We define the lightlike [Formula: see text]-magnetic curves, lightlike [Formula: see text]-magnetic curves and lightlike [Formula: see text]-magnetic curves in Minkowski 3-space. Then we discuss the characteristics of the corresponding magnetic fields of these lightlike magnetic curves. Furthermore, we also investigate the geometric properties of the magnetic flux surfaces formed by the magnetic fields along lightlike magnetic curves.

Theoretical analysis of effects of ferrite geometrical parameters on tokamak magnetic field perturbations

In this paper, a magneto-dynamic approach has been developed to investigate the probable enhancing effect of a magnetizing material like ferrite on the mitigation of the edge-localized mode instabilities. It has been assumed that the ferrite is placed on the first wall of the tokamak. The Landau–Lifschitz–Gilbert equation has been used to calculate the ferrite contribution to the equilibrium and the perturbed magnetic fields in the plasma. Simulations have shown that the ferrite can alter the magnetic field topology. Investigating the effects of the ferrite geometrical parameters such as the length, the width and the angle have shown that increasing the values of these parameters elevates the ferrite magnetization contribution in the plasma. Because of the type of the perturbation initially defined, this contribution was especially strong at the bottom of the ferrite. Since the ferrite can increase the total perturbation field in the plasma, it can help to mitigate the edge-localized modes by the resonant magnetic perturbation phenomenon. The simulation has shown that for an optimum geometry of the ferrite, the magnetic flux surfaces near the edge can be modified from an opened-line topology to a closed-line topology. Thus, the ferrite can prevent the magnetic flux from touching the wall which is necessary to have a good confinement and to reduce the edge instabilities.

A new flux coordinates-based solver for fixed-boundary tokamak equilibrium with toroidal flow

The plasma equilibrium plays a crucial role in nuclear fusion studies, serving as the foundation for various aspects of fusion research, including plasma stability, transport, and current drive. In this paper, a new Grad–Shafranov equation solver is developed for the fixed-boundary plasma equilibria with toroidal flow. This solver utilizes the pressure profile, safety factor profile (not current profile), and any two profiles of the toroidal angular velocity, plasma temperature, and square of the Mach number as inputs. The numerical results obtained by this solver exhibit good agreement with known analytic solution under identical parameters, and the potential applications of the solver are demonstrated through several numerical equilibria with toroidal flow. It is very convenient to apply this code to simulate the tokamak equilibrium with a smooth plasma shape. In addition, the effect of toroidal flow on the plasma equilibria is investigated as a simple application. The results reveal a notable outward shift in the contour profiles of magnetic flux surface, density, pressure, and temperature induced by toroidal flow.

Interplay among turbulence, flow and impurities for sustaining magnetic island

As ubiquitous structures in magnetized fusion plasmas, magnetic islands (MIs) would short-circuit adjacent magnetic flux surfaces and result in a reduced pressure gradient and fluctuations inside the island; it is widely accepted that due to the stabilizing of drift wave instability, the turbulence intensity inside MIs is much lower for larger islands. Here, we provide the first observations that strong turbulence could be generated inside a large radiation MI, which is probably driven by the electron temperature dip due to strongly localized impurity radiation. Moreover, the flow velocity inside the MI is strongly correlated with the turbulence intensity, and the impurity concentration rate suddenly increases as the flow velocity reaches a threshold value, strongly suggesting that turbulence and flow inside the island play important roles in trapping heavy impurities and sustaining radiative MIs.

Estimation of plasma parameter profiles and their derivatives from linear observations by using Gaussian processes

Gaussian process regression (GPR) has been utilized to provide fast and robust estimates of plasma parameter profiles and their derivatives. We present an alternative GPR technique that performs profile regression analyses based on arbitrary linear observations. This method takes into account finite spatial resolution of diagnostics by introducing a sensitivity matrix. In addition, the profiles of interest and their derivatives can be estimated in the form of a multivariate normal distribution even when only integrated quantities are observable. We show that this GPR provides meaningful measurements of the electron density profile and its derivative in a toroidal plasma by utilizing only ten line-integrated data points given that the locations of magnetic flux surfaces are known.

Magnetic spherical indicatrix of a ruled surface

The spherical indicatrix of a ruled surface is the image curve of its rulings on the sphere [Formula: see text]. In this paper, we present Lorentz forces and magnetic curves produced by the Frenet frame [Formula: see text] of the spherical indicatrix of a ruled surface in a magnetic field. We calculate magnetic vector fields of magnetic curves for [Formula: see text]. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatrix. We obtain developability conditions for these surfaces. Finally, we give some examples to show magnetic flux surfaces.

A new approach to find the magnetic flux surfaces

Magnetic flux surfaces allow us to examine surfaces in a laboratory environment. In this paper, we obtain differential equations to find the magnetic flux surfaces. Differential equations are needed for defining, modeling and solving engineering problems. They are particularly widely used in systems and control theory. First, we thought of linear combinations of Killing vector fields in three-dimensional Euclidean space [Formula: see text]. We obtain partial differential equations by using these vectors. We find magnetic flux surface families with the solutions of these equations. Examples of the magnetic flux surface families members found by solving each equation are given.

Electromagnetic turbulence simulation of tokamak edge plasma dynamics and divertor heat load during thermal quench

The edge plasma turbulence and transport dynamics, as well as the divertor power loads during the thermal-quench phase of tokamak disruptions, are numerically investigated with BOUT++’s flux-driven six-field electromagnetic turbulence model. Here, transient yet intense particle and energy sources are applied at the pedestal top to mimic the plasma power drive at the edge induced by a core thermal collapse, which flattens the core temperature profile. Interesting features, such as surging of divertor heat load (up to 50 times) and broadening of heat-flux width (up to four times) on the outer-divertor target plate, are observed in the simulation, in qualitative agreement with experimental observations. The dramatic changes in divertor heat load and width are due to the enhanced plasma turbulence activities inside the separatrix. Two cross-field transport mechanisms, namely, the E × B turbulent convection and the stochastic parallel advection/conduction, are identified to play important roles in this process. First, an elevated edge pressure gradient drives instabilities and subsequent turbulence in the entire pedestal region. The enhanced turbulence not only transports particles and energy radially across the separatrix via the E × B convection, which causes the initial divertor heat-load burst, but it also induces amplified magnetic fluctuation . Once themagnetic fluctuation is large enough to break the magnetic flux surface, magnetic flutter effect provides an additional radial transport channel. In the late stage of our simulation, reaches to 10−4 level that completely breaks magnetic flux surfaces such that stochastic field lines are directly connecting pedestal top plasma to the divertor target plates or first wall, further contributing to the divertor heat-flux width broadening.

Waveform distortion of off-axis fishbone instability in the nonlinear magnetohydrodynamic evolution in tokamak plasmas

We have investigated the waveform distortion of energetic particle driven off-axis fishbone mode (OFM) in tokamak plasmas with kinetic magnetohydrodynamic (MHD) hybrid simulations. We extended our previous simulations (Li et al 2022 Nucl. Fusion 62 026013) by considering higher-n harmonics in the MHD fluid, where n is toroidal mode number. The waveform distortion is successfully reproduced in the simulation for both magnetic fluctuations and temperature fluctuations. It is clarified that the waveform distortion arises from the superposition of the n = 2 harmonics on the fundamental n = 1 harmonics of OFM, where the n = 2 harmonics are generated by the MHD nonlinearity from the n = 1 OFM. Two types of waveform distortion can occur depending on the phase relationship between the n = 1 and n = 2 harmonics and the relative amplitude of the n = 2 harmonics to the n = 1 harmonics. Lissajous curve analyses indicate that the wave couplings between the n = 1 and n = 2 harmonics with phase-lock and lead to ‘rising distortion’ and ‘falling distortion’, respectively. The two types of waveform distortion can be attributed to the strong shearing profile of radial MHD velocity with n = 2 around the q = 2 magnetic flux surface. The dependence of waveform distortion on viscosity is investigated. It is found that the viscosity which is needed to reproduce the waveform distortion is larger than that in the experiment.

Improved flux-surface parameterization through constrained nonlinear optimization

Parameterization of magnetic flux-surfaces is often used for magnetohydrodynamic stability analysis and microturbulence modeling in tokamaks. Shape parameters for such local parameterization of a (numerical) equilibrium are traditionally computed analytically using geometrically derived quantities. However, often the shape is approximated by the average of values for different sections of the flux-surface contour or a truncated series, which does not guarantee an optimal fit. Here, instead nonlinear least squares optimization is used to compute these parameters, with a weighted sum of squared error cost function that is robust to outliers. This method results in a lower total absolute error for both the parameterization of the flux-surface contour and the poloidal magnetic field density than current methods for several parameterizations based on the well-known “Miller geometry.” Furthermore, rapid convergence of shape parameters is achieved, no approximate geometric measurements of the contour are needed, and the method is applicable to any analytical shape parameterization. Validation with local, linear gyrokinetic simulations using these optimized shape parameters showed reduced root mean square errors in both the growth rate and frequency spectra when compared with simulations based on numerical equilibria. In particular, the popular Turnbull–Miller parameterization benefits from this approach, extending its usability closer toward the last-closed flux-surface for cases with minor up-down asymmetry.

Magnetic spherical indicatricies in Minkowski 3-space

In this paper, we examine curves drawn by a timelike or spacelike curve[Formula: see text] on the pseudo-hyperbolic sphere [Formula: see text] and the unit Lorentzian sphere [Formula: see text]in three-dimensional Minkowski space [Formula: see text] We present Lorentz forces and magnetic curves produced by the Frenet frame [Formula: see text] on spherical indicatricies in a magnetic field of the curve [Formula: see text]. We calculate magnetic vector fields of magnetic curves for [Formula: see text]. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatricies. We obtain developability conditions for these surfaces. Finally, we illustrate magnetic flux surfaces with some interesting examples.

Drift surface solver for runaway electron current dominant equilibria during the current quench

Runaway electron current generated during the current quench phase of tokamak disruptions could result in severe damage to future high performance devices. To control and mitigate such runaway electron current, it is important to accurately describe the runaway electron current dominated equilibrium, based on which further stability analysis could be carried out. In this paper, we derive a Grad–Shafranov-like equation solving for the axisymmetric drift surfaces of the runaway electrons instead of the magnetic flux surfaces for the simple case that all runaway electrons share the same parallel momentum. This new equilibrium equation is then numerically solved with simple rectangular wall with ITER-like and MAST-like geometry parameters. The deviation between the drift surfaces and the flux surfaces is readily obtained, and runaway electrons are found to be well confined even in regions with open field lines. The change of the runaway electron parallel momentum is found to result in a horizontal current center displacement without any changes in the total current or the external field. The runaway current density profile is found to affect the susceptibility of such displacement, with flatter profiles result in more displacement by the same momentum change. With up–down asymmetry in the external poloidal field, such displacement is accompanied by a vertical displacement of runaway electron current. It is found that this effect is more pronounced in smaller, compact device and weaker poloidal field cases. The above results demonstrate the dynamics of current center displacement caused by the momentum space change in the runaway electrons, and pave a way for more sophisticated runaway current equilibrium theory in the future with more realistic consideration on the runaway electron momentum distribution. This new equilibrium theory also provides foundation for future stability analysis of the runaway electron current.

Determination of MHD mode structures using soft x-ray diagnostics in TCV

A forward modeling technique is developed for determining the characteristic features of observed MHD modes from the line-of-sight data of the soft x-ray (SXR) diagnostics with 64 vertical lines-of-sight in theTokamak à Configuration Variable tokamak. Using diagnostics with excellent spatial resolution, this technique is shown to evaluate the poloidal mode numbers m, radial location and ballooning character of the MHD modes. In the first stage, the poloidal mode structures are modeled by the radially localized Gaussian-shaped emission regions rotating along the magnetic flux surfaces. In the second stage, the space structures of observed 95–100 kHz toroidal Alfvén eigenmodes (TAEs) are investigated numerically using HELENA, CSCAS and MISHKA codes. The calculated TAE eigenfunctions are then used in our forward modeling similarly to the approach of contrast imaging diagnostics (Edlund et al 2009 Phys. Rev. Lett. 102 165003) and SXR diagnostics (Piovesan et al 2008 Nucl. Fusion 48 065001). A ballooning structure of the observed n= 1 TAE mode can be analyzed more easily due to the low poloidal mode number m in contrast to the high-m modes analyzed by a similar SXR technique in the W7-X stellarator (Dreval et al 2021 Plasma Phys. Control. Fusion 63 065006).

Antimony Hall sensor testing at ITER and DEMO relevant temperatures

Measurements of the steady-state magnetic field of ITER and DEMO reactors will be performed by Hall sensors that will be part of the magnetic diagnostics. The sensors will contribute to the measurement of the plasma current, plasma-wall clearance, and local perturbations of the magnetic flux surfaces near the wall. The paper describes new ceramic-metal Hall sensors with an antimony sensing layer. The results of temperature tests of second-generation antimony sensors based on thick printed copper technology are presented, including testing of the long-term stability of the sensors.

Energetic particle optimization of quasi-axisymmetric stellarator equilibria

An important goal of stellarator optimization is to achieve good confinement of energetic particles such as, in the case of a reactor, alphas created by deuterium–tritium fusion. In this work, a fixed-boundary stellarator equilibrium was re-optimized for energetic particle confinement via a two-step process: first, by minimizing deviations from quasi-axisymmetry (QA) on a single flux surface near the mid-radius, and secondly by maintaining this improved QA while minimizing the analytical quantity , which represents the angle between magnetic flux surfaces and contours of , the second adiabatic invariant. This was performed multiple times, resulting in a group of equilibria with significantly reduced energetic particle losses, as evaluated by Monte Carlo simulations of alpha particles in scaled-up versions of the equilibria. This is the first time that energetic particle losses in a QA stellarator have successfully been reduced by optimizing . The relationship between energetic particle losses and metrics such as QA error () and in this set of equilibria were examined via statistical methods and a nearly linear relationship between volume-averaged and prompt particle losses was found.

NON-AXISYMMETRIC NEOCLASSICAL TRANSPORT FROM MIS-ALIGNMENT OF EQUIPOTENTIAL AND MAGNETIC SURFACES

Coefficients of neoclassical transport in tokamaks and stellarators with non-axisymmetric equipotential surfaces mis-aligned with magnetic flux surfaces are derived for the 1/ν regime. In the general case, they are given by integral expressions including field line integrals similar to those defining the effective helical ripple [V.V. Nemov et al. // Phys. Plasmas. 1999, v. 6, p. 4622]. For small mis-alignments in a tokamak with a simplified geometry, they are reduced to simple analytical expressions.

Account of non-standard orbits in computations of neoclassical toroidal viscous torque in the resonant plateau regime of a tokamak

This article extends theoretical details based on a short paper originally submitted to the 2022 EPS conference in plasma physics [1]. The quasilinear theory of resonant transport regimes in a tokamak is developed for the general case of orbits forming various classes separated in phase space by homoclinic orbits with infinite bounce time. Beyond standard orbits (banana and passing orbits) also all types of non-standard orbits (e.g. “potato” orbits) are taken into account. In case of a weak radial electric field, such orbits are usually present only near the magnetic axis. If the radial electric field cannot be treated as weak, there can be arbitrary many classes, located elsewhere. The present approach covers all such cases and is demonstrated on a specific example of a radial electric field profile. The resulting quasilinear kinetic equation is applicable to compute neoclassical toroidal viscous (NTV) torque in a tokamak with non-axisymmetric magnetic field perturbations. A fully non-local approach to NTV computation has been realized in the upgraded version of the code NEO-RT. Based on a generalization of magnetic flux surfaces to drift surfaces, the notion of a local thermodynamic equilibrium is extended for our purpose. We obtain an expression for the integral toroidal torque within a chosen flux surface and dicuss means to compute such integrals taking singularities in bounce and precession frequencies into account.