Gyro-kinetic analysis of electromagnetic geodesic acoustic modes in tokamak plasmas

Within the ideal magnetohydrodynamic (MHD) model, the geodesic acoustic modes (GAMs) in tokamaks derived by Winsor et al (1968 Phys. Fluids 11 2448) belong to the continuous spectrum, characterised by unbounded non-square integrable eigenfunctions (delta functions) at the singular surfaces (Goedbloed 1975 Phys. Fluids 18 1258). The eigenfunctions of the MHD continua in cylindrical as well as toroidal plasmas include, in addition, components that exist outside the singular surfaces and have singularities of type or where is a flux function that labels the magnetic surfaces, and defines the singular surface (Pao 1975 Nucl. Fusion 15 631). Using a large aspect ratio approximation of tokamak plasmas it is shown in this paper that the GAMs indeed include such singular components. Hence, in addition to the non-square integrable and components of the plasma flow and of the density and pressure perturbations at the GAM surface, the GAM continua also include accompanying and singular components varying as This gives the and components of each GAM in the continuum radially extended profiles and a global character also within ideal theory. To the same order in the expansion, effects of a finite aspect ratio and a non-circular plasma cross section on the GAM frequency are also calculated, and we recover the dependence on inverse aspect ratio and Shafranov shift of the real GAM frequency previously calculated within gyrokinetic theory by Gao (2010 Phys. Plasmas 17 092503). Furthermore, while the dominating shaping effect on the GAM frequency comes from plasma elongation, as shown previously, it is shown in this paper that there is a higher-order triangularity effect that can also be significant. The calculated triangularity effect predicts a nearly linearly increasing GAM frequency with increasing triangularity, a phenomenon observed also in the TCV tokamak.

We analytically investigate geodesic acoustic modes (GAMs) in tokamak plasmas with up-down asymmetric and non-circular cross-sections using magnetohydrodynamics (MHD) and a Miller-like flux surface model. Explicit expressions for GAM frequency, magnetic field perturbations, and Lagrangian displacement are presented. Our results reveal that (I) up-down asymmetry (σ) slightly increases the GAM frequency and introduces additional sin or cos components (opposite to the dominant component) to the perturbations; (II) the inverse aspect ratio (ɛ), the gradient of the Shafranov shift ( Δ ′ ), triangularity (δ), and its gradient (s δ ) can induce additional subdominant components of perturbations. The poloidal mode numbers of the dominant and subdominant components differ, and in certain cases, the amplitude of the subdominant component can approach or even exceed that of the dominant component. These results provide analytical explanations for previous MHD and gyro-kinetic simulation outcomes, and offer useful guidance for measuring multiple components of perturbations.

Electron phase‐space holes, sometimes referred to as electron holes, are Debye‐scale plasma structures commonly observed in space plasmas. Although they are usually thought of as electrostatic structures, recent observations have shown that these structures are often accompanied by magnetic field perturbations. Such magnetic perturbations may originate from Lorentz transformation of the electric field, although it has been also suggested that localized currents caused by the electron E × B motion may also play a role. In this paper, we consider the contribution of electron polarization drift to the current and therefore the magnetic field perturbations, which is validated by comparison to Magnetospheric MultiScale observations of 69 electron holes based on a superposed epoch technique. Our results show that the magnetic perturbations caused by the electron polarization drift can be significant compared to Lorentz transformation effect, especially in regions with low magnetic strength and large electron density.

An external resonant magnetic perturbation (RMP) field, which is an effective method to mitigate or suppress the edge localized mode (ELM), has been planned to be applied on the ELM control issue in ITER. A new set of magnetic perturbation coils, named as high m coils, has been developed for the EAST tokamak. The magnetic perturbation field of the high m coils is localized in the midplane of the low field side, with the spectral characteristic of high m and wide n, where m and n are the poloidal and toroidal mode numbers, respectively. The high m coils generate a strong localized perturbation field. Edge magnetic topology under the application of high m coils should have either a small or no stochastic region. With the combination of the high m coils and the current RMP coils in the EAST, flexible working scenarios of the magnetic perturbation field are available, which is beneficial for ELM control exploration on EAST. Numerical simulations have been carried out to characterize the high m coil system, including the magnetic spectrum and magnetic topology, which shows a great flexibility of magnetic perturbation variation as a tool to investigate the interaction between ELM and external magnetic perturbation.

The results of theoretical and experimental studies on excitation of magnetic field perturbations correlated with large amplitude lower-hybrid (LH) wave bursts in a high-voltage linear plasma discharge are presented. It is shown that the magnetic field perturbations, which are excited in the experiments mostly in the paramagnetic sense, are associated with the development of the magneto-modulational processes. The equations describing the relationship between the quasistationary magnetic field perturbations, plasma density perturbations, and the fields of LH waves are derived using the general nonlinear formalism developed for description of the modulational effects in arbitrary media. The results obtained on the basis of these equations are compared with the experimental data using the following parameters: the values of the magnitude of the magnetic field perturbations; the direction of the vector of the magnetic field perturbation; the correlation between the magnetic field perturbations and the electron density perturbations. It is shown that for LH waves propagating in one plane theoretical predictions are in a good agreement with the experimental results.

Small three-dimensional (3D) magnetic field perturbations have many interesting and possibly useful effects on tokamak and quasi-symmetric stellarator plasmas. Plasma transport equations that include these effects, most notably on diamagnetic-level toroidal plasma flows, have recently been developed. The 3D field perturbations and their plasma effects can be classified according to their toroidal mode number n: low n (say 1–5) resonant (with field line pitch, q = m/n) and non-resonant fields, medium n (∼20, due to toroidal field ripple) and high n (due to microturbulence). Low n non-resonant fields induce a neoclassical toroidal viscosity (NTV) that damps toroidal rotation throughout the plasma towards an offset rotation in the counter-current direction. Recent tokamak experiments have generally confirmed and exploited these predictions by applying external low n non-resonant magnetic perturbations. Medium n toroidal field ripple produces similar effects plus possible ripple-trapping NTV effects and ion direct losses in the edge. A low n (e.g. n = 1) resonant field is mostly shielded by the toroidally rotating plasma at and inside the resonant (rational) surface. If it is large enough it can stop plasma rotation at the rational surface, facilitate magnetic reconnection there and lead to a growing stationary magnetic island (locked mode), which often causes a plasma disruption. Externally applied 3D magnetic perturbations usually have many components. In the plasma their lowest n (e.g. n = 1) externally resonant components can be amplified by kink-type plasma responses, particularly at high β. Low n plasma instabilities (e.g. resistive wall modes, neoclassical tearing modes) cause additional 3D magnetic perturbations in tokamak plasmas. Tearing modes in their nonlinear (Rutherford) regime bifurcate the topology and form magnetic islands. Finally, multiple resonant magnetic perturbations (RMPs) can, if not shielded by plasma rotation effects, cause local magnetic stochasticity and increase plasma transport in the edge of H-mode plasmas. These various effects of 3D fields can be used to modify directly the plasma toroidal rotation (and possibly transport via multiple RMPs for controlling edge localized modes) and indirectly anomalous plasma transport. The present understanding and modelling of these various 3D magnetic field perturbation effects including for test blanket modules in ITER are summarized. Finally, implications of the present understanding and key open issues for developing a predictive capability of them for ITER are discussed.

We study the impact of external magnetic perturbations on the stability of ballooning modes. A unique feature of our analysis is the two-step parametric process [Chaturvedi and Kaw, J. Geophys. Res. 81, 3257 (1976)], which enables us to calculate contributions from all the modes with high toroidal mode numbers. The analysis shows that the externally applied magnetic field perturbations can modify the linear dispersion characteristics of the ballooning mode. Specifically, the growth rate spectrum of the ballooning modes becomes broader in poloidal wavenumber (kθ) space, implying the generation of high-k fluctuations. The increase of high-k fluctuations (micro-turbulence) can lead to the mitigation of an edge localized mode crash by increasing turbulent transport in the pedestal. In addition to this, a new nonlinear instability is found even below the threshold of the ballooning mode instability when the amplitude of magnetic perturbation is sufficiently large (i.e., δB/B0≥1.0×10−4). A discussion is given of the implication of this new finding.

A numerical model to evaluate the effects of the non-axisymmetric magnetic perturbations on magnetic topology and magnetic field ripple in tokamaks is presented in this paper. It is illustrated by using an example magnetic field perturbation induced by a coil system on the EAST tokamak. The influence of the choice of the coordinates on the spectrum is presented. The amplitude of resonant components of the spectrum are found to be independent of the coordinates system, while that of the non-resonant components are not. A better way to describe the edge topology by using the Chirikov parameter profile is proposed and checked by the numerical Poincaré plot results. The contribution of the magnetic perturbation on local toroidal field ripple can be significant. One approximate method to model the helical ripple on the perturbed flux surface induced by a given non-axisymmetric magnetic field perturbation is presented. All of the spectrum analysis is applicable in case the plasma response is taken into account in the input of perturbed magnetic field.

Asymptotical and mapping methods to study the structure of magnetic field perturbations and magnetic field line dynamics in a tokamak ergodic divertor in toroidal geometry are developed. The investigation is applied to the Dynamic Ergodic Divertor under construction for the Torus Experiment for the Technology Oriented Research (TEXTOR-94) Tokamak at Jülich [Fusion Eng. Design 37, 337 (1997)]. An ideal coil configuration designed to create resonant magnetic perturbations at the plasma edge is considered. In cylindrical geometry, the analytical expressions for the vacuum magnetic field perturbations of such a coil system are derived, and its properties are studied. Corrections to the magnetic field due to the toroidicity are presented. The asymptotical analysis of transformation of magnetic perturbation into the Hamiltonian perturbation in toroidal geometry is carried out, and the asymptotic formulas for the spectrum of the Hamiltonian perturbations are found. A new method of integration of Hamiltonian equations is developed. It is based on a canonical transformation of variables that replaces the dynamics of a continuous Hamiltonian system by a symplectic mapping. The form of the mapping is established in the first order of perturbation theory. It is shown that the mapping well reproduces Poincaré sections of field lines, as well as their statistical properties in an ergodic zone obtained by the numerical integration of field line equations. The mapping is applied to study, in particular, the formation of a stochastic layer and the statistical properties of field lines at the plasma edge.

The mechanisms by which rotation influences zonal flows (ZFs) in plasma are incompletely understood, presenting a significant challenge in the study of plasma dynamics. This research addresses this gap by investigating the role of non-inertial effects—specifically centrifugal and Coriolis forces—on Geodesic Acoustic Modes (GAMs) and ZFs in rotating tokamak plasmas. While previous studies have linked centrifugal convection to plasma toroidal rotation, they often overlook the Coriolis effects or inconsistently incorporate non-inertial terms into magneto-hydrodynamic (MHD) equations. In this work, we derive self-consistent drift-ordered two-fluid equations from the collisional Vlasov equation in a non-inertial frame, and we modify the Hermes cold ion code to simulate the impact of rotation on GAMs and ZFs. Our simulations reveal that toroidal rotation enhances ZF amplitude and GAM frequency, with Coriolis convection playing a critical role in GAM propagation and the global structure of ZFs. Analysis of simulation outcomes indicates that centrifugal drift drives parallel velocity growth, while Coriolis drift facilitates radial propagation of GAMs. This work may provide valuable insights into momentum transport and flow shear dynamics in tokamaks, with implications for turbulence suppression and confinement optimization.

The excitation and time evolution of magnetohydrodynamic (MHD) density waves are studied in a differentially rotating thin gaseous disc embedded with an azimuthal magnetic field. This analysis shows that both fast and slow MHD density waves are amplified when they swing from leading to trailing configurations, but the amplification factors of fast and slow MHD density waves depend differently on the disc differential rotation. Fast MHD density waves tend to be excited in discs of strong differential rotation, while slow MHD density waves are expected to manifest preferentially in discs of almost rigid rotation. Surface mass density and magnetic field perturbations associated with fast MHD density waves are roughly in phase. A distinct feature of slow MHD density waves is that at a fixed spatial point, there is a significant phase difference δ (i.e. π/2 < δ < π) between the azimuthal magnetic field and surface mass density perturbations during almost the entire evolutionary period. For trailing slow MHD density waves propagating inwards from the corotation, the azimuthal magnetic field perturbation leads spatially the surface mass density perturbation by δ in phase. This feature of slow MHD density waves may explain the approximate anticorrelation between the magnetic field spiral arms seen in polarized radio emissions and the optical spiral arms of the nearby galaxy NGC 6946.

Transport bursts in simulations of edge-localized modes (ELMs) in tokamaks are suppressed by the application of magnetic field perturbations. The amplitude of the applied magnetic field perturbations is characterized by a stochasticity parameter S. When S&amp;gt;1, magnetic flux surfaces are destroyed and the magnetic field lines diffuse in minor radius. As S increases in the simulations, the magnitude of the ELM bursts decreases. The size of bursts is reduced to a very small value while S is still less than unity and most of the magnetic flux surfaces are still preserved. Magnetic field line stochasticity is not a requirement for the stabilization of ELMs by the magnetic field perturbations. The magnetic field perturbations act by suppressing the growth of the resistive ballooning instability that underlies the ELM bursts.

This paper describes a new approach for modelling the pedestal energy transport in the presence of a small radial magnetic perturbation. The cases of a ballooning instability leading to Type I edge localized modes (ELMs) and a magnetic perturbation generated by external coils are treated. The model for Type I ELMs is based on the linear ideal MHD code MISHKA coupled with the non-linear energy transport code TELM in a realistic tokamak geometry. The main mechanism of the increased transport through the external transport barrier in this model of ELMs is due to the appearance of a radial velocity and a radial magnetic field perturbation due to the MHD mode. Both lead to additional transport perpendicular to the magnetic surface and hence to a relaxation of the pressure profile in the unstable zone. The typical Type I ELM time-cycle was reproduced numerically including the destabilization of the ballooning modes leading to the fast (250 μ s) collapse of the pedestal pressure followed by the edge pressure profile re-building on a diffusive time scale. A possible mechanism for the control of Type I ELMs using a stochastic plasma boundary created by external coils is modelled in this paper using data on ELM suppression by I-coils from the DIII-D experiment. In the stochastic layer the transverse transport is effectively increased by diffusion of the magnetic field lines. The modelling results demonstrate the possibility of decreasing the edge pressure gradient to a value that is just below the ideal ballooning limit, leading to a high confinement regime without Type I ELMs.

The fast ion dynamics and the associated heat load on the plasma facing components in the KSTAR tokamak were investigated with the orbit following Monte-Carlo (OFMC) code in several magnetic field configurations and realistic wall geometry. In particular, attention was paid to the effect of resonant magnetic perturbation (RMP) fields. Both the vacuum field approximation as well as the self-consistent field that includes the response of a stationary plasma were considered. In both cases, the magnetic perturbation (MP) is dominated by the toroidal mode number n = 1, but otherwise its structure is strongly affected by the plasma response. The loss of fast ions increased significantly when the MP field was applied. Most loss particles hit the poloidal limiter structure around the outer mid-plane on the low field side, but the distribution of heat loads across the three limiters varied with the form of the MP. Short-timescale loss of supposedly well-confined co-passing fast ions was also observed. These losses started within a few poloidal transits after the fast ion was born deep inside the plasma on the high-field side of the magnetic axis. In the configuration studied, these losses are facilitated by the combination of two factors: (i) the large magnetic drift of fast ions across a wide range of magnetic surfaces due to a low plasma current, and (ii) resonant interactions between the fast ions and magnetic islands that were induced inside the plasma by the external RMP field. These effects are expected to play an important role in present-day tokamaks.

Results of simulation and parametric analysis of magnetic island production by helical magnetic perturbation generated under non-axisymmetric halo current are presented. Predictions are made for a cylindrical ITER-size plasma in conditions of disruption. Calculations are carried out with the TEAR code based on the visco-resistive MHD approximation. The radial distribution of the magnetic flux perturbation is calculated with account of the external helical field produced by halo current. The equations for the magnetic flux perturbation describe the dynamics of the tearing mode depending on plasma rotation. In sequence, this rotation is affected by electromagnetic forces depending on the tearing mode magnetic field and external magnetic perturbation. The coupled diffusion-type equations for the helical flux function and for the plasma rotation velocity are numerically treated in a similar way. The magnetic island behavior is analyzed for different plasma parameters expected at the Current Quench stage of disruption. The calculated width of the produced magnetic islands extends to a significant part of plasma minor radius.