Geodesic Acoustic Mode Frequency Research Articles

Destabilization of geodesic acoustic-like mode in the presence of poloidally inhomogeneous heat sources in tokamak plasmas

The effects of poloidally inhomogeneous heat sources are investigated through a gyrokinetic formula in collisionless toroidal plasmas. A gyrokinetic dispersion relation is newly derived under the assumption that equilibrium parallel heat flows are generated to remove the injected poloidally nonuniform heat source. The dispersion relation is numerically solved, considering both inboard and outboard heat source injections. In the case of the inboard source injection, both Stringer spin-up and geodesic acoustic mode (GAM) are excited. Conversely, outboard injection leads to the emergence of a heat source-driven GAM (referred to as Q-GAM), featuring a frequency around half that of the standard GAM. Various physical quantities of the Q-GAM, such as mode frequency and source threshold, are analyzed through parametric scans. The Q-GAM exhibits similarities with the energetic-particle-driven GAM (EGAM), particularly in its frequency range, and both belong to one of the strong Landau damped poles. Despite having distinct driving mechanisms and structural differences in parallel velocity and poloidal coordinates, the response function of the perturbed parallel pressure to the potential, mainly contributing to the destabilization of each mode around half of the GAM frequency, is derived to have a similar form for both the Q-GAM and EGAM cases.

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.

Shaping effects on the geodesic acoustic mode in tokamaks

The geodesic acoustic mode (GAM) is investigated with gyro-kinetic equations in Miller local equilibrium model for shaped tokamak plasmas with an arbitrary elongation κ and a finite triangularity δ. In particular, the effects of triangularity on GAM frequency and damping rate are analyzed both analytically and numerically. The asymptotic analytical and exact numerical results both show that the frequency almost linearly increases with the triangularity but increases relatively more slowly for a negative δ, which agree well with the TCV observation on the trend. The analytical results clearly claim that the triangularity effect strength is dependent on the inverse aspect ratio and Shafranov shift gradient , while the numerical results indicate that the safety factor q also has a significant impact on the triangularity effects. In addition, the damping rate increases rapidly with triangularity when q is not too large and then saturates when δ is above about 0.3.

Effect of temperature anisotropy on the dynamics of geodesic acoustic modes

In this work, we revisit the linear gyro-kinetic theory of geodesic acoustic modes (GAMs) and derive a general dispersion relation for an arbitrary equilibrium distribution function of ions. A bi-Maxwellian distribution of ions is then used to study the effects of ion temperature anisotropy on GAM frequency and growth rate. We find that ion temperature anisotropy yields sensible modifications to both the GAM frequency and growth rate as both tend to increase with anisotropy and these results are strongly affected by the electron to ion temperature ratio.

Nonlinear excitation of geodesic acoustic mode by reversed shear Alfvén eigenmodes in non-uniform plasmas

Effects of plasma non-uniformities and kinetic dispersiveness on the spontaneous excitation of geodesic acoustic mode (GAM) by reversed shear Alfvén eigenmode (RSAE) are investigated numerically. It is found that, due to the turning points induced by the shear Alfvén continuum structure, the nonlinear excitation of GAM is a quasiexponentially growing absolute instability. As the radial dependence of GAM frequency and pump RSAE mode structure are accounted for, the radially inward propagating GAM is preferentially excited, leading to core localized thermal plasma heating by GAM collisionless damping. Our work, thus, suggests that GAM excitation plays a crucial role in not only RSAE nonlinear saturation, but also anomalous fuel ion heating in future reactors.

Energetic particle-induced geodesic acoustic modes on DIII-D

Various properties of the energetic particle-induced geodesic acoustic mode (EGAM) are explored in this large database analysis of DIII-D experimental data. EGAMs are n = 0 modes with m = 0 electrostatic potential fluctuations (where n/m = toroidal/poloidal mode number), m = 1 density fluctuations, and m = 2 magnetic fluctuations. The fundamental frequency (∼20–40 kHz) of the mode is typically below that of the traditional geodesic acoustic mode frequency. EGAMs are most easily destabilized by beams in the counter plasma current (counter-I p) direction as compared to co-I p and off-axis beams. During counter beam injection, the mode frequency is found to have the strongest linear dependence (correlation coefficient r = −0.71) with the safety factor (q). The stability of the mode in the space of q and poloidal beta (β p) shows a clear boundary for the mode stability. The stability of the mode depends more strongly on damping rate than on fast-ion drive for a given injection geometry.

Experimental investigation of electromagnetic GAMs under the influence of 3D magnetic topological structure in EAST

The relationship between geodesic acoustic modes (GAMs) and tearing modes (TMs), or locked modes (LMs), have been investigated in EAST. Firstly, multi-branch electromagnetic GAMs are observed synchronously after the excitation of TMs. Broadened radial coverage is observed for these branches, and these frequencies can be described by the dispersion relation of GAM. Secondly, multi-branch electromagnetic GAMs are generated after the excitation of LMs. The toroidal asymmetric structure of GAMs is excited due to the existence of 3D intrinsic error fields, and the radial distribution of those branches is altered accordingly. Four branches of electromagnetic GAMs are observed in the region that is closer to the ‘X’-point of the locked magnetic islands, which should be merged into two branches for the region that is closer to the ‘O’-point, as confirmed by XUV array measurement. Thirdly, a similar phenomenon is reproduced by the modulation of resonant magnetic perturbation (RMP). The frequency of GAM increases with the current of the RMP coils, and multi-branches of similar modes are generated when the current achieves kA, where the electrostatic and electromagnetic components are altered accordingly. The features of those branches are modulated by the rotation of the RMP. Therefore, the structures of the pair of GAMs can be different, which should be influenced by the different magnetic topological structure.

Nonlinear excitation of geodesic acoustic mode by drift waves in anisotropic tokamak plasmas with toroidal rotation

Nonlinear excitation of geodesic acoustic mode (GAM) by ions drift waves is investigated in anisotropic toroidally rotating tokamak plasmas by using an ideal magnetohydrodynamic model and standard double adiabatic equations. It is found that the radial wave number kx of the effectively excited GAM and the GAM frequency increase with the ratio of the perpendicular pressure to the parallel pressure χ as well as the toroidal rotation Mach number. The equilibrium anisotropy and the ratio of the radial scale length of the equilibrium density to the tokamak major radius are shown to play a significant role in the growth rate and frequency shift of GAM. The growth rate increases with the normalized GAM frequency first and then decreases to zero as the GAM frequency increases. When the GAM frequency increases again, the growth rate continues to increase but with a relatively smaller magnitude.

Relation Between the Low Frequency GAM and the Neoclassical Flow

Since the magnetic field strength is not constant on the magnetic flux surface, the flow also varies so that the density compression occurs along the poloidal direction. Since the inhomogeneous flow causes the density compression in the poloidal direction, the parallel flow is also perturbed. In this study, we investigate the effects of the parallel flow perturbation on the geodesic acoustic mode (GAM) when it is described by the kinetic approach. Using the continuity equation, it is shown that the flow perturbation in the geodesic curvature direction is balanced by the lowest-order term of the density perturbation in , and the flow perturbation in the parallel direction is balanced by the higher-order terms of the density perturbation. Since the density perturbation includes both the perpendicular and parallel flow perturbation contributions, the GAM frequency obtained by the kinetic approach has the parallel flow perturbation contribution, which is 1/ term in the GAM frequency equation. The low frequency branch of the dispersion relation is also discussed to demonstrate the connection between the GAM theory and neoclassical theory for the first time. It is shown that the flow perturbation in the geodesic curvature direction is balanced mostly by the parallel flow perturbation. It means that the flow in the flux surface is divergence free approximately as in the usual transport ordering. Thus, the poloidal flow goes to the neoclassical flow when the low frequency branch is taken.

Toroidal component of velocity for geodesic acoustic modes in the edge plasmas of the J-TEXT tokamak

The toroidal component of the velocity for geodesic acoustic modes (GAMs) is first demonstrated. Multiple Langmuir probe arrays set up near the top tokamak of the J-TEXT were utilized for this study. A significant peak at the GAM frequency is observed in Mach number fluctuations. The toroidal velocity for the GAMs is estimated as ∼10–100 m s−1 and increases with the poloidal velocity. The ratio of toroidal component to the poloidal one of the velocity is mainly located in the interval between 0.3 and 1.0. With higher safety factors q, the ratio almost does not change with decreasing the safety factor, whereas it goes up sharply at low q. The coherencies between poloidal electric fields and Mach number fluctuations in turbulence frequency bands are also evaluated, and are higher than those between radial electric fields and Mach number fluctuations.

Evidence of ‘two plasmon’ decay of energetic particle induced geodesic acoustic mode

Secondary low frequency mode generation by energetic particle induced geodesic acoustic mode (EGAM) observed in LHD experiment is studied using nonlinear gyrokinetic theory. It is found that the EGAM frequency can be significantly higher than local geodesic acoustic mode (GAM) frequency in low collisionality plasmas, and it can decay into two GAMs as its frequency approaches twice GAM frequency, in a process analogous to the well-known two plasmon decay instability. The condition for this process to occur is also discussed.

Nonlinear oscillations of geodesic acoustic modes due to E×B convection in edge pedestal

An analytic study is conducted on the impact of poloidal E×B flow on geodesic acoustic mode (GAM) in the edge pedestal of a tokamak plasma. A set of coupled nonlinear equations is derived from a reduced magnetohydrodynamic model. Analytic solutions to the set of coupled equations reveal that the non-geodesic component of a poloidally asymmetric pressure perturbation begins to contribute to GAM when the E×B flow is accounted for in the analysis. The full nonlinear solution shows that the sinusoidal GAM oscillation changes into the cnoidal one which is represented by the Jacobi elliptic function. The GAM frequency increases in proportion to the initial radial electric field (Er). The potential impact of this pedestal Er oscillation on the transport process in the externally perturbed edge transport barrier is discussed.

Geodesic modes driven by untrapped resonances of NB energetic ions in tokamaks

Geodesic modes are typically excited by a minor concentration of energetic ions, but unstable mode frequencies are substantially different from Geodesic Acoustic Modes (GAMs) and are named EGAM (Energetic particle GAM). The EGAM instability driven by Neutral Beam Injection (NBI) has been observed in DIII-D tokamak experiments. The problem of the geodesic mode instability is analytically studied using a full drift kinetic equation. To analyze the instability condition, an ionization NBI location is assumed to be on the high field side of tokamaks. A minority NBI ion distribution is modeled by an energetic ion tail in the untrapped-passing region that remains between a magnetic axis and the trapped NBI boundary. The EGAM instability condition is defined by the parallel NBI ion velocity v||≈(1.2−1.5)ωR0q0 that has to be above the effective EGAM phase velocity. In this case, the EGAM frequency is ≈50% below the standard stable GAM frequency, which is reduced by a small concentration of energetic NBI ions. Qualitative comparison of the developed geodesic mode theory with NBI heating experiments in the midregion of the tokamak plasma is discussed.

Singular global components and frequency shift of the geodesic acoustic continuum modes in shaped tokamaks

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.

Global turbulence features across marginality and non-local pedestal-core interactions

Spatially non-local aspects of turbulent transport in tokamak plasmas are examined with global gyrokinetic simulations using the ORB5 code. Inspired by very accurate measurements in the TCV tokamak in L-mode, we initialise plasma profiles with constant logarithmic gradients in the core and constant linear gradients in the ‘pedestal’ (). The main finding is that transport in the core is strongly affected by the presence of pedestal gradients. This non-local pedestal-core coupling appears to be correlated with the appearance of repetitive avalanches that propagate across both pedestal and core regions. Below a certain threshold value in pedestal gradient, no well-defined frequency is found for avalanches. Above this threshold, a well-defined frequency shows up, which roughly matches that of the local geodesic acoustic mode (GAM) frequency near the plasma edge and is thus well below the local GAM frequency in the core: this behaviour is very similar to the global coherent mode structure observed in TCV. Above this threshold in pedestal gradient, the core transport increases sharply: there is therefore a non-locality in marginality. The probability density functions (PDFs) of density, temperature, temperature gradient and potential are found to have nearly Gaussian distributions, whereas the heat flux can have, in the presence of avalanches, a more or less strongly positively skewed PDF, which could be fitted by a log-normal distribution. The skewness of the heat flux is found to be radially and non-locally dependent: its value in the plasma core critically depends on the presence of gradients in the pedestal. The relation flux versus gradient is examined in detail. The local instantaneous flux versus gradient relation shows a hysteresis behaviour during an avalanche but no clear correlation, unlike the flux and zonal flow (ZF) shearing rate, which are anti-correlated: flux is higher when shearing rate is lower. This leads to corrugated time-averaged radial profiles of transport, heat and temperature gradient, with heat diffusivity having local maxima where the ZF shearing rate goes to zero and temperature gradient has local minima. Finally, we show how the flux versus gradient relation can be analysed locally for series of simulations with different averaged gradients.

Electromagnetic effect on geodesic acoustic mode with adiabatic electrons

The geodesic acoustic mode (GAM) is analytically investigated by taking into account the finite-orbit-width (FOW) resonance effect to the second order and the finite β effect. The general dispersion relation is derived from the gyro-kinetic equations in the presence of nonzero δA∥, the parallel component of the perturbed magnetic vector potential. Transparent and concise expressions for the GAM frequency and Landau damping rate in the presence of the second order FOW effect and finite β effect are first presented. It is clearly shown that the m = ±2 harmonics dominant δA∥ and the kinetic expression of δA∥ have the same form as the fluid one. For the real frequency, the electromagnetic effect introduces a term on the order of q2β, which is comparable to the second order electrostatic terms, namely, the terms introduced by the second order FOW resonance effect. While for the collisionless damping rate, δA∥ does not directly introduce β–dependent terms, but affects the damping rate via modifying the real frequency. Besides, our analytical result shows good agreement with the numerical examinations.

Observation of geodesic acoustic mode in EAST using Doppler backscattering system

This paper presents an investigation of the geodesic acoustic mode (GAM) using two poloidally separated Doppler backscattering systems in Experimental Advanced Superconducting Tokamak. Each system allows for the simultaneous measurement of turbulence rotation velocity and density fluctuations with high precision. With the Doppler backscattering systems, the GAM frequency, the symmetric feature of poloidal flow fluctuations, and the interaction between GAM and turbulence are surveyed. The results of bispectral analysis show a clear interaction of GAM with the ambient turbulence. A clear measurement of GAM in the envelope of plasma density fluctuations using Doppler backscatter system is shown, and the modulations of the turbulence perpendicular velocity on the density turbulence are correlated at two different poloidal position.

Investigation of the geodesic acoustic mode in EAST H-mode operation using Doppler backscattering systems

The geodesic acoustic mode (GAM) has been observed for the first time in EAST H-mode operation using Doppler backscattering (DBS) systems. The poloidal and toroidal symmetries of the radial electric field (Er) fluctuation (m = 0 and n = 0) have also been demonstrated. Using the multichannel DBS system, the GAM frequencies at different radial locations have been investigated, with observations showing that H-mode GAM acts as a radial coherent eigenmode, rather than a continuum of frequencies in L-mode. The intermittency of GAM is observed in both L-mode and H-mode. The interplay between the background density fluctuations and the GAM intensity is revealed. The bicoherence analysis is also used to verify the presence of the nonlinear interaction between the GAM and background turbulence.

Geodesic mode instability driven during ion cyclotron heating in tokamaks

The effect of a minor concentration of energetic particles produced by ion cyclotron resonance (ICR) heating on the stability of Geodesic Acoustic Modes (GAM) is studied using fully drift kinetic equation. Assuming the ICR layer position on the high field side of the tokamak magnetic axis, a minority ion distribution is modeled by the energetic ion tail and “triangle”-like pitch angle distribution with the maximum at the trapped-passing ion boundary. The resulting GAM instability is found when the hot ion thermal velocity is larger than the effective parallel GAM phase velocity Rqω and the geodesic frequency has the order of the energetic ion bounce frequency. The standard GAM frequency may be strongly reduced by some small concentration of the energetic ions allowing the mode to be unstable. Respective neoclassical polarization corrections (residual flow) driven by energetic ions are found to be negative and larger than that calculated by Rosenbluth and Hinton [Phys. Rev. Lett. 80, 724 (1998)]. Qualitative comparison of the theory with ICR heating experiments in tokamak is discussed.

Global MHD GAMs in toroidal plasmas with reversed magnetic shear

A formalism is developed for axisymmetric (n = 0) perturbations of toroidal plasmas in the framework of ideal magnetohydrodynamics, and is applied successfully to the explanation of the global structures of geodesic acoustic modes (GAMs) that have been revealed by the numerical simulations of tokamak plasmas with an off-axis maximum of local GAM frequency. It turns out that the global structures of GAM can emerge as the consequence of balancing the momenta associated with the second poloidal harmonics introduced by the coupling between the curvatures of equilibrium magnetic field lines and the pressure perturbations via Alfvénic perturbations. The global GAM frequency is allowed to reside in the narrow gap between the maxima of the modified local GAM frequencies due to Alfvénic perturbations. This mode features a distinct ‘singular’ layer centered around the local GAM maxima, where both the acoustic/geodesic acoustic and Alfvénic perturbations are equally important and the eigenfunction resembles the symbol . This layer is sandwiched between two wider regions where the Alfvénic activities dominate. The negative magnetic shear plays an indispensable role in establishing the global mode structures, in that it shapes the eigenfunctions in the right way in the outer regions and allows them to match with the inner segment smoothly. Many features of the global GAM predicted by our theory are confirmed by running our shooting code for both weakly and strongly reversed magnetic shear profiles.