Finite Orbit Width Research Articles

The formation of an radial edge electric field due to finite ion orbit width effects is the possible root cause of the H-mode edge

Abstract Full orbit-following simulations of thermal ions show that finite ion-orbit width effects create charge separation near the last closed flux surface (LCFS) which generates a localized radial electric field. Experimentally, edge electric fields are observed in H-mode plasmas and they are necessary for the edge turbulence suppression via the E × B flow shear mechanism. Confined trapped (and to a lesser extent co-passing) ions near the plasma edge form a positive charge distribution outside the LCFS, while thermal electrons are tied more tightly to field lines owing to their small mass and are poorly confined outside the LCFS, hence charge neutrality is violated outside the LCFS. A large number of reported observations from spherical and conventional tokamaks support the results from the simulations although the simulations were not performed fully self consistently. The results suggest ways to lower the H-mode power threshold and optimize the H-mode plasma edge.

Finite orbit width effects on turbulent transport of ion parallel momentum

Abstract A kinetic model for ion turbulent parallel momentum transport is developed with finite orbit width effects for Tokamak plasmas. It is shown that the curvature and gradient drifts of ions can introduce pressure perturbations into the transport equation of ion parallel momentum, which leads to a new source term. And the source term can be understood as a Coriolis force and can play a key role in the toroidal symmetry breaking during the spontaneous spin-up process.

Transport characteristic evaluation of runaway electrons in an ITER disruption simulation

In previous studies, it has been observed that the transport of energetic electrons decreases with increasing energy. This observation is a global and long-time-scale result, attributed to the space-averaged perturbations. In this work, we focus on the local and instantaneous transport characteristics of runaway electrons (REs) during the phase of tokamak disruption, with REs speeds close to the speed of light. To simulate the dynamics of REs, we utilize a particle tracing code called PTC. By coupling PTC with the MHD code JOREK, we are able to study the energy and spatial dependence of RE transport. Our investigations reveal that the finite orbit width (FOW) effect plays an important role in RE transport. This effect is influenced by the relative direction of the electron drift and the magnetic field line. Specifically, the FOW effect strengthens the transport when the drift direction aligns with the deflecting direction of the field line. And we compare the transport profiles among three time slices: at the beginning of the thermal quench, during the thermal quench, and at the beginning of the current quench. In this ITER disruption simulation, the perturbation scale is strong and δB/B is up to 0.05 at developed thermal quench stage. It is reasonable that the influence of FOW effect on transport is less than that of magnetic perturbation even if the energy of REs is about hundreds MeV and the orbit width is equal to or greater than the perturbation length. These analyses provide insights into the mechanisms of RE transport based on magnetic perturbations.

On the energetic particle-induced geodesic acoustic modes with finite-orbit-width effects

This study presents an analytical investigation of energetic particle-induced geodesic acoustic modes (EGAMs) within a gyro-kinetic model, incorporating finite-orbit-width (FOW) effects up to the second order. The inclusion of second-order FOW effects introduces two distinct types of energetic particle-wave resonances, occurring at ω=ωth and ω=2ωth , respectively, where ωth denotes the transit frequency of energetic particles (EPs). It is found that two unstable EGAM branches coexist: a low frequency branch (LFB) characterized by 0<ωLFB<ωt,maxh , and a high frequency branch (HFB) marked by ωt,maxh<ωHFB<2ωt,maxh . The instability of LFB primarily arises from the resonance ω=ωth , mainly introduced by first-order FOW effects. As a result, the instability of LFB always exists regardless of the presence or absence of second-order FOW effects, and is barely modified by these effects. In contrast, the instability of HFB is exclusively attributed to the resonance ω=2ωth induced by second-order FOW effects. Consequently, the HFB exhibits instability in the presence of these effects.

DRIFT-KINETIC EQUATIONS IN MAGNETIZED CURRENT-CARRYING PLASMAS

Kinetic models of magnetized current-carrying plasma have been developed to study the influence of magnetic drift effects on the wave-particle interactions in tokamaks and cylindrical plasma columns. The drift-kinetic equations are derived for the perturbed distribution functions of trapped and untrapped particles in a two-dimensional axisymmetric toroidal plasma, taking into account their bounce oscillations and the finite orbit-widths of their banana trajectories.

Finite orbit width effects in large aspect ratio stellarators

New orbit-averaged equations for low collisionality neoclassical fluxes in large aspect ratio stellarators with mirror ratios close to unity are derived. The equations retain finite orbit width effects by employing the second adiabatic invariant $J$ as a velocity-space coordinate and they have been implemented in the orbit-averaged neoclassical code KNOSOS (Velasco et al., J. Comput. Phys., vol. 418, 2020, 109512; Velasco et al., Nucl. Fusion, vol. 61, 2021, 116013). The equations are used to study the $1/\nu$ regime and the lower collisionality regimes. For generic large aspect ratio stellarators with mirror ratios close to unity, as the collision frequency decreases, the $1/\nu$ regime transitions directly into the $\nu$ regime, without passing through a $\sqrt {\nu }$ regime. An explicit formula for the neoclassical fluxes in the $\nu$ regime is obtained. The formula includes the effect of particles that transition between different types of wells. While these transitions produce stochastic scattering independent of the value of the collision frequency in velocity space, the diffusion in real space remains proportional to the collision frequency. The $\sqrt {\nu }$ regime is only recovered in large aspect ratio stellarators close to omnigeneity: large aspect ratio stellarators with large mirror ratios and optimized large aspect ratio stellarators with mirror ratios close to unity. Neoclassical transport in large aspect ratio stellarators with large mirror ratios can be calculated with the orbit-averaged equations derived by Calvo et al. (Plasma Phys. Control. Fusion, vol. 59, 2017, 055014). In these stellarators, the $\sqrt {\nu }$ regime exists in the collisionality interval $(a/R) \ln (R/a) \ll \nu _{ii} R a/\rho _i v_{ti} \ll R/a$ . In optimized large aspect ratio stellarators with mirror ratios close to unity, the $\sqrt {\nu }$ regime occurs in an interval of collisionality that depends on the deviation from omnigeneity $\delta$ : $\delta ^{2} |\ln \delta | \ll \nu _{ii} R a/\rho _i v_{ti} \ll 1$ . Here, $\nu _{ii}$ is the ion–ion collision frequency, $\rho _i$ and $v_{ti}$ are the ion gyroradius and thermal speed, and $a$ and $R$ are the minor and major radii.

Effect of anisotropic fast ions on internal kink stability in DIII-D negative and positive triangularity plasmas

Recent DIII-D experiments show that sawtooth stability is strongly affected by anisotropic fast ions from neutral beam injection (NBI) in both negative and positive triangularity plasmas. Fast ions from co-current NBI are stabilizing for the sawtooth stability, resulting in longer sawtooth periods. On the other hand, fast ions from counter-current NBI are destabilizing, leading to small and frequent sawteeth. The relative change of sawtooth period and amplitude is more than a factor of two. These observations appear to hold in both plasma shapes. Non-perturbative toroidal modeling, utilizing the magnetohydrodynamic-kinetic hybrid stability code MARS-K (Liu et al 2008 Phys. Plasmas 15 112503), reveals an asymmetric dependence of the stability of the n = 1 (n is the toroidal mode number) internal kink mode on the injection direction of NBI, being qualitatively consistent with the experimentally observed sawtooth behavior. The MARS-K modeling results suggest that anisotropic fast ions affect the mode growth rate and frequency through both adiabatic and non-adiabatic contributions. The asymmetry of the internal kink mode instability relative to the NBI direction is mainly due to the non-adiabatic contribution of passing fast ions, which stabilize (destabilize) the internal kink with the co-(counter-) current NBI as compared to the fluid counterpart. However, finite orbit width (FOW) correction to passing particles partially cancels the asymmetry. Trapped particles are always stabilizing due to precessional drift resonance. Modeling also shows that fast ions affect the internal kink in a similar manner in both negative and positive triangularity plasmas, although being slightly more unstable in the negative triangularity configuration already in the fluid limit. The similarity is mainly attributed to the fact that the mode is localized in the plasma core region, with very similar eigenmode structures in both negative and positive configurations. Furthermore, MARS-K modeling indicates that other factors, such as the plasma rotation and the drift kinetic effects of thermal plasmas, weakly modify the mode stability as compared to the drift kinetic resonance effects and FOW correction of fast ions.

Characterization of beam ion loss in high poloidal beta regime on EAST

A critical issue for achieving the integrated operation of steady-state long-pulse high-confinement (H-mode) plasmas on experimental advanced superconducting tokamak (EAST) is to improve beam ion population confinement during neutral beam injection (NBI). To study the characterization of beam ion loss and improve beam ion confinement, the steady-state long pulse scenario discharges were conducted on EAST (β p ⩾ 2.0, β N ⩾ 1.7, q 95 ⩾ 6.7 and H 98y2 ⩾ 1.1) with NBI heating. Based on neutron yield, the beam voltage and line-averaged electron density were adjusted from 50 kV to 60 kV and 4.4 × 1019 m−3 to 5.0 × 1019 m−3, respectively. The results show that the dominant mechanisms of beam ion loss are shine-through loss, prompt loss, and stochastic ripple loss. The shine-through loss fraction is determined by initial velocity, flight time and entire beam path. The change in prompt loss fraction is caused by the change in the deposition of beam ions. The change in stochastic ripple loss fraction is caused by the change in the initial fraction of trapped-confined ions. Detailed physics shows that the prompt loss fraction during counter-Ip injections (∼45%) is far larger than during co-Ip injections (∼5%) due to the finite orbit width. The lost ions are mainly deposited on the lower divertor or below the midplane since the direction of magnetic drift is vertical down. The orbit types of prompt loss during counter-Ip injections are mainly trapped-lost and ctr-passing lost. To minimize the prompt loss fraction during counter-Ip injections, a reversed Ip configuration (rev-Ip) discharge #94758 was conducted. The result suggests that the beam ion wall load fraction during counter-Ip tangential injection (∼3%) is far lower than that in normal Ip configuration (nor-Ip) discharge #94820. It is also found that the confinement of beam ion population in the counter-Ip injection #94758 was greatly improved when compared to #94820. This study can provide unique support for the improvement of beam ion population confinement and for the performance evaluation of the NBI system on EAST and future tokamaks.

Influence of adiabatic response of passing energetic ions on high-frequency fishbone

Adiabatic response effects on high-frequency fishbone instability driven by passing energetic ions are studied. With finite orbit width effects, the adiabatic contribution is derived analytically for purely passing energetic ions. By approximating the adiabatic contribution to the first order of the reverse aspect ratio we derive one of its analytic expressions, which is expected to be more accurate than that in a previous work (Graves J P 2004 Phys. Rev. Lett. 92 185003). For high-frequency fishbone instability, nonadiabatic response is usually dominant over adiabatic response, but under certain circumstances the latter plays an important role, comparable to the former. With a more generalized distribution function by introducing Gaussian-type factors representing their pitch and radial dependences and using a slowing-down equilibrium distribution for the energy of energetic ions, numerical analysis indicates that the adiabatic contribution is conducive to stabilization of the mode and causes a decrease in mode frequency. In addition, we find that the adiabatic contribution has a weak stabilizing effect on the fishbone instability when the finite orbit width effect is taken into account. We further analyze the dependence of the adiabatic contribution on the characteristic parameters of the distribution function. When the neutral beam has either a larger deviation from the plasma axis or a larger radial profile, the adiabatic contribution has a more evident effect on the fishbone instability. When the neutral beam has a relatively small critical energy, the adiabatic contribution has a greater effect on the mode instability.

Toroidal modeling of energetic passing particle drift kinetic effects on tearing mode stability

Drift kinetic effects of the neutral beam injection induced passing energetic particles (EPs) on the linear stability of the n = 1 tearing mode (TM) (with the dominant poloidal harmonic of m = 2) are numerically investigated utilizing the MARS-K code (Liu et al 2008 Phys. Plasmas 15 112503), in a tokamak plasma with finite equilibrium pressure and anisotropic thermal transport. In the low plasma pressure regime, it is found that co- (counter-) passing EPs stabilize (destabilize) the TM, agreeing with previous studies. However, as the plasma pressure increases beyond a critical value, it is found that co-passing EPs also destabilize the mode. An in-depth analysis reveals that the net effect of co-passing EPs is a result of competition between the stabilizing contribution from the non-adiabatic drift kinetic terms and the destabilizing contribution associated with adiabatic terms, with the latter becoming more dominant at higher equilibrium pressure. Non-perturbative magnetohydrodynamic-kinetic hybrid modeling also finds that co- and counter-passing EPs modify the TM eigenfunction differently, with the counter-passing EPs enhancing the sideband harmonics. Furthermore, effects of the plasma resistivity and toroidal rotation, as well as that of the equilibrium distribution of EPs in the particle pitch angle space, are also investigated, showing asymmetric results on the TM stability between the co- and counter-passing EPs. The first order finite orbit width correction is found to be stabilizing with co-passing EPs and destabilizing with counter-passing particles. Finally, drift resonances between passing EPs and the TM induce finite frequency to the mode and generate finite net torques inside the plasma, due to the neoclassical toroidal viscosity and the Reynolds stress associated with 3D perturbations.

Prediction of the energetic particle redistribution by an improved critical gradient model and analysis of the transport threshold

Based on the theory of critical gradient model (CGM) and following the simulation method proposed by Waltz et al. [Nucl. Fusion 55, 123012 (2015)], a combination of TGLFEP and EPtran code is employed to predict the energetic particle (EP) transport induced by Alfvén eigenmodes (AEs). To be consistent with the experiment, recent improvements to the simulation method include consideration of threshold evolution and orbit loss due to finite orbit width. The revised CGM is applied to simulate two DIII-D experimental discharges (#142111 and #153071). It well reproduces the experimental profiles with multiple unstable AEs and large-scale EP transport. Discharge #142111 had previously been simulated using a nonlinear MHD-kinetic code MEGA [Todo et al., Nucl. Fusion 55, 073020 (2015)] with a transport mechanism based on stochasticity induced by overlapping AE. By comparing the simulated EP profiles, we find that the AE transport threshold is approximated by both the MEGA nonlinear stability threshold and the proposed CGM threshold (error &amp;lt;5% for single n and &amp;lt;17% for multiple n simulation). Both of them are larger than the linear stability threshold of the most unstable AE mode by a quantity of the order of the flux needed to sustain EP transport by the background turbulence. We have also applied the improved CGM to simulate the α particle redistribution for a China Fusion Engineering Test Reactor steady state scenario. Because of the clear separation between the AE unstable region and the loss cone, only a moderate α particle loss of ∼9.6% is predicted.

Generation of E × B flow shear by finite orbit width effects from heat sources in tokamaks

We present a possible mechanism for the generation of strong E × B flow shear relevant to internal transport barrier formation in tokamak plasmas. From gyrokinetic calculations, we show that strong E × B flow shear can be generated by finite orbit width (FOW) effects associated with a non-uniform heat source and is sufficient to lead to transport barrier formation in the core region with a moderate power level. Two FOW effects inducing neoclassical polarization are shown to be responsible for this: (1) the radial drift of particle orbit center due to the variation of the heat source within orbit width and (2) the non-uniformly evolved orbit width by the non-uniform heating.

Sheath collapse at critical shallow angle due to kinetic effects

The Debye sheath is known to vanish completely in magnetised plasmas for a sufficiently small electron gyroradius and small angle between the magnetic field and the wall. This angle depends on the current onto the wall. When the Debye sheath vanishes, there is still a potential drop between the wall and the plasma across the magnetic presheath. The magnetic field angle corresponding to the predicted sheath collapse is shown to be much smaller than previous estimates, scaling with the electron-ion mass ratio and not with the square root of the mass ratio. This is shown to be a consequence of the kinetic electron and finite ion orbit width effects, which are not captured by fluid models. The wall potential with respect to the bulk plasma at which the Debye sheath vanishes is calculated. Above this wall potential, it is possible that the Debye sheath will invert.

Drift kinetic theory of neoclassical tearing modes in a low collisionality tokamak plasma: magnetic island threshold physics

A new drift kinetic theory for the plasma response to the neoclassical tearing mode (NTM) magnetic perturbation is presented. Small magnetic islands of width, (a is the tokamak minor radius) are assumed, retaining the limit w ∼ ρ bi (ρ bi is the ion banana orbit width) to include finite orbit width effects. When collisions are small, the ions/electrons follow streamlines in phase space; for passing particles, these lie in surfaces that reproduce the magnetic island structure but have a radial shift by an amount, proportional to , where is the ion/electron poloidal Larmor radius. This shift is associated with the curvature and ∇B drifts and is found to be in opposite directions for , where is the component of velocity parallel to the magnetic field. The particle distribution function is then found to be flattened across these shifted or drift islands rather than the magnetic island. This results in the pressure gradient being sustained across the magnetic island for and hence reduces the neoclassical drive for NTMs when w is small. This provides a physics basis for the NTM threshold, which is quantified. In Imada et al (2019 Nucl. Fusion 59 046016, and references therein), a 4D drift kinetic non-linear code has been applied to describe these modes. In the present paper, the drift island formalism is employed. Valid at low collisionality, it allows a dimensionality reduction to a 3D problem, simplifying the numerical task and efficiently resolving the collisional boundary layer across the trapped-passing boundary. An improved model is adopted for the magnetic drift frequency. This decreases the NTM threshold, compared to the results shown in Imada et al (2019 Nucl. Fusion 59 046016, and references therein), making it in quantitative agreement with experimental observations, with , where w c is the threshold magnetic island half-width, or 2.85ρ bi for the full threshold island width, predicted for our equilibrium.

Simulation of the loss of passing fast ions induced by magnetic islands in EAST tokamak plasmas

The loss of beam ions due to magnetic islands is investigated in a tokamak. The perturbed guiding-center drifts of passing particles including the effect of the finite orbit width are demonstrated. The widths of the drift islands under resonant conditions are studied theoretically and numerically. The ORBIT code is used to simulate the action of the neoclassical tearing mode with a toroidal mode number n = 1 and poloidal mode number m = 2 on passing fast ions generated by neutral beam injection in the Experimental Advanced Superconducting Tokamak. Two loss channels for passing fast ions are identified as the resonant interaction and the stochastic interaction. The lost fast ions in the loss detector zone (LDZ) to simulate the fast-ion loss detector assemble around two regions in phase space, namely, (i) a pitch angle of θ = 28° both with and without the mode and (ii) θ = 59° when the mode amplitude is large enough, where θ=arccosv∥/v. The number of these lost ions in the LDZ evolves in the period of the mode. The fraction of the total lost ions evolves in the period of the n = 1 oscillation in the toroidal direction. The fraction of lost beam ions has a linear relationship with the mode amplitude in first 10 µs and a quadratic one thereafter. The corresponding characteristics of the lost beam ions in phase space are also discussed.

Assessing energy dependence of the transport of relativistic electrons in perturbed magnetic fields with orbit-following simulations

Experimental observations, as well as theoretical predictions, indicate that the transport of energetic electrons decreases with energy. This reduction in transport is attributed to finite orbit width (FOW) effects. Using orbit-following simulations in perturbed tokamak magnetic fields that have an ideal homogeneous stochastic layer at the edge, we quantify the energy dependence of energetic electrons transport and confirm previous theoretical estimates. However, using magnetic configurations characteristic of JET disruptions, we find no reduction in runaway electron transport at higher energies, which we attribute to the mode widths being comparable to the minor radius, making the FOW effects negligible. Instead, the presence of islands and non-uniform magnetic perturbations are found to be more important. The diffusive-advective transport coefficients calculated in this work, based on simulations for electron energies 10 keV–100 MeV, can be used in integrated disruption modelling to account for the transport due to the magnetic field perturbations.

Drift Alfvén energetic particle stability with circulating particles

We develop from scratch a comprehensive linear stability eigenvalue code based on a finite element method, namely, the drift Alfvén energetic particle stability (DAEPS) code, to investigate the physics of various stable and unstable modes observed in toroidal fusion plasmas, which has the advantage of accurate calculation of the mode characteristics near marginal stability. The DAEPS code is dedicated to providing a thorough understanding of low frequency modes in collisionless plasmas, e.g., shear Alfvén wave SAW and drift Alfvén wave physics with an energetic particle (EP) effect. DAEPS can calculate the linear frequency and growth rate for these modes by keeping correct asymptotic behavior in ballooning space. In this work, we demonstrate that the DAEPS code is able to analyze linear electromagnetic modes excited by circulating particles, including the thermal particle excited beta-induced Alfvén eigenmode and EP excited toroidicity-induced Alfvén eigenmode, where the verifications are performed successfully with other codes and theories, where the finite orbit width is discovered to play an important stabilizing role, which are usually ignored by traditional theory.

Excitation of zonal flow by nonlinear geodesic acoustic mode

The zero frequency zonal flow (ZFZF) excitation due to the nonlinear geodesic acoustic mode (GAM) is investigated in the framework of gyro-kinetic equations, which were first investigated by Chen et al. [Europhys. Lett. 107, 15003 (2014)]. We show that after integrating over the velocity space, the first order finite orbit width (FOW) term of nonlinear GAM disappears, and hence, the second order FOW term should be taken into account to generate the ZFZF. The anisotropy of the equilibrium is also considered.

Low-frequency fishbone driven by passing fast ions in tokamak plasmas

Low-frequency fishbone instability driven by passing fast ions via wave-particle resonance is studied within the framework of energetic particle mode (EPM), where and are toroidal transit frequency and poloidal transit frequency of passing fast ions respectively. The effects of finite orbit width of energetic particles are responsible for the low-frequency fishbone of EPM type. It is found that magnetic shear at the q = 1 radius plays an important role in the instability whereas the effect of background plasma beta is weak. In particular, for the case of co-injection of beam ions, there exists a critical magnetic shear below which the beam ion beta threshold for EPM excitation is very small. For moderate or higher magnetic shear, the beam ion beta threshold is much higher, on order of a few percent. These results are consistent with the observation of the low-frequency fishbone in the HL-2A tokamak.

Finite ion orbit width effect on the neoclassical tearing mode threshold in a tokamak plasma

A new drift kinetic theory for the response of ions to small magnetic islands in toroidal plasma is presented. Islands whose width w is comparable to the ion poloidal Larmor radius are considered, expanding the ion response solution in terms of , where r is the minor radius. In this limit, the ion distribution can be represented as a function of toroidal canonical momentum, . With effects of grad-B and curvature drifts taken into account, the ion distribution function is a constant on a ‘drift island’ structure, which is identical to the magnetic island but radially shifted by . The distribution is then flattened across the drift island, rather than the magnetic island. For small islands , the pressure gradient is maintained across the magnetic island, suppressing the bootstrap current drive for the neoclassical tearing mode (NTM) growth. As , the ions are largely unperturbed. However, the electrons respond to the electrostatic potential required for quasi-neutrality and this provides a stabilizing contribution to the NTM evolution. This gives a new physical understanding of the NTM threshold mechanism, with implications for the design of NTM control systems for future tokamaks such as ITER.