Collisionless Trapped Electron Mode Research Articles

Linear analysis and crossphase dynamics in the CTEM fluid model

Collisionless trapped-electron mode (CTEM) turbulence is an important contributor to heat and particle transport in fusion devices. The ion-temperature gradient (ITG)/trapped-electron mode (TEM) fluid models are rarely treated analytically, due to the large number of transport channels involved, e.g., particle and ion/electron heat transport. The CTEM fluid model [Anderson et al., Plasma Phys. Controlled Fusion 48, 651 (2006)] provides a simplified model, in the regime where the density gradient drive (∇n) is negligible compared to the electron temperature gradient drive (∇Te). This provides a starting point to study mechanisms associated with linear waves, such as crossphase dynamics, and its possible role in the formation of E × B staircase. Here, an extended CTEM fluid model (with both ∇n and ∇Te drive) is derived from the more general ITG/TEM model, using a simplified ion density response, and its linear dynamics is first analyzed and compared with CTEM gyrokinetic simulations with bounce-averaged kinetic electrons, while nonlinear analysis is left for future work. The wave action density is derived for this CTEM model. Comparisons of linear ITG spectrum are also made with other analytical models.

Global gyrokinetic simulation of edge coherent mode in EAST

Linear and nonlinear simulations are carried out for the edge coherent mode (ECM) using the global gyrokinetic code GEM based on the EAST experimental parameters. The linear simulation results show that ECM is an electrostatic mode with dominant toroidal mode number n = 18 and frequency about 48 kHz, and propagates along the direction of electron diamagnetic drift, which are consistent with the experimental results. In addition, the density and electron temperature gradients destabilize the mode, while the collision stabilizes the mode. The nonlinear simulation results show that the saturated particle and heat fluxes induced by ECM are mainly due to the perturbed electrostatic ExB drift, and the fluxes of electrons and ions are almost equal. The ECM drives significant outward particle and heat fluxes, thus greatly promoting the maintenance of the long pulse H-mode. The Fourier decomposition of fluxes and potentials demonstrate that the intermediate-n modes of n = 14, 18 grow fastest in the linear phase, while in the nonlinear saturation phase, the low-n modes such as n = 4, 6 dominate and the fluxes are mainly contributed by the mode of n = 10. It is found that zonal flow is not the dominant saturation mechanism of the turbulence. The inverse spectral cascade of turbulence is inevitably observed in the nonlinear saturation process, indicating that it is a more universal turbulence saturation mechanism. It is also found that radial electric field can greatly reduce the turbulence intensity and transport level. From the analyses of frequency and transport channels, it can be concluded that ECM appears to be the collisionless trapped electron mode.

Effects of alpha particles on the transport of helium ash driven by collisionless trapped electron mode turbulence

The effects of alpha (α) particles on the transport of helium ash driven by collisionless trapped electron mode (CTEM) turbulence are analytically studied using quasi-linear theory in tokamak deuterium (D) and tritium (T) plasmas. Under the parameters used in this work, the transport of helium ash is mainly determined by the diffusion due to very weak convection. It is found that the ratio between helium ash diffusivity and effective electron thermal conductivity (D He/χ eff) driven by CTEM turbulence, which is a proper normalized parameter for quantifying the efficiency of helium ash removal, is smaller than unity. This indicates the less efficient removal of helium ash through CTEM turbulence as compared with ion temperature gradient (ITG) turbulence in [Angioni et al 2009 Nucl. Fusion 49 055013]. However, the efficiency of helium ash removal is increased 55% by the presence of 3% α particles with their density gradient being equivalent to that of electrons, and this enhancement can be further strengthened by steeper profile of α particles. This is mainly because the enhancement of helium ash diffusivity by α particles is stronger than that of the effective electron thermal conductivity. Moreover, the higher fraction of T ions, higher temperature ratio between electrons and thermal ions as well as flatter electron density profile, the stronger enhancement of D He/χ eff, and α particles further strengthen the favorable effects of these parameters on the removal of helium ash.

Multi-scale structures of electric current generated by collisionless trapped-electron-mode turbulence

The spatial structure and amplitude of the current induced by collisionless trapped-electron-mode (CTEM) turbulence are investigated by gyrokinetic simulations. It is shown that the barely passing electrons play a crucial role in determining the magnitude and direction of the current density. Two characteristic radial scales of the current density are found. The fine structure (a few ion Larmor radius) of the turbulence-induced current is observed near the rational surfaces. Furthermore, the mesoscale structure (tens of ion Larmor radii) of the turbulence-induced current related to the zonal flow shear is confirmed, especially for the high toroidal mode number (n) CTEM. For the strongly driven CTEM, the zonal flow shear effect on the turbulence-induced current is significant, while it is not visible for the weakly driven CTEM. We show that the magnitude of the CTEM turbulence-induced current density features a moderate local magnitude comparable to the bootstrap current density near rational surfaces, as demonstrated by nonlinear simulations with multi-n modes.

The synergetic effects of three-dimensional magnetic perturbations and finite beta on collisionless trapped electron mode in tokamak plasmas

The effects of three-dimensional (3D) magnetic perturbations (MPs) and finite beta (β, i.e., the ratio of plasma kinetic pressure to magnetic pressure) on the instability of collisionless trapped electron mode (CTEM) have been studied. Based on the local 3D equilibrium model, we have derived general expressions for longitudinal invariant and the corresponding precession drift frequency of trapped electrons, which include the synergetic effects of MPs and finite β. It is found that 3D effects can either stabilize or destabilize CTEM instability by analytically solving the linear dispersion relation of CTEM. These effects depend on the poloidal and toroidal mode numbers as well as the phase of 3D MPs. Specially, for the destabilizing phase of MPs, the stabilizing effect of finite β on CTEM can be even reversed when the displacement of magnetic flux surface exceeds a critical value. Moreover, the synergetic effects of 3D MPs with stabilizing phase and finite β can further reduce the required absolute value of negative magnetic shear to completely stabilize CTEM instability. This indicates that 3D MPs might be used as an actuator for lowing the level of anomalous electron heat transport, and thus facilitate the formation of electron internal transport barrier (eITB).

Extended bounce-kinetic model for trapped electron mode turbulence

The bounce-kinetic model based on the modern nonlinear bounce-kinetic theory [Fong and Hahm, Phys. Plasmas 6, 188 (1999)] has been developed and used for simulations previously. This work reports on an extension of the bounce-kinetic model including more accurate treatment of barely trapped particles and its implementation in the Gyro-Kinetic Plasma Simulation Program gyrokinetic code [Kwon et al., Comp. Phys. Commun. 215, 81 (2017)]. This leads to more accurate gyrokinetic simulations of the collisionless trapped electron mode at low magnetic shear.

Effects of alpha particles on the CTEM driven zonal flow in deuterium–tritium tokamak plasmas

The effects of fusion-born alpha (α) particles on zonal flow (ZF) driven by collisionless trapped electron mode (CTEM) turbulence are analytically investigated, using gyrokinetic and bounce kinetic theories in the deuterium–tritium (D–T) tokamak plasmas. It is found that ZF growth rate is increased by α particles because of the reduction of polarization shielding as well as enhancement of CTEM instability. The results of this paper are qualitatively consistent with the enhancement of the level of residual ZF by α particles in (Cho and Hahm 2019 Nucl. Fusion 59 066026). The parametric dependence of ZF growth rate is also analyzed. The increment of ZF growth rate is further enhanced by α particles with higher fraction and steeper density profile. Besides, the dependence of ZF growth rate on electron temperature T e could be changed qualitatively by the presence of α particles when T i = T e. Moreover, the difference of ZF growth rates in the presence of α particles with slowing down and equivalent Maxwellian distribution functions is very weak. These results could be very important for accurate prediction of the confinement in the future burning plasmas such as International Thermonuclear Experimental Reactor and China Fusion Engineering Test Reactor.

Study of turbulence in the high β P discharge using only RF heating on EAST

High poloidal beta scenarios with a favorable energy confinement (, have been achieved on the Experimental Advanced Superconducting Tokamak using only radio frequency wave heating. Gyrokinetic simulations are carried out with experimental plasma parameters and tokamak equilibrium data of a typical high discharge using the gyrokinetic toroidal code. Linear simulations show that electron-temperature scale length and electron-density scale length destabilize the turbulence, collision effects stabilize the turbulence, and the instability propagates in the electron diamagnetic direction. These properties indicate that the dominant instability in the core of high plasma is a collisionless trapped electron mode. Ion thermal diffusivity, calculated by nonlinear gyrokinetic simulations, is consistent with the experimental value, in which the electron collision effects play an important role. Further analyses show that instabilities with are suppressed by collision effects and collision effects reduce the radial correlation length of turbulence, resulting in the suppression of the turbulence.

The synergetic effects of three-dimensional magnetic perturbations and finite beta on collisionless trapped electron mode in tokamak plasmas

Abstract The effects of three-dimensional (3D) magnetic perturbations (MPs) and finite beta (β, i.e., the ratio of plasma kinetic pressure to magnetic pressure) on the instability of collisionless trapped electron mode (CTEM) have been studied. Based on the local 3D equilibrium model, we have derived general expressions for longitudinal invariant and the corresponding precession drift frequency of trapped electrons, which include the synergetic effects of MPs and finite β. It is found that 3D effects can either stabilize or destabilize CTEM instability by analytically solving the linear dispersion relation of CTEM. These effects depend on the poloidal and toroidal mode numbers as well as the phase of 3D MPs. Specially, for the destabilizing phase of MPs, the stabilizing effect of finite β on CTEM can be even reversed when the displacement of magnetic flux surface exceeds a critical value. Moreover, the synergetic effects of 3D MPs with stabilizing phase and finite β can further reduce the required absolute value of negative magnetic shear to completely stabilize CTEM instability. This indicates that 3D MPs might be used as an actuator for lowing the level of anomalous electron heat transport, and thus facilitate the formation of electron internal transport barrier (eITB).

Limit cycle oscillations, response time, and the time-dependent solution to the Lotka–Volterra predator–prey model

In this work, the time-dependent solution for the Lotka–Volterra predator–prey model is derived with the help of the Lambert W function. This allows an exact analytical expression for the period of the associated limit cycle oscillations and also for the response time between predator and prey population. These results are applied to the predator–prey interaction of zonal density corrugations and turbulent particle flux in gyrokinetic simulations of the collisionless trapped-electron mode turbulence. In the turbulence simulations, the response time is shown to increase when approaching the linear threshold, and the same trend is observed in the Lotka–Volterra model.

Role of wave-particle resonance in turbulent transport in toroidal plasmas

A clear understanding of wave-particle interaction and associated transport mechanisms of different particle species in the drift wave instabilities is important for accurate modeling and predictions of plasma confinement properties in tokamaks. In particular, the roles of linear resonance and nonlinear scattering in turbulent transport need to be delineated when constructing reduced transport models. First-principle, global gyrokinetic simulations find that electron particle and heat transport decreases to a very low level, while ion heat transport level has no dramatic change when wave-particle resonance is suppressed in the collisionless trapped electron mode (CTEM) turbulence. On the other hand, ion heat transport in the self-consistent ion temperature gradient (ITG) turbulence simulation is qualitatively similar to that in the test-particle simulation using the static ITG turbulence fields. These simulation results show that electron transport is primarily driven by the wave-particle resonance in the CTEM turbulence, and the ion transport is mostly driven by the nonlinear wave-particle scattering in both the CTEM and ITG turbulence.

How Zonal Flow Affects Trapped-Electron-Driven Turbulence in Tokamak Plasmas.

The role of self-generated zonal flows in the collisionless trapped-electron-mode (CTEM) turbulence is a long-standing open issue in tokamak plasmas. Here, we show, for the first time, that the zonal flow excitation in the CTEM turbulence is formally isomorphic to that in the ion temperature gradient turbulence. Trapped electrons contribute implicitly only via linear dynamics. Theoretical analyses further suggest that, for short wavelength CTEMs, the zonal flow excitation is weak and, more importantly, not an effective saturation mechanism. Corresponding controlling parameters are also identified theoretically. These findings not only offer a plausible explanation for previous seemingly contradictory simulation results, but can also facilitate controlling the CTEM instability and transport with experimentally accessible parameters.

Effects of energetic particles on the density-gradient-driven collisionless trapped electron mode instability in tokamak plasmas

We investigate analytically the effects of energetic particles (EPs) on the instability of the density-gradient-driven collisionless trapped electron mode (CTEM) through linear gyrokinetic theory and bounce kinetic theory in tokamak plasmas. The effects of EPs, including fusion-born alpha particles and neutral-beam-injection-driven beam ions, on the CTEM instability are compared for the dynamic model with slowing-down (SD) and equivalent Maxwellian (EM) equilibrium EP distribution functions and dilution model. It is found that the density-gradient-driven CTEM instability in the long wavelength regime can be further destabilized by EPs mainly due to the downshift in the real frequency of the mode by dilution effects. This is attributed to more resonant electrons around the smaller phase velocity of the drift wave and the consequent stronger excitation of CTEM instability. The growth rate is slightly higher for the dilution model as compared to that for the dynamic model since the Landau damping effects of EPs are neglected in the dilution model. Moreover, there is no significant difference in the growth rate between the cases of SD and EM equilibrium EP distribution functions, except for the case of alpha particles and with relatively higher electron temperature.

Effects of Micro-turbulence on the Removal of Helium Ash in Deuterium–Tritium Plasmas

Avoiding the accumulation of helium ash in the plasma core is a critical issue for future fusion reactors such as International Thermonuclear Experimental Reactor and China Fusion Engineering Test Reactor. The effects of micro-turbulence, including ion temperature gradient (ITG), parallel velocity shear (PVS) and collisionless trapped electron mode (CTEM) turbulence, on the removal of helium ash are briefly reviewed. We study how helium ash affects ITG and PVS instabilities based on our previous theoretical works, and compare the corresponding results with CTEM instability. The parametric dependence of ash flux is illustrated by calculating the turbulent flux and the corresponding transport coefficients. It indicates that long wavelength electrostatic micro-turbulence is favorable for removing helium ash, especially when its density profile is steeper than that of electrons. The outward flux of helium ash becomes larger for the temperature of helium ash being slightly higher than that of background plasmas ( $${T}_{z}>{T}_{e}={T}_{i}$$ ). Isotopic effects are favorable (unfavorable) for exhausting helium ash through PVS and CTEM (ITG) turbulence. In addition, the ambipolarity of turbulent transport fluxes between electrons, ions and helium ash is self-consistently verified, and its implication on the simultaneous transport of both helium ash and D–T ions is discussed.

Group velocity in spatiotemporal representation of collisionless trapped electron mode in tokamak

The multiple scale derivative expansion method is used to manipulate the electron drift kinetic equation, following the theoretical framework of drift wave–zonal flow system developed by Zhang et al. [Zhang Y Z, Liu Z Y, Mahajan S M, Xie T, Liu J <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="http://doi.org/10.1063/1.4995302"> 2017 <i>Phys. Plasmas</i> <b>24</b> 122304 </ext-link>]. At the zeroth order it is the linear eigenmode equation describing the trapped electron mode on a mirco-scale. At the first order it is the envelop equation for trapped electron mode modulated by the zonal flow on a meso-scale. The eigenmode equation has been solved by Xie et al. [Xie T, Zhang Y Z, Mahajan S M, Wu F, He Hongda, Liu Z Y <ext-link ext-link-type="uri" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1063/1.5048538"> 2019 <i>Phys. Plasmas</i> <b>26</b> 022503 </ext-link>] to obtain the eigenvalue and two-dimensional mode structure of trapped electron mode. These are essential components in calculating group velocities contained in the envelop equation. The radial group velocity arises from the geodesic curvature of magnetic field in tokamak. The poloidal group velocity stems from the normal curvature and diamagnetic drift velocity, which yields the mapping between the poloidal angle and time. Since the radial group velocity is also a function of poloidal angle, it is mapped to a periodic function of time with a period of milliseconds. The numerical results indicate the rapid zero-crossing, which is significant in the drift wave – zonal flow system and provides a sound foundation for studying zonal flow driven by trapped electron mode.

The role of collisionless trapped electron mode turbulence on removal of helium ash and transport of deuterium-tritium ions

Removal of helium ash and the anomalous transport of deuterium (D) and tritium (T) ions driven by collisionless trapped electron mode (CTEM) turbulence in tokamak plasmas with weak magnetic shear are studied. We derive the eigenvalue of CTEM with helium ash, and calculate the quasi-linear turbulent fluxes of helium ash, D and T ions simultaneously. Based on the analytical results, the parametric dependence of CTEM instability as well as the anomalous transport of helium ash and D-T ions is investigated, in order to explore the parameter region that is favorable for expelling more helium ash than D and T ions. It is found that helium ash with higher temperature and steeper density profile plays a role of destabilizing CTEM instability, and has higher transport level than that of T ions. We also find that increasing electron temperature and flattening electron density profile are favorable for exhausting helium ash. Isotopic effects (i.e. increasing the fraction of T ions) enhance the transport of both helium ash and D-T ions. Moreover, the trend of stronger transport level of helium ash than that of D-T ions is enhanced by raising electron temperature and flattening electron density profile as well as isotopic effects. Besides, the diffusivity is much larger than the convection. This indicates that the CTEM turbulence driven helium ash transport is favorable for removing helium ash under the parameter region used in the present paper. The possible relevance of our theoretical results to experimental observations is also discussed.

The dominant micro-turbulence instabilities in the lower q95 high βp plasmas on DIII-D and predict-first extrapolation

Large-radius internal transport barriers (ITB) are the signature of high scenarios on DIII-D. Previous studies show that a large Shafranov shift, rather the shear, suppresses the turbulence and helps in the formation of the large-radius ITB. New gyrokinetic simulations suggest that the remaining micro-instabilities in lower q95 (<7.0), high ITB plasmas are drift wave instabilities, including the collisionless trapped electron mode in the core and ITB peak gradient region, the electron temperature gradient mode in the ITB peak gradient region and at the ITB foot and the ion temperature gradient mode at the ITB foot. Gyrokinetic simulation results qualitatively agree with the density fluctuation analysis from beam emission spectroscopy, which suggests the existence of a low-k ion mode at the ITB foot. The gyrokinetic simulations also predict that a larger Shafranov shift can overwhelm the driving sources for turbulence from the profile gradients at higher , leading to stronger turbulence suppression and stronger ITBs in lower q95, high plasmas.

On the cascading of collisionless trapped-electron mode turbulence in tokamak plasmas

Cascading of collisionless trapped-electron mode turbulence via particle-induced scatterings is analytically investigated for axisymmetric tokamak plasmas. Our results demonstrate that the nonlinear mode-coupling processes depend crucially on both the parallel and radial mode structures associated with the realistic toroidal geometry. As a consequence, the scattering coefficients are found to be significantly suppressed with respect to those derived for slab-like models. For typical tokamak parameters, the wave energy cascades inversely towards lower toroidal mode numbers.

The two-dimensional kinetic ballooning theory for trapped electron mode in tokamak

The two-dimensional (2D) kinetic theory for a collisionless trapped electron mode is developed based on the Fourier-ballooning transform in an up-down symmetric equilibrium (illustrated via concentric circular magnetic surfaces). The system consists of two equations: the ballooning (integral) equation with a parameterized Floquet phase and a second order differential equation for the distribution of the Floquet phase. The coupled equations are, then, numerically solved as an eigenvalue problem yielding the 2D mode structure (in real space) as well as the global (phase-independent) eigenvalue for an L-mode parameter set. The 2D mode structure exhibits apparent radial-poloidal asymmetry; due to the poloidal coupling, the radial correlation length is found to be, at least, twice as large as the poloidal one. The global (phase-independent) eigenvalue of the mode differs considerably from the conventional local (phase-dependent) estimate. This paper shares many technical aspects with a published paper that works out the 2D kinetic theory for the ion temperature gradient mode [Xie et al., Phys. Plasmas 24, 102506 (2017)].

Gyrokinetic theory of turbulent acceleration and momentum conservation in tokamak plasmas

Understanding the generation of intrinsic rotation in tokamak plasmas is crucial for future fusion reactors such as ITER. We proposed a new mechanism named turbulent acceleration for the origin of the intrinsic parallel rotation based on gyrokinetic theory. The turbulent acceleration acts as a local source or sink of parallel rotation, i.e., volume force, which is different from the divergence of residual stress, i.e., surface force. However, the order of magnitude of turbulent acceleration can be comparable to that of the divergence of residual stress for electrostatic ion temperature gradient (ITG) turbulence. A possible theoretical explanation for the experimental observation of electron cyclotron heating induced decrease of co-current rotation was also proposed via comparison between the turbulent acceleration driven by ITG turbulence and that driven by collisionless trapped electron mode turbulence. We also extended this theory to electromagnetic ITG turbulence and investigated the electromagnetic effects on intrinsic parallel rotation drive. Finally, we demonstrated that the presence of turbulent acceleration does not conflict with momentum conservation.