Excitation Of Geodesic Acoustic Mode Research Articles

Nonlinear excitation of energetic-particle-driven geodesic acoustic mode by ions drift waves

Nonlinear excitation of geodesic acoustic mode (GAM) by ions drift waves in the presence of energetic particles (EPs) is investigated through the hybrid model of fluid description and gyro-kinetic description, while the bulk ions temperature anisotropy and toroidal rotation are taken into account simultaneously. The coupling relation of EPs and the nonlinear effects on GAM is analytically derived and discussed. It is found that the nonlinear excitation can split inherent energetic-particle-driven GAM (EGAM) branches into different growth rates. For the intrinsic stable high frequency EGAM branch, the nonlinear effect generates a growth/damping rate. For the low frequency branches with inherent growth/damping rates due to the EPs, the nonlinearity cooperates with EPs in the excitation process.

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.

Nonlinear mode couplings between geodesic acoustic mode and toroidal Alfvén eigenmodes in the EAST tokamak

Multiple toroidal Alfvén eigenmodes (TAEs) driven unstable by energetic electrons and a geodesic acoustic mode (GAM) have been successively observed in the Experimental Advanced Superconducting Tokamak (EAST) low-density Ohmic discharges. Nonlinear mode couplings among these modes are conclusively identified. Theoretical analysis suggests that the coupling of simultaneously driven TAEs is the mechanism for GAM excitation. These results experimentally show the potential role of nonlinear mode coupling to the saturation of energetic-particle driven TAE instability, which may nonlinearly transfer energy of energetic particles to bulk plasma and heat fuel ions via low frequency mode such as GAM.

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.

The intermittent excitation of geodesic acoustic mode by resonant Instanton of electron drift wave envelope in L-mode discharge near tokamak edge

There are two distinct phases in the evolution of drift wave envelope in the presence of zonal flow. A long-lived standing wave phase, which we call the Caviton, and a short-lived traveling wave phase (in radial direction) we call the Instanton. Several abrupt phenomena observed in tokamaks, such as intermittent excitation of geodesic acoustic mode (GAM) shown in this paper, could be attributed to the sudden and fast radial motion of Instanton. The composite drift wave — zonal flow system evolves at the two well-separate scales: the micro-scale and the meso-scale. The eigenmode equation of the model defines the zero-order (micro-scale) variation; it is solved by making use of the two-dimensional (2D) weakly asymmetric ballooning theory (WABT), a theory suitable for modes localized to rational surface like drift waves, and then refined by shifted inverse power method, an iterative finite difference method. The next order is the equation of electron drift wave (EDW) envelope (containing group velocity of EDW) which is modulated by the zonal flow generated by Reynolds stress of EDW. This equation is coupled to the zonal flow equation, and numerically solved in spatiotemporal representation; the results are displayed in self-explanatory graphs. One observes a strong correlation between the Caviton-Instanton transition and the zero-crossing of radial group velocity of EDW. The calculation brings out the defining characteristics of the Instanton: it begins as a linear traveling wave right after the transition. Then, it evolves to a nonlinear stage with increasing frequency all the way to 20 kHz. The modulation to Reynolds stress in zonal flow equation brought in by the nonlinear Instanton will cause resonant excitation to GAM. The intermittency is shown due to the random phase mixing between multiple central rational surfaces in the reaction region.

Soliton generation and drift wave turbulence spreading via geodesic acoustic mode excitation

The two-field equations that govern fully nonlinear dynamics of the drift wave (DW) and geodesic acoustic mode (GAM) interaction in toroidal geometry are derived within the nonlinear gyrokinetic framework. Two stages with distinctive features are identified and analyzed using both analytical and numerical approaches. In the ‘linear’ growth stage, the derived set of nonlinear equations can be reduced to the intensively studied parametric decay instability, accounting for the spontaneous resonant excitation of GAM by the DW. The main results of previous works on spontaneous GAM excitation, e.g. the greatly enhanced GAM group velocity and the nonlinear growth rate of GAM, are reproduced from the numerical solution of the two-field equations. In the fully nonlinear stage, soliton structures are observed to form due to the balancing of the self-trapping effect by the spontaneously excited GAM and kinetic dispersiveness of the DW. The soliton structures enhance turbulence spreading from the DW linearly unstable region to the stable region, exhibiting convective propagation instead of a typical linear dispersive process, and are thus expected to induce core-edge interaction and nonlocal transport.

Geodesic modes driven by plasma fluxes during oblique NB heating in tokamaks

Some relevant aspects of the instability of Geodesic Acoustic Modes (GAMs) driven by Neutral Beam (NB) injection are studied, in particular its dependence on the injection direction, that is, co- or counter-injection, and on the pitch angle distribution of the beam particles in velocity space. In this paper, we further investigate these and other related issues considering the excitation of GAMs by energetic ions created during NB injection and modeled by a bump-on-tail distribution function with a sharp Gaussian dependence over the pitch angle at the injection angle. The bump is considered to have an energetic ion tail with temperature of the order of the third part of the critical energy that appears due to the slowing down effect on electrons. It is found that the maximum frequency of the GAM instability stays below the particle circulation frequency at the critical energy, and it is substantially reduced to be closer to parallel injection conditions. The instability may be preferentially driven for counter NB injection due to the interaction of plasma rotation and/or diamagnetic drift with electron current velocity.

Short wavelength geodesic acoustic mode excitation by energetic particles

Taking the collisionless damping of the geodesic acoustic mode (GAM) in the short wavelength limit originally investigated in [Z. Qiu et al., Plasma Phys. Controlled Fusion 51, 012001 (2009)] as an example, the physics processes underlying wave particle resonances in the short wavelength limit are clarified. As an illustrative application, GAM excitation by energetic particles in the short wavelength limit is investigated assuming a single pitch angle slowing-down fast ion equilibrium distribution function. Conditions for this energetic particle-induced GAM to be unstable are discussed.

Observation of subcritical geodesic acoustic mode excitation in the large helical device

The abrupt and strong excitation of the geodesic acoustic mode (GAM) has been found in the large helical device (LHD), when the frequency of a chirping energetic particle-driven GAM (EGAM) approaches twice that of the GAM frequency. The temporal evolution of the phase relation between the abrupt GAM and the chirping EGAM is common in all events. The result indicates a coupling between the GAM and the EGAM. In addition, the nonlinear evolution of the growth rate of the GAM is observed, and there is a threshold in the amplitude of the GAM for the appearance of nonlinear behavior. A threshold in the amplitude of the EGAM for the abrupt excitation of the GAM is also observed. According to one theory (Lesur et al 2016 Phys. Rev. Lett. 116 015003, Itoh et al 2016 Plasma Phys. Rep. 42 418) the observed abrupt phenomenon can be interpreted as the excitation of the subcritical instability of the GAM. The excitation of a subcritical instability requires a trigger and a seed with sufficient amplitude. The observed threshold in the amplitude of the GAM seems to correspond with the threshold in the seed, and the threshold in the amplitude of the EGAM seems to correspond with the threshold in the magnitude of the trigger. Thus, the observed threshold supports the interpretation that the abrupt phenomenon is the excitation of a subcritical instability of the GAM.

Fast excitation of geodesic acoustic mode by energetic particle beams

A new mechanism for geodesic acoustic mode (GAM) excitation by a not fully slowed down energetic particle (EP) beam is analyzed to explain experimental observations in Large Helical Device. It is shown that the positive velocity space gradient near the lower-energy end of the EP distribution function can strongly drive the GAM unstable. The new features of this EP-induced GAM (EGAM) are: (1) no instability threshold in the pitch angle; (2) the EGAM frequency can be higher than the local GAM frequency; and (3) the instability growth rate is much larger than that driven by a fully slowed down EP beam.

Analytic dispersion relation of energetic particle driven geodesic acoustic modes and simulations with NEMORB

Recent progress regarding the excitation of energetic-particle driven geodesic acoustic modes (EGAMs) in particle-in-cell simulations is presented in this paper. The exact dispersion relation with adiabatic electrons is derived and solved. The origin of the so-called EGAM is briefly analysed and we show that its nature changes, at least, with the safety factor. A simple expression for the GAM frequency modified in the presence of a small concentration of energetic particles is given in the fluid limit. We show that gyrokinetic simulations with Nemorb in the presence of adiabatic electrons are able to reproduce the analytic predictions. Also, different energy channels are analysed by means of dedicated energy diagnostics characterizing the wave-particle interaction. Finite Larmor radius and finite orbit width effects are studied regarding the excitation of geodesic acoustic modes, showing that these effects are likely to be negligible for sufficiently high concentration of energetic particles, but significant when approaching the threshold of excitation.

Nonlinear excitation of geodesic acoustic mode by collisionless trapped electron mode

Excitation of geodesic acoustic mode (GAM) by collisionless trapped electron mode (CTEM) is studied within the framework of nonlinear gyrokinetic theory. The dispersion relation describing GAM excitation by drift wave turbulence valid for arbitrary k⊥ρi is derived. Here, k⊥ and ρi are, respectively, perpendicular wave vector and thermal ion Larmor radius. It is found that the parametric decay process of CTEM into a GAM and a CTEM sideband is unstable in both small and large k⊥ρi regimes due to, respectively, ion and trapped electron nonlinearity. For moderate k⊥ρi, however, there exists a negligible-GAM-excitation state where ion and trapped electron nonlinearities cancel each other.

Excitation of kinetic geodesic acoustic modes by drift waves in nonuniform plasmas

Effects of system nonuniformities and kinetic dispersiveness on the spontaneous excitation of Geodesic Acoustic Mode (GAM) by Drift Wave (DW) turbulence are investigated based on nonlinear gyrokinetic theory. The coupled nonlinear equations describing parametric decay of DW into GAM and DW lower sideband are derived and then solved both analytically and numerically to investigate the effects on the parametric decay process due to system nonuniformities, such as nonuniform diamagnetic frequency, finite radial envelope of DW pump, and kinetic dispersiveness. It is found that the parametric decay process is a convective instability for typical tokamak parameters when finite group velocities of DW and GAM associated with kinetic dispersiveness and finite radial envelope are taken into account. When, however, nonuniformity of diamagnetic frequency is taken into account, the parametric decay process becomes, time asymptotically, a quasi-exponentially growing absolute instability.

Parametric excitation of axisymmetric toroidal electrostatic mode by drift wave turbulences

The axisymmetric toroidal electrostatic mode discussed in this paper refers collectively to the nearly ideal electrostatic fluid mode with zero toroidal mode number in magnetically confined toroidal plasmas like tokamak, including geodesic acoustic mode, sound waves and the so-called nearly zero-frequency zonal flow. Use is made of cold ion fluid model in the toroidal coordinate system with a circular cross section to develop the theory of parametric excitation for the three above mentioned modes systematically to the first order of inverse large aspect ratio, which ends up with the four following observations: (1) The density zonal flow is only associated with the excitation of the first harmonic cosine sound wave and is independent of the potential zonal flow. (2) The geodesic acoustic mode is the high frequency branch of the dispersion in the form of coupling between the first harmonic sine sound wave and the nearly zero-frequency zonal flow due to geodesic curvature, while the low frequency branch of the same dispersion is identified to be the ‘toroidally modified nearly zero-frequency zonal flow’. (3) Only a weak coupling exists between the second harmonic sine sound wave and the nearly zero-frequency zonal flow. (4) All cosine sound waves and sine sound waves beyond the second harmonic are decoupled to the nearly zero-frequency zonal flow. A Gaussian type of drift wave energy spectrum with only a few parameters is introduced for calculation. Emphasis is laid on the effects resulting from the finite radial spectrum width such as double Landau-singularity, which reveal a significant modification to the δ -spectrum, thus resulting in serious restriction to the parametric excitation of geodesic acoustic mode and nearly zero-frequency zonal flow. Also discussed is the possibility of excitation of density zonal flow in the high q region. Numerical results are presented graphically and discussed in the reasonable physical regime. It is indicated that the geodesic acoustic mode and the nearly zero-frequency zonal flow cannot be parametrically excited at the same radii, and that if the geodesic acoustic mode is parametrically excited, the density zonal flow is expectedly to be observed.

Geodesic acoustic mode excitation by drift waves in tokamak plasmas with toroidal rotation

Nonlinear excitation of geodesic acoustic mode by drift waves is investigated in toroidally rotating tokamak plasmas using the ideal magnetohydrodynamic equations. It is found that both the radial wave number and the nonlinear growth rate of the driven geodesic acoustic mode increase with the speed of the toroidal rotation, where the decisive factor is the frequency shift of the geodesic acoustic mode induced by the toroidal rotation. The additional nonlinear terms induced by the toroidal rotation velocity make little effect, even though they promote the nonlinear excitation at all times.

Nonlinear dynamics of multiple neoclassical tearing modes in tokamaks

Numerical simulations investigating the interaction of co-existent 2/1 and 3/1 neoclassical tearing modes are presented. The results obtained from an initial value 3D toroidal code that solves a set of generalized reduced MHD equations exhibit a host of complex phenomena arising from the coupling of the two modes. These include a modification of the island saturation widths of the two modes, a significant modification in the perpendicular flow patterns in the vicinity of the islands, and the excitation of geodesic acoustic mode like oscillations that lead to concomitant oscillations in the kinetic and magnetic energies of the islands. These oscillations only occur in the presence of the neoclassical stress tensor contribution and are absent for two coupled classical tearing modes.

Spontaneous excitation of geodesic acoustic mode by toroidal Alfvén eigenmodes

Spontaneous nonlinear excitation of geodesic acoustic mode (GAM) by toroidal Alfvén eigenmodes (TAE) is studied within the framework of gyrokinetic theory. The dispersion relation for the parametric decays of a pump TAE mode into a TAE lower sideband and a GAM is derived. It is shown that, in the ideal MHD first stability region, the condition for spontaneous excitation of GAM by TAEs is ω20 > V2A/(4q2R20), in which, ω0 is the pump TAE real frequency, VA is the Alfvén speed, q is the safety factor and R0 is the torus major radius. The corresponding threshold condition is also derived and suggests the decay process as an effective saturation mechanism for TAE.

Excitation of Geodesic Acoustic Modes by External Fields

It is planned to use external magnetic perturbations at acoustic frequencies at the DIII-D tokamak to attempt to drive geodesic acoustic modes (GAM) to modify the turbulent transport. We show that this might not only be possible--despite the well-known electrostatic nature of the GAMs--but might be a viable and efficient method to generate GAMs in magnetically confined plasmas, by developing an elegant analytic method which allows us to couple numerical dynamic equilibrium calculations with massively parallel non-Boussinesq turbulence code runs and yields practical estimates of the effectivity of the method.

Geodesic acoustic mode excitation by a spatially broad energetic particle beam

Global radial eigenmodes of energetic-particle-induced geodesic acoustic mode are systematically studied, and their properties are found to depend on the nonuniformities of both the geodesic acoustic mode continuous spectrum and the energetic particle (EP) radial density profile. For a broad EP drive, the eigenmode equation valid for arbitrary EP drift orbit width is derived, and the excited energetic-particle-induced geodesic acoustic mode is shown to be strongly coupled to the geodesic acoustic mode continuous spectrum; resulting in a finite drive threshold in EP density. The cross-scale couplings between micro-, meso-, and macro-scales, discussed in this work, are mediated by the EP dynamics and have many interesting similarities with complex behaviors, expected in burning plasmas of fusion interest.

The nonlinear dispersion relation of geodesic acoustic modes

The energy input and frequency shift of geodesic acoustic modes (GAMs) due to turbulence in tokamak edge plasmas are investigated in numerical two-fluid turbulence studies. Surprisingly, the turbulent GAM dispersion relation is qualitatively equivalent to the linear GAM dispersion but can have drastically enhanced group velocities. As a consequence radially broad (∼cm) GAM eigenmodes may form. Those may lead to experimentally observable deviations from the expected scaling of the GAM frequency with the square root of the plasma temperature. In up-down asymmetric geometry, the energy input due to turbulent transport may favor the excitation of GAMs with one particular sign of the radial phase velocity relative to the magnetic drifts. Including the radial gradient of the GAM frequency may lead to periodic bursts of the GAM and the turbulence intensity.