Drift Kinetic Equation Research Articles

Neoclassical transport computations in non-isothermal tokamak plasmas

Neoclassical transport in a non-isothermal plasma in which each plasma species has different equilibrium temperature is investigated by solving the drift kinetic equation with a Fokker–Planck (FP) collision operator in a circular tokamak model. Since it is known that a linearized FP operator does not have a self-adjoint property in a non-isothermal plasma, approximate models are developed for comparison to intend to have the self-adjoint property in the non-isothermal case. To achieve this, we set a common temperature that the system should reach after a long time, and the individual temperature of each particle species is expressed by a parameter to measure a shift of the individual temperature from the common one. Then, both the Vlasov part and the collision term of the kinetic equation are expanded around the common temperature, taking the temperature shift parameter up to the first order. It is found that the lowest order collision term of expansion preserves the self-adjointness while the first-order, nonlinear FP term does not. A large difference of the ion heat neoclassical transport is found in comparison between the developed models with and without the self-adjointness and the original FP collision term in the non-isothermal plasma, especially in a strong temperature equilibration regime, showing that a contribution of the collision term without the self-adjointness seems to be significant. Furthermore, when an impurity species is included, the result is complicated where the usual enhancement in the main ion particle transport coefficient, due to the impurity effect, is rather suppressed with the increase in the ion heat transport coefficients by the non-isothermal effects.

1D drift-kinetic numerical model based on semi-implicit particle-in-cell method

The paper presents a new one-dimensional drift-kinetic electrostatic model based on the particle-in-cell method and capable of simulating the processes of plasma heating and confinement in mirror traps. The most of particle and energy losses in these traps occur along the magnetic field lines. The key role in limiting these losses is played by the ambipolar electric potential which creates a potential barrier for electrons and significantly reduces the heat flux that, without this barrier, would go to wall due to the classical electron thermal conductivity. However, modeling the formation of such a potential on real spatial and temporal scales of experiments is a challenging problem, since it requires a detailed description of not only ion, but also electron kinetics. In this work, we propose to solve the problem of taking into account electron kinetic effects on the time scale of plasma confinement in a mirror trap using the particle-in-cell method adapted to the approximate drift-kinetic equations of plasma motion. Unlike other electrostatic particle-in-cell models, which use fully implicit schemes to solve the nonlinear system of Vlasov-Poisson and Vlasov-Ampere equations, we propose a semi-implicit approach. By analogy with the Energy Conserving Semi-Implicit Method (ECSIM), it allows for precise conservation of energy and reduces the procedure for finding the electric field to inverting a tridiagonal matrix without multiple nonlinear iterations. Such a model will be useful for simulating not only collisional losses of hot plasma in fusion experiments, but also for studying the features of creating cold starting plasma in mirror traps using plasma or electron guns.

Wave–particle interactions in tokamaks

Transport consequences of the wave–particle interactions in the quasilinear plateau (QP) regime are presented. Eulerian approach is adopted to solve the drift kinetic equation that includes the physics of the nonlinear trapping (NT) and QP regimes. The localization of the perturbed distribution simplifies the test particle collision operator. It is shown that a mirror force like term responsible for the flattening of the distribution in the NT regime is subdominant in the QP regime, and controls the transition between these two regimes. Transport fluxes, flux-power relation, and nonlinear damping or growth rate are all calculated. There is no explicit collision frequency dependence in these quantities; however, the width of the resonance does. Formulas that join the asymptotic results of these two regimes to facilitate thermal and energetic particle transport, and nonlinear wave evolution of a single mode are presented.

Nonlinear dynamics of multiple Alfvén modes driven by trapped energetic ions in tokamaks: A triplet paradigm

Non-linear dynamics of multiple infernal Alfvén eigenmodes—a subset of global Alfvén eigenmodes in tokamak plasmas with extended low-shear central core [Marchenko et al., Phys. Plasmas 16, 092502 (2009)]—is studied. The analysis is carried out for a mode triplet with toroidal mode-numbers n=1, 2, 3. It was assumed that the n = 1 mode was linearly unstable due to precession resonance with trapped fast ions, whereas the other modes were linearly damped. The modes were coupled due to a non-linearity in a bounce-averaged drift kinetic equation for the distribution function of fast ions. Nonlinear equations for the mode amplitudes and phases are derived and solved numerically. It is found that the temporal evolution of the amplitudes and the phase (responsible for the frequency chirping) of the modes exhibit Hopf bifurcations to stable limit cycles. This can explain a synchronous cyclic destabilization of multiple modes in Alfvén avalanches (sudden growth of amplitudes of the mode cluster with different n and approximately equal frequency spacing) in NSTX and bursting modes in MAST—events, which resulted in enhanced loss of fast ions.

ALLIANCE: Spectral solver for kinetic plasma turbulence

The ALLIANCE (ALLIANCE - spectrAL soLver for kInetic plAsma turbuleNCE) code is developed to solve a new set of four-dimensional electromagnetic drift-kinetic equations in slab geometry [1]. The nonlinear equations are useful for the study of magnetized plasma systems at scales comparable to, or larger than the ion gyroradius. In particular, it is suited for the study the kinetic turbulent cascade in astrophysical plasma, while preserving finite Larmor radius effects at the fluid-kinetic transition. The equations solved are in spectral Fourier-Laguerre-Hermite form, a pseudo-spectral approach is used for the nonlinear terms, and the code is parallelised over multiple directions. After a presentation of the code, validation runs are shown, and benchmarks for serial and parallel computations are presented.

A quasi-linear model of particle, momentum and heat transport in a stochastic magnetic field

We present a quasi-linear treatment of the drift-kinetic equation in the presence of a stochastic magnetic field, which provides a self-contained description of particle, parallel momentum and heat transport. Explicit analytical expressions, which satisfy the Onsager reciprocal relations, are obtained by approximating the distribution function by a local shifted Maxwellian. This theory completes previous formulations (Harvey et al., Phys. Rev. Lett., vol. 47, 1981, p. 102) by including the momentum transport and by generalizing the derivation from the cylindrical tokamak configuration to an arbitrary cylindrical pinch. Application to the reversed field pinch provides satisfactory results.

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.

Influence of the beam-beam collisions on the energetic particle distribution in large helical device

The influence of Coulomb collisions between energetic particles in a neutral beam injection (NBI)-heated plasma is numerically investigated utilizing the improved collision operator for the orbit-following type of Monte Carlo codes. This nonlinear collision operator enables the collisional effect by non-Maxwellian plasmas including energetic particles to be taken into account. We have implemented it into GNET, one of the orbit-following Monte Carlo codes solving drift kinetic equation in the 5D phase-space, and performed simulations for tangentially injected NBI cases (beam energy: 180 keV), taking account of beam–beam Coulomb collisions. It is found that the beam–beam collisions cause energetic particle orbits to transition from passing to trapped and enhance the collisional transport, which results in the deterioration of the energetic particle confinement.

Neoclassical transport in strong gradient regions of large aspect ratio tokamaks

We present a new neoclassical transport model for large aspect ratio tokamaks where the gradient scale lengths are of the size of the ion poloidal gyroradius. Previous work on neoclassical transport across transport barriers assumed large density and potential gradients but a small temperature gradient, or neglected the gradient of the mean parallel flow. Using large aspect ratio and low collisionality expansions, we relax these restrictive assumptions. We define a new set of variables based on conserved quantities, which simplifies the drift kinetic equation whilst keeping strong gradients, and derive equations describing the transport of particles, parallel momentum and energy by ions in the banana regime. The poloidally varying parts of density and electric potential are included. Studying contributions from both passing and trapped particles, we show that the resulting transport is dominated by trapped particles. We find that a non-zero neoclassical particle flux requires parallel momentum input which could be provided through interaction with turbulence or impurities. We derive upper and lower bounds for the energy flux across a transport barrier in both temperature and density and present example profiles and fluxes.

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.

Moment-Fourier approach to ion parallel fluid closures and transport for a toroidally confined plasma

A general method of solving the drift kinetic equation is developed for an axisymmetric magnetic field. Expanding a distribution function in general moments, a set of ordinary differential equations is obtained. Successively expanding the moments and magnetic-field involved quantities in Fourier series, a set of linear algebraic equations is obtained. The set of full (Maxwellian and non-Maxwellian) moment equations is solved to express the first-order density, temperature, and flow velocity in terms of radial gradients of the zeroth-order pressure and temperature. Closure relations that connect parallel heat flux density and viscosity to the radial gradients and parallel gradients of temperature and flow velocity are also obtained by solving the non-Maxwellian moment equations. The closure relations combined with the linearized fluid equations reproduce the same solution obtained directly from the full moment equations. The method can be generalized to derive closures and transport for an electron-ion plasma and a multi-ion plasma in a general magnetic field.

Impact of magnetic ripple on neoclassical equilibrium in gyrokinetic simulations

The effect of magnetic field ripple on tokamak plasma without turbulence is studied numerically and augmented with a reduced analytical model that includes neoclassical processes in the presence of non-axisymmetric perturbation and stochastic transport. For this study, a magnetic field ripple perturbation has been implemented in the GYSELA gyrokinetic code. This implementation has been verified thanks to a test of toroidal angular momentum conservation. The GYSELA code was then successfully benchmarked against the NEO code, which solves the drift kinetic equation, and against the reduced model in the collisionality range [0.05–0.5] for several amplitudes of the magnetic ripple. An observation, shared by the model, the NEO code and GYSELA simulations is that the thermal drive of the mean poloidal velocity—measured by the coefficient—decreases sharply for large yet experimentally relevant magnetic ripple amplitudes, and may even change sign.

Neoclassical toroidal plasma viscosity in bounce-transit and drift resonance regimes in tokamaks

Neoclassical toroidal plasma viscosity in the bounce-transit and drift resonance regimes is calculated using a version of the drift kinetic equation that encompasses the physics of the nonlinear trapping and quasilinear plateau regimes in tokamaks. It is demonstrated that the mirror-force like term controls the transition between these two regimes. When the effective collision frequency is larger than the mirroring or the nonlinear bounce frequency, the quasilinear regime prevails; otherwise, the nonlinear trapping regime reigns. The demonstration is accomplished by using the Eulerian approach and is beyond the grasp of the method of the integration along the unperturbed orbit in solving the drift kinetic equation. The neoclassical toroidal plasma viscosity in the quasilinear plateau regime is calculated. Approximate analytic expressions for the neoclassical toroidal plasma viscosity that include the asymptotic limits of the nonlinear trapping and quasilinear regimes are presented to facilitate thermal and energetic alpha particle transport modeling in tokamaks.

Nonlinear saturation of reverse shear Alfvén eigenmodes induced by kinetic electrons

A strong nonlinear saturation mechanism of the reverse shear Alfvén eigenmode (RSAE) induced by thermal electrons is observed in gyrokinetic particle-in-cell simulation. This nonlinear effect occurs at moderate mode amplitude, , and is associated with the electron parallel streaming along the perturbed magnetic field lines, the term , in the electron drift-kinetic equation. In the case of an n = 4 RSAE, where n is the toroidal mode number, this magnetic fluttering nonlinearity leads to strong mode coupling and broadening of the saturation spectrum. The nonlinear components, which are highly damped through the interaction between thermal electrons and the parallel electric field, modifies the RSAE mode structure and suppresses energetic particle drive significantly. The split-weight scheme (Chen and Parker 2007 J. Comput. Phys. 220 839–55) is used to simulate kinetic electrons.

ASCOT5 simulations of neutral beam heating and current drive in the TJ-II stellarator

The TJ-II stellarator neutral-beam injection (NBI) system, vacuum vessel and magnetic configuration have been included in the orbit-following Monte Carlo code ASCOT5 to simulate neutral-beam heating and current drive for high-density NBI plasmas. Co- and counter-injection beams are simulated separately. A scan in both electron density and temperature is carried out within the range of values corresponding to realistic high-density NBI plasmas, for which a low level of fast-ion losses due to charge-exchange reactions is expected, since the version of ASCOT5 used in the paper does not include such processes. The rest of the kinetic profiles (ion temperature, radial electric field and effective charge) are kept fixed. The initial distribution of markers shows that the amount of available power in the plasma carried by the beam ions depends slightly on the electron temperature and on the injection direction (co/counter). The steady-state fast-ion distribution function is obtained and used to calculate the three-dimensional fast-ion density, the neutral-beam driven current and the amount of power deposited to the plasma in the two injection scenarios. These three quantities are higher in the counter-injected case due to a lower amount of promptly lost particles. The neutral-beam current drive (NBCD) has been calculated using the fast-ion beam current given by ASCOT5 and the electron return current, which is computed with the analytic solution of the drift kinetic equation for electrons in the presence of fast ions in the low-collisionality regime. Neither the calculated fast-ion density nor the NBCD are flux functions, in consistency with the fact that fast-ion drift surfaces and flux surfaces are generally not aligned.

Verification of a fully kinetic ion model for electromagnetic simulations of high-frequency waves in toroidal geometry

For the study of high-frequency electromagnetic waves in tokamaks, an electromagnetic simulation model, in which the ion dynamics is described by a six-dimensional Vlasov equation and the electron dynamics is described by a drift kinetic equation, is formulated and implemented in the global gyrokinetic toroidal code (GTC). Analytic dispersion relations are derived in reduced systems and compared with various theories to verify the model. Linear simulations of a generalized ion Bernstein wave and ion cyclotron emission are verified by comparing the GTC simulation results with analytic dispersion relation theory and magnetoacoustic cyclotron instability theory, respectively, in cylindrical geometry.

Description of global EGAM in the maximum of local frequency during current ramp-up discharges in DIII-D

Energetic-particle-induced geodesic acoustic modes, EGAMs (Fu, Phys. Rev. Let., vol. 101, 2008, pp. 185002), driven by neutral beam injection (NBI), have been observed in many DIII-D tokamak experiments (Nazikian et al., Phys. Rev. Lett., vol. 101, 2008, pp. 185001). This mechanism has been theoretically investigated in (Qiu et al., Plasma Phys. Control. Fusion, vol. 52, 2010, pp. 095003), using a sharp energetic particle distribution function, and in (Qu et al., Plasma Phys. Control. Fusion, vol. 59, 2017, pp. 055018), where the dispersion relation and eigenmode behaviour were obtained for the situation of early beam scenario, that is, for times smaller than the beam slowing down time. In this work, we extend these studies determining the eigenmode for beyond the slowing down time, in a scenario with reverse safety factor $q$ profile, where a small concentration of energetic ions can produce an off-axis maximum in the GAM dispersion relation. The characteristics of EGAM are analytically studied with the drift kinetic equation together with the MHD code NOVA. The toroidal energetic ion transit frequency, coupled with the GAM frequency, produces the maximum in the dispersion relation where the eigenmode can be found. The quantitative correspondence of experimental results with the predictions of the proposed model is analysed.

A model for the fast evaluation of prompt losses of energetic ions in stellarators

A good understanding of the confinement of energetic ions in non-axisymmetric magnetic fields is key for the design of reactors based on the stellarator concept. In this work, we develop a model that, based on the radially-local bounce-averaged drift-kinetic equation, classifies orbits and succeeds in predicting configuration-dependent aspects of the prompt losses of energetic ions in stellarators. Such a model could in turn be employed in the optimization stage of the design of new devices.

An asymptotic-preserving 2D-2P relativistic drift-kinetic-equation solver for runaway electron simulations in axisymmetric tokamaks

We propose an asymptotic-preserving (AP), uniformly convergent numerical scheme for the relativistic collisional Drift-Kinetic Equation (rDKE) to simulate runaway electrons in axisymmetric toroidal magnetic field geometries typical of tokamak devices. The approach is derived from an exact Green's function solution with numerical approximations of quantifiable impact, and results in a simple, two-step operator-split algorithm, consisting of a collisional Eulerian step, and a Lagrangian orbit-integration step with analytically prescribed kernels. The AP character of the approach is demonstrated by analysis of the dominant numerical errors, as well as by numerical experiments. We demonstrate the ability of the algorithm to provide accurate answers regardless of plasma collisionality on a circular axisymmetric tokamak geometry.

Effects of electron cyclotron heating on the toroidal flow in LHD plasmas

The toroidal force related to electron cyclotron heating (ECH) is investigated in large helical device (LHD) plasmas. When we apply the ECH to the plasma kept by neutral beam injection (NBI) heating, the radial profile of the toroidal flow velocity changes drastically in LHD. ECH-generated supra-thermal electrons can apply forces on the plasma through radial electron current and collisions. We investigate the perturbed electron distribution due to ECH by using the GNET code, which can solve the 5D drift kinetic equation. We also evaluate the electromagnetic force due to radial current and the collisional force driven by ECH. As a result, we find a comparable force driven by ECH to that by NBI heating. The direction of the force is the counter (co) direction radially inside (outside) from the ECH heating location, and these directions correspond with that of experiment results. Finally, we evaluate toroidal flows in ECH and NBI heated plasma solving the radial diffusion equation and compare them with that of experimental observations. We reproduce the co-rotating toroidal flow quantitatively in the balanced-NBI+ECH heated case, but we see a difference in the toroidal flow profiles in the co-NBI+ECH heated case.