Gyrokinetic Theory Research Articles

Effects of plasma nonuniformity on zero frequency zonal structure generation by drift Alfvén wave instabilities in toroidal plasmas

Abstract The effects of plasma nonuniformity on zero frequency zonal structure (ZFZS) excitation by drift Alfvén wave (DAW) instabilities in toroidal plasmas are investigated using nonlinear gyrokinetic theory. The governing equations describing nonlinear interactions among ZFZSs and DAWs are derived, with the contribution of DAW self-beating and radial modulation accounted for on the same footing, and the physics picture contributing to both channels clarified. The obtained equations are then used to derive the nonlinear dispersion relation, which is then applied to investigate ZFZS generation in several scenarios. In particular, it is found that the condition for zonal flow excitation by the kinetic ballooning mode (KBM) could be sensitive to plasma parameters and a more detailed investigation is needed to understand KBM nonlinear saturation, crucial for bulk plasma transport in future reactors.

Kinetic Theory of Drift‐Mirror Mode

AbstractWe present a nonlocal gyrokinetic theory for the drift‐mirror mode in high‐ anisotropic plasmas. Here, represents the ratio of plasma pressure to magnetic pressure. The equilibrium distribution is established self‐consistently via guiding‐center Hamiltonian theory. To keep the nonuniformity and finite Larmor radius (FLR) effects on an equal footing, we derive the three‐field nonlocal eigenmode equations in Fourier space under the assumption of weak nonuniformity. It is found that the drift‐mirror mode is essentially a dissipative instability confined within the potential well due to the FLR effect. Numerical analyses demonstrate that the drift‐mirror mode originates from the coupling between shear Alfvén and mirror branch, where the ion‐sound wave component is negligible. Furthermore, the threshold of this nonlocal drift‐mirror mode is significantly lower than that of the conventional drift‐mirror mode.

Validity of gyrokinetic theory in magnetized plasmas

Gyrokinetics, as a reduced kinetic theory derived from adiabaticity, provides a general framework for the long-term dynamics of magnetized plasmas. While its validity limits are stated in terms of formal expansion parameters, more quantitative test of such is not widely mentioned even if it existed. Here we show, by detailed analyses of the Hamiltonian map with a test particle model, that gyrokinetic theory rests on the inherent nature of particle dynamics as a boundary layer problem. For low-frequency fluctuations, we demonstrate the existence of a frequency-independent threshold in the normalized amplitude, below which gyrokinetics is generally applicable. However, this threshold becomes sensitive to wave parameters in the high-frequency regime, which raises concerns about the generality of high-frequency gyrokinetic theory. Further analyses indicate that constructing a reduced kinetic equation based on superadiabaticity is not feasible. These findings contribute to a deeper understanding of the basic physics behind gyrokinetic theory.

Turbulence Transition in Magnetically Confined Hydrogen and Deuterium Plasmas.

In this study, we discovered a turbulence transition in a large helical device. The turbulence level and turbulence-driven energy transport decrease to a specific transition density and increase above it. The ruling turbulences below and above the transition density were ion-temperature gradient (ITG) and resistive-interchange (RI) turbulences, consistent with the predictions of gyrokinetic theory and two-fluid MHD model, respectively. Isotope experiments on hydrogen (H) and deuterium (D)clarified the role of transitions. In the ITG regime, turbulence levels and energy transport were comparable in the H and D plasmas. In contrast, in the RI regime, they were clearly suppressed in the D plasma. The results provide crucial knowledge for understanding isotope effects and future optimization of stellarator and heliotron devices.

Nonlinear dynamics of the reversed shear Alfvén eigenmode in burning plasmas

In a tokamak fusion reactor operated at steady state, the equilibrium magnetic field is likely to have reversed shear in the core region, as the noninductive bootstrap current profile generally peaks off-axis. The reversed shear Alfvén eigenmode (RSAE) as a unique branch of the shear Alfvén wave in this equilibrium, can exist with a broad spectrum in wavenumber and frequency, and be resonantly driven unstable by energetic particles (EP). After briefly discussing the RSAE linear properties in burning plasma condition, we review several key topics of the nonlinear dynamics for the RSAE through both wave-EP resonance and wave-wave coupling channels, and illustrate their potentially important role in reactor-scale fusion plasmas. By means of simplified hybrid MHD-kinetic simulations, the RSAEs are shown to have typically broad phase space resonance structure with both circulating and trapped EP, as results of weak/vanishing magnetic shear and relatively low frequency. Through the route of wave-EP nonlinearity, the dominant saturation mechanism is mainly due to the transported resonant EP radially decoupling with the localized RSAE mode structure, and the resultant EP transport generally has a convective feature. The saturated RSAEs also undergo various nonlinear couplings with other collective oscillations. Two typical routes as parametric decay and modulational instability are studied using nonlinear gyrokinetic theory, and applied to the scenario of spontaneous excitation by a finite amplitude pump RSAE. Multiple RSAEs could naturally couple and induce the spectral energy cascade into a low frequency Alfvénic mode, which may effectively transfer the EP energy to fuel ions via collisionless Landau damping. Moreover, zero frequency zonal field structure could be spontaneously excited by modulation of the pump RSAE envelope, and may also lead to saturation of the pump RSAE by both scattering into stable domain and local distortion of the continuum structure.

Simulation of ion temperature gradient driven modes with 6D kinetic Vlasov code

With the increase in computational capabilities over the last few years, it becomes possible to simulate more and more complex and accurate physical models. Gyrokinetic theory has been introduced in the 1960s and 1970s in the need of describing a plasma with more accurate models than fluid equations but eliminating the complexity of the fast gyration about the magnetic field lines. Although results from current gyrokinetic computer simulations are in fair agreement with experimental results in core physics, crucial assumptions made in the derivation make it unreliable in regimes of higher fluctuations and stronger gradient, such as the tokamak edge. With our novel optimized and scalable semi-Lagrangian solver, we are able to simulate ion temperature gradient modes with the 6D kinetic model including the turbulent saturation. After thoroughly testing our simulation code against analytical computations and gyrokinetic simulations (with the gyrokinetic code GYRO), it has been possible to show first plasma properties that go beyond standard gyrokinetic simulations. This includes the explicit description of the complete perpendicular energy fluxes and the excitation of high-frequency waves (around the Larmor frequency) in the nonlinear saturation phase.

Drift wave soliton formation via beat-driven zonal flow and implication on plasma confinement

In this work, gyrokinetic theory of drift waves (DWs) self-regulation via the beat-driven zonal flow (ZF) is presented, and finite diamagnetic drift frequency due to plasma nonuniformity is shown to play a dominant role in the ZF beat generation. The obtained nonlinear DW equation is a nonlinear Schrödinger equation, in which the linear dispersiveness, linear growth, nonuniformity of diamagnetic drift frequency, and cubic nonlinearity induced by the feedback of beat-driven ZF to DWs are self-consistently included. The nonlinear DW equation is solved numerically in both uniform and nonuniform plasmas. It is shown that the DW envelope soliton may form due to the balance of linear dispersiveness and nonlinearity and lead to turbulence spreading to linearly stable region. It is further found that though the threshold on the DW amplitude for soliton formation is well within the relevant parameter regimes of realistic tokamak experiments, solitons cannot extend beyond the range bounded by the turning points of the wave packet when plasma nonuniformity is self-consistently accounted for.

Resonant decay of kinetic Alfvén waves and implication on spectral cascading

A general equation describing the resonant nonlinear mode-coupling among kinetic Alfvén waves (KAWs) is derived using nonlinear gyrokinetic theory, which can be applied to study the potentially strong spectral energy transfer of KAWs. As a first application, the parametric decay of a pump KAW into two sideband KAWs is studied, with particular emphasis on cascading in the perpendicular wavenumber. It is found that, for the “co-propagating” cases with all three KAWs propagating in the same direction along the equilibrium magnetic field line, it exhibits a dual cascading character in the perpendicular wavenumber space; while for the “counter-propagating” cases with one sideband propagating in the opposite direction with respect to the pump wave, it instead, can exhibit both dual and inverse cascading behaviors. Furthermore, the implications of these findings on shear Alfvén waves regarding instability, nonlinear saturation, and charged particle transport in fusion plasmas are also discussed.

The residual flow in well-optimized stellarators

The gyrokinetic theory of the residual flow, in the electrostatic limit, is revisited, with optimized stellarators in mind. We consider general initial conditions for the problem, and identify cases that lead to a non-zonal residual electrostatic potential, i.e. one having a significant component that varies within a flux surface. We investigate the behaviour of the ‘intermediate residual’ in stellarators, a measure of the flow that remains after geodesic acoustic modes have damped away, but before the action of the slower damping that is caused by unconfined particle orbits. The case of a quasi-isodynamic stellarator is identified as having a particularly large such residual, owing to the small orbit width achieved by optimization.

A performance portable implementation of the semi-Lagrangian algorithm in six dimensions

This paper describes our approach to developing a simulation software application for the fully kinetic 6D-Vlasov equation, which will be used to explore physics beyond the reduced gyrokinetic model. Simulating the fully kinetic Vlasov equation requires efficient utilization of compute and storage capabilities due to the high dimensionality of the problem. In addition, the implementation needs to be extensible regarding the physical model and flexible regarding the hardware for production runs. We start on the algorithmic background to simulate the 6-D Vlasov equation using a semi-Lagrangian algorithm. The performance portable software stack, which enables production runs on pure CPU as well as AMD or Nvidia GPU accelerated nodes, is presented. The extensibility of our implementation is guaranteed through the described software architecture of the main kernel, which achieves a memory bandwidth of almost 500 GB/s on a V100 Nvidia GPU and around 100 GB/s on an Intel Xeon Gold CPU using a single code base. We provide performance data on multiple node-level architectures discussing utilized and further available hardware capabilities. Finally, the network communication bottleneck of 6-D grid-based algorithms is quantified. A verification of physics beyond gyrokinetic theory, for the example of ion Bernstein waves, concludes the work.

Comparison of momentum transport in matched hydrogen and deuterium H-mode plasmas in ASDEX Upgrade

Advanced momentum transport analysis is used to study matched hydrogen (H) and deuterium (D) plasmas in the core of ASDEX Upgrade. The aim is to validate gyrokinetic theory and assess a possible isotope dependence. The methodology extracts momentum diffusion, convection, and intrinsic torque as a function of time from experiments employing neutral beam injection (NBI) modulation. H and D plasma scenarios with comparable ion heat fluxes, NBI torque, electron densities, and several dimensionless parameters were designed to highlight any mass dependency. Linear gyrokinetic simulations predict that, for similar background gradients, the Prandtl and pinch numbers should be similar for H and D. This was confirmed by the experimental momentum transport analyses. The assessed intrinsic torques were found to be similar between H and D, co-current directed and located near the outermost region of the plasma core. The strength of the intrinsic torque is correlated with the amplitude of the plasma pressure gradient in the pedestal. Finally, a robust error analysis demonstrates the uniqueness of the parameters obtained together with their uncertainties. Neglecting the intrinsic torque, or its time dependence, systematically distorts the assessed momentum diffusion and convection. This is the first method to separate all three transport mechanisms from experimental data by retaining their time dependencies, that is found to match, quantitatively, the gyrokinetic predictions for Prandtl and pinch numbers, within experimental uncertainties.

Numerical study of impurity effects on ion temperature gradient modes in tokamak edge plasmas based on the Euler matrix eigenvalue method

Low-Z impurity injection is frequently used for divertor detachment operations in current tokamaks; however, the impurity effects on the main plasma are yet to be fully understood. In this paper, the impurity effects on the ion temperature gradient (ITG) modes in tokamak edge plasmas are investigated based on the Euler matrix eigenvalue method. The eigen-equations with multiple ion species are established from the fundamental gyrokinetic theory, in which each ion species is treated equally. A novel and efficient gyro-kinetic code is developed for this numerical study, and the code’s availability to examine quasi-linear ITG modes is demonstrated by its comparison with existing results. At the pedestal top parameters in Experimental Advanced Superconducting Tokamak high-β p H-mode plasmas, the ITG mode behavior is investigated in pure deuterium plasmas and with impurities. Impurities can induce destabilizing or stabilizing effects on ITG modes, which are determined by the impurity density scale length. The inwardly peaked impurity density profile tends to reduce the ITG growth rate. The effect strength also increases with the impurity charge concentration. The effects of impurity species, including boron, carbon, neon and argon, are also evaluated. Numerical results show that the strength of destabilizing or stabilizing effect inverses with impurity ion charge at the same effective charge.

MAS: a versatile Landau-fluid eigenvalue code for plasma stability analysis in general geometry

We have developed a new global eigenvalue code, multiscale analysis for plasma stabilities (MAS), for studying plasma problems with wave toroidal mode number (n) and frequency (ω) in a broad range of interest in general tokamak geometry, based on a five-field Landau-fluid description of thermal plasmas. Beyond keeping the necessary plasma fluid response, we further retain the important kinetic effects including diamagnetic drift, ion finite Larmor radius, finite parallel electric field (), ion and electron Landau resonances in a self-consistent and non-perturbative manner without sacrificing the attractive efficiency in computation. The physical capabilities of the code are evaluated and examined in terms of both theory and simulation. In theory, the comprehensive Landau-fluid model implemented in MAS can be reduced to the well-known ideal magnetohydrodynamic (MHD) model, electrostatic ion-fluid model, and drift-kinetic model in various limits, which clearly delineates the physics validity regime. In simulation, MAS has been well benchmarked with theory and other gyrokinetic and kinetic-MHD hybrid codes in a manner of adopting the unified physical and numerical framework, which covers the kinetic Alfvén wave, ion sound wave, low-n kink, high-n ion temperature gradient mode and kinetic ballooning mode. Moreover, MAS is successfully applied to model the Alfvén eigenmode (AE) activities in DIII-D discharge #159243, which faithfully captures the frequency sweeping of reversed shear AE, the tunneling damping of toroidal AE, as well as the polarization characteristics of kinetic beta-induced AE and beta-induced Alfvén-acoustic eigenmode being consistent with former gyrokinetic theory and simulation. With respect to the key progress contributed to the community, MAS has the advantage of combining rich physics ingredients, realistic global geometry and high computation efficiency together for plasma stability analysis in the linear regime.

Self-generated vortex flows in a tokamak magnetic island with a background flow

We present a gyrokinetic theory of self-generated E × B vortex flows in a magnetic island in a collisionless tokamak plasma with a background vortex flow. We find that the long-term evolution of the self-generated vortex flows can be classified into two regimes by the background vortex flow potential Φ, with an asymptotic criterion given by , where T is temperature, ε is the inverse aspect ratio and r is the radial coordinate. We find that the background vortex flow above the criterion significantly weakens the toroidal precession-induced long-term damping and structure change of the self-generated vortex flows. That is, the finite background vortex flow is beneficial to maintain the self-generated vortex flows, favorable to an internal transport barrier formation. Our result indicates that the island boundary region is a prominent location for triggering the transition to an enhanced confinement state of the magnetic island.

Quantifying the role of higher order neoclassical corrections to gyrokinetics in tokamak plasmas

We implement the higher order gyrokinetic theory developed in Dudkovskaia et al (2023 Plasma Phys. Control. Fusion 65 045010), reduced to the limit of , where B 0 is the tokamak equilibrium magnetic field, and B ϑ is its poloidal component, in the local gyrokinetic turbulence code, GS2. The principal motivation for this extension is to quantify the importance of neoclassical flows in electromagnetic gyrokinetics, with a particular interest in sharp pressure gradient regions where the bootstrap current becomes dominant. To incorporate neoclassical equilibrium physics, GS2 is coupled to NEO, a multi-species drift kinetic solver. It is found that the regions where microinstabilities are most likely to be influenced by neoclassical equilibrium effects are in a pedestal plasma and a spherical tokamak core plasma.

A drift-kinetic perturbed Lagrangian for low-frequency nonideal MHD applications

We find that the perturbed Lagrangian derived from the drift-kinetic equation in [Porcelli F et al 1994 Phys. Plasmas 1 470] is inconsistent with the ordering for the low-frequency large-scale magnetohydrodynamic (MHD). Here, we rederive the expression for the perturbed Lagrangian within the framework of nonideal MHD using the ordering system for the low-frequency large-scale MHD in a low-beta plasma. The obtained perturbed Lagrangian is consistent with Chen’s gyrokinetic theory [Chen L and Zonca F 2016 Rev. Mod. Phys. 88 015008], where the terms related to the field curvature and gradient are small quantities of higher order and thus negligible. As the perturbed Lagrangian has been widely used in the literature to calculate the plasma nonadiabatic response in low-frequency MHD applications, this finding may have a significant impact on the understanding of the kinetic driving and dissipative mechanisms of MHD instabilities and the plasma response to electromagnetic perturbations in fusion plasmas.

Stabilization of trapped electron mode through effective diffusion in electron temperature gradient turbulence

Multi-scale gyrokinetic theory and simulations of a toroidal magnetized plasma have revealed the existence of cross-scale interactions of the trapped electron mode (TEM) and the electron temperature gradient (ETG) turbulence. Reduction of the TEM instability growth rate in the ETG turbulence is clearly identified, and is well represented in the form of effective diffusion. A theoretical model based on the stochastic forcing by the ETG turbulence well describes the turbulent diffusion coefficient observed in multi-scale turbulence simulations.

Nonlinear second order electromagnetic gyrokinetic theory for a tokamak plasma

The steep plasma pressure gradient that forms at the edge of the high confinement, H-mode regime of tokamak operation provides free energy to drive electromagnetic micro-instabilities that are widely believed to influence the transport processes in this so-called pedestal region. This high pressure gradient also provides a high current density (bootstrap current), known to influence ballooning mode stability and to be important for driving kink modes in the ideal magneto-hydrodynamic plasma model (so-called peeling-ballooning modes). Furthermore, efficient, steady state future tokamak power plants must operate with a large bootstrap current in the core and especially concerning spherical tokamaks, confinement will be influenced by electromagnetic turbulence. To accommodate these important situations, conventional electromagnetic gyrokinetic theory is extended to incorporate neoclassical effects in the equilibrium drives, allowing (B 0 is the confining magnetic field, and B ϑ is its poloidal component). This provides a global gyrokinetic model that self-consistently captures the consequences of large bootstrap current fractions on the equilibrium distribution functions.

Relationship between drift kinetics, gyrokinetics and magnetohydrodynamics in the long-wavelength limit

Gyrokinetic theories, even ‘global’ models, typically rely on a separation in scale (in perpendicular wavelength) between the fluctuations and the system scale. In such models direct simulation of system-scale dynamics like magnetohydrodynamic (MHD) motion is formally not consistent. Drift-kinetic theory, on the other hand, may be directly applied to model system-scale MHD-ordered behaviour. I review the long-wavelength limit of standard gyrokinetics and drift kinetics to present the relationships between these theories in an elementary fashion. This provides a pathway to global gyrokinetic modelling, resulting in an approach that is structurally similar to kinetic MHD, and I present dynamical equations for solving global field evolution in this framework. Departures from certain earlier global gyrokinetic theories include the appearance of magnetosonic (fast) modes, and the cross-coupling of the parallel and perpendicular currents with perpendicular and parallel magnetic field components. A periodic two-dimensional testcase is outlined as a benchmarking and implementation target, to help clarify practical aspects of these theories, with minimal complexity in terms of boundary conditions, and a proof-of-principle implementation of a field-solver is exhibited. To motivate this work, I first illustrate certain limitations of existing global gyrokinetic frameworks and directly identify how scale separation approximations lead to certain ‘missing’ system-scale field terms in global gyrokinetics, largely as a result of simplifications associated with the field representation in terms of $A_{\|}$ and $B_{\|}$ . As a result, the currents in the gyrokinetic Ampère's law resulting from a gyrokinetic equilibrium distribution do not match the currents implied by Ampère's law in a force-balance MHD equilibrium. I present a simple choice of equilibrium distribution function whose drift-kinetic currents are consistent with MHD currents; a specific Grad–Shafranov equilibrium is used to illustrate the size of the components of the parallel currents.

Effect of temperature anisotropy on the dynamics of geodesic acoustic modes

In this work, we revisit the linear gyro-kinetic theory of geodesic acoustic modes (GAMs) and derive a general dispersion relation for an arbitrary equilibrium distribution function of ions. A bi-Maxwellian distribution of ions is then used to study the effects of ion temperature anisotropy on GAM frequency and growth rate. We find that ion temperature anisotropy yields sensible modifications to both the GAM frequency and growth rate as both tend to increase with anisotropy and these results are strongly affected by the electron to ion temperature ratio.