Trapped Particle Modes Research Articles

Trapped-particle precession and modes in quasisymmetric stellarators and tokamaks: a near-axis perspective

This paper presents the calculation of the bounce-averaged drift of trapped particles in a near-axis framework for axisymmetric and quasisymmetric magnetic fields that possess up-down and stellarator symmetry, respectively. This analytic consideration provides important insight on the dependence of the bounce-averaged drift on the geometry and stability properties of the field. In particular, we show that although the maximum- $\mathcal {J}$ property is unattainable in quasisymmetric stellarators, one may approach it through increased plasma $\beta$ and triangular shaping, albeit going through a reduced precession scenario with potentially higher particle losses. The description of trapped particles allows us to calculate the available energy of trapped electrons analytically in two asymptotic regimes, providing insight into the behaviour of this measure of turbulence. It is shown that the available energy is intimately related to magnetohydrodynamics (MHD) stability, providing a potential synergy between this measure of gyrokinetic turbulence and MHD stability.

Bounce-averaged drifts: Equivalent definitions, numerical implementations, and example cases

In this article, we provide various analytical and numerical methods for calculating the average drift of magnetically trapped particles across field lines in complex geometries, and we compare these methods against each other. To evaluate bounce integrals, we use a generalization of the trapezoidal rule which is able to circumvent integrable singularities. We contrast this method with more standard quadrature methods in a parabolic magnetic well and find that the computational cost is significantly lower for the trapezoidal method, though at the cost of accuracy. With numerical routines in place, we next investigate conditions on particles which cross the computational boundary, and we find that important differences arise for particles affected by this boundary, which can depend on the specific implementation of the calculation. Finally, we investigate the bounce-averaged drifts in the optimized stellarator NCSX. From investigating the drifts, one can readily deduce important properties, such as what subset of particles can drive trapped-particle modes and in what regions radial drifts are most deleterious to the stability of such modes.

The universal instability in optimised stellarators

In tokamaks and neoclassically optimised stellarators, like Wendelstein 7-X (W7-X) and the Helically Symmetric Experiment, turbulent transport is expected to be the dominant transport mechanism. Among the electrostatic instabilities that drive turbulence, the trapped-electron mode (TEM) has been shown both analytically and in simulations to be absent over large ranges of parameter space in quasi-isodynamic stellarator configurations with the maximum-$J$property. It has been proposed that the reduction of the linear TEM growth rate in such configurations may lead to the passing-electron-driven universal instability, which is often subdominant to the TEM, becoming the fastest-growing instability over some range of parameter space. Here, we show through gyrokinetic simulations using theGenecode, that the universal instability is dominant in a variety of stellarator geometries over a range of parameter space typically occupied by the TEM, but most consequentially in devices which possess beneficial TEM stability properties like W7-X, which locally satisfies the maximum-$J$property for deeply trapped particles in the regions of worst curvature. We find that the universal instability exists at long perpendicular wavelengths and, as a result, dominates the potential fluctuation amplitude in nonlinear simulations. In W7-X, universal modes are found to differ in parallel mode structure from trapped-particle modes, which may impact turbulence localisation in experiments.

Global linear stability analysis of kinetic trapped ion mode (TIM) in tokamak plasma using the spectral method

Trapped ion modes (TIMs) belong to the family of ion temperature gradient (ITG) modes, which are one of the important ingredients in heat turbulent transport at the ion scale in tokamak plasmas. A global linear analysis of a reduced gyro-bounce kinetic model for trapped particle modes is performed, and a spectral method is proposed to solve the dispersion relation. Importantly, the radial profile of the particle drift velocity is taken into account in the linear analysis by considering the magnetic flux ψ dependency of the equilibrium Hamiltonian in both the quasi-neutrality equation and equilibrium gyro-bounce averaged distribution function . Using this spectral method, linear growth rates of TIM instability in the presence of different temperature profiles and precession frequencies of trapped ions, with an approximated constant Hamiltonian and the exact ψ dependent equilibrium Hamiltonian, are investigated. The growth rate depends on the logarithmic gradient of temperature κ T , density κ n and equilibrium Hamiltonian . With the exact ψ dependent Hamiltonian, the growth rates and potential profiles are modified significantly, compared to the cases with an approximated constant Hamiltonian. All the results from the global linear analysis agree with a semi-Lagrangian based linear Vlasov solver with good accuracy. This spectral method is very fast and requires much less computation resources compared to a linear version of the Vlasov-solver based on a semi-Lagrangian scheme.

The model of particles modes. II. Transition to a fishbone-like state triggered by global synchronization and energetic particles

The interplay between kinetic aspects induced by energetic particles on turbulence is analyzed with a simplified model of ion-temperature-gradient-driven turbulence in magnetically confined plasmas. These topics are presented within an unified Hamiltonian framework in light of a new approach based on global phase synchronization between trapped particle modes and energetic particle modes. Numerical experiments have been carried out to elucidate concepts and physical processes of transition to a fishbone-like state triggered by energetic particles.

Transport Barrier Triggered by Resonant Three-Wave Processes Between Trapped-Particle-Modes and Zonal Flow

We address the mechanisms underlying low-frequency zonal flow generation in a turbulent system through the parametric decay of collisionless trapped particle modes and its feedback on the stabilization of the system. This model is in connection with the observation of barrier transport in reduced gyrokinetic simulations (A. Ghizzo et al., Euro. Phys. Lett. 119(1), 15003 (2017)). Here the analysis is extended with a detailed description of the resonant mechanism. A key role is also played by an initial polarisation source that allows the emergence of strong initial shear flow. The parametric decay leads to the growth of a zonal flow which differs from the standard zero frequency zonal flow usually triggered by the Reynolds stress in fluid drift-wave turbulence. The resulting zonal flow can oscillate at low frequency close to the ion precession frequency, making it sensitive to strong amplification by resonant kinetic processes. The system becomes then intermittent. These new findings, obtained from numerical experiments based on reduced semi-Lagrangian gyrokinetic simulations, shed light on the underlying physics coming from resonant wave-particle interactions for the formation of transport barriers. Numerical simulations are based on a Hamiltonian reduction technique, including magnetic curvature and interchange turbulence, where both fastest scales (cyclotron and bounce motions) are gyro-averaged.

Kinetic ballooning modes in tokamaks and stellarators

Kinetic ballooning modes (KBMs) are investigated by means of linear electromagnetic gyrokinetic (GK) simulations in the stellarator Wendelstein 7-X (W7-X), for high-$\unicode[STIX]{x1D6FD}$ plasmas, where $\unicode[STIX]{x1D6FD}$ is the ratio of thermal to magnetic plasma pressure. The analysis shows suppression of ion-temperature-gradient (ITG) and trapped particle modes (TEM) by finite-$\unicode[STIX]{x1D6FD}$ effects and destabilization of KBMs at high $\unicode[STIX]{x1D6FD}$. The results are compared with a generic tokamak case. We show that, for large pressure gradients, the frequency of KBMs evaluated by the GENE code is in agreement with the analytical prediction of the diamagnetic modification of the ideal magnetohydrodynamic limit in W7-X general geometry. Thresholds for destabilization of the KBM are predicted for different W7-X equilibrium configurations. We discuss the relation of these thresholds to the ideal magnetohydrodynamic (MHD) stability properties of the corresponding equilibria.

Linear electrostatic gyrokinetics for electron–positron plasmas

Gyrokinetic stability of plasmas in different magnetic geometries is studied numerically using the GENE code. We examine the stability of plasmas, varying the mass ratio between the positive and negative charge carriers, from conventional hydrogen plasmas through to electron–positron plasmas. Stability is studied for prescribed temperature and density gradients in different magnetic geometries: (i) An axisymmetric, circular flux surface, low$\unicode[STIX]{x1D6FD}$(tokamak) configuration. (ii) A non-axisymmetric quasi-isodynamic (optimised stellarator) configuration using the geometry of the stellarator Wendelstein 7-X. We also present the analytic theory of trapped particle modes in electron–positron plasmas. We found similar behaviour of the growth rate and real frequency compared to previous studies on the tokamak case. We are able to identify two distinct regimes dominated by modes propagating in the electron diamagnetic direction and modes propagating in the ion/positron diamagnetic direction, depending on the mass ratio. In both the tokamak and the stellarator case we observe that the real frequency tends to zero as the mass ratio approaches unity and are able to explain this using gyrokinetic theory.

Adiabatic electron response and solitary wave generation by trapped particle nonlinearity in a hydrogen plasma

The finite amplitude ion acoustic waves that trap electrons modify the structure of the evolving nonlinear soliton solutions. In the numerical simulations, self-consistently generated solitary waves are studied that emerge as a result of a current driven microinstability growing the ion acoustic mode in a collisionless Vlasov plasma. The growth saturates as a result of nonlinear effects governed by a combination of nonlinearities originating from the hydrodynamic model and kinetic particle trapping effects. The resulting solitary waves also coexist with a finite current and an electron plasma wave capable of perturbing the trapping potential. The results of multiscale simulation are analyzed and characterized following the kinetic prescription of undamped trapped particle mode in the form of phase space vortex solutions that are generalized form of Sagdeev's solitons and obey the solutions of a modified Korteweg-de Vries equation, accounting for a stronger nonlinearity originating from the electron trapping.

Collisionless microinstabilities in stellarators. I. Analytical theory of trapped-particle modes

This is the first of two papers about collisionless, electrostatic micro-instabilities in stellarators, with an emphasis on trapped-particle modes. It is found that, in so-called maximum-$J$ configurations, trapped-particle instabilities are absent in large regions of parameter space. Quasi-isodynamic stellarators have this property (approximately), and the theory predicts that trapped electrons are stabilizing to all eigenmodes with frequencies below the electron bounce frequency. The physical reason is that the bounce-averaged curvature is favorable for all orbits, and that trapped electrons precess in the direction opposite to that in which drift waves propagate, thus precluding wave-particle resonance. These considerations only depend on the electrostatic energy balance, and are independent of all geometric properties of the magnetic field other than the maximum-$J$ condition. However, if the aspect ratio is large and the instability phase velocity differs greatly from the electron and ion thermal speeds, it is possible to derive a variational form for the frequency showing that stability prevails in a yet larger part of parameter space than what follows from the energy argument. Collisionless trapped-electron modes should therefore be more stable in quasi-isodynamic stellarators than in tokamaks.

Gyrokinetic TEM stability calculations for quasi-isodynamic stellarators

Recent theoretical findings suggest that in perfectly quasi-isodynamic stellarators, the trapped-particle instability as well as the ordinary electron-density-gradient-driven trapped-electron mode are stable in the electrostatic and collisionless approximation. In these configurations contours of constant magnetic field strength B are poloidally closed, the second adiabatic invariant J is constant on flux surfaces and peaks in the centre. It follows that the diamagnetic drift frequency ω*α and the bounce-averaged magnetic drift frequency are in opposite directions, 0, everywhere on the flux surface. This is the signature of average "good curvature" for trapped particles and, thanks to this property, particles that bounce faster than the frequency of any unstable mode must draw energy from it near marginal stability. Consequently, the point of marginal stability cannot exist for the collisionless trapped-particle mode, and hence this mode will be absent. By a similar argument, the ordinary trapped-electron mode is also stable. Because perfect quasi-isodynamicity can never be reached, it is necessary to test configurations approaching quasi-isodynamicity numerically in order to probe their resilience against trapped-particle modes. Progress has been made to extend gyrokinetic simulations to stellarator geometries, enabling us to perform the first linear gyrokinetic simulations of density-gradient-driven modes in stellarator configurations approaching quasi-isodynamicity, such as Wendelstein 7-X and more recently found configurations. These simulations appear to confirm the analytical predictions.

Trapped particle stability for the kinetic stabilizer

A kinetically stabilized axially symmetric tandem mirror (KSTM) uses the momentum flux of low-energy, unconfined particles that sample only the outer end-regions of the mirror plugs, where large favourable field-line curvature exists. The window of operation is determined for achieving magnetohydrodynamic (MHD) stability with tolerable energy drain from the kinetic stabilizer. Then MHD stable systems are analysed for stability of the trapped particle mode. This mode is characterized by the detachment of the central-cell plasma from the kinetic-stabilizer region without inducing field-line bending. Stability of the trapped particle mode is sensitive to the electron connection between the stabilizer and the end plug. It is found that the stability condition for the trapped particle mode is more constraining than the stability condition for the MHD mode, and it is challenging to satisfy the required power constraint. Furthermore, a severe power drain may arise from the necessary connection of low-energy electrons in the kinetic stabilizer to the central region.

Global energy confinement scaling predictions for the kinetically stabilized tandem mirror

Transport is studied for the kinetically stabilized tandem mirror, an attractive magnetic confinement device for achieving a steady-state burning plasma. For a magnetohydrodynamic stable system, three different radial transport models with Bohm, gyro-Bohm, and electron temperature gradient (ETG) scaling are derived. As a conservative estimate, numerical coefficients in the models are taken to be consistent with tokamak and stellarator databases. The plug mirrors create an ambipolar potential that controls end losses, whereas radial losses are driven by drift wave turbulence, which lowers the electron temperature through radially trapped particle modes and ETG transport losses. The radial transport equations are analyzed, taking into account the Pastukhov energy and particle end losses. For mirror ratio Rm=9 and a large density ratio between plug and central cell regions, there is a high axial ion confinement potential ϕi∕Ti⪢1, as demonstrated in the GAMMA-10 by Cho et al. [Nucl. Fusion 45, 1650 (2005)]. Profiles and total energy confinement times are calculated for a proof-of-principle experiment (length L=7m, central cell magnetic field B=0.28T, and radius a=1m) and for a test reactor facility (L=30m, B=3T, a=1.5m). For these parameter sets, radial loss dominates the end losses except in the low temperature periphery. In the limit of negligible radial losses, ideal ignition occurs at Ti=7.6keV from the two-body power end losses. The transport suppressing rotation rate is well below the sonic value and scales similarly to biased wall rotation rates in the Large Plasma Device experiments [Horton et al., Phys. Plasmas 12, 022303 (2005)]. Simulation results show that the positive dependence of electron radial transport with increasing electron temperature stabilizes the thermal instabilities giving steady state with Ti=30–60keV and Te=50–150keV with a fusion amplification Q of order 1.5 to 5.0.

Non-Linear Growth of Trapped Particle Modes in Linearly Stable, Current-Carrying Plasmas – A Fundamental Process in Plasma Turbulence and Anomalous Transport

The two-stream instability as a fundamental process in a current-carrying plasma is reconsidered. Its well-established linear version, based on kinetic Landau theory, predicts a threshold for the drift velocity between both species below which the plasma should be stable. We report on simulations which, however, show that a plasma as a non-linearly responding medium can be destabilized well below this threshold. Responsible for this unexpected behaviour are coherent, electrostatic, trapped particle structures such as phase space vortices or holes which can grow non-linearly out of thermal noise receiving their energy from the net imbalance of loss of electron kinetic energy and gain of ion kinetic energy. The birth of predominantly zero-energy holes is shown numerically being associated with initial, non-topological fluctuations. The latter are not subject to Landau damping, as they lie outside the realm of linear wave theory. For a pair plasma a typical scenario is presented, which encompasses several regimes such as non-linear growth of multiple holes, saturation and fully developed structural turbulence as well as an asymptotic approach to a new collisionless equilibrium. During the transient, structural state the plasma transport appears to be highly anomalous.

Trapped-electron solitary wave structures in a magnetized plasma

The classic one-dimensional (1D) BGK-mode (trapped-particle) analysis is extended to 3D plasmas in which a background magnetic field is present. A theoretical analysis is made of such nonlinear states from the coupled plasma equations for a magnetized plasma, which yields a generalized BGK-like equation for trapped-particle distributions that produce this structure in the guiding-center approximation, and the resulting conditional requirement for electron trapping as a function of the electric-field components, and magnetic gradient and polarization drifts. This condition shows that the amount of trapping decreases as the angle of the electric field with respect to the background magnetic field increases, often ceasing at a critical finite angle. The full solution to the generalized BGK-type equation for the trapped-electron distribution is obtained and reduced to the extent possible, thereby extending the classic 1D solution for electron holes to that for the 3D magnetized plasmas for cylindrical symmetry. This solution is very easily generalized for the inclusion of any cross-tail drift for the case of interest. For the case that the E×B drift dominates, the solution shows that these trapped-particle modes usually are restricted to angles θ<sin−1−2B2W∕meE2, where −W is the available trapping energy for the electrons, although in rarer cases slightly higher angles for these modes may be allowed. In no case can θ be as large as sin−12−B2W∕meE2.

Trapped-particle diocotron modes

Recent experiments have characterized trapped-particle modes on a non-neutral plasma column [A. A. Kabantsev, C. F. Driscoll, T. J. Hilsabeck, T. M. O’Neil, and J. H. Yu, Phys. Rev. Lett. 87, 225002 (2001)], and in this paper we present a theoretical model of the modes. Theoretical predictions for the mode frequency, damping rate, and eigenmode structure are compared to experimental observation. The modes are excited on a non-neutral plasma column in which classes of trapped and passing particles have been created by the application of a potential barrier. The column resides in a Malmberg–Penning trap, and the barrier is created by applying a voltage to an azimuthally symmetric section of the wall near the axial mid-point of the column. Low energy particles near the edge of the column (where the barrier is strong) are trapped in one end or the other, while high energy particles near the center of the column transit the entire length. The modes have azimuthal variation l=1,2,…, and odd z-symmetry. The trapped particles on either side of the barrier execute E×B drift oscillations producing density perturbations that are 180° out of phase with each other, while passing particles run back and forth along the field lines attempting to Debye shield the perturbed charge density. The mode is damped by collisional scattering across the separatrix between trapped and passing particles. The damping rate is calculated using a boundary layer analysis of the Fokker–Planck equation. It is also shown that the damping is associated with the radial transport of plasma particles.

Damping of the trapped-particle diocotron mode.

The damping mechanism of a recently discovered trapped-particle mode is identified as collisional velocity scattering of marginally trapped particles. The mode exists on non-neutral plasma columns that are partially divided by an electrostatic potential. This damping mechanism is similar to that responsible for damping of the dissipative trapped-ion mode. The damping rate is calculated using a Fokker-Planck analysis and agrees with measurement to within 50%. Also, an experimental signature confirms a causal relation between scattering of marginally trapped particles and damping.

Trapped particles and asymmetry-induced transport

Trapped particle modes and the associated asymmetry-induced transport are characterized experimentally in cylindrical electron plasmas. Axial variations in the electric or magnetic confinement fields cause the particle trapping, and enable the E×B drift trapped-particle modes. Collisional diffusion across the trapping separatrix causes the modes to damp, and causes bulk radial transport when the confinement fields also have θ asymmetries. The measured asymmetry-induced transport rates are directly proportional to the measured mode damping rates, with simple scalings for all other plasma parameters. Significant transport is observed for even weak trapping fields (δB/B∼10−3), possibly explaining the “anomalous” background transport observed so ubiquitously in single species plasmas.

Diagnosing the velocity-space separatrix of trapped particle modes

Trapped particle modes in pure electron plasmas are similar to modes in neutral plasmas and exhibit damping due to velocity diffusion across the separatrix between trapped and untrapped particles, as commonly occurs in neutral plasmas. Applied rf voltages cause resonant perturbation of particle velocities near the separatrix, giving a greatly enhanced mode damping. This diagnostic technique can determine the velocity-space separatrices for either electrostatic or magnetic trapping, or determine the particle distribution function along the separatrix.

Trapped-particle modes and asymmetry-induced transport in single-species plasmas.

A new asymmetry-induced transport mechanism in pure electron plasmas is shown to be proportional to the damping rate of the corresponding trapped-particle mode, with simple scalings for all other parameters. This transport occurs when axial particle trapping exists due to variations in the electric or magnetic confinement fields. This new transport is strong for even weak unintentional trapping (deltaB/B approximately 10(-3)), and may be prevalent in transport experiments with magnetic or electrostatic theta asymmetries.