4 pages, 4 figures. Submitted for publication in AIP Conference Proceedings (Proc. 23rd Topical Conference on Radiofrequency Power in Plasmas, Hefei, China, May 14-17, 2019.)
New integral kernels describing the full-wave dielectric response of Maxwellian tokamak plasmas are presented. They realistically account for the rotational transform and for wave dispersion in presence of equilibrium magnetic field parallel gradients. These kernels rely on special functions of three variables that generalize the standard plasma dispersion function; their main analytical properties are given, leading to efficient evaluation. This approach is free from the poloidal Fourier mode expansion of the HF fields which appears in earlier formulations and gives complete freedom for the numerical resolution of the wave equation: it will typically be applied to 2D finite element discretizations, allowing local mesh refinements as required near cyclotron resonance layers and in regions of rapid HF field variations. This first presentation is to lowest order in the Larmor radius for the sake of clarity but will readily generalize to all orders in ($ρ_{LT}/λ_\perp$).
The effect of small deviations from a Maxwellian equilibrium on turbulent momentum transport in tokamak plasmas is considered. These non-Maxwellian features, arising from diamagnetic effects, introduce a strong dependence of the radial flux of co-current toroidal angular momentum on collisionality: As the plasma goes from nearly collisionless to weakly collisional, the flux reverses direction from radially inward to outward. This indicates a collisionality-dependent transition from peaked to hollow rotation profiles, consistent with experimental observations of intrinsic rotation.
Vlasov equilibria of axisymmetric plasmas with vacuum toroidal magnetic field can be reduced, up to a selection of ions and electrons distributions functions, to a Grad-Shafranov-like equation. Quasineutrality narrow the choice of the distributions functions. In contrast to two-dimensional translationally symmetric equilibria whose electron distribution function consists of a displaced Maxwellian, the toroidal equilibria need deformed Maxwellians. In order to be able to carry through the calculations, this deformation is produced by means of either a Heaviside step function or an exponential f…
The formula for dielectric function of non-degenerate and maxwellian collisional plasmas is transformed to the form, convenient for research. Graphic comparison of longitudinal dielectric functions of quantum and classical non-degenerate collisional plasmas is made.
Formulas for the longitudinal dielectric permeability in quantum non-degenerate and maxwellian collisional plasma with the frequency of collisions proportional to the module of the wave vector, in Mermin's approach, are received. Equation of Shrödinger - Boltzmann with integral of collisions relaxation type in Mermin's approach is applied. It is spent numerical and graphic comparison of the real and imaginary parts of dielectric function of non-degenerate and maxwellian collisional quantum plasma with a constant and a variable frequencies of collisions. It is shown, that the longitudinal diele…
Formulas for transverse conductance and dielectric permeability in quantum non-degenerate and Maxwellian collisional plasma with arbitrary variable collision frequency in Mermin's approach are deduced. Frequency of collisions of particles depends arbitrarily on a wave vector. The special case of frequency of collisions proportional to the module of a wave vector is considered. The graphic analysis of the real and imaginary parts of dielectric function is made.
Formulas for transverse conductance in quantum Maxwellian collisional plasma are deduced. The kinetic Von Neumann equation in momentum space with collision integral in the relaxation form is used. It is shown, that at Planck's constant tends to zero the derived formula transfers to the classical one. It is shown also, that when values of dimensionless wave numbers are small, the conductance formula transfers in the known formula for classical Maxwellian plasmas.
By choosing appropriate deformed Maxwellian ion and electron distribution functions depending on the two particle constants of motion, i.e. the energy and toroidal angular momentum, we reduce the Vlasov axisymmetric equilibrium problem for quasineutral plasmas to a transcendental Grad-Shafranov-like equation. This equation is then solved numerically under the Dirichlet boundary condition for an analytically prescribed boundary possessing a lower X-point to construct tokamak equilibria with toroidal sheared ion flow and anisotropic pressure. Depending on the deformation of the distribution func…
In a tokamak pedestal, radial scale lengths can become comparable to the ion orbit width, invalidating conventional neoclassical calculations of flow and bootstrap current. In this work we illustrate a non-local approach that allows strong radial density variation while maintaining small departures from a Maxwellian distribution. Non-local effects alter the magnitude and poloidal variation of the flow and current. The approach is implemented in a new global delta-f continuum code using the full linearized Fokker-Planck collision operator. Arbitrary collisionality and aspect ratio are allowed a…
Non-thermal electron cyclotron emission during runaway plateau in tokamak disruptions from a highly anisotropic dielectric tensor
Seoul National University（首尔大学）；Nuclear Research Institute for Future Technology and Policy, Seoul National University（首尔大学未来技术与政策核研究所）；POSTECH（浦项工科大学）
Extending the ideal MHD stability code MISHKA, a new code, MISHKA-A, is developed to study the impact of pressure anisotropy on plasma stability. Based on full anisotropic equilibrium and geometry, the code can provide normal mode analysis with three fluid closure models: the single adiabatic model (SA), the double adiabatic model (CGL) and the incompressible model. A study on the plasma continuous spectrum shows that in low beta, large aspect ratio plasma, the main impact of anisotropy lies in the modification of the BAE gap and the sound frequency, if the q profile is conserved. The SA model…
This paper presents a complete theoretical framework for plasma turbulence and transport in tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio of the gyroradius to the equilibrium scale length. Proceeding order-by-order in this expansion, a framework for plasma turbulence is developed. It comprises an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the fluctuations. The equilibrium d…
Lower-hybrid current drive (LHCD) actuators are important components of modern day fusion experiments as well as proposed fusion reactors. However, simulations of LHCD often differ substantially from experimental results, and from each other, especially in the inferred power deposition profile shape. Here we investigate some possible causes of this discrepancy; "full-wave" effects such as interference and diffraction, which are omitted from standard raytracing simulations and the breakdown of the raytracing near reflections and caustics. We compare raytracing simulations to state-of-the-art fu…
We present particle-in-cell (PIC) simulations of minority energetic protons in deuterium plasmas, which demonstrate a collective instability responsible for emission near the lower hybrid frequency and its harmonics. The simulations capture the lower hybrid drift instability in a regime relevant to tokamak fusion plasmas, and show further that the excited electromagnetic fields collectively and collisionlessly couple free energy from the protons to directed electron motion. This results in an asymmetric tail antiparallel to the magnetic field. We focus on obliquely propagating modes under cond…
The incomplete plasma dispersion function is a generalization of the plasma dispersion function in which the defining integral spans a semi-infinite, rather than infinite, domain. It is useful for describing the linear dielectric response and wave dispersion in non-Maxwellian plasmas when the distribution functions can be approximated as Maxwellian over finite, or semi-infinite, intervals in velocity phase-space. A ubiquitous example is the depleted Maxwellian electron distribution found near boundary sheaths or double layers, where the passing interval can be modeled as Maxwellian with a lowe…
A class of parametric distribution functions has been proposed in [C.DiTroia, Plasma Physics and Controlled Fusion,54,2012] as equilibrium distribution functions (EDFs) for charged particles in fusion plasmas, representing supra-thermal particles in anisotropic equilibria for Neutral Beam Injection, Ion Cyclotron Heating scenarios. Moreover, the EDFs can also represent nearly isotropic equilibria for Slowing-Down $alpha$ particles and core thermal plasma populations. These EDFs depend on constants of motion (COMs). Assuming an axisymmetric system with no equilibrium electric field, the EDF dep…
Parallel permittivity elements are derived for radio frequency waves in an axisymmetric D-shaped tokamak with Soloviev type equilibrium under arbitrary aspect ratio, arbitrary elongation and moderate triangularity. The drift-kinetic equation is solved separately for untrapped (passing or circulating) and three groups of trapped particles as a boundary-value problem, accounting for the transit-time and bounce resonances, respectively. It is shown that the new two groups of trapped particles can appear only in tokamaks with large elongation and large central beta. Our dielectric characteristics …
Formulas for the longitudinal dielectric permeability in quantum degenerate collisional plasma with the frequency of collisions proportional to the module of the wave vector, in Mermin's approach, are received. Equation of Shrödinger - Boltzmann with integral of collisions relaxation type in Mermin's approach is applied. It is spent numerical and graphic comparison of the real and imaginary parts of dielectric function of non-degenerate and maxwellian collisional quantum plasma with a constant and a variable frequencies of collisions. It is shown, that the longitudinal dielectric function weak…
We are developing a new continuum gyrokinetic code, Gkeyll, for use in edge plasma simulations, and here present initial simulations of turbulence on open field lines with model sheath boundary conditions. The code implements an energy conserving discontinuous Galerkin scheme, applicable to a general class of Hamiltonian equations. Several applications to test problems have been done, including a calculation of the parallel heat-flux on divertor plates resulting from an ELM crash in JET, for a 1x/1v SOL scenario explored previously, where the ELM is modeled as a time-dependent intense upstream…
A largely unsolved theoretical issue in controlled fusion research is the consistent \textit{kinetic} treatment of slowly-time varying plasma states occurring in collisionless and magnetized axisymmetric plasmas. The phenomenology may include finite pressure anisotropies as well as strong toroidal and poloidal differential rotation, characteristic of Tokamak plasmas. Despite the fact that physical phenomena occurring in fusion plasmas depend fundamentally on the microscopic particle phase-space dynamics, their consistent kinetic treatment remains still essentially unchalleged to date. The goal…
Formulas for calculation of transverse dielectric function and transverse electric conductivity in quantum collisional Maxwellian plasma are obtained. The Wigner -- Vlasov -- Boltzmann kinetic equation with collision integral in BGK (Bhatnagar, Gross and Krook) form in coordinate space is used. Various special cases are in\-ves\-ti\-gated. Comparison with Lindhard's formula has been carried out.
Scattering phenomena between charged particles and highly excited Rydberg atoms are of critical importance in many processes in plasma physics and astrophysics. While a Maxwell-Boltzmann (MB) energy distribution for the charged particles is often assumed for calculations of collisional rate coefficients, in this contribution we relax this assumption and use two different energy distributions, a bimodal MB distribution and a $κ$-distribution. Both variants share a high-energy tails occurring with higher probability than the corresponding MB distribution. The high energy tail may significantly a…
We propose a new and effective method to find plasma oscillatory and wave modes. It implies searching a pair of poles of two-dimensional (in coordinate $x$ and time $t$) Laplace transform of self-consistent plasma electric field $E(x,t) \to E_{p_1p_2}$, where $p_1 \equiv -i ω$, $p_2 \equiv i k$ are Laplace transform parameters, that is determining a pair of zeros of the following equation $$\frac1{E_{p_1p_2}} = 0 .$$ This kind of conditional equation for searching double poles of $E_{p_1p_2}$ we call ``general dispersion equation'', so far as it is used to find the pair values ($ω^{(n)}, k^{(n…
Using kinetic theory for homogeneous collisionless magnetized plasmas, we present an extended review of the plasma waves and instabilities and discuss the anisotropic response of generalized relativistic dielectric tensor and Onsager symmetry properties for arbitrary distribution functions. In general, we observe that for such plasmas only those electromagnetic modes whose magnetic field perturbations are perpendicular to the ambient magneticeld, i.e.,B1 \perp B0, are effected by the anisotropy. However, in oblique propagation all modes do show such anisotropic effects. Considering the non-rel…
Energetic particle (EP) driven instabilities will be of strongly increased relevance in future burning plasmas as the EP pressure will be very large compared to the thermal plasma. Understanding the interaction of EPs and bulk plasma is crucial for developing next-generation fusion devices. In this work, the JOREK MHD code is extended to allow for the simulation of EP instabilities at high EP pressures using realistic plasma and EP parameter in a full-f formulation with anisotropic pressure coupling to the bluid background. The code is first benchmarked linearly for the ITPA-TAE as well as the…
In this paper the experimental fact of presence of ionic - sound oscillations in plasma of tokamak DIII-D is given for plasma with additional heating by means of a neutral hydrogen beam. It contradicts modern theoretical representations about an opportunity of the build-up of such type of oscillations in plasma, when an ionic temperature is biger than an electronic one. Inasmuch as electrons provide the own screening by means of pushing aside forcibly around of itself of electrons and by baring an ionic skeleton their polarizing cloud should be of bigger radius than radius of Debye. Therefore …
Collisionless plasmas, such as those encountered in tokamaks, exhibit a rich variety of instabilities. The physical origin, triggering mechanisms and fundamental understanding of many plasma instabilities, however, are still open problems. We investigate the stability properties of a collisionless Vlasov plasma in a stationary homogeneous magnetic field. We narrow the scope of our investigation to the case of Maxwellian plasma. For the first time using a fully kinetic approach we show the emergence of the local instability, a transient growth, followed by classical Landau damping in a stable m…
The decays of the Langmuir waves in dense plasmas are computed using the dielectric function theory widely used in the solid state physics. Four cases are considered: a classical plasma, a Maxwellian plasma, a degenerate quantum plasma, and a partially degenerate plasma. The result is considerably different from the conventional Landau damping theory.
Transverse and longitudinal dielectric permittivity elements are derived for radio frequency waves in a laboratory dipole magnetic field plasma (LDMFP) accounting for the finite ring current radius. The main feature of LDMFP is the fact that, in contrast to planetary magnetospheric plasmas, the untrapped (circulating) particles can exist/circulate (together with the trapped particles) along the equilibrium magnetic field lines. To estimate the wave power absorbed in LDMFP the perturbed electric field and current density components are decomposed in a Fourier series over the poloidal angle. The…
Anisotropic Maxwell neural operator for rapid parametric full-wave modelling of ion cyclotron resonance heating
Chongqing University of Posts and Telecommunications（重庆邮电大学）；University of Science and Technology of China（中国科学技术大学）；Xidian University（西安电子科技大学）；Institute of Plasma Physics, Hefei Institutes of Physical Science, Chinese Academy of Sciences（中国科学院合肥物质科学研究院等离子体物理研究所）
In this paper, a reduced model of quasilinear diffusion by a small Larmor radius approximation is derived to couple the Maxwell's equations and the Fokker-Planck equation self-consistently for ion cyclotron range of frequency waves in a tokamak. The reduced model ensures the important properties of the full model by Kennel-Engelmann diffusion, such as diffusion directions, wave polarizations, and H-theorem. The kinetic energy change (W-dot) is used to derive the reduced model diffusion coefficients for the fundamental damping and the second harmonic damping to the lowest order of the finite La…
This work proposes a parametric equilibrium distribution function F_eq to be applied to the gyrokinetic studies of the Finite Orbit Width behavior of guiding centers representing several species encountered in axisymmetric tokamak plasma, as fusion products, thermal bulk and energetic particles from Ion Cyclotron Radiation Heating and Negative Neutral Beam Injections. After the analysis of the basic results of orbit theory obtained with a particularly convenient orbit coordinates set, it is shown how the proposed F_eq satisfies the two conditions that make it an equilibrium distribution functi…
The propagation of radio frequency (RF) waves in tokamaks can be affected by filamentary structures present in the edge plasma. The waves are reflected, refracted, and diffracted, leading to a decrease in efficiency of heating and/or current generation. The scattering of RF waves (lower hybrid, helicon, and ion cyclotron waves) by a single cylindrical filament is studied using a full-wave analytical theory. Analysis reveals a variety of common scattering features among the three different RF waves which can be inferred by examining the cold plasma dispersion relation. The physical intuition is…
Velocity distribution functions (VDFs) that exhibit a power-law dependence on the high-energy tail have been the subject of intense research by the plasma physics community. Such functions, known as kappa or superthermal distributions, have been found to provide a better fitting to the VDFs measured by spacecraft in the solar wind. One of the problems that is being addressed on this new light is the temperature anisotropy of solar wind protons and electrons. In the literature, the general treatment for waves excited by (bi-)Maxwellian plasmas is well-established. However, for kappa distributio…
