Trapped Ion Modes Research Articles

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.

Staircase formation by resonant and non-resonant transport of potential vorticity

The E × B staircase is a quasi-periodic pattern of pressure profile corrugations. In this work, we present a new mechanism for E × B staircase formation that involves resonant transport versus non-resonant transport. We start from a potential vorticity evolution system and use quasi-linear theory, a model dispersion relation, and a bi-Lorentzian spectrum approximation, to construct the relation between the fluxes and the profiles. With these fluxes, we close the profile evolution equations and the extended turbulence intensity evolution equation, which together constitute a turbulence-profile evolution system. In this system, the Doppler effect from the E × B mean flow can cause resonance between trapped ion precession motion and the trapped ion mode, which drives a resonant transport contribution to the fluxes. The profiles will be flattened where the resonant transport is switched on. In contrast, for the regions of non-resonant transport, profiles are steeper. A quasi-periodic pattern of profile corrugation (the E × B staircase) spontaneously emerges in this system, which is the two states mentioned above, arranged as alternating layers in space. The feedback processes during the staircase pattern formation are identified. An estimate of the critical value of the boundary heat flux is obtained, above which the staircase formation will be triggered. An estimate scaling of the step size in the staircase pattern is obtained. The resonant turbulent transport is also a mechanism for collisionless saturation of zonal flow. This work is related to internal transport barrier formation and suggests some new scenarios, such as an enhanced confined L mode.

The model of particles modes. I. A paradigm for phase synchronization in tokamak turbulence

Superthermal energetic particles may alter the kinetic and resonant nature of zonal flows by leading to new types of instabilities. Here, we study the effects induced by superthermal energetic ions on trapped-ion modes (TIM) by using a reduced Hamiltonian gyrokinetic model, where both fast scales (cyclotron and bounce or transit motions) are gyro-averaged. In particular, we analyze the enhancement of resonant processes induced by energetic ions associated with nonlinear phase synchronization, in an extended version of the TIM model including circulating ions. Once an energetic particle mode is driven unstable, a rich nonlinear dynamics is observed, which encompasses a frequency chirping associated with a synchronization process driven by TIM and a transition scenario. An equivalence with the classic Kuramoto model—the paradigm describing the synchronization of a system of coupled oscillators—explains much of this phenomenology.

Machine learning for prediction with missing dynamics

This article presents a general framework for recovering missing dynamical systems using available data and machine learning techniques. The proposed framework reformulates the prediction problem as a supervised learning problem to approximate a map that takes the memories of the resolved and identifiable unresolved variables to the missing components in the resolved dynamics. We demonstrate the effectiveness of the proposed framework with a strong convergence error bound of the resolved variables up to finite time and numerical tests on prototypical models in various scientific domains. These include the 57-mode barotropic stress models with multiscale interactions that mimic the blocked and unblocked patterns observed in the atmosphere, the nonlinear Schrödinger equation which found many applications in physics such as optics and Bose-Einstein-Condense, the Kuramoto-Sivashinsky equation which spatiotemporal chaotic pattern formation models trapped-ion modes in plasma and phase dynamics in reaction-diffusion systems. While many machine learning techniques can be used to validate the proposed framework, we found that recurrent neural networks outperform kernel regression methods in terms of recovering the trajectory of the resolved components and the equilibrium one-point and two-point statistics. This superb performance suggests that a recurrent neural network is an effective tool for recovering the missing dynamics that involves approximation of high-dimensional functions.

Impurity pinch generated by trapped particle driven turbulence

Impurity turbulent transport in tokamak plasmas is investigated in the case of trapped ion mode (TIM) and trapped electron mode (TEM) using a non-linear bounce-averaged gyrokinetic model. In particular, we focus on the thermo-diffusive pinch and the curvature-driven pinch observed in non-linear simulations. Thermodiffusion is analyzed independently of curvature pinch by disabling the latter artificially. A parameter scan shows that the direction of thermodiffusion depends mainly on two factors. First, it depends on the sign of the turbulence phase velocity, and the resulting transport is inward in the case of TEM whereas it is outward in the case of TIM. Second, it depends on the sign of the impurity temperature gradient. Moreover, the magnitude of the thermodiffusion tends to decrease with respect to the impurity charge Z, while it is proportional to the intensity of background turbulence. Lastly, the direction of the curvature pinch is investigated as a function of the magnetic shear. The results of non-linear simulations are compared with quasi-linear theory. The Kubo number is calculated for our non-linear simulations to justify this comparison. Quasi-linear theory is found to be in qualitative agreement with our gyrokinetic numerical observations.

Test particle dynamics in low-frequency tokamak turbulence

We study the evolution of one million test particles in a turbulent plasma simulation, using the gyrokinetic code Trapped Element REduction in Semi-Lagrangian Approach (TERESA), as a method to get insights into the type of transport governing the plasma. TERESA (Trapped Element REduction in Semi-Lagrangian Approach) is a collisionless global 4D code which treats the trapped particles kinetically, while the passing particles are considered adiabatic. The Vlasov-Poisson system of equations is averaged over the cyclotron and the trapped particle's bounce motion, and thus, the model focuses on slow phenomena of the order of the toroidal precession motion of the banana orbits. We initialize the test particles, which are de facto “test banana-centers,” at a time of the simulation when the plasma is turbulent. We impose an initial temperature and density gradients, and only the Trapped Ion Mode (TIM) instability can develop in this system. We then calculate the Mean Squared Displacement of the test particles as a function of time in order to obtain a random walk diffusion coefficient. We observe that the radial diffusion of the test particles depends on their toroidal precession kinetic energy (E), in such a way that the transport of particles is dominated by a strong, relatively narrow peak at the resonant energies. A radial particle diffusion flux is then calculated and compared to the total radial particle flux accounting for all the transport processes such as diffusion and advection which is obtained directly from the TERESA code. We can thus compare the diffusive contribution to the particle flux against the nondiffusive contributions. The results show that the total flux is essentially diffusive which is consistent with our simulation setup aiming for “global turbulence.” Both fluxes present a peak around a resonance energy ER≈1.74Ti between the TIM and the particles. Both thermal and high-energy particles do not contribute significantly to radial transport.

Diffusive impurity transport driven by trapped particle turbulence in tokamak plasmas

The diffusive impurity transport as a function of charge and mass numbers is investigated in ion driven or electron driven turbulence, in the limit of zero impurity temperature gradient. It is found that the impurity transport decreases slightly with increasing mass number and depends much more strongly on the charge number. Moreover, this transport depends on the nature of the instability that drives turbulence. The impurity flux due to Trapped Electron Mode (TEM) turbulence increases with the charge number Z. In contrast, it is found to decrease with Z when the Trapped Ion Mode (TIM) dominated. In order to explain these observations, the quasilinear flux is derived and is compared with the results obtained from the nonlinear simulations. Quasilinear theory qualitatively reproduces the gyrokinetic numerical observations.

Radial density and heat fluxes description in the velocity space: Nonlinear simulations and quasi-linear calculations

In the context of temperature gradient-driven, collisionless trapped-ion modes in magnetic confinement fusion, we perform and analyse numerical simulations to explore the turbulent transport of density and heat, with a focus on the velocity dimension (without compromising the details in the real space). We adopt the bounce-averaged gyrokinetic code TERESA, which focuses on trapped particles dynamics and allows one to study low frequency phenomena at a reduced computational cost. We focus on a time in the simulation where the trapped-ion modes have just saturated in amplitude. We present the structure in velocity space of the fluxes. Both density and heat fluxes present a narrow (temperature-normalized energy width ΔE/T ≈ 0.15) resonance peak with an amplitude high enough for resonant particles to contribute for 90% of the heat flux. We then compare these results obtained from a nonlinear simulation to the prediction from the quasi-linear theory and we find a qualitative agreement throughout the whole energy dimension: from thermal particles to high-energy particles. Quasi-linear theory over-predicts the fluxes by about 15%; however, this reasonable agreement is the result of a compensation between two larger errors of about 50%, both at the resonant energy and at the thermal energy.

Logarithmically discretized model of bounce averaged gyrokinetics and its implications on tokamak turbulence

A logarithmically discretized model, which consists of writing the system in log polar coordinates in wave-number domain and reducing the nonlinear interactions to a sum over neighboring scales that satisfy the triad conditions, is proposed for bounce averaged gyrokinetics, where the energy dependence is kept over a semi-regular grid that allows quadrature calculations in order to guarantee quasi-neutrality. The resulting model is a cheaper implementation of nonlinear multi-scale physics involving trapped electron modes, trapped ion modes, and zonal flows, which can handle anisotropy. The resulting wave-number spectrum is anisotropic at large scales, where the energy injection is clearly anisotropic, but is isotropised rapidly, leading generally towards an isotropic k−4 spectrum for spectral potential energy density for fully kinetic system and a k−5 spectrum for the system with one adiabatic species. Zonal flow damping, which is necessary for reaching a steady state in this model, plays an important role along with electron adiabaticity. Interesting dynamics akin to predator-prey evolution is observed among zonal flows and similarly large scale but radially elongated structures.

Impurity density gradient influence on trapped particle modes

The effect of the presence of an impurity species on the trapped particle turbulence is studied using the gyro-bounce kinetic code TERESA, which allows the study of Trapped Electron Modes and Trapped Ion Modes. The impurity species is treated self-consistently and its influence on the nature of the turbulence, ion driven or electron driven, is investigated. It is found that the presence of heavy impurities with a flat density profile tends to stabilize the both electron and ion modes, whereas a peaked or hollow impurity density profile can change the turbulence from an electron driven turbulence to an ion driven turbulence. The effect of the turbulence regime on impurity transport is studied.

Stimulated zonal flow generation in the case of TEM and TIM microturbulence

In this paper, we show that in some parameter range in gyrokinetic simulations, it is possible to apply a control method to stimulate the appearance of zonal flows while minimizing the duration of the control process and the impact on plasma parameters. For this purpose, a gyrokinetic code considering only trapped particles is used. The starting point of our work is a situation where zonal flows transiently appear after the nonlinear phase of saturation of trapped electron modes or trapped ion modes' micro-instabilities. These are observed to be strongly reduced in a later phase, permitting streamers to govern the plasma behavior in the steady-state. By intervening during this latter state (after this transient growth and decay of zonal flow), i.e., by increasing the ion/electron temperature ratio for a short time, it is found to be possible to bifurcate to a new steady-state, in which zonal flows are strongly present and are maintained indefinitely, thereby allowing a significant reduction in radial heat fluxes.

Effects of transition particles on plasma instabilities in a quasi-isodynamic stellarator

We consider the impact of transition particles on the Alfvén waves and electrostatic trapped ion modes in a stellarator with closed contours of the second adiabatic invariant, which do not perfectly coincide with magnetic surfaces (a so-called quasi-isodynamic (QI) stellarator). Due to orbit transitions, the damping rate of Alfvén eigenmodes on localized electrons becomes independent of the collision frequency in the broad range of low-collision rates. Dissipative trapped ion modes (Kadomtsev–Pogutse (KP) modes) induced by the ion temperature gradient (ITG) are strongly stabilized by collisionless orbit transitions.

Self-generated zonal flows in the plasma turbulence driven by trapped-ion and trapped-electron instabilities

This paper presents a study of zonal flows generated by trapped-electron mode and trapped-ion mode micro turbulence as a function of two plasma parameters—banana width and electron temperature. For this purpose, a gyrokinetic code considering only trapped particles is used. First, an analytical equation giving the predicted level of zonal flows is derived from the quasi-neutrality equation of our model, as a function of the density fluctuation levels and the banana widths. Then, the influence of the banana width on the number of zonal flows occurring in the system is studied using the gyrokinetic code. Finally, the impact of the temperature ratio Te/Ti on the reduction of zonal flows is shown and a close link is highlighted between reduction and different gyro-and-bounce-average ion and electron density fluctuation levels. This reduction is found to be due to the amplitudes of gyro-and-bounce-average density perturbations ne and ni gradually becoming closer, which is in agreement with the analytical results given by the quasi-neutrality equation.

Shear-flow trapped-ion-mode interaction revisited. II. Intermittent transport associated with low-frequency zonal flow dynamics

We address the mechanisms underlying low-frequency zonal flow generation in turbulent system and the associated intermittent regime of ion-temperature-gradient (ITG) turbulence. This model is in connection with the recent observation of quasi periodic zonal flow oscillation at a frequency close to 2 kHz, at the low-high transition, observed in the ASDEX Upgrade [Conway et al., Phys. Rev. Lett. 106, 065001 (2011)] and EAST tokamak [Xu et al., Phys. Rev. Lett 107, 125001 (2011)]. Turbulent bursts caused by the coupling of Kelvin-Helmholtz (KH) driven shear flows with trapped ion modes (TIMs) were investigated by means of reduced gyrokinetic simulations. It was found that ITG turbulence can be regulated by low-frequency meso-scale zonal flows driven by resonant collisionless trapped ion modes (CTIMs), through parametric-type scattering, a process in competition with the usual KH instability.

Shear-flow trapped-ion-mode interaction revisited. I. Influence of low-frequency zonal flow on ion-temperature-gradient driven turbulence

Collisionless trapped ion modes (CTIMs) turbulence exhibits a rich variety of zonal flow physics. The coupling of CTIMs with shear flow driven by the Kelvin-Helmholtz (KH) instability has been investigated. The work explores the parametric excitation of zonal flow modified by wave-particle interactions leading to a new type of resonant low-frequency zonal flow. The KH-CTIM interaction on zonal flow growth and its feedback on turbulence is investigated using semi-Lagrangian gyrokinetic Vlasov simulations based on a Hamiltonian reduction technique, where both fast scales (cyclotron plus bounce motions) are gyro-averaged.

Shear flow instabilities induced by trapped ion modes in collisionless temperature gradient turbulence

One important issue in turbulence self-organization is the interplay between the Kelvin–Helmholtz (KH) instability and streamers and/or zonal flows. This question has been debated for a long time. The effects of the KH instability and its position in the sequence of events between streamers, turbulence, and zonal flows have been investigated with a reduced gyro-bounce averaged kinetic code devoted to study the primary ion temperature gradient (ITG) instability linked to trapped ion modes (TIM). In toroidal geometry, the specific dynamics of TIM linked to trapped particles becomes important when the frequency of ITG modes falls below the ion bounce frequency, allowing one to average on both the cyclotron and bounce motion fast time scales. This reduction of the number of degrees of freedom leads to a strong reduction of computer resources (memory and computation time). Bounce-averaged gyrokinetic code can be considered as a toy model able to describe basic structures of turbulent transport in tokamak devices. In particular, by means of this code, we have observed that the energy injected in the system by the TIM instability is exchanged between streamers and zonal flows by means of KH vortices that grow along these structures in the nonlinear phase. The energy transfer occurs throughout the relaxation phase of the streamer growth leading to a modification of the KH modes and to the generation of the zonal flows.

Bicoherence analysis of streamer dynamics induced by trapped ion modes

High order spectral analyses have recently attracted a great deal of attention in the context of experimental studies for magnetic fusion. Among these techniques, bicoherence analysis plays an important role as it allows distinguishing between spontaneously excited waves and waves that arise from a coupling between different modes linked to specific physical mechanisms. Here, we describe and apply bicoherence analysis to kinetic simulations performed with reduced “bounce-averaged gyrokinetic” code, in order to investigate the nonlinear dynamics of trapped ion modes. This analysis shows a nonlinear wave coupling process linked to the formation of convective cells, thus furthers our understanding of the nonlinear energy transfer between turbulence structures in tokamaks.

Staircase temperature profiles and plasma transport self-organisation in a minimum kinetic model of turbulence based on the trapped ion mode instability

A minimum model of plasma turbulence in a kinetic framework is presented. It is based on trapped ion turbulence, gyro and bounce averaged, and implemented in the versatile and efficient code TERESA. Zonal flow - streamer interplay are readily shown to be key players that govern the confinement properties of the model. The parameter space of the model is explored with brute force numerics. A generic result is either a streamer dominated pattern with large transport, or a staircase temperature profile with very marked corrugations and quenched transport. A case with off-axis heating is found to exhibit quasiperiodic relaxation events relevant to investigate dynamical turbulence self-organisation.

Global Gyrokinetic Stability of Temperature-Gradient-Driven Trapped Ion Modes with Magnetic Shear

Global gyrokinetic Vlasov simulations for trapped ion modes indicate that the ion temperature gradient (ITG) instability saturates via nonlinear toroidal coupling. Trapped Ion modes were studied by solving a Vlasov equation averaged over the cyclotron and bounce motion of trapped ions. The distribution function, for trapped ions, is then calculated in a two-dimensional phase space, parametrized by two adiabatic invariants, the longitudinal action and the magnetic moment, in presence of a magnetic shear. Our model can be viewed as a kinetic version of the standard one-field Hasegawa-Mima-type and a broad frequency spectrum can originate from model for trapped-ion driven turbulence. Some interesting new features, which appear in the nonlinear regime of the instability, are then discussed.

Streamer-induced transport in the presence of trapped ion modes in tokamak plasmas

Global gyrokinetic Vlasov simulations for trapped ion modes are performed by solving a Vlasov equation averaged over the cyclotron and bounce motions of trapped ions. The distribution function, for trapped ions, is then calculated in a two-dimensional phase space, parametrized by the longitudinal action (energy) and the magnetic moment in presence of magnetic shear. The physical mechanism of the saturation processes between streamerlike structures and zonal flows in relation to the suppression of turbulent transport is discussed. The magnetic shear is identified to play a key role in the dominant streamer-induced transport regime, which exhibits a Bohm-like scaling. The interaction of streamerlike structures with plasma turbulence is shown to produce the inverse cascade that condenses onto long-wavelength trapped ion structures, on the basis on wave triad interactions.