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A one-dimensional Vlasov-Poisson system is considered to elucidate how the information entropies of the probability distribution functions of the electron position and velocity variables evolve in the Landau damping process. Considering the initial condition given by the Maxwellian velocity distribution with the spatial density perturbation in the form of the cosine function of the position, we derive linear and quasilinear analytical solutions that accurately describe both early and late time behaviors of the distribution function and the electric field. The validity of these solutions is confirmed by comparison with numerical simulations based on contour dynamics. Using the quasilinear analytical solution, the time evolutions of the velocity distribution function and its kurtosis indicating deviation from the Gaussian distribution are evaluated with the accuracy of the squared perturbation amplitude. We also determine the time evolutions of the information entropies of the electron position and velocity variables and their mutual information. We further consider Coulomb collisions which relax the state in the late-time limit in the collision less process to the thermal equilibrium state. In this collisional relaxation process, the mutual information of the position and velocity variables decreases to zero while the total information entropy of the phase-space distribution function increases by the decrease in the mutual information and demonstrates the validity of Boltzmann's H-theorem.

Microturbulence in magnetic confined plasmas contributes to energy exchange between particles of different species as well as the particle and heat fluxes. Although the effect of turbulent energy exchange has not been considered significant in previous studies, it is anticipated to have a greater impact than collisional energy exchange in low collisional plasmas such as those in future fusion reactors. In this study, gyrokinetic simulations are performed to evaluate the energy exchange in ion temperature gradient (ITG) turbulence. The energy exchange due to the ITG turbulence mainly consists of the cooling of ions in the $\nabla B$-curvature drift motion and the heating of electrons streaming along a field line. It is found that the ITG turbulence transfers energy from ions to electrons regardless of whether the ions or electrons are hotter, which is in marked contrast to the energy transfer by Coulomb collisions. This implies that the ITG turbulence should be suppressed from the viewpoint of sustaining the high ion temperature required for fusion reactions since it prevents energy transfer from alpha-heated electrons to ions as well as enhancing ion heat transport toward the outside of the reactor. Furthermore, linear and nonlinear simulation analyses confirm the feasibility of quasilinear modeling for predicting the turbulent energy exchange in addition to the particle and heat fluxes., Comment: 13pages, 4 figures

The Eulerian variational formulation is presented to obtain governing equations of the electromagnetic turbulent gyrokinetic system. A local momentum balance in the system is derived from the invariance of the Lagrangian of the system under an arbitrary spatial coordinate transformation by extending the previous work [H. Sugama et al., Phys. Plasmas 28, 022312 (2021)]. Polarization and magnetization due to finite gyroradii and electromagnetic microturbulence are correctly described by the gyrokinetic Poisson equation and Amp\`{e}re's law which are derived from the variational principle. Also shown is how the momentum balance is influenced by including collisions and external sources. Momentum transport due to collisions and turbulence is represented by a symmetric pressure tensor which originates in a variational derivative of the Lagrangian with respect to the metric tensor. The relations of the axisymmetry and quasi-axisymmetry of the toroidal background magnetic field to a conservation form of the local momentum balance equation are clarified. In addition, an ensemble-averaged total momentum balance equation is shown to take the conservation form even in the background field with no symmetry when a constraint condition representing the macroscopic Amp\`{e}re's law is imposed on the background field. Using the WKB representation, the ensemble-averaged pressure tensor due to the microturbulence is expressed in detail and it is verified to reproduce the toroidal momentum transport derived in previous works for axisymmetric systems. The local momentum balance equation and the pressure tensor obtained in this work present a useful reference for elaborate gyrokinetic simulation studies of momentum transport processes., Comment: 25 pages, submitted to Phys. Plasmas

Expressions of polarization and magnetization in magnetically confined plasmas are derived, which include full expansions in the gyroradius to treat effects of both equilibrium and microscopic electromagnetic turbulence. Using the obtained expressions, densities and flows of particles are related to those of gyrocenters. To the first order in the normalized gyroradius expansion, the mean part of the particle flow is given by the sum of the gyrocenter flow and the magnetization flow, which corresponds to the so-called magnetization law in drift kinetics, while the turbulent part contains the polarization flow as well. Collisions make an additional contribution to the second-order particle flow. The mean particle flux across the magnetic surface is of the second-order and it contains classical, neoclassical, and turbulent transport processes. The Lagrangian variational principle is used to derive the gyrokinetic Poisson and Amp\`{e}re equations which properly include mean and turbulent parts so as to be useful for full-$f$ global electromagnetic gyrokinetic simulations. It is found that the second-order Lagrangian term given by the inner product of the turbulent vector potential and the drift velocity consisting of the curvature drift and the $\nabla B$ drift should be retained in order for the derived Amp\`{e}re equation to correctly include the diamagnetic current which is necessary especially for the full-$f$ high-beta plasma simulations. The turbulent parts of these gyrokinetic Poisson and Amp\`{e}re equations are confirmed to agree with the results derived from the WKB representation in earlier works., Comment: 16 pages, submitted to Physics of Plasmas

The Eulerian variational formulation of the gyrokinetic system with electrostatic turbulence is presented in general spatial coordinates by extending our previous work [H. Sugama, {\it et al}., Phys.\ Plasmas {\bf 25}, 102506 (2018)]. The invariance of the Lagrangian of the system under an arbitrary spatial coordinate transformation is used to derive the local momentum balance equation satisfied by the gyrocenter distribution functions and the turbulent potential which are given as solutions of the governing equations. This derivation is in contrast with the conventional method using the spatial translation in which the asymmetric canonical pressure tensor generally enters the momentum balance equation. In the present study, the variation of the Lagrangian density with respect to the metric tensor is taken to directly obtain the symmetric pressure tensor which includes the effect of turbulence on the momentum transport. In addition, it is shown in this work how the momentum balance is modified when the collision and/or external source terms are added to the gyrokinetic equation. The results obtained here are considered useful for global gyrokinetic simulations investigating both neoclassical and turbulent transport processes even in general non-axisymmetric toroidal systems.

The Eulerian variational principle for the Vlasov-Poisson-Amp\`{e}re system of equations in a general coordinate system is presented. The invariance of the action integral under an arbitrary spatial coordinate transformation is used to obtain the momentum conservation law and the symmetric pressure in a more direct way than using the translational and rotational symmetries of the system. Next, the Eulerian variational principle is given for the collisionless drift kinetic equation, where particles' phase-space trajectories in given electromagnetic fields are described by Littlejohn's guiding center equations~[R. G. Littlejohn, J. Plasma Phys.\ {\bf 29}, 111 (1983)]. Then, it is shown that, in comparison with the conventional moment method, the invariance under a general spatial coordinate transformation yields a more convenient way to obtain the momentum balance as a three-dimensional vector equation in which the symmetric pressure tensor, the Lorentz force, and the magnetization current are properly expressed. Furthermore, the Eulerian formulation is presented for the extended drift kinetic system, for which, in addition to the drift kinetic equations for the distribution functions of all particle species, the quasineutrality condition and Amp\`{e}re's law to determine the self-consistent electromagnetic fields are given. Again, the momentum conservation law for the extended system is derived from the invariance under the general spatial coordinate transformation. Besides, the momentum balances are investigated for the cases where the collision and/or external source terms are added into the Vlasov and drift kinetic equations.

The benchmarks of the neoclassical transport codes based on the several local drift-kinetic models are reported here. Here, the drift-kinetic models are ZOW, ZMD, DKES-like, and global, as classified in [Matsuoka et al., Physics of Plasmas 22, 072511 (2015)]. The magnetic geometries of HSX, LHD, and W7-X are employed in the benchmarks. It is found that the assumption of $\boldsymbol E \times \boldsymbol B$ incompressibility causes discrepancy of neoclassical radial flux and parallel flow among the models, when $\boldsymbol E \times \boldsymbol B$ is sufficiently large compared to the magnetic drift velocities. On the other hand, when $\boldsymbol E \times \boldsymbol B$ and the magnetic drift velocities are comparable, the tangential magnetic drift, which is included in both the global and ZOW models, fills the role of suppressing unphysical peaking of neoclassical radial-fluxes found in the other local models at $E_r \simeq 0$. In low collisionality plasmas, in particular, the tangential drift effect works well to suppress such unphysical behavior of the radial transport caused in the simulations. It is demonstrated that the ZOW model has the advantage of mitigating the unphysical behavior in the several magnetic geometries, and that it also implements evaluation of bootstrap current in LHD with the low computation cost compared to the global model.

Achieving impurity and helium ash control is a crucial issue in the path towards fusion-grade magnetic confinement devices, and this is particularly the case of helical reactors, whose low-collisionality ion-root operation scenarios usually display a negative radial electric field which is expected to cause inwards impurity pinch. In these work we discuss, based on experimental measurements and standard predictions of neoclassical theory, how plasmas of very low ion collisionality, similar to those observed in the impurity hole of the Large Helical Device, can be an exception to this general rule, and how a negative radial electric field can coexist with an outward impurity flux. This interpretation is supported by comparison with documented discharges available in the International Stellarator-Heliotron Profile Database, and it can be extrapolated to show that achievement of high ion temperature in the core of helical devices is not fundamentally incompatible with low core impurity content.

A novel radially local approximation of the drift kinetic equation is presented. The new drift kinetic equation that includes both ${\bf E} \times {\bf B}$ and tangential magnetic drift terms is written in the conservative form and it has favorable properties for numerical simulation that any additional terms for particle and energy sources are unnecessary for obtaining stationary solutions under the radially local approximation. These solutions satisfy the intrinsic ambipolarity condition for neoclassical particle fluxes in the presence of quasisymmetry of the magnetic field strength. Also, another radially local drift kinetic equation is presented, from which the positive definiteness of entropy production due to neoclassical transport and Onsager symmetry of neoclassical transport coefficients are derived while it sacrifices the ambipolarity condition for neoclassical particle fluxes in axisymmetric and quasi-symmetric systems.

Effects of collisions on conservation laws for toroidal plasmas are investigated based on the gyrokinetic field theory. Associating the collisional system with a corresponding collisionless system at a given time such that the two systems have the same distribution functions and electromagnetic fields instantaneously, it is shown how the collisionless conservation laws derived from Noether's theorem are modified by the collision term. Effects of the external source term added into the gyrokinetic equation can be formulated similarly with the collisional effects. Particle, energy, and toroidal momentum balance equations including collisional and turbulent transport fluxes are systematically derived using a novel gyrokinetic collision operator, by which the collisional change rates of energy and canonical toroidal angular momentum per unit volume in the gyrocenter space can be given in the conservative forms. The ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work are shown to include classical, neoclassical, and turbulent transport fluxes which agree with those derived from conventional recursive formulations., Comment: 19 pages

The structural properties of an economical model for a confined plasma turbulence governor are investigated through bifurcation and stability analyses. A close relationship is demonstrated between the underlying bifurcation framework of the model and typical behavior associated with low- to high-confinement transitions such as shear flow stabilization of turbulence and oscillatory collective action. In particular, the analysis evinces two types of discontinuous transition that are qualitatively distinct. One involves classical hysteresis, governed by viscous dissipation. The other is intrinsically oscillatory and non-hysteretic, and thus provides a model for the so-called dithering transitions that are frequently observed. This metamorphosis, or transformation, of the system dynamics is an important late side-effect of symmetry-breaking, which manifests as an unusual non-symmetric transcritical bifurcation induced by a significant shear flow drive., Comment: 17 pages, revtex text, 9 figures comprised of 16 postscript files. Submitted to Phys. Rev. E