The coupled fluid plasma and kinetic neutral physics equations are analyzed through theory and simulation of benchmark cases.
It is shown that coupling methods that do not treat the coupling rates implicitly are restricted to short time steps for stability.
Fast charge exchange, ionization and recombination coupling rates exist, even after constraining the solution by requiring that the neutrals are at equilibrium.
For explicit coupling, the present implementation of Monte Carlo correlated sampling techniques does not allow for complete convergence in slab geometry.
For the benchmark case, residuals decay with particle number and increase with grid size, indicating that they scale in a manner that is similar to the theoretical prediction for nonlinear bias error.
Progress is reported on implementation of a fully implicit Jacobian-free Newton–Krylov coupling scheme.
The present block Jacobi preconditioning method is still sensitive to time step and methods that better precondition the coupled system are under investigation.
Astrophysical particle acceleration mechanisms in colliding magnetized laser-produced plasmas
(abstract)
Significant particle energization is observed to occur in numerous astrophysical environments, and in the standard models this acceleration occurs alongside energy conversion processes including collisionless shocks or magnetic reconnection. Recent platforms for laboratory experiments using magnetized laser-produced plasmas have opened opportunities to study these particle acceleration processes in the laboratory. Through fully kinetic particle-in-cell simulations, we investigate acceleration mechanisms in experiments with colliding magnetized laser-produced plasmas, with geometry and parameters matched to recent high-Mach number reconnection experiments with externally-controlled magnetic fields. 2-D simulations demonstrate significant particle acceleration with three phases of energization: first a “direct” Fermi acceleration driven by approaching magnetized plumes; second, x-line acceleration during magnetic reconnection of anti-parallel fields; and finally an additional Fermi energization of particles trapped in contracting and relaxing magnetic islands produced by reconnection. The relative effectiveness of these mechanisms depends on plasma and magnetic field parameters of the experiments.
A brief critique is presented of some different classes of magnetohydrodynamic equilibrium
solutions based on their continuity properties and whether the magnetic field is
integrable or not. A generalized energy functional is introduced that is comprised of alternating
ideal regions, with nested flux surfaces with irrational rotational-transform, and
Taylor-relaxed regions, possibly with magnetic islands and chaos. The equilibrium states
have globally continuous magnetic fields, and may be constructed for arbitrary three dimensional
plasma boundaries and appropriately prescribed pressure and rotational-transform
profiles.
Overview of NSTX Upgrade Initial Results and Modeling Highlights
(abstract)
The hybrid regime with beta, collisionality, safety factor and plasma shape relevant to the ITER steady-state mission has been successfully integrated with ELM suppression by applying an odd parity $n=3$ resonant magnetic perturbation (RMP).
Fully non-inductive hybrids in the DIII-D tokamak with high beta ( $<\beta>≤ 2.8$%) and high confinement ($H_{98y2} ≤ 1.4$) in the ITER similar shape have achieved zero surface loop voltage for up to two current relaxation times using efficient central current drive from ECCD and NBCD. The $n=3$ RMP causes surprisingly little increase in thermal transport during ELM suppression. Poloidal magnetic flux pumping in hybrid plasmas maintains $q$ above 1 without loss of current drive efficiency, except that experiments show that extremely peaked ECCD profiles can create sawteeth.
During ECCD, Alfvén eigenmode (AE) activity is replaced by a more benign fishbone-like mode, reducing anomalous beam ion diffusion by a factor of 2. While the
electron and ion thermal diffusivities substantially increase with higher ECCD power, the loss of confinement can be offset by the decreased fast ion transport resulting from AE suppression.
Extrapolations from DIII-D along a dimensionless parameter scaling path as well as those using self-consistent theory-based modeling show that these ELM-suppressed, fully non-inductive hybrids can achieve the $Q_{fus} = 5$ ITER steady-state mission.
Hybrid simulation of fishbone instabilities in the EAST tokamak
Hybrid simulations with the global kinetic-magnetohydrodynamic (MHD) code M3D-K have been carried out to investigate the linear stability and nonlinear dynamics of beam-driven fishbone in the Experimental Advanced Superconducting Tokamak (EAST) experiment. Linear simulations show that a low frequency fishbone instability is excited at experimental value of beam ion pressure. The mode is mainly driven by low energy beam ions via precessional resonance. The results are consistent with the experimental measurement with respect to mode frequency and mode structure. When the beam ion pressure is increased to exceed a critical value, the low frequency mode transits to a beta-induced Alfvén eigenmode (BAE) with much higher frequency. This BAE is driven by higher energy beam ions. Nonlinear simulations show that the frequency of the low frequency fishbone chirps up and down with corresponding hole-clump structures in phase space, consistent with the Berk-Breizman theory. In addition to the low frequency mode, the high frequency BAE is excited during the nonlinear evolution. For the transient case of beam pressure fraction where the low and high frequency modes are simultaneously excited in the linear phase, only one dominant mode appears in the nonlinear phase with frequency jumps up and down during nonlinear evolution.
Generation and Evolution of High-Mach-Number Laser-Driven Magnetized Collisionless Shocks in the Laboratory
We present the first laboratory generation of high-Mach-number magnetized collisionless shocks created through the interaction of an expanding laser-driven plasma with a magnetized ambient plasma.
Time-resolved, two-dimensional imaging of plasma density and magnetic fields shows the formation and evolution of a supercritical shock propagating at magnetosonic Mach number $M_{ms}≈12$.
Particle-in-cell simulations constrained by experimental data further detail the shock formation and separate dynamics of the multi-ion-species ambient plasma.
The results show that the shocks form on time scales as fast as one gyroperiod, aided by the efficient coupling of energy, and the generation of a magnetic barrier between the piston and ambient ions.
The development of this experimental platform complements present remote sensing and spacecraft observations, and opens the way for controlled laboratory investigations of high-Mach number collisionless shocks, including the mechanisms and efficiency of particle acceleration.
An often-neglected portion of the radial $\boldsymbol{E}\times \boldsymbol{B}$ drift is shown to drive an outward flux of co-current momentum when free energy is transferred from the electrostatic potential to ion parallel flows. This symmetry breaking is fully nonlinear, not quasilinear, necessitated simply by free-energy balance in parameter regimes for which significant energy is dissipated via ion parallel flows. The resulting rotation peaking is counter-current and has a scaling and order of magnitude that are comparable with experimental observations. The residual stress becomes inactive when frequencies are much higher than the ion transit frequency, which may explain the observed relation of density peaking and counter-current rotation peaking in the core.
Kinetic simulations of X-B and O-X-B mode conversion and its deterioration at high input power
Spherical tokamak plasmas are typically overdense and thus inaccessible to externally-injected microwaves in
the electron cyclotron range. The electrostatic electron Bernstein wave (EBW), however, provides a method
to access the plasma core for heating and diagnostic purposes. Understanding the details of the coupling
process to electromagnetic waves is thus important both for the interpretation of microwave diagnostic data
and for assessing the feasibility of EBW heating and current drive. While the coupling is reasonably well–understood in the linear regime, nonlinear physics arising from high input power has not been previously
quantified. To tackle this problem, we have performed one- and two-dimensional fully kinetic particle-in-cell
simulations of the two possible coupling mechanisms, namely X-B and O-X-B mode conversion. We find that
the ion dynamics has a profound effect on the field structure in the nonlinear regime, as high amplitude shortscale
oscillations of the longitudinal electric field are excited in the region below the high-density cut-off prior
to the arrival of the EBW. We identify this effect as the instability of the X wave with respect to resonant
scattering into an EBW and a lower-hybrid wave. We calculate the instability rate analytically and find this
basic theory to be in reasonable agreement with our simulation results.
The effects of recycled neutral atoms on tokamak ion temperature gradient (ITG) driven turbulence have been investigated in a steep edge pedestal, magnetic separatrix configuration, with the full-$f$ edge gryokinetic code XGC1. An adiabatic electron model has been used; hence, the impacts of neutral particles and turbulence on the density gradient are not considered, nor are electromagnetic turbulence effects. The neutral atoms enhance the ITG turbulence, first, by increasing the ion temperature gradient in the pedestal via the cooling effects of charge exchange and, second, by a relative reduction in the ${\bf E} \times {\bf B}$ shearing rate.
Suppression of Alfvén Modes on the National Spherical Torus Experiment Upgrade with Outboard Beam Injection
In this Letter we present data from experiments on the National Spherical Torus Experiment Upgrade, where it is shown for the first time that small amounts of high pitch-angle beam ions can strongly suppress the counterpropagating global Alfvén eigenmodes (GAE). GAE have been implicated in the redistribution of fast ions and modification of the electron power balance in previous experiments on NSTX. The ability to predict the stability of Alfvén modes, and developing methods to control them, is important for fusion reactors like the International Tokamak Experimental Reactor, which are heated by a large population of nonthermal, super-Alfvénic ions consisting of fusion generated α’s and beam ions injected for current profile control. We present a qualitative interpretation of these observations using an analytic model of the Doppler-shifted ion-cyclotron resonance drive responsible for GAE instability which has an important dependence on $k_\perp \rho_L$. A quantitative analysis of this data with the hym stability code predicts both the frequencies and instability of the GAE prior to, and suppression of the GAE after the injection of high pitch-angle beam ions.
Local energy conservation law for a spatially-discretized Hamiltonian Vlasov-Maxwell system
Because of the unparalleled long-term conservative property, the structure-preserving geometric algorithm for the Vlasov-Maxwell (VM) equations is currently an active research topic. We show that spatially discretized Hamiltonian systems for the VM equations admit a local energy conservation law in space-time. This is accomplished by proving that a sum-free and only locally non-zero scalar field can always be written as the divergence of a vector field that is only locally non-zero. The result demonstrates that the Hamiltonian discretization of Vlasov-Maxwell system can preserve local conservation laws, in addition to the symplectic structure, both of which are the intrinsic physical properties of infinite dimensional Hamiltonian systems in physics.
Bifurcation of quiescent H-mode to a wide pedestal regime in DIII-D and advances in the understanding of edge harmonic
oscillations
New experimental studies and modelling of the coherent edge harmonic oscillation (EHO), which regulates the conventional Quiescent H-mode (QH-mode) edge, validate the proposed hypothesis of edge rotational shear in destabilizing the low-$n$ kink-peeling mode as the additional drive mechanism for the EHO.
The observed minimum edge ${\bf E}\times{\bf B}$ shear required for the EHO decreases linearly with pedestal collisionality $\nu _{\text{e}}^{\ast}$, which is favorable for operating QH-mode in machines with low collisionality and low rotation such as ITER.
In addition, the QH-mode regime in DIII-D has recently been found to bifurcate into a new 'wide-pedestal' state at low torque in double-null shaped plasmas, characterized by increased pedestal height, width and thermal energy confinement (Burrell 2016 Phys. Plasmas 23 056103, Chen 2017 Nucl. Fusion 57 022007). This potentially provides an alternate path for achieving high performance ELM-stable operation at low torque, in addition to the low-torque QH-mode sustained with applied 3D fields.
Multi-branch low-$k$ and intermediate-$k$ turbulences are observed in the 'wide-pedestal'.
New experiments support the hypothesis that the decreased edge ${\bf E}\times{\bf B}$ shear enables destabilization of broadband turbulence, which relaxes edge pressure gradients, improves peeling-ballooning stability and allows a wider and thus higher pedestal. The ability to accurately predict the critical ${\bf E}\times{\bf B}$ shear for EHO and maintain high performance QH-mode at low torque is an essential requirement for projecting QH-mode operation to ITER and future machines.
Investigation of Neutral Particle Dynamics in Aditya Tokamak Plasma with
DEGAS2 Code
Neutral particle behavior in Aditya tokamak, which has a circular poloidal ring limiter at one particular toroidal location, has been investigated using DEGAS2 code. The code is based on the calculation using Monte Carlo algorithms and is mainly used in tokamaks with divertor configuration. This code has been successfully implemented in Aditya tokamak with limiter configuration. The penetration of neutral hydrogen atom is studied with various atomic and molecular contributions and it is found that the maximum contribution comes from the dissociation processes. For the same, H α spectrum is also simulated and matched with the experimental one. The dominant contribution around 64% comes from molecular dissociation processes and neutral particle is generated by those processes have energy of ~2.0 eV. Furthermore, the variation of neutral hydrogen density and H α emissivity profile are analysed for various edge temperature profiles and found that there is not much changes in H α emission at the plasma edge with the variation of edge temperature (7–40 eV).
Full-f XGC1 gyrokinetic study of improved ion energy confinement
from impurity stabilization of ITG turbulence
Flux-driven full-$f$ gyrokinetic simulations are performed to study carbon impurity effects on the
ion temperature gradient (ITG) turbulence and ion thermal transport in a toroidal geometry.
Employing the full-$f$ gyrokinetic code XGC1, both main ions and impurities are evolved
self-consistently including turbulence and neoclassical physics. It is found that the carbon impurity
profile self-organizes to form an inwardly peaked density profile, which weakens the ITG instabilities
and reduces the overall fluctuations and ion thermal transport. A stronger reduction appears in
the low frequency components of the fluctuations. The global structure of ${\bf E}\times{\bf B}$ flow also changes,
resulting in the reduction of global avalanche like transport events in the impure plasma. Detailed
properties of impurity transport are also studied, and it is revealed that both the inward neoclassical
pinch and the outward turbulent transport are equally important in the formation of the steady state
impurity profile.
A model of energetic ion effects on pressure driven tearing modes in tokamaks
The effects that energetic trapped ions have on linear resistive magnetohydrodynamic (MHD) instabilities are studied in a reduced model that captures the essential physics driving or damping the modes through variations in the magnetic shear. The drift-kinetic orbital interaction of a slowing down distribution of trapped energetic ions with a resistive MHD instability is integrated to a scalar contribution to the perturbed pressure, and entered into an asymptotic matching formalism for the resistive MHD dispersion relation. Toroidal magnetic field line curvature is included to model trapping in the particle distribution, in an otherwise cylindrical model. The focus is on a configuration that is driven unstable to the $m/n = 2/1$ mode by increasing pressure, where $m$ is the poloidal mode number and $n$ is the toroidal. The particles and pressure can affect the mode both in the core region where there can be low and reversed shear and outside the resonant surface in significant positive shear. The results show that the energetic ions damp and stabilize the mode when orbiting in significant positive shear, increasing the marginal stability boundary. However, the inner core region contribution with low and reversed shear can drive the mode unstable. This effect of shear on the energetic ion pressure contribution is found to be consistent with the literature. These results explain the observation that the $2/1$ mode was found to be damped and stabilized by energetic ions in $\delta f$-MHD simulations of tokamak experiments with positive shear throughout, while the $2/1$ mode was found to be driven unstable in simulations of experiments with weakly reversed shear in the core. This is also found to be consistent with related experimental observations of the stability of the $2/1$ mode changing significantly with core shear.
Kinetic simulations of ladder climbing by electron plasma waves
The energy of plasma waves can be moved up and down the spectrum using chirped modulations of plasma
parameters, which can be driven by external fields. Depending on whether the wave spectrum is discrete (bounded
plasma) or continuous (boundless plasma), this phenomenon is called ladder climbing (LC) or autoresonant
acceleration of plasmons. It was first proposed by Barth et al. [Phys. Rev. Lett. 115, 075001 (2015)] based on a
linear fluid model. In this paper, LC of electron plasma waves is investigated using fully nonlinear Vlasov-Poisson
simulations of collisionless bounded plasma. It is shown that, in agreement with the basic theory, plasmons survive
substantial transformations of the spectrum and are destroyed only when their wave numbers become large enough
to trigger Landau damping. Since nonlinear effects decrease the damping rate, LC is even more efficient when
practiced on structures like quasiperiodic Bernstein-Greene-Kruskal (BGK) waves rather than on Langmuir
waves per se.
Verification of long wavelength electromagnetic modes with a gyrokinetic-fluid hybrid model in the XGC code
As an alternative option to kinetic electrons, the gyrokinetic total-f particle-in-cell (PIC) code XGC1 has been extended to the MHD/fluid type electromagnetic regime by combining gyrokinetic PIC ions with massless drift-fluid electrons analogous to Chen and Parker [Phys. Plasmas 8, 441 (2001)]. Two
representative long wavelength modes, shear Alfvén waves and resistive tearing modes, are verified in cylindrical and toroidal magnetic field geometries.
Nonlinear reconnecting edge localized modes in current-carrying plasmas
Nonlinear edge localized modes in a tokamak are examined using global three-dimensional resistive magnetohydrodynamics simulations. Coherent current-carrying filament (ribbon-like) structures wrapped around the torus are nonlinearly formed due to nonaxisymmetric reconnecting current sheet instabilities, the so-called peeling-like edge localized modes. These fast growing modes saturate by breaking axisymmetric current layers isolated near the plasma edge and go through repetitive relaxation cycles by expelling current radially outward and relaxing it back. The local bi-directional fluctuation-induced electromotive force (emf) from the edge localized modes, the dynamo action, relaxes the axisymmetric current density and forms current holes near the edge. The three-dimensional coherent current-carrying filament structures (sometimes referred to as 3-D plasmoids) observed here should also have strong implications for solar and astrophysical reconnection.
Conductivity tensor for anisotropic plasma in gyrokinetic theory
It has been argued that oblique firehose and mirror instabilities are important candidates for the regulation of temperature anisotropy in solar wind. To quantify the role of anisotropy driven instabilities, global kinetic simulations of the solar wind would be extremely useful. However, due to long time scales involved, such simulations are prohibitively expensive. Gyrokinetic theory and simulations have proven to be valuable tools for the study of low frequency phenomena in nonuniform plasmas; however, there are discrepancies between the anisotropy driven instabilities appearing in the gyrokinetic theory and those of a fully kinetic one. We present a derivation of the conductivity tensor based on the arbitrary frequency gyrokinetics and show that relaxing the condition ω/Ω ≪ 1, where ω is the wave frequency, and the Ω is the cyclotron frequency, eliminates these discrepancies, while preserving the advantages of the gyorkinetic theory for global kinetic simulations.
M3D-C1 simulations of the plasma response to RMPs in NSTX-U single-null and snowflake divertor configurations
n this work, single- and two-fluid resistive magnetohydrodynamic calculations of the plasma response to $n=3$ magnetic perturbations in single-null (SN) and snowflake (SF) divertor configurations are compared with those based on the vacuum approach. The calculations are performed using the code M3D-C1 and are based on simulated NSTX-U plasmas. Significantly different plasma responses were found from these calculations, with the difference between the single- and two-fluid plasma responses being caused mainly by the different screening mechanism intrinsic to each of these models. Although different plasma responses were obtained from these different plasma models, no significant difference between the SN and SF plasma responses were found. However, due to their different equilibrium properties, magnetic perturbations cause the SF configuration to develop additional and longer magnetic lobes in the null-point region than the SN, regardless of the plasma model used. The intersection of these longer and additional lobes with the divertor plates are expected to cause more striations in the particle and heat flux target profiles. In addition, the results indicate that the size of the magnetic lobes, in both single-null and snowflake configurations, are more sensitive to resonant magnetic perturbations than to non-resonant magnetic perturbations.
Explicit symplectic methods for solving charged particle trajectories
In this paper, we consider the Lorentz force system based on its Hamiltonian formulation. We decompose the Lorentz force system into four subsystems which can be solved with the help of coordinate transformations. Via the coordinate transformations, three kinds of explicit symplectic numerical methods have been established for simulating the motion of charged particles under the time-independent electromagnetic field. We generalize our methods to solve the system with time-dependent external electromagnetic fields, and also the system with a relativistic effect. In numerical experiments, the computing efficiency and accuracy over a long time for the newly derived methods are demonstrated. Also, the long-term simulation for the dynamics of runaway electrons is performed.
Migration of a carbon adatom on a charged single-walled carbon nanotube
We find that negative charges on an armchair single-walled carbon nanotube (SWCNT) can significantly enhance the migration of a carbon adatom on the external surfaces of SWCNTs, along the direction of the tube axis. Nanotube charging results in stronger binding of adatoms to SWCNTs and consequent longer lifetimes of adatoms before desorption, which in turn increases their migration distance several orders of magnitude. These results support the hypothesis of diffusion enhanced SWCNT growth in the volume of arc plasma. This process could enhance effective carbon flux to the metal catalyst.
Gas Puff Imaging Diagnostics of Edge Plasma Turbulence in Magnetic
Fusion Devices
Gas puff imaging (GPI) is a diagnostic of plasma turbulence which uses a puff of neutral gas at the plasma edge to increase the local visible light emission for improved space-time resolution of plasma fluctuations. This paper reviews gas puff imaging diagnostics of edge plasma turbulence in magnetic fusion research, with a focus on the instrumentation, diagnostic cross-checks, and interpretation issues. The gas puff imaging hardware, optics, and detectors are described for about 10 GPI systems implemented over the past ∼15 years. Comparison of GPI results with other edge turbulence diagnostic results is described, and many common features are observed. Several issues in the interpretation of GPI measurements are discussed, and potential improvements in hardware and modeling are suggested.
Energetic particle modes of q = 1 high-order harmonics in tokamak plasmas with monotonic weak magnetic shear
Linear and nonlinear simulations of high-order harmonics $q=1$ energetic particle modes excited by trapped energetic particles in tokamaks are carried out using kinetic/magnetohydrodynamic hybrid code M3D-K. It is found that with a flat safety factor profile in the core region, the linear growth rate of high-order harmonics $(m=n>1$) driven by energetic trapped particles can be higher than the $m/n=1/1$ component. The high $m=n>1$ modes become more unstable when the pressure of energetic particles becomes higher. Moreover, it is shown that there exist multiple resonant locations satisfying different resonant conditions in the phase space of energetic particles for the high-order harmonics modes, whereas there is only one precessional resonance for the $m/n=1/1$ harmonics. The fluid nonlinearity reduces the saturation level of the $n=1$ component, while it hardly affects those of the high $n$ components, especially the modes with $m=n=3,4$. The frequency of these modes does not chirp significantly, which is different with the typical fishbone driven by trapped particles. In addition, the flattening region of energetic particle distribution due to high-order harmonics excitation is wider than that due to $m/n=1/1$ component, although the $m/n=1/1$ component has a higher saturation amplitude.
Modeling of lithium granule injection in NSTX with M3D-C1
In this paper we present initial simulations of pedestal control by Lithium Granule Injection (LGI) in NSTX. A model for small granule ablation has been implemented in the M3D-C1 code [1], allowing the simulation of realistic Lithium granule injections. 2D simulations in NSTX L-mode and H-mode plasmas are done and the effect of granule size, injection angle and velocity on the pedestal gradient increase are studied. For H-mode cases, the amplitude of the local pressure perturbation caused by the granules is highly dependent on the solid granule size. In our simulations, reducing the granule injection velocity allows one to inject more particles at the pedestal top.
Exact collisional moments for plasma fluid theories
The velocity-space moments of the often troublesome nonlinear Landau collision operator are
expressed exactly in terms of multi-index Hermite-polynomial moments of distribution functions.
The collisional moments are shown to be generated by derivatives of two well-known functions,
namely, the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The
resulting formula has a nonlinear dependency on the relative mean flow of the colliding species
normalised to the root-mean-square of the corresponding thermal velocities and a bilinear dependency
on densities and higher-order velocity moments of the distribution functions, with no restriction
on temperature, flow, or mass ratio of the species. The result can be applied to both the classic
transport theory of plasmas that relies on the Chapman-Enskog method, as well as to derive collisional
fluid equations that follow Grad’s moment approach. As an illustrative example, we provide
the collisional ten-moment equations with exact conservation laws for momentum- and energy-transfer
rates.
Modeling of lithium granule injection in NSTX using M3D-C1
In this paper, we present simulations of pedestal control by lithium granule injection (LGI) in NSTX.
A model for small granule ablation has been implemented in the M3D-C1 code [Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)], allowing the simulation of realistic lithium granule injections. 2D and 3D simulations of Li injections in NSTX H-mode plasmas are performed and the effect of granule size, injection angle and velocity on the pedestal gradient increase is studied.
The amplitude of the local pressure perturbation caused by the granules is found to be highly dependent on the solid granule size.
Adjusting the granule injection velocity allows one to inject more particles at the pedestal top.
3D simulations show the destabilization of high order MHD modes whose amplitude is directly linked to the localized pressure perturbation, which is found to depend on the toroidal localization of the granule density source.
Multi-region relaxed magnetohydrodynamics in plasmas with slowly changing boundaries --- resonant response of a plasma slab
The adiabatic limit of a recently proposed dynamical extension of Taylor relaxation, multi-region relaxed magnetohydrodynamics (MRxMHD), is summarized, with special attention to the appropriate definition of a relative magnetic helicity. The formalism is illustrated using a simple two-region, sheared-magnetic-field model similar to the Hahm–Kulsrud–Taylor (HKT) rippled-boundary slab model. In MRxMHD, a linear Grad–Shafranov equation applies, even at finite ripple amplitude. The adiabatic switching on of boundary ripple excites a shielding current sheet opposing reconnection at a resonant surface. The perturbed magnetic field as a function of ripple amplitude is calculated by invoking the conservation of magnetic helicity in the two regions separated by the current sheet. At low ripple amplitude, “half islands” appear on each side of the current sheet, locking the rotational transform at the resonant value. Beyond a critical amplitude, these islands disappear and the rotational transform develops a discontinuity across the current sheet.
Nonlinear simulations of beam-driven compressional Alfvén eigenmodes in NSTX
Results of 3D nonlinear simulations of neutral-beam-driven compressional Alfvén eigenmodes (CAEs) in the National Spherical Torus Experiment (NSTX) are presented. Hybrid MHD-particle simulations for the H-mode NSTX discharge (shot 141398) using the HYM code show unstable CAE modes for a range of toroidal mode numbers, $n=4$−$9$, and frequencies below the ion cyclotron frequency. It is found that the essential feature of CAEs is their coupling to kinetic Alfvén wave (KAW) that occurs on the high-field side at the Alfvén resonance location. High-frequency Alfvén eigenmodes are frequently observed in beam-heated NSTX plasmas, and have been linked to flattening of the electron temperature profiles at high beam power. Coupling between CAE and KAW suggests an energy channeling mechanism to explain these observations, in which beam-driven CAEs dissipate their energy at the resonance location, therefore significantly modifying the energy deposition profile. Nonlinear simulations demonstrate that CAEs can channel the energy of the beam ions from the injection region near the magnetic axis to the location of the resonant mode conversion at the edge of the beam density profile. A set of nonlinear simulations show that the CAE instability saturates due to nonlinear particle trapping, and a large fraction of beam energy can be transferred to several unstable CAEs of relatively large amplitudes and absorbed at the resonant location. Absorption rate shows a strong scaling with the beam power.
Pedestal-to-wall 3D fluid transport simulations on DIII-D
The 3D fluid-plasma edge transport code EMC3-EIRENE is used to test several magnetic field models with and without plasma response against DIII-D experimental data for even and odd-parity $n = 3$ magnetic field perturbations. The field models include ideal and extended MHD equilibria, and the vacuum approximation. Plasma response is required to reduce the stochasticity in the pedestal region for even-parity fields, however too much screening suppresses the measured splitting of the downstream $T_e$ profile. Odd-parity perturbations result in weak tearing and only small additional peaks in the downstream measurements. In this case plasma response is required to increase the size of the lobe structure. No single model is able to simultaneously reproduce the upstream and downstream characteristics for both odd and even-parity perturbations.
Parametric decay of plasma waves near the upper-hybrid resonance
An intense X wave propagating perpendicularly to dc magnetic field is unstable with respect to a parametric decay into an electron Bernstein wave and a lower-hybrid wave. A modified theory of this effect is proposed that extends to the high-intensity regime, where the instability rate $\gamma$ ceases to be a linear function of the incident-wave amplitude. An explicit formula for $\gamma$ is derived and expressed in terms of cold-plasma parameters. Theory predictions are in reasonable agreement
with the results of the particle-in-cell simulations presented in a separate publication.
Improving fast-ion confinement in high-performance discharges by suppressing Alfvén eigenmodes
We show that the degradation of fast-ion confinement in steady-state DIII-D discharges is
quantitatively consistent with predictions based on the effects of multiple unstable Alfvén
eigenmodes on beam-ion transport. Simulation and experiment show that increasing the
radius where the magnetic safety factor has its minimum is effective in minimizing beam-ion
transport. This is favorable for achieving high performance steady-state operation in DIII-D
and future reactors. A comparison between the experiments and a critical gradient model, in
which only equilibrium profiles were used to predict the most unstable modes, show that in a
number of cases this model reproduces the measured neutron rate well.
High frequency fishbone driven by passing energetic ions in tokamak plasmas
High frequency fishbone instability driven by passing energetic ions was first reported in the Princeton beta experiment with tangential neutral-beam-injection (Heidbrink et al 1986 Phys. Rev. Lett. 57 835–8). It could play an important role for ITER-like burning plasmas, where α particles are mostly passing particles. In this work, a generalized energetic ion distribution function and finite drift orbit width effect are considered to improve the theoretical model for passing particle driving fishbone instability. For purely passing energetic ions with zero drift orbit width, the kinetic energy $\delta {{W}_{k}}$ is derived analytically. The derived analytic expression is more accurate as compared to the result of previous work (Wang 2001 Phys. Rev. Lett. 86 5286–8). For a generalized energetic ion distribution function, the fishbone dispersion relation is derived and is solved numerically. Numerical results show that broad and off-axis beam density profiles can significantly increase the beam ion beta threshold ${{\beta}_{c}}$ for instability and decrease mode frequency.
Prediction of nonlinear evolution character of energetic-particle-driven instabilities
A general criterion is proposed and found to successfully predict the emergence of chirping
oscillations of unstable Alfvénic eigenmodes in tokamak plasma experiments. The model
includes realistic eigenfunction structure, detailed phase-space dependences of the instability
drive, stochastic scattering and the Coulomb drag. The stochastic scattering combines the
effects of collisional pitch angle scattering and micro-turbulence spatial diffusion. The latter
mechanism is essential to accurately identify the transition between the fixed-frequency mode
behavior and rapid chirping in tokamaks and to resolve the disparity with respect to chirping
observation in spherical and conventional tokamaks.
Extending geometrical optics: A Lagrangian theory for vector waves
Even when neglecting diffraction effects, the well-known equations of geometrical optics (GO) are
not entirely accurate. Traditional GO treats wave rays as classical particles, which are completely
described by their coordinates and momenta, but vector-wave rays have another degree of freedom,
namely, their polarization. The polarization degree of freedom manifests itself as an effective (classical)
“wave spin” that can be assigned to rays and can affect the wave dynamics accordingly. A
well-known manifestation of polarization dynamics is mode conversion, which is the linear
exchange of quanta between different wave modes and can be interpreted as a rotation of the wave
spin. Another, less-known polarization effect is the polarization-driven bending of ray trajectories.
This work presents an extension and reformulation of GO as a first-principle Lagrangian theory,
whose effective Hamiltonian governs the aforementioned polarization phenomena simultaneously.
As an example, the theory is applied to describe the polarization-driven divergence of right-hand
and left-hand circularly polarized electromagnetic waves in weakly magnetized plasma.
Ponderomotive dynamics of waves in quasiperiodically modulated media
Similarly to how charged particles experience time-averaged ponderomotive forces in high-frequency fields,
linear waves also experience time-averaged refraction in modulated media. Here we propose a covariant
variational theory of this ponderomotive effect on waves for a general nondissipative linear medium. Using
the Weyl calculus, our formulation accommodates waves with temporal and spatial period comparable to that of
the modulation (provided that parametric resonances are avoided). Our theory also shows that any wave is, in
fact, a polarizable object that contributes to the linear dielectric tensor of the ambient medium. The dynamics
of quantum particles is subsumed as a special case. As an illustration, ponderomotive Hamiltonians of quantum
particles and photons are calculated within a number of models. We also explain a fundamental connection
between these results and the well-known electrostatic dielectric tensor of quantum plasmas.
Total fluid pressure imbalance in the scrape-off layer of tokamak plasmas
Simulations using the fully kinetic neoclassical code XGCa ( X-point included guiding- center
axisymmetric) were undertaken to explore the impact of kinetic effects on scrape-off layer
(SOL) physics in DIII-D H-mode plasmas. XGCa is a total-f, gyrokinetic code which selfconsistently
calculates the axisymmetric electrostatic potential and plasma dynamics, and
includes modules for Monte Carlo neutral transport.
Previously presented XGCa results showed several noteworthy features, including large
variations of ion density and pressure along field lines in the SOL, experimentally relevant
levels of SOL parallel ion flow (Mach number ∼ 0.5), skewed ion distributions near the sheath
entrance leading to subsonic flow there, and elevated sheath potentials (Churchill 2016 Nucl.
Mater. Energy 1–6).
In this paper, we explore in detail the question of pressure balance in the SOL, as it was
observed in the simulation that there was a large deviation from a simple total pressure balance
(the sum of ion and electron static pressure plus ion inertia). It will be shown that both the
contributions from the ion viscosity (driven by ion temperature anisotropy) and neutral source
terms can be substantial, and should be retained in the parallel momentum equation in the
SOL, but still falls short of accounting for the observed fluid pressure imbalance in the XGCa
simulation results.
Recent progress in understanding electron thermal transport in NSTX
The anomalous level of electron thermal transport inferred in magnetically confined configurations is one of the most challenging problems for the ultimate realization of fusion power using toroidal devices: tokamaks, spherical tori and stellarators. It is generally believed that plasma instabilities driven by the abundant free energy in fusion plasmas are responsible for the electron thermal transport. The National Spherical Torus eXperiment (NSTX) [Ono et al., Nucl. Fusion 40, 557 (2000)] provides a unique laboratory for studying plasma instabilities and their relation to electron thermal transport due to its low toroidal field, high plasma beta, low aspect ratio and large ${\bf E} \times {\bf B}$ flow shear. Recent findings on NSTX have shown that multiple instabilities are required to explain observed electron thermal transport, given the wide range of equilibrium parameters due to different operational scenarios and radial regions in fusion plasmas. Here we review the recent progresses in understanding anomalous electron thermal transport in NSTX and focus on mechanisms that could drive electron thermal transport in the core region. The synergy between experiment and theoretical/numerical modeling is essential to achieving these progresses. The plans for newly commissioned NSTX-Upgrade will also be discussed.
Total Fluid Pressure Imbalance in the Scrape-Off
Layer of Tokamak Plasmas
Simulations using the fully kinetic neoclassical code XGCa (X-point included guiding- center axisymmetric) were undertaken to explore the impact of kinetic effects on scrape-off layer (SOL) physics in DIII-D H-mode plasmas. XGCa is a total-f, gyrokinetic code which self-consistently calculates the axisymmetric electrostatic potential and plasma dynamics, and includes modules for Monte Carlo neutral transport.
Previously presented XGCa results showed several noteworthy features, including large variations of ion density and pressure along field lines in the SOL, experimentally relevant levels of SOL parallel ion flow (Mach number ~ 0.5), skewed ion distributions near the sheath entrance leading to subsonic flow there, and elevated sheath potentials (Churchill 2016 Nucl. Mater. Energy 1–6).
In this paper, we explore in detail the question of pressure balance in the SOL, as it was observed in the simulation that there was a large deviation from a simple total pressure balance (the sum of ion and electron static pressure plus ion inertia). It will be shown that both the contributions from the ion viscosity (driven by ion temperature anisotropy) and neutral source terms can be substantial, and should be retained in the parallel momentum equation in the SOL, but still falls short of accounting for the observed fluid pressure imbalance in the XGCa simulation results.
Saturation of Alfvén modes in tokamak plasmas investigated by Hamiltonian mapping techniques
Nonlinear dynamics of single toroidal number Alfvén eigenmodes destabilised by the
the resonant interaction with fast ions is investigated, in tokamak equilibria, by means of
Hamiltonian mapping techniques. The results obtained by two different simulation codes,
XHMGC and HAGIS, are presented for $n = 2$ Beta induced Alfvén eigenmodes and,
respectively $n = 6$ toroidal Alfvén eigenmodes. Simulations of the bump-on-tail instability
performed by a 1-dimensional code, PIC1DP, are also analysed for comparison. As a general
feature, modes saturate as the resonant-particle distribution function is flattened over the whole
region where mode-particle power transfer can take place in the linear phase. Such region
is limited by the narrowest of resonance width and mode width. In the former case, mode
amplitude at saturation exhibits a quadratic scaling with the linear growth rate; in the latter
case, the scaling is linear. These results are explained in terms of the approximate analytic
solution of a nonlinear pendulum model. They are also used to prove that the radial width of
the single poloidal harmonic sets an upper limit to the radial displacement of circulating fast
ions produced by a single-toroidal-number gap mode in the large $n$ limit, irrespectively of the
possible existence of a large global mode structure formed by many harmonics.
A geometrical correction to the ${\bf E}\times{\bf B}$ drift causes an outward flux of co-current momentum whenever
electrostatic potential energy is transferred to ion parallel flows. The robust, fully nonlinear
symmetry breaking follows from the free-energy flow in phase space and does not depend on any
assumed linear eigenmode structure. The resulting rotation peaking is counter-current and scales as
temperature over plasma current. This peaking mechanism can only act when fluctuations are low-frequency
enough to excite ion parallel flows, which may explain some recent experimental observations
related to rotation reversals.
Photon polarizability and its effect on the dispersion of plasma waves
High-frequency photons travelling in plasma exhibit a linear polarizability that can influence the dispersion of linear plasma waves. We present a detailed calculation of this effect for Langmuir waves as a characteristic example. Two alternative formulations are given. In the first formulation, we calculate the modified dispersion of Langmuir waves by solving the governing equations for the electron fluid, where the photon contribution enters as a ponderomotive force. In the second formulation, we provide a derivation based on the photon polarizability. Then, the calculation of ponderomotive forces is not needed, and the result is more general.
Kinetic simulations of scrape-off layer physics in the DIII-D tokamak
Simulations using the fully kinetic code XGCa were undertaken to explore the impact of kinetic effects on scrape-off layer (SOL) physics in DIII-D H-mode plasmas. XGCa is a total-f, gyrokinetic code which self-consistently calculates the axisymmetric electrostatic potential and plasma dynamics, and includes modules for Monte Carlo neutral transport. Fluid simulations are normally used to simulate the SOL, due to its high collisionality. However, depending on plasma conditions, a number of discrepancies have been observed between experiment and leading SOL fluid codes (e.g. SOLPS), including underestimating outer target temperatures, radial electric field in the SOL, parallel ion SOL flows at the low field side, and impurity radiation. Many of these discrepancies may be linked to the fluid treatment, and might be resolved by including kinetic effects in SOL simulations.
The XGCa simulation of the DIII-D tokamak in a nominally sheath-limited regime show many noteworthy features in the SOL. The density and ion temperature are higher at the low-field side, indicative of ion orbit loss. The SOL ion Mach flows are at experimentally relevant levels (Mi ∼ 0.5), with similar shapes and poloidal variation as observed in various tokamaks. Surprisingly, the ion Mach flows close to the sheath edge remain subsonic, in contrast to the typical fluid Bohm criterion requiring ion flows to be above sonic at the sheath edge. Related to this are the presence of elevated sheath potentials, eΔΦ/Te∼3−4, over most of the SOL, with regions in the near-SOL close to the separatrix having eΔΦ/Te > 4. These two results at the sheath edge are a consequence of non-Maxwellian features in the ions and electrons there.
Proposals to reach the next generation of laser intensities through Raman or Brillouin backscattering have centered on optical frequencies. Higher frequencies are beyond the range of such methods mainly due to the wave damping that accompanies the higher-density plasmas necessary for compressing higher frequency lasers. However, we find that an external magnetic field transverse to the direction of laser propagation can reduce the required plasma density. Using parametric interactions in magnetized plasmas to mediate pulse compression, both reduces the wave damping and alleviates instabilities, thereby enabling higher frequency or lower intensity pumps to produce pulses at higher intensities and longer durations. In addition to these theoretical advantages, our method in which strong uniform magnetic fields lessen the need for high-density uniform plasmas also lessens key engineering challenges or at least exchanges them for different challenges.
Variational principles for dissipative (sub)systems, with applications to the theory of linear dispersion and geometrical optics
Applications of variational methods are typically restricted to conservative systems. Some extensions to dissipative systems have been reported too but require ad hoc techniques such as the artificial doubling of the dynamical variables. Here, a different approach is proposed. We show that, for a broad class of dissipative systems of practical interest, variational principles can be formulated using constant Lagrange multipliers and Lagrangians nonlocal in time, which allow treating reversible and irreversible dynamics on the same footing. A general variational theory of linear dispersion is formulated as an example. In particular, we present a variational formulation for linear geometrical optics in a general dissipative medium, which is allowed to be nonstationary, inhomogeneous, anisotropic, and exhibit both temporal and spatial dispersion simultaneously.
On the correspondence between classical geometric phase of gyro-motion and quantum Berry phase
We show that the geometric phase of the gyro-motion of a classical charged particle in a uniform time-dependent magnetic field described by Newton's equation can be derived from a coherent Berry phase for the coherent states of the Schrödinger equation or the Dirac equation. This correspondence is established by constructing coherent states for a particle using the energy eigenstates on the Landau levels and proving that the coherent states can maintain their status of coherent states during the slow varying of the magnetic field. It is discovered that the orbital Berry phases of the eigenstates interfere coherently to produce an observable effect (which we termed “coherent Berry phase”), which is exactly the geometric phase of the classical gyro-motion. This technique works for the particles with and without spin. For particles with spin, on each of the eigenstates that make up the coherent states, the Berry phase consists of two parts that can be identified as those due to the orbital and the spin motion. It is the orbital Berry phases that interfere coherently to produce a coherent Berry phase corresponding to the classical geometric phase of the gyro-motion. The spin Berry phases of the eigenstates, on the other hand, remain to be quantum phase factors for the coherent states and have no classical counterpart.
Continuum kinetic and multi-fluid simulations of classical sheaths
The kinetic study of plasma sheaths is critical, among other things, to understand the deposition of
heat on walls, the effect of sputtering, and contamination of the plasma with detrimental impurities.
The plasma sheath also provides a boundary condition and can often have a significant global
impact on the bulk plasma. In this paper, kinetic studies of classical sheaths are performed with the
continuum kinetic code, Gkeyll, which directly solves the Vlasov-Maxwell equations. The code
uses a novel version of the finite-element discontinuous Galerkin scheme that conserves energy in
the continuous-time limit. The fields are computed using Maxwell equations. Ionization and scattering
collisions are included; however, surface effects are neglected. The aim of this work is to
introduce the continuum kinetic method and compare its results with those obtained from an
already established finite-volume multi-fluid model also implemented in Gkeyll. Novel boundary
conditions on the fluids allow the sheath to form without specifying wall fluxes, so the fluids and
fields adjust self-consistently at the wall. The work presented here demonstrates that the kinetic
and fluid results are in agreement for the momentum flux, showing that in certain regimes, a multi-
fluid model can be a useful approximation for simulating the plasma boundary. There are differences
in the electrostatic potential between the fluid and kinetic results. Further, the direct solutions
of the distribution function presented here highlight the non-Maxwellian distribution of electrons in
the sheath, emphasizing the need for a kinetic model. The densities, velocities, and the potential
show a good agreement between the kinetic and fluid results. However, kinetic physics is
highlighted through higher moments such as parallel and perpendicular temperatures which provide
significant differences from the fluid results in which the temperature is assumed to be isotropic.
Besides decompression cooling, the heat flux is shown to play a role in the temperature differences
that are observed, especially inside the collisionless sheath.
Effect of rotation zero-crossing on single-fluid plasma response to three-dimensional magnetic perturbations
In order to understand the effect of rotation on the response of a plasma to three-dimensional
magnetic perturbations, we perform a systematic scan of the zero-crossing of the rotation profile
in a DIII-D ITER-similar shape equilibrium using linear, time-independent modeling with the
M3D-C1 extended magnetohydrodynamics code. We confirm that the local resonant magnetic
field generally increases as the rotation decreases at a rational surface. Multiple peaks in the
resonant field are observed near rational surfaces, however, and the maximum resonant field does
not always correspond to zero rotation at the surface. Furthermore, we show that non-resonant
current can be driven at zero-crossings not aligned with rational surfaces if there is sufficient
shear in the rotation profile there, leading to amplification of near-resonant Fourier harmonics of
the perturbed magnetic field and a decrease in the far-off-resonant harmonics. The quasilinear
electromagnetic torque induced by this non-resonant plasma response provides drive to flatten
the rotation, possibly allowing for increased transport in the pedestal by the destabilization of
turbulent modes. In addition, this torque acts to drive the rotation zero-crossing to dynamically
stable points near rational surfaces, which would allow for increased resonant penetration. By
one or both of these mechanisms, this torque may play an important role in bifurcations into
suppression of edge-localized modes. Finally, we discuss how these changes to the plasma
response could be detected by tokamak diagnostics. In particular, we show that the changes to
the resonant field discussed here have a significant impact on the external perturbed magnetic
field, which should be observable by magnetic sensors on the high-field side of tokamaks but not
on the low-field side. In addition, TRIP3D-MAFOT simulations show that none of the changes
to the plasma response described here substantially affects the divertor footprint structure.
Explicit K-symplectic algorithms for charged particle dynamics
We study the Lorentz force equation of charged particle dynamics by considering its K-symplectic structure. As the Hamiltonian of the system can be decomposed as four parts, we are able to construct the numerical methods that preserve the K-symplectic structure based on Hamiltonian splitting technique. The newly derived numerical methods are explicit, and are shown in numerical experiments to be stable over long-term simulation. The error convergency as well as the long term energy conservation of the numerical solutions is also analyzed by means of the Darboux transformation.
Photons, phonons, and plasmons with orbital angular momentum in plasmas
Exact eigen modes with orbital angular momentum (OAM) in the complex media of unmagnetized homogeneous plasmas are studied. Three exact eigen modes with OAM are derived, i.e., photons, phonons, and plasmons. The OAM of different plasma components are closely related to the charge polarities. For photons, the OAM of electrons and ions are of the same magnitude but opposite direction, and the total OAM is carried by the field. For the phonons and plasmons, their OAM are carried by the electrons and ions. The OAM modes in plasmas and their characteristics can be explored for potential applications in plasma physics and accelerator physics.
Investigation of instabilities and rotation alteration in high beta KSTAR plasmas
H-mode plasma operation of the Korea Superconducting Tokamak Advanced Research (KSTAR) device has been expanded to significantly surpass the ideal MHD no-wall beta limit. Plasmas with high normalized beta, $\beta_N$, up to 4.3 have been achieved with reduced plasma internal inductance, $l_i$, to near 0.7, exceeding the computed $n=1$ ideal no-wall limit by a factor of 1.6. Pulse lengths at maximum $\beta_N$ were extended to longer pulses by new, more rapid control.
The stability of the observed $m/n$$=$$2/1$ tearing mode that limited the achieved high $\beta_N$ is computed by the M3D-$C^1$ code, and the effect of sheared toroidal rotation to tearing stability is examined.
As a method to affect the mode stability in high $\beta_N$ plasmas, the non-resonant alteration of the rotation profile by non-axisymmetric magnetic fields has been used, enabling a study of the underlying neoclassical toroidal viscosity (NTV) physics and stability dependence on rotation. Non-axisymmetric field spectra were applied using in-vessel control coils (IVCCs) with varied $n=2$ field configurations to alter the plasma toroidal rotation profile in high beta H-mode plasmas and to analyze their effects on the rotation. The rotation profile was significantly altered with rotation reduced by more than 60% without tearing activity or mode locking. To investigate the physical characteristics and scaling of the measured rotation braking by NTV, changes in the rotation profile are analytically examined in steady state. The expected NTV scaling with the square of the normalized applied field perturbation agrees with the measured profile change $\delta B^{2.1-2.3}$. The NTV is also found to scale as $T_i^{2.1-2.4}$, in general agreement with the low collisionality “1/$\nu$” regime scaling of the NTV theory $(T_{NTV . (1/\nu)} \propto T_i^{2.5})$.
Simulations of ion velocity distribution functions taking into account both elastic and charge exchange collisions
Based on accurate representation of the He+–He angular differential scattering cross sections consisting of both elastic and charge exchange collisions, we performed detailed numerical simulations of the ion velocity distribution functions (IVDF) by Monte Carlo collision method (MCC). The results of simulations are validated by comparison with the experimental data of the ion mobility and the transverse diffusion. The IVDF simulation study shows that due to significant effect of scattering in elastic collisions IVDF cannot be separated into product of two independent IVDFs in the transverse and parallel to the electric field directions.
Ion velocity distribution functions in argon and helium discharges: detailed comparison of numerical simulation results and experimental data
Using the Monte Carlo collision method, we have performed simulations of ion velocity distribution functions (IVDF) taking into account both elastic collisions and charge exchange collisions of ions with atoms in uniform electric fields for argon and helium background gases. The simulation results are verified by comparison with the experiment data of the ion mobilities and the ion transverse diffusion coefficients in argon and helium. The recently published experimental data for the first seven coefficients of the Legendre polynomial expansion of the ion energy and angular distribution functions are used to validate simulation results for IVDF. Good agreement between measured and simulated IVDFs shows that the developed simulation model can be used for accurate calculations of IVDFs.
Kinetic Neoclassical Calculations of Impurity Radiation Profiles
Modifications of the drift-kinetic transport code XGC0 to include the transport, ionization, and recombination of individual charge states, as well as the associated radiation, are described. The code is first applied to a simulation of an NSTX H-mode discharge with carbon impurity to demonstrate the approach to coronal equilibrium. The effects of neoclassical phenomena on the radiated power profile are examined sequentially through the activation of individual physics modules in the code. Orbit squeezing and the neoclassical inward pinch result in increased radiation for temperatures above a few hundred eV and changes to the ratios of charge state emissions at a given electron temperature. Analogous simulations with a neon impurity yield qualitatively similar results.
Lorentz covariant canonical symplectic algorithms for dynamics of charged particles
In this paper, the Lorentz covariance of algorithms is introduced. Under Lorentz transformation, both the form and performance of a Lorentz covariant algorithm are invariant. To acquire the advantages of symplectic algorithms and Lorentz covariance, a general procedure for constructing Lorentz covariant canonical symplectic algorithms (LCCSAs) is provided, based on which an explicit LCCSA for dynamics of relativistic charged particles is built. LCCSA possesses Lorentz invariance as well as long-term numerical accuracy and stability, due to the preservation of a discrete symplectic structure and the Lorentz symmetry of the system. For situations with time-dependent electromagnetic fields, which are difficult to handle in traditional construction procedures of symplectic algorithms, LCCSA provides a perfect explicit canonical symplectic solution by implementing the discretization in 4-spacetime. We also show that LCCSA has built-in energy-based adaptive time steps, which can optimize the computation performance when the Lorentz factor varies.
A family of new explicit, revertible, volume-preserving numerical schemes for the system of Lorentz force
The Lorentz system underlies the fundamental rules for the motion of charged particle in electromagnetic field, which is proved volume-preserving. In this paper, we construct a family of new revertible numerical schemes for general autonomous systems, which in particular, are explicit and volume-preserving for Lorentz systems. These new schemes can prevent the extra numerical errors caused by mismatched initial half-step values in the Boris-like algorithm. Numerical experiments demonstrate the superiorities of our second-order methods in long-term simulations and energy preservation over the Boris algorithm and a higher order Runge-Kutta method (RK3). We also apply these new methods to the guiding center system and find that they behave much better than RK3.
Kinetic Simulations of Scrape-Off Layer Physics in the DIII-D
Tokamak
Simulations using the fully kinetic code XGCa were undertaken to explore the impact of kinetic effects on scrape-off layer (SOL) physics in DIII-D H-mode plasmas. XGCa is a total-f, gyrokinetic code which self-consistently calculates the axisymmetric electrostatic potential and plasma dynamics, and includes modules for Monte Carlo neutral transport. Fluid simulations are normally used to simulate the SOL, due to its high collisionality. However, depending on plasma conditions, a number of discrepancies have been observed between experiment and leading SOL fluid codes (e.g. SOLPS), including underestimating outer target temperatures, radial electric field in the SOL, parallel ion SOL flows at the low field side, and impurity radiation. Many of these discrepancies may be linked to the fluid treatment, and might be resolved by including kinetic effects in SOL simulations.
The XGCa simulation of the DIII-D tokamak in a nominally sheath-limited regime show many noteworthy features in the SOL. The density and ion temperature are higher at the low-field side, indicative of ion orbit loss. The SOL ion Mach flows are at experimentally relevant levels (Mi ∼ 0.5), with similar shapes and poloidal variation as observed in various tokamaks. Surprisingly, the ion Mach flows close to the sheath edge remain subsonic, in contrast to the typical fluid Bohm criterion requiring ion flows to be above sonic at the sheath edge. Related to this are the presence of elevated sheath potentials, eΔΦ/Te∼3−4, over most of the SOL, with regions in the near-SOL close to the separatrix having eΔΦ/Te > 4. These two results at the sheath edge are a consequence of non-Maxwellian features in the ions and electrons there.
Effect of collisions on the two-stream instability in a finite length plasma
The instability of a monoenergetic electron beam in a collisional one-dimensional plasma bounded between grounded walls is considered both analytically and numerically. Collisions between electrons and neutrals are accounted for the plasma electrons only. Solution of a dispersion equation shows that the temporal growth rate of the instability is a decreasing linear function of the collision frequency which becomes zero when the collision frequency is two times the collisionless growth rate. This result is confirmed by fluid simulations. Practical formulas are given for the estimate of the threshold beam current which is required for the two-stream instability to develop for a given system length, neutral gas pressure, plasma density, and beam energy. Particle-in-cell simulations carried out with different neutral densities and beam currents demonstrate a good agreement with the fluid theory predictions for both the growth rate and the threshold beam current.
Zonal-flow dynamics from a phase-space perspective
The wave kinetic equation (WKE) describing drift-wave (DW) turbulence is widely used in the
studies of zonal flows (ZFs) emerging from DW turbulence. However, this formulation neglects the
exchange of enstrophy between DWs and ZFs and also ignores effects beyond the geometricaloptics
limit. We derive a modified theory that takes both of these effects into account, while still
treating DW quanta (“driftons”) as particles in phase space. The drifton dynamics is described by
an equation of the Wigner–Moyal type, which is commonly known in the phase-space formulation
of quantum mechanics. In the geometrical-optics limit, this formulation features additional terms
missing in the traditional WKE that ensure exact conservation of the total enstrophy of the system,
in addition to the total energy, which is the only conserved invariant in previous theories based on
the WKE. Numerical simulations are presented to illustrate the importance of these additional
terms. The proposed formulation can be considered as a phase-space representation of the second order
cumulant expansion, or CE2.
Large-scale dynamo action precedes turbulence in shearing box simulations of the magnetorotational instability
We study the dynamo generation (exponential growth) of large-scale (planar averaged) fields in unstratified shearing box simulations of the magnetorotational instability (MRI). In contrast to previous studies restricted to horizontal (x–y) averaging, we also demonstrate the presence of large-scale fields when vertical (y–z) averaging is employed instead. By computing space–time planar averaged fields and power spectra, we find large-scale dynamo action in the early MRI growth phase – a previously unidentified feature. Non-axisymmetric linear MRI modes with low horizontal wavenumbers and vertical wavenumbers near that of expected maximal growth, amplify the large-scale fields exponentially before turbulence and high wavenumber fluctuations arise. Thus the large-scale dynamo requires only linear fluctuations but not non-linear turbulence (as defined by mode–mode coupling). Vertical averaging also allows for monitoring the evolution of the large-scale vertical field and we find that a feedback from horizontal low wavenumber MRI modes provides a clue as to why the large-scale vertical field sustains against turbulent diffusion in the non-linear saturation regime. We compute the terms in the mean field equations to identify the individual contributions to large-scale field growth for both types of averaging. The large-scale fields obtained from vertical averaging are found to compare well with global simulations and quasi-linear analytical analysis from a previous study by Ebrahimi & Blackman. We discuss the potential implications of these new results for understanding the large-scale MRI dynamo saturation and turbulence.
Local properties of magnetic reconnection in nonlinear resistive- and extended-magnetohydrodynamic toroidal simulations of the sawtooth crash
We diagnose local properties of magnetic reconnection during a sawtooth crash employing the three-dimensional toroidal, extended-magnetohydrodynamic (MHD) code M3D-C1. To do so, we sample simulation data in the plane in which reconnection occurs, the plane perpendicular to the helical $(m,n)=(1,1)$ mode at the $q = 1$ surface, where $m$ and $n$ are the poloidal and toroidal mode numbers and $q$ is the safety factor. We study the nonlinear evolution of a particular test equilibrium in a non-reduced field representation using both resistive-MHD and extended-MHD models. We find growth rates for the extended-MHD reconnection process exhibit a nonlinear acceleration and greatly exceed that of the resistive-MHD model, as is expected from previous experimental, theoretical, and computational work. We compare the properties of reconnection in the two simulations, revealing the reconnecting current sheets are locally different in the two models and we present the first observation of the quadrupole out-of-plane Hall magnetic field that appears during extended-MHD reconnection in a 3D toroidal simulation (but not in resistive-MHD). We also explore the dependence on toroidal angle of the properties of reconnection as viewed in the plane perpendicular to the helical magnetic field, finding qualitative and quantitative effects due to changes in the symmetry of the reconnection process. This study is potentially important for a wide range of magnetically confined fusion applications, from confirming simulations with extended-MHD effects are sufficiently resolved to describe reconnection, to quantifying local reconnection rates for purposes of understanding and predicting transport, not only at the $q = 1$ rational surface for sawteeth, but also at higher order rational surfaces that play a role in disruptions and edge-confinement degradation.
Dynamo-driven plasmoid formation from a current-sheet instability
Axisymmetric current-carrying plasmoids are formed in the presence of nonaxisymmetric fluctuations
during nonlinear three-dimensional resistive MHD simulations in a global toroidal geometry.
We utilize the helicity injection technique to form an initial poloidal flux in the presence of a toroidal
guide field. As helicity is injected, two types of current sheets are formed from (1) the oppositely
directed field lines in the injector region (primary reconnecting current sheet), and (2) the
poloidal flux compression near the plasma edge (edge current sheet). We first find that nonaxisymmetric
fluctuations arising from the current-sheet instability isolated near the plasma edge have
tearing parity but can nevertheless grow fast (on the poloidal Alfven time scale). These modes saturate
by breaking up the current sheet. Second, for the first time, a dynamo poloidal flux amplification
is observed at the reconnection site (in the region of the oppositely directed magnetic field).
This fluctuation-induced flux amplification increases the local Lundquist number, which then triggers
a plasmoid instability and breaks the primary current sheet at the reconnection site. The plasmoids
formation driven by large-scale flux amplification, i.e., a large-scale dynamo, observed here
has strong implications for astrophysical reconnection as well as fast reconnection events in laboratory
plasmas.
Structure of nonlocal gradient-drift instabilities in Hall E x B discharges
Gradient-drift (collisionless Simon-Hoh) instability is a robust instability often considered to be important for Hall plasma discharges supported by the electron current due to the ${\bf E} \times {\bf B}$ drift. Most of the previous studies of this mode were based on the local approximation. Here, we consider the nonlocal model which takes into account the electron inertia as well as the effects of the entire profiles of plasma parameters such as the electric, magnetic fields, and plasma density. Contrary to local models, nonlocal analysis predicts multiple unstable modes, which exist in the regions, where local instability criteria are not satisfied. This is especially pronounced for the long wavelength modes which provide larger contribution to the anomalous transport.
Local properties of magnetic reconnection in nonlinear resistive- and extended-magnetohydrodynamic toroidal simulations of the sawtooth crash
We diagnose local properties of magnetic reconnection during a sawtooth crash employing the three-dimensional toroidal, extended-magnetohydrodynamic (MHD) code M3D-$C^1$.
To do so, we sample simulation data in the plane in which reconnection occurs, the plane perpendicular to the helical $(m,n)$$=$$(1,1)$ mode at the $q=1$ surface, where $m$ and $n$ are the poloidal and toroidal mode numbers and $q$ is the safety factor.
We study the nonlinear evolution of a particular test equilibrium in a non-reduced field representation using both resistive-MHD and extended-MHD models.
We find growth rates for the extended-MHD reconnection process exhibit a nonlinear acceleration and greatly exceed that of the resistive-MHD model, as is expected from previous experimental, theoretical, and computational work.
We compare the properties of reconnection in the two simulations, revealing the reconnecting current sheets are locally different in the two models and we present the first observation of the quadrupole out-of-plane Hall magnetic field that appears during extended-MHD reconnection in a 3D toroidal simulation (but not in resistive-MHD).
We also explore the dependence on toroidal angle of the properties of reconnection as viewed in the plane perpendicular to the helical magnetic field, finding qualitative and quantitative effects due to changes in the symmetry of the reconnection process.
This study is potentially important for a wide range of magnetically confined fusion applications, from confirming simulations with extended-MHD effects are sufficiently resolved to describe reconnection, to quantifying local reconnection rates for purposes of understanding and predicting transport, not only at the $q=1$ rational surface for sawteeth, but also at higher order rational surfaces that play a role in disruptions and edge-confinement degradation.
Modeling of reduced effective secondary electron emission yield from a velvet surface
Complex structures on a material surface can significantly reduce total secondary electron emission from that surface. A velvet is a surface that consists of an array of vertically standing whiskers. The reduction occurs due to the capture of low-energy, true secondary electrons emitted at the bottom of the structure and on the sides of the velvet whiskers. We performed numerical simulations and developed an approximate analytical model that calculates the net secondary electron emission yield from a velvet surface as a function of the velvet whisker length and packing density, and the angle of incidence of primary electrons. We found that to suppress secondary electrons, the following condition on dimensionless parameters must be met: (π/2)DA tan θ≫1(π/2)DA tan θ≫1, where θ is the angle of incidence of the primary electron from the normal, D is the fraction of surface area taken up by the velvet whisker bases, and A is the aspect ratio, A ≡ h/r, the ratio of height to radius of the velvet whiskers. We find that velvets available today can reduce the secondary electron yield by 90% from the value of a flat surface. The values of optimal velvet whisker packing density that maximally suppresses the secondary electron emission yield are determined as a function of velvet aspect ratio and the electron angle of incidence.
Magnetorotational Turbulence and Dynamo in a Collisionless Plasma
We present results from the first 3D kinetic numerical simulation of magnetorotational turbulence and
dynamo, using the local shearing-box model of a collisionless accretion disk. The kinetic magnetorotational
instability grows from a subthermal magnetic field having zero net flux over the computational
domain to generate self-sustained turbulence and outward angular-momentum transport. Significant
Maxwell and Reynolds stresses are accompanied by comparable viscous stresses produced by field-aligned
ion pressure anisotropy, which is regulated primarily by the mirror and ion-cyclotron instabilities through
particle trapping and pitch-angle scattering. The latter endow the plasma with an effective viscosity that is
biased with respect to the magnetic-field direction and spatiotemporally variable. Energy spectra suggest an
Alfvén-wave cascade at large scales and a kinetic-Alfvén-wave cascade at small scales, with strong smallscale
density fluctuations and weak nonaxisymmetric density waves. Ions undergo nonthermal particle
acceleration, their distribution accurately described by a κ distribution. These results have implications for
the properties of low-collisionality accretion flows, such as that near the black hole at the Galactic center.
Validation and benchmarking of two particle-in-cell codes for a glow discharge
The two particle-in-cell codes EDIPIC and LSP are benchmarked and validated for a
parallel-plate glow discharge in helium, in which the axial electric field had been carefully
measured, primarily to investigate and improve the fidelity of their collision models. The
scattering anisotropy of electron-impact ionization, as well as the value of the secondaryelectron
emission yield, are not well known in this case. The experimental uncertainty for the
emission yield corresponds to a factor of two variation in the cathode current. If the emission
yield is tuned to make the cathode current computed by each code match the experiment,
the computed electric fields are in excellent agreement with each other, and within about
10% of the experimental value. The non-monotonic variation of the width of the cathode fall
with the applied voltage seen in the experiment is reproduced by both codes. The electron
temperature in the negative glow is within experimental error bars for both codes, but the
density of slow trapped electrons is underestimated. A more detailed code comparison done
for several synthetic cases of electron-beam injection into helium gas shows that the codes are
in excellent agreement for ionization rate, as well as for elastic and excitation collisions with
isotropic scattering pattern. The remaining significant discrepancies between the two codes are
due to differences in their electron binary-collision models, and for anisotropic scattering due
to elastic and excitation collisions.
Validation and benchmarking of two particle-in-cell codes for a glow discharge
The two particle-in-cell codes EDIPIC and LSP are benchmarked and validated for a parallel-plate glow discharge in helium, in which the axial electric field had been carefully measured, primarily to investigate and improve the fidelity of their collision models. The scattering anisotropy of electron-impact ionization, as well as the value of the secondary-electron emission yield, are not well known in this case. The experimental uncertainty for the emission yield corresponds to a factor of two variation in the cathode current. If the emission yield is tuned to make the cathode current computed by each code match the experiment, the computed electric fields are in excellent agreement with each other, and within about 10% of the experimental value. The non-monotonic variation of the width of the cathode fall with the applied voltage seen in the experiment is reproduced by both codes. The electron temperature in the negative glow is within experimental error bars for both codes, but the density of slow trapped electrons is underestimated. A more detailed code comparison done for several synthetic cases of electron-beam injection into helium gas shows that the codes are in excellent agreement for ionization rate, as well as for elastic and excitation collisions with isotropic scattering pattern. The remaining significant discrepancies between the two codes are due to differences in their electron binary-collision models, and for anisotropic scattering due to elastic and excitation collisions.
Generalized Kapchinskij-Vladimirskij Distribution and Beam Matrix for Phase-Space Manipulations of High-Intensity Beams
In an uncoupled linear lattice system, the Kapchinskij-Vladimirskij (KV) distribution formulated on the basis of the single-particle Courant-Snyder invariants has served as a fundamental theoretical basis for the analyses of the equilibrium, stability, and transport properties of high-intensity beams for the past several decades. Recent applications of high-intensity beams, however, require beam phase-space manipulations by intentionally introducing strong coupling. In this Letter, we report the full generalization of the KV model by including all of the linear (both external and space-charge) coupling forces, beam energy variations, and arbitrary emittance partition, which all form essential elements for phase-space manipulations. The new generalized KV model yields spatially uniform density profiles and corresponding linear self-field forces as desired. The corresponding matrix envelope equations and beam matrix for the generalized KV model provide important new theoretical tools for the detailed design and analysis of high-intensity beam manipulations, for which previous theoretical models are not easily applicable.
Generalized Kapchinskij-Vladimirskij Distribution and Beam Matrix for Phase-Space Manipulations of High-Intensity Beams
In an uncoupled linear lattice system, the Kapchinskij-Vladimirskij (KV) distribution formulated on the
basis of the single-particle Courant-Snyder invariants has served as a fundamental theoretical basis for the
analyses of the equilibrium, stability, and transport properties of high-intensity beams for the past several
decades. Recent applications of high-intensity beams, however, require beam phase-space manipulations
by intentionally introducing strong coupling. In this Letter, we report the full generalization of the KV
model by including all of the linear (both external and space-charge) coupling forces, beam energy
variations, and arbitrary emittance partition, which all form essential elements for phase-space manipulations.
The new generalized KV model yields spatially uniform density profiles and corresponding linear
self-field forces as desired. The corresponding matrix envelope equations and beam matrix for the
generalized KV model provide important new theoretical tools for the detailed design and analysis of
high-intensity beam manipulations, for which previous theoretical models are not easily applicable.
Band structure of the growth rate of the two-stream instability of an electron beam propagating in a bounded plasma
This paper presents a study of the two-stream instability of an electron beam propagating in a finite-size plasma placed between two electrodes. It is shown that the growth rate in such a system is much smaller than that of an infinite plasma or a finite size plasma with periodic boundary conditions.
Even if the width of the plasma matches the resonance condition for a standing wave, a spatially growing wave is excited instead with the growth rate small compared to that of the standing wave in a periodic system.
The approximate expression for this growth rate is
$γ$ $\approx$ $(1/13)$ $\omega_{pe}(n_b/n_p)$ $(L\omega_{pe}/v_b)$ $\ln(L \omega_{pe}/v_b)$ $[1−0.18 \cos (L \omega_{pe}/v_b+\pi/2)]$ $γ$ $≈$ $(1/13)$ $\omega_{pe}(n_b/n_p)$ $(L \omega_{pe}/v_b)$ $ln(L \omega_{pe}/v_b)$ $[1−0.18 cos (L \omega_{pe}/v_b+π/2)]$, where $\omega_{pe}$ is the electron plasma frequency, $n_b$ and $n_p$ are the beam and the plasma densities, respectively, $v_b$ is the beam velocity, and $L$ is the plasma width. The frequency, wave number, and the spatial and temporal growth rates, as functions of the plasma size, exhibit band structure. The amplitude of saturation of the instability depends on the system length, not on the beam current. For short systems, the amplitude may exceed values predicted for infinite plasmas by more than an order of magnitude.
Verification of the SPEC code in stellarator geometries
We present the first calculations performed with the Stepped-Pressure Equilibrium Code (SPEC) in stellarator geometry. Provided a boundary magnetic surface, stellarator vacuum fields with islands are computed and verified to machine precision, for both a classical $l = 2$ stellarator field and a Wendelstein 7-X limiter configuration of the first experimental campaign. Beyond verification, a detailed comparison of SPEC solutions to Biot-Savart solutions for the corresponding coil currents is shown. The level of agreement is quantified, and the error is shown to be dominated by the accuracy with which the boundary representation is given. Finally, partially relaxed stellarator equilibria are computed with SPEC, and verification is presented with force-balance down to machine precision.
Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E x B discharges
Partially-magnetized plasmas with magnetized electrons and non-magnetized ions are common in Hall thrusters for electric propulsion and magnetron material processing devices. These plasmas are usually in strongly non-equilibrium state due to presence of crossed electric and magnetic fields, inhomogeneities of plasma density, temperature, magnetic field and beams of accelerated ions. Free energy from these sources make such plasmas prone to various instabilities resulting in turbulence, anomalous transport, and appearance of coherent structures as found in experiments. This paper provides an overview of instabilities that exist in such plasmas. A nonlinear fluid model has been developed for description of the Simon-Hoh, lower-hybrid and ion-sound instabilities. The model also incorporates electron gyroviscosity describing the effects of finite electron temperature. The nonlinear fluid model has been implemented in the BOUT++ framework. The results of nonlinear simulations are presented demonstrating turbulence, anomalous current and tendency toward the formation of coherent structures.
Explicit high-order noncanonical symplectic algorithms for ideal two-fluid systems
An explicit high-order noncanonical symplectic algorithm for ideal two-fluid systems is developed. The fluid is discretized as particles in the Lagrangian description, while the electromagnetic fields and internal energy are treated as discrete differential form fields on a fixed mesh. With the assistance of Whitney interpolating forms [H. Whitney, Geometric Integration Theory (Princeton University Press, 1957); M. Desbrun et al., Discrete Differential Geometry (Springer, 2008); J. Xiao et al., Phys. Plasmas 22, 112504 (2015)], this scheme preserves the gauge symmetry of the electromagnetic field, and the pressure field is naturally derived from the discrete internal energy. The whole system is solved using the Hamiltonian splitting method discovered by He et al. [Phys. Plasmas 22, 124503 (2015)], which was been successfully adopted in constructing symplectic particle-in-cell schemes [J. Xiao et al., Phys. Plasmas 22, 112504 (2015)]. Because of its structure preserving and explicit nature, this algorithm is especially suitable for large-scale simulations for physics problems that are multi-scale and require long-term fidelity and accuracy. The algorithm is verified via two tests: studies of the dispersion relation of waves in a two-fluid plasma system and the oscillating two-stream instability.
Energetic particle-driven compressional Alfvén eigenmodes and prospects for ion cyclotron emission studies in fusion plasmas
As a fundamental plasma oscillation the compressional Alfvén waves(CAWs) are interesting for
plasma scientists both academically and in applications for fusion plasmas. They are believed to be
responsible for the ion cyclotron emission (ICE) observed in many tokamaks. The theory of CAW and
ICE was significantly advanced at the end of 20th century in particular motivated by first DT
experiments on TFTR and subsequent JET DT experimental studies. More recently, ICE theory was
advanced by ST (or spherical torus) experiments with the detailed theoretical and experimental studies
of the properties of each instability signal. There the instability responsible for ICE signals previously
indistinguishable in high aspect ratio tokamaks became the subjects of experimental studies. We
discuss further the prospects of ICE theory and its applications for future burning plasma experiments
such as the ITER tokamak-reactor prototype being build in France where neutrons and gamma rays
escaping the plasma create extremely challenging conditions for fusion alpha particle diagnostics.
Impact of ideal MHD stability limits on high-beta hybrid operation
The hybrid scenario is a candidate for stationary high-fusion gain tokamak operation in
ITER and DEMO. To obtain such performance, the energy confinement and the normalized
pressure $\beta_N$ must be maximized, which requires operating near or above ideal MHD no-wall
limits. New experimental findings show how these limits can affect hybrid operation. Even
if hybrids are mainly limited by tearing modes, proximity to the no-wall limit leads to 3D
field amplification that affects plasma profiles, e.g. rotation braking is observed in ASDEX
Upgrade throughout the plasma and peaks in the core. As a result, even the small ASDEX
Upgrade error fields are amplified and their effects become visible. To quantify such effects,
ASDEX Upgrade measured the response to 3D fields applied by 8×2 non-axisymmetric coils
as $\beta_N$ approaches the no-wall limit. The full n = 1 response profile and poloidal structure were
measured by a suite of diagnostics and compared with linear MHD simulations, revealing
a characteristic feature of hybrids: the n = 1 response is due to a global, marginally-stable
n = 1 kink characterized by a large m = 1, n = 1 core harmonic due to qmin being just above
1. A helical core distortion of a few cm forms and affects various core quantities, including
plasma rotation, electron and ion temperature, and intrinsic W density. In similar experiments,
DIII-D also measured the effect of this helical core on the internal current profile, providing
information useful to understanding of the physics of magnetic flux pumping, i.e. anomalous
current redistribution by MHD modes that keeps qmin>1. Thanks to flux pumping, a broad
current profile is maintained in DIII-D even with large on-axis current drive, enabling fully
non-inductive operation at high $\beta_N$ up to 3.5–4.
Nonlinear asymmetric tearing mode evolution in cylindrical geometry
The growth of a tearing mode is described by reduced MHD equations.
For a cylindrical equilibrium, tearing mode growth is governed by the modified Rutherford equation, i.e., the
nonlinear $\Delta^\prime(\omega)$.
For a low beta plasma without external heating, $\Delta^\prime(\omega)$ can be approximately
described by two terms, $\Delta^\prime_{ql}(\omega)$, $\Delta^\prime_A(\omega)$ [White et al., Phys. Fluids 20, 800 (1977); Phys. Plasmas
22, 022514 (2015)].
In this work, we present a simple method to calculate the quasilinear stability
index $\Delta^\prime_{ql}$ rigorously, for poloidal mode number $m \ge 2$.
$\Delta^\prime_{ql}(\omega)$ is derived by solving the outer equation through the Frobenius method.
$\Delta^\prime_{ql}$ is composed of four terms proportional to: constant $\Delta^\prime_{0}$, $\omega$, $\omega \ln \omega$ and $\omega^2$.
$\Delta^\prime_{A}(\omega)$ is proportional to the asymmetry of island that is roughly proportional to $\omega$.
The sum of $\Delta^\prime_{ql}(\omega)$ and $\Delta^\prime_{A}(\omega)$ is consistent with the more accurate expression calculated perturbatively [Arcis et al., Phys. Plasmas 13, 052305 (2006)].
The reduced MHD equations are also solved numerically through a 3D MHD code M3D-C1 [Jardin et al., Comput. Sci. Discovery 5, 014002 (2012)].
The analytical expression of the perturbed helical flux and the saturated island width agree with the simulation results.
It is also confirmed by the simulation that the $\Delta^\prime_{A}(\omega)$ has to be considered in calculating island saturation.
Linear Vlasov theory of a magnetised, thermally stratified atmosphere
The stability of a collisionless, magnetised plasma to local convective disturbances is examined, with a focus on kinetic and finite-Larmor-radius effects. Specific application is made to the outskirts of galaxy clusters, which contain hot and tenuous plasma whose temperature increases in the direction of gravity. At long wavelengths (the ‘drift-kinetic’ limit), we obtain the kinetic version of the magnetothermal instability (MTI) and its Alfvénic counterpart (Alfvénic MTI), which were previously discovered and analysed using a magnetofluid (i.e. Braginskii) description. At sub-ion-Larmor scales, we discover an overstability driven by the electron-temperature gradient of kinetic-Alfvén drift waves – the electron MTI (eMTI) – whose growth rate is even larger than the standard MTI. At intermediate scales, we find that ion finite-Larmor-radius effects tend to stabilise the plasma. We discuss the physical interpretation of these instabilities in detail, and compare them both with previous work on magnetised convection in a collisional plasma and with temperature-gradient-driven drift-wave instabilities well known to the magnetic-confinement-fusion community. The implications of having both fluid and kinetic scales simultaneously driven unstable by the same temperature gradient are briefly discussed.
Approach to Chandrasekhar-Kendall-Woltjer state in a chiral plasma
We study the time evolution of the magnetic field in a plasma with a chiral magnetic current. The vector spherical harmonic (VSH) functions are used to expand all fields. We define a measure for the Chandrasekhar-Kendall-Woltjer (CKW) state, which has a simple form in VSH expansion. We propose the conditions for a general class of initial momentum spectra that will evolve into the CKW state. For this class of initial conditions, to approach the CKW state, (i) a nonvanishing chiral magnetic conductivity is necessary, and (ii) the time integration of the product of the electric resistivity and chiral magnetic conductivity must grow faster than the time integration of the resistivity. We give a few examples to test these conditions numerically, which work very well.
Evidence of Toroidally Localized Turbulence with Applied 3D Fields in the DIII-D Tokamak
New evidence indicates that there is significant 3D variation in density fluctuations near the boundary
of weakly 3D tokamak plasmas when resonant magnetic perturbations are applied to suppress transient
edge instabilities. The increase in fluctuations is concomitant with an increase in the measured density
gradient, suggesting that this toroidally localized gradient increase could be a mechanism for turbulence
destabilization in localized flux tubes. Two-fluid magnetohydrodynamic simulations find that, although
changes to the magnetic field topology are small, there is a significant 3D variation of the density gradient
within the flux surfaces that is extended along field lines. This modeling agrees qualitatively with the
measurements. The observed gradient and fluctuation asymmetries are proposed as a mechanism by which
global profile gradients in the pedestal could be relaxed due to a local change in the 3D equilibrium. These
processes may play an important role in pedestal and scrape-off layer transport in ITER and other future
tokamak devices with small applied 3D fields.
High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields
We construct high order symmetric volume-preserving methods for the relativistic dynamics of a charged particle by the splitting technique with processing. By expanding the phase space to include the time t, we give a more general construction of volume-preserving methods that can be applied to systems with time-dependent electromagnetic fields. The newly derived methods provide numerical solutions with good accuracy and conservative properties over long time of simulation. Furthermore, because of the use of an accuracy-enhancing processing technique, the explicit methods obtain high-order accuracy and are more efficient than the methods derived from standard compositions. The results are verified by the numerical experiments. Linear stability analysis of the methods shows that the high order processed method allows larger time step size in numerical integrations.
Hamiltonian particle-in-cell methods for Vlasov-Maxwell equations
In this paper, we study the Vlasov-Maxwell equations based on the Morrison-Marsden-Weinstein bracket. We develop Hamiltonian particle-in-cell methods for this system by employing finite element methods in space and splitting methods in time. In order to derive the semi-discrete system that possesses a discrete non-canonical Poisson structure, we present a criterion for choosing the appropriate finite element spaces. It is confirmed that some conforming elements, e.g., Nédélec's mixed elements, satisfy this requirement. When the Hamiltonian splitting method is used to discretize this semi-discrete system in time, the resulting algorithm is explicit and preserves the discrete Poisson structure. The structure-preserving nature of the algorithm ensures accuracy and fidelity of the numerical simulations over long time.
Multi-region relaxed Hall magnetohydrodynamics with flow
The recent formulations of multi-region relaxed magnetohydrodynamics (MRxMHD) have generalized the famous Woltjer-Taylor states by incorporating a collection of “ideal barriers” that prevent global relaxation and flow. In this paper, we generalize MRxMHD with flow to include Hall effects, and thereby obtain the partially relaxed counterparts of the famous double Beltrami states as a special subset. The physical and mathematical consequences arising from the introduction of the Hall term are also presented. We demonstrate that our results (in the ideal MHD limit) constitute an important subset of ideal MHD equilibria, and we compare our approach against other variational principles proposed for deriving the partially relaxed states.
Nonlinear Simulations of Coalescence Instability Using a Flux Difference Splitting Method
A flux difference splitting numerical scheme based on the finite volume method is applied to study ideal/resistive magnetohydrodynamics. The ideal/resistive MHD equations are cast as a set of hyperbolic conservation laws, and we develop a numerical capability to solve the weak solutions of these hyperbolic conservation laws by combining a multi-state Harten-Lax-Van Leer approximate Riemann solver with the hyperbolic divergence cleaning technique, high order shock-capturing reconstruction schemes, and a third order total variance diminishing Runge-Kutta time evolving scheme. The developed simulation code is applied to study the long time nonlinear evolution of the coalescence instability. It is verified that small structures in the instability oscillate with time and then merge into medium structures in a coherent manner. The medium structures then evolve and merge into large structures, and this trend continues through all scale-lengths. The physics of this interesting nonlinear dynamics is numerically analyzed.
Effective-action approach to wave propagation in scalar QED plasmas
A relativistic quantum field theory with nontrivial background fields is developed and applied to study waves in plasmas. The effective action of the electromagnetic 4-potential is calculated ab initio from the standard action of scalar QED using path integrals. The resultant effective action is gauge invariant and contains nonlocal interactions, from which gauge bosons acquire masses without breaking the local gauge symmetry. To demonstrate how the general theory can be applied, we give two examples: a cold unmagnetized plasma and a cold uniformly magnetized plasma. Using these two examples, we show that all linear waves well known in classical plasma physics can be recovered from relativistic quantum results when taking the classical limit. In the opposite limit, classical wave dispersion relations are modified substantially. In unmagnetized plasmas, longitudinal waves propagate with nonzero group velocities even when plasmas are cold. In magnetized plasmas, anharmonically spaced Bernstein waves persist even when plasmas are cold. These waves account for cyclotron absorption features observed in spectra of x-ray pulsars. Moreover, cutoff frequencies of the two nondegenerate electromagnetic waves are red-shifted by different amounts. These corrections need to be taken into account in order to correctly interpret diagnostic results in laser plasma experiments.
Impact of magnetic topology on radial electric field profile and comparisons with models of edge transport in the Large Helical Device
The radial electric field in the plasma edge is studied in the Large Helical Device (LHD) experiments. When magnetic field lines become stochastic or open at the plasma edge and connected to the vessel, electrons are lost faster than ions along these field lines. Then, a positive electric field appears in the plasma edge. The radial electric field profile can be used to detect the effective plasma boundary. Magnetic topology is an important issue in stellarator and tokamak research because the 3D boundary has the important role of controlling MHD edge stability with respect to ELMs, and plasma detachment. Since the stochastic magnetic field layer can be controlled in the LHD by changing the preset vacuum magnetic axis, this device is a good platform to study the properties of the radial electric field that appear with the different stochastic layer width. Two magnetic configurations with different widths of the stochastic layer as simulated in vacuum are studied for low-β discharges. It has been found that a positive electric field appeared outside of the last closed flux surface. In fact the positions of the positive electric field are found in the boundary between of the stochastic layer and the scrape-off layer. To understand where is the boundary of the stochastic layer and the scrape-off layer, the magnetic field lines are analyzed statistically. The variance of the magnetic field lines in the stochastic layer is increased outwards for both configurations. However, the skewness, which means the asymmetry of the distribution of the magnetic field line, increases for only one configuration. If the skewness is large, the connection length becomes effectively short. Since that is consistent with the experimental observation, the radial electric field can be considered as an index of the magnetic topology..
Magnetohydrodynamics for collisionless plasmas from the gyrokinetic perspective
The effort to obtain a set of MagnetoHydroDynamic (MHD) equations for a magnetized collisionless plasma was started nearly 60 years ago by Chew et al. [Proc. R. Soc. London, Ser. A 236(1204), 112–118 (1956)]. Many attempts have been made ever since. Here, we will show the derivation of a set of these equations from the gyrokinetic perspective, which we call it gyrokinetic MHD, and it is different from the conventional ideal MHD. However, this new set of equations still has conservation properties and, in the absence of fluctuations, recovers the usual MHD equilibrium. Furthermore, the resulting equations allow for the plasma pressure balance to be further modified by finite-Larmor-radius effects in regions with steep pressure gradients. The present work is an outgrowth of the paper on “Alfven Waves in Gyrokinetic Plasmas” by Lee and Qin [Phys. Plasmas 10, 3196 (2003)].
Envelope Hamiltonian for charged-particle dynamics in general linear coupled systems
Dynamics of a charged particle in the canonical coordinates is a Hamiltonian system, and the well-known symplectic algorithm has been regarded as the de facto method for numerical integration of Hamiltonian systems due to its long-term accuracy and fidelity.
For long-term simulations with high efficiency, explicit symplectic algorithms are desirable.
However, it is generally believed that explicit symplectic algorithms are only available for sum-separable Hamiltonians, and this restriction limits the application of explicit symplectic algorithms to charged particle dynamics.
To overcome this difficulty, we combine the familiar sum-split method and a generating function method to construct second- and third-order explicit symplectic algorithms for dynamics of charged particle.
The generating function method is designed to generate explicit symplectic algorithms for product-separable Hamiltonian with form of
$H({\bf x},{\bf p})=p_i f({\bf x})$ or $H(x,p)=x_i g({\bf p})$.
Applied to the simulations of charged particle dynamics, the explicit symplectic algorithms based on generating functions demonstrate superiorities in conservation and efficiency.
Improved kinetic neoclassical transport calculation for a low-collisionality QH-mode pedestal
The role of neoclassical, anomalous and neutral transport to the overall H-mode pedestal and scrape-off layer (SOL) structure in an ELM-free QH-mode discharge on DIII-D is explored using XGC0, a 5D full-f multi-species particle-in-cell drift-kinetic solver with self-consistent neutral recycling and sheath potentials. The work in this paper builds on previous work aimed at achieving quantitative agreement between the flux-driven simulation and the experimental electron density, impurity density and orthogonal measurements of impurity temperature and flow profiles. Improved quantitative agreement is achieved by performing the calculations with a more realistic electron mass, larger neutral density and including finite-Larmor-radius corrections self-consistently in the drift-kinetic motion of the particles. Consequently, the simulations provide stronger evidence that the radial electric field (${{E}_{\text{r}}}$ ) in the pedestal is primarily established by the required balance between the loss of high-energy tail main ions against a pinch of colder main ions and impurities. The kinetic loss of a small population of ions carrying a large proportion of energy and momentum leads to a separation of the particle and energy transport rates and introduces a source of intrinsic edge torque. Ion orbit loss and finite orbit width effects drive the energy distributions away from Maxwellian, and describe the anisotropy, poloidal asymmetry and local minimum near the separatrix observed in the ${{T}_{i}}$ profile.
On the structure of the two-stream instability -- complex G-Hamiltonian structure and Krein collisions between positive- and negative-action modes
The two-stream instability is probably the most important elementary example of collective instabilities in plasma physics and beam-plasma systems. For a warm plasma with two charged particle species, the instability diagram of the two-stream instability based on a 1D warm-fluid model exhibits an interesting band structure that has not been explained. We show that the band structure for this instability is the consequence of the Hamiltonian nature of the warm two-fluid system. Interestingly, the Hamiltonian nature manifests as a complex G-Hamiltonian structure in wave-number space, which directly determines the instability diagram. Specifically, it is shown that the boundaries between the stable and unstable regions are locations for Krein collisions between eigenmodes with different Krein signatures. In terms of physics, this rigorously implies that the system is destabilized when a positive-action mode resonates with a negative-action mode, and that this is the only mechanism by which the system can be destabilized. It is anticipated that this physical mechanism of destabilization is valid for other collective instabilities in conservative systems in plasma physics, accelerator physics, and fluid dynamics systems, which admit infinite-dimensional Hamiltonian structures.
Design of geometric phase measurement in EAST Tokamak
The optimum scheme for geometric phase measurement in EAST Tokamak is proposed in this paper. The theoretical values of geometric phase for the probe beams of EAST Polarimeter-Interferometer (POINT) system are calculated by path integration in parameter space. Meanwhile, the influences of some controllable parameters on geometric phase are evaluated. The feasibility and challenge of distinguishing geometric effect in the POINT signal are also assessed in detail.
Multi-Species Measurements of the Firehose and Mirror Instability Thresholds
The firehose and mirror instabilities are thought to arise in a variety of space and astrophysical plasmas, constraining the pressure anisotropies and drifts between particle species. The plasma stability depends on all species simultaneously, meaning that a combined analysis is required. Here, we present the first such analysis in the solar wind, using the long-wavelength stability parameters to combine the anisotropies and drifts of all major species (core and beam protons, alphas, and electrons). At the threshold, the firehose parameter was found to be dominated by protons (67%), but also to have significant contributions from electrons (18%) and alphas (15%). Drifts were also found to be important, contributing 57% in the presence of a proton beam. A similar situation was found for the mirror, with contributions of 61%, 28%, and 11% for protons, electrons, and alphas, respectively. The parallel electric field contribution, however, was found to be small at 9%. Overall, the long-wavelength thresholds constrain the data well (${\text{}}\lt 1 \% $ unstable), and the implications of this are discussed.
Linear and nonlinear kinetic-MHD hybrid simulations have been carried out to investigate linear stability and nonlinear dynamics of beam-driven fishbone instability in spherical tokamak plasmas. Realistic NSTX parameters with finite toroidal rotation were used. The results show that the fishbone is driven by both trapped and passing particles. The instability drive of passing particles is comparable to that of trapped particles in the linear regime. The effects of rotation are destabilizing and a new region of instability appears at higher q min (>1.5) values, q min being the minimum of safety factor profile. In the nonlinear regime, the mode saturates due to flattening of beam ion distribution, and this persists after initial saturation while mode frequency chirps down in such a way that the resonant trapped particles move out radially and keep in resonance with the mode. Correspondingly, the flattening region of beam ion distribution expands radially outward. A substantial fraction of initially non-resonant trapped particles become resonant around the time of mode saturation and keep in resonance with the mode as frequency chirps down. On the other hand, the fraction of resonant passing particles is significantly smaller than that of trapped particles. Our analysis shows that trapped particles provide the main drive to the mode in the nonlinear regime.
Numerical simulations of the Princeton magnetorotational instability experiment with conducting axial boundaries
We investigate numerically the Princeton magnetorotational instability (MRI) experiment and the effect of conducting axial boundaries or endcaps. MRI is identified and found to reach a much higher saturation than for insulating endcaps. This is probably due to stronger driving of the base flow by the magnetically rather than viscously coupled boundaries. Although the computations are necessarily limited to lower Reynolds numbers ($Re$) than their experimental counterparts, it appears that the saturation level becomes independent of $Re$ when $Re$ is sufficiently large, whereas it has been found previously to decrease roughly as $Re^{-1/4}$ with insulating endcaps. The much higher saturation levels will allow for the positive detection of MRI beyond its theoretical and numerical predictions.
Stabilization of the vertical instability by non-axisymmetric coils
In a published Physical Review Letter (Reiman 2007 Phys. Rev. Lett. 99 135007), it
was shown that axisymmetric (or vertical) stability can be improved by placing a set of
parallelogram coils above and below the plasma oriented at an angle to the constant toroidal
planes. The physics of this stabilization can be understood as providing an effective additional
positive stability index. The original work was based on a simplified model of a straight
tokamak and is not straightforwardly applicable to a finite aspect ratio, strongly shaped plasma
such as in DIII-D. Numerical calculations were performed in a real DIII-D -like configuration
to provide a proof of principal that 3-D fields can, in fact raise the elongation limits as
predicted. A four field period trapezioid-shaped coil set was developed in toroidal geometry
and 3D equilibria were computed using trapezium coil currents of 10 kA, 100 kA, and 500 kA.
The ideal magnetohydrodynamics growth rates were computed as a function of the conformal
wall position for the n = 0 symmetry-preserving family. The results show an insignificant
relative improvement in the stabilizing wall location for the two lower coil current cases, of
the order of 10−3
and less. In contrast, the marginal wall position is increased by 7% as the
coil current is increased to 500 kA, confirming the main prediction from the original study in
a real geometry case. In DIII-D the shift in marginal wall position of 7% would correspond
to being able to move the existing wall outward by 5 to 10 cm. While the predicted effect
on the axisymmetric stability is real, it appears to require higher coil currents than could be
provided in an upgrade to existing facilities. Additional optimization over the pitch of the
coils, the number of field periods and the coil positions, as well as plasma parameters, such
as the internal inductivity $l_i$, $\beta$, and $q_{95}$ would mitigate this but seem unlikely to change the
conclusion
Pressure driven currents near magnetic islands in 3D MHD equilibria: Effects
of pressure variation within flux surfaces and of symmetry
In toroidal, magnetically confined plasmas, the heat and particle transport is strongly anisotropic,
with transport along the field lines sufficiently strong relative to cross-field transport that the equilibrium
pressure can generally be regarded as constant on the flux surfaces in much of the plasma.
The regions near small magnetic islands, and those near the X-lines of larger islands, are exceptions,
having a significant variation of the pressure within the flux surfaces. It is shown here that
the variation of the equilibrium pressure within the flux surfaces in those regions has significant
consequences for the pressure driven currents. It is further shown that the consequences are
strongly affected by the symmetry of the magnetic field if the field is invariant under combined
reflection in the poloidal and toroidal angles. (This symmetry property is called “stellarator
symmetry.”) In non-stellarator-symmetric equilibria, the pressure-driven currents have logarithmic
singularities at the X-lines. In stellarator-symmetric MHD equilibria, the singular components
of the pressure-driven currents vanish. These equilibria are to be contrasted with equilibria having ${\bf B}\cdot \nabla p = 0$; where the singular components of the pressure-driven currents vanish regardless of the
symmetry. They are also to be contrasted with 3D MHD equilibrium solutions that are constrained
to have simply nested flux surfaces, where the pressure-driven current goes like $1/x$ near rational
surfaces, where x is the distance from the rational surface, except in the case of quasi-symmetric
flux surfaces. For the purpose of calculating the pressure-driven currents near magnetic islands, we
work with a closed subset of the MHD equilibrium equations that involves only perpendicular force
balance, and is decoupled from parallel force balance. It is not correct to use the parallel component
of the conventional MHD force balance equation, ${\bf B}\cdot \nabla p = 0 $; near magnetic islands. Small but
nonzero values of ${\bf B}\cdot \nabla p$ are important in this region, and small non-MHD contributions to the parallel
force balance equation cannot be neglected there. Two approaches are pursued to solve our
equations for the pressure driven currents. First, the equilibrium equations are applied to an analytically
tractable magnetic field with an island, obtaining explicit expressions for the rotational transform
and magnetic coordinates, and for the pressure-driven current and its limiting behavior near
the X-line. The second approach utilizes an expansion about the X-line to provide a more general
calculation of the pressure-driven current near an X-line and of the rotational transform near a separatrix.
The study presented in this paper is motivated, in part, by tokamak experiments with nonaxisymmetric
magnetic perturbations, where significant differences are observed between the
behavior of stellarator-symmetric and non-stellarator-symmetric configurations with regard to
stabilization of edge localized modes by resonant magnetic perturbations. Implications for the coupling
between neoclassical tearing modes, and for magnetic island stability calculations, are also
discussed.
Mitigation of Alfvénic activity by 3D magnetic perturbations on NSTX
Observations on the National Spherical Torus Experiment (NSTX) indicate that externally
applied non-axisymmetric magnetic perturbations (MP) can reduce the amplitude of toroidal
Alfvén eigenmodes (TAE) and global Alfvén eigenmodes (GAE) in response to pulsed n = 3
non-resonant fields. From full-orbit following Monte Carlo simulations with the one- and
two-fluid resistive MHD plasma response to the magnetic perturbation included, it was found
that in response to MP pulses the fast-ion losses increased and the fast-ion drive for the GAEs
was reduced. The MP did not affect the fast-ion drive for the TAEs significantly but the Alfvén
continuum at the plasma edge was found to be altered due to the toroidal symmetry breaking
which leads to coupling of different toroidal harmonics. The TAE gap was reduced at the
edge creating enhanced continuum damping of the global TAEs, which is consistent with the
observations. The results suggest that optimized non-axisymmetric MP might be exploited to
control and mitigate Alfvén instabilities by tailoring the fast-ion distribution function and/or
continuum structure.
A fully non-linear multi-species Fokker-Planck-Landau collision operator for simulation of fusion plasma
Fusion edge plasmas can be far from thermal equilibrium and require the use of a non-linear collision operator for accurate numerical simulations. In this article, the non-linear single-species Fokker–Planck–Landau collision operator developed by Yoon and Chang (2014) [9] is generalized to include multiple particle species. The finite volume discretization used in this work naturally yields exact conservation of mass, momentum, and energy. The implementation of this new non-linear Fokker–Planck–Landau operator in the gyrokinetic particle-in-cell codes XGC1 and XGCa is described and results of a verification study are discussed. Finally, the numerical techniques that make our non-linear collision operator viable on high-performance computing systems are described, including specialized load balancing algorithms and nested OpenMP parallelization. The collision operator's good weak and strong scaling behavior are shown.
A new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma
In order to enable kinetic simulation of non-thermal edge plasmas at a reduced computational cost, a new hybrid-Lagrangian δf scheme has been developed that utilizes the phase space grid in addition to the usual marker particles, taking advantage of the computational strengths from both sides. The new scheme splits the particle distribution function of a kinetic equation into two parts. Marker particles contain the fast space-time varying, δf, part of the distribution function and the coarse-grained phase-space grid contains the slow space-time varying part. The coarse-grained phase-space grid reduces the memory-requirement and the computing cost, while the marker particles provide scalable computing ability for the fine-grained physics. Weights of the marker particles are determined by a direct weight evolution equation instead of the differential form weight evolution equations that the conventional delta-f schemes use. The particle weight can be slowly transferred to the phase space grid, thereby reducing the growth of the particle weights. The non-Lagrangian part of the kinetic equation – e.g., collision operation, ionization, charge exchange, heat-source, radiative cooling, and others – can be operated directly on the phase space grid. Deviation of the particle distribution function on the velocity grid from a Maxwellian distribution function – driven by ionization, charge exchange and wall loss – is allowed to be arbitrarily large. The numerical scheme is implemented in the gyrokinetic particle code XGC1, which specializes in simulating the tokamak edge plasma that crosses the magnetic separatrix and is in contact with the material wall.
Multi-scale full-orbit analysis on phase-space behavior of runaway electrons in tokamak fields with synchrotron radiation
In this paper, the secular full-orbit simulations of runaway electrons with synchrotron radiation in tokamak fields are carried out using a relativistic volume-preserving algorithm. Detailed phase-space behaviors of runaway electrons are investigated in different dynamical timescales spanning 11 orders. In the small timescale, i.e., the characteristic timescale imposed by Lorentz force, the severely deformed helical trajectory of energetic runaway electron is witnessed. A qualitative analysis of the neoclassical scattering, a kind of collisionless pitch-angle scattering phenomena, is provided when considering the coupling between the rotation of momentum vector and the background magnetic field. In large timescale up to 1 s, it is found that the initial condition of runaway electrons in phase space globally influences the pitch-angle scattering, the momentum evolution, and the loss-gain ratio of runaway energy evidently. However, the initial value has little impact on the synchrotron energy limit. It is also discovered that the parameters of tokamak device, such as the toroidal magnetic field, the loop voltage, the safety factor profile, and the major radius, can modify the synchrotron energy limit and the strength of neoclassical scattering. The maximum runaway energy is also proved to be lower than the synchrotron limit when the magnetic field ripple is considered.
Impact of resistive MHD plasma response on perturbation field sidebands
Single fluid linear simulations of a KSTAR RMP ELM suppressed discharge with the M3D-C1
resistive magnetohydrodynamic code have been performed for the first time. The simulations
show that the application of the $n = 1$ perturbation using the KSTAR in-vessel control coils
(IVCC), which apply modest levels of $n = 3$ sidebands (~20% of the $n = 1$), leads to levels of
$n = 3$ sideband that are comparable to the $n = 1$ when plasma response is included. This is due
to the reduced level of screening of the rational-surface-resonant $n = 3$ component relative to the
rational-surface-resonant $n = 1$ component. The $n = 3$ sidebands could play a similar role in ELM
suppression on KSTAR as the toroidal sidebands ($n = 1, 2, 4$) in DIII-D $n = 3$ ELM suppression
with missing I-coil segments (Paz Soldan et al 2014 Nucl. Fusion 54 073013). This result may
help to explain the uniqueness of ELM suppression with $n = 1$ perturbations in KSTAR since the
effective perturbation is a mixed $n = 1/n = 3$ perturbation similar to $n = 3$ ELM suppression in
DIII-D.
Multi-region approach to free-boundary three-dimensional tokamak equilibria and resistive wall instabilities
Free-boundary 3D tokamak equilibria and resistive wall instabilities are calculated using a new
resistive wall model in the two-fluid M3D-C1 code. In this model, the resistive wall and surrounding
vacuum region are included within the computational domain. This implementation contrasts
with the method typically used in fluid codes in which the resistive wall is treated as a boundary
condition on the computational domain boundary and has the advantage of maintaining purely local
coupling of mesh elements. This new capability is used to simulate perturbed, free-boundary nonaxisymmetric
equilibria; the linear evolution of resistive wall modes; and the linear and nonlinear
evolution of axisymmetric vertical displacement events (VDEs). Calculated growth rates for a
resistive wall mode with arbitrary wall thickness are shown to agree well with the analytic theory.
Equilibrium and VDE calculations are performed in diverted tokamak geometry, at physically realistic
values of dissipation, and with resistive walls of finite width. Simulations of a VDE disruption
extend into the current-quench phase, in which the plasma becomes limited by the first wall, and
strong currents are observed to flow in the wall, in the SOL, and from the plasma to the wall.
Application of Lie Algebra in Constructing Volume-Preserving Algorithms for Charged Particles Dynamics
Volume-preserving algorithms (VPAs) for the charged particles dynamics is preferred because of their long-term accuracy and conservativeness for phase space volume. Lie algebra and the Baker-Campbell-Hausdorff (BCH) formula can be used as a fundamental theoretical tool to construct VPAs. Using the Lie algebra structure of vector fields, we split the volume-preserving vector field for charged particle dynamics into three volume-preserving parts (sub-algebras), and find the corresponding Lie subgroups. Proper combinations of these subgroups generate volume preserving, second order approximations of the original solution group, and thus second order VPAs. The developed VPAs also show their significant effectiveness in conserving phase-space volume exactly and bounding energy error over long-term simulations.
Collisionless Pitch-Angle Scattering of Runaway Electrons
It is discovered that the tokamak field geometry generates a toroidicity induced broadening of
the pitch-angle distribution of runaway electrons. This collisionless pitch-angle scattering is
much stronger than the collisional scattering and invalidates the gyro-center model for runaway
electrons. As a result, the energy limit of runaway electrons is found to be larger than the
prediction of the gyro-center model and to depend heavily on the background magnetic field.
Evaluation of Thermal Helium Beam and Line-Ratio
Fast Diagnostic on the National Spherical Torus Experiment-Upgrade
A 1-D kinetic collisional radiative model with state-of-the-art atomic data is developed and employed to simulate line emission to evaluate the Thermal Helium Beam (THB) diagnostic on NSTX-U. This diagnostic is currently in operation on RFX-mod, and it is proposed to be installed on NSTX-U. The THB system uses the intensity ratios of neutral helium lines 667.8, 706.5, and 728.1 nm to derive electron temperature (eV) and density ($cm^{−3}$) profiles. The purpose of the present analysis is to evaluate the applications of this diagnostic for determining fast (∽4 μs) electron temperature and density radial profiles on the scrape-off layer and edge regions of NSTX-U that are needed in turbulence studies. The diagnostic is limited by the level of detection of the 728.1 nm line, which is the weakest of the three. This study will also aid in future design of a similar 2-D diagnostic system on the divertor.
An explanation is provided for the disruptive instability in diverted tokamaks when the safety factor $q$ at the 95% poloidal flux surface, $q_{95}$, is driven below 2.0. The instability is a resistive kink counterpart to the current-driven ideal mode that traditionally explained the corresponding disruption in limited cross-sections (Shafranov, Sov. Phys. Tech. Phys., vol. 15, 1970, p. 175) when $q_{edge}$, the safety factor at the outermost closed flux surface, lies just below a rational value $m/n$. Experimentally, external kink modes are observed in limiter configurations as the current in a tokamak is ramped up and $q_{edge}$ decreases through successive rational surfaces. For $q_{edge}<2$ , the instability is always encountered and is highly disruptive. However, diverted plasmas, in which $q_{edge}$ is formally infinite in the magnetohydrodynamic (MHD) model, have presented a longstanding difficulty since the theory would predict stability, yet, the disruptive limit occurs in practice when $q_{95}$ reaches 2. It is shown from numerical calculations that a resistive kink mode is linearly destabilized by the rapidly increasing resistivity at the plasma edge when $q_{95}<2$, but $q_{edge}>>2$. The resistive kink behaves much like the ideal kink with predominantly kink or interchange parity and no real sign of a tearing component. However, the growth rates scale with a fractional power of the resistivity near the $q=2$ surface. The results have a direct bearing on the conventional edge cutoff procedures used in most ideal MHD codes, as well as implications for ITER and for future reactor options.
Ion Cyclotron Emission Studies: Retrospects and Prospects
Ion Cyclotron Emission (or ICE) studies emerged in part from the papers by A.B. Mikhailovskii published in s. Among the discussed subjects were electromagnetic compressional Alfvénic cyclotron instabilities with the linear growth rate driven by fusion products, -particles which draw a lot of attention to energetic particle physics. The theory of ICE excited by energetic particles was significantly advanced at the end of century motivated by first DT experiments on TFTR and subsequent JET experimental studies which we highlight. More recently ICE theory was advanced by detailed theoretical and experimental studies on ST (or spherical torus) fusion devices where the instability signals previously indistinguishable in high aspect ratio tokamaks due to high toroidal magnetic field became the subjects of experiments. We discuss further prospects of ICE theory applications for future burning plasma (BP) experiments such as those to be conducted in ITER device in France where neutron and gamma rays escaping the plasma create extremely challenging conditions fuison alpha particle diagnostics.
The unified ballooning theory with weak up-down asymmetric mode structure and the numerical studies
A unified ballooning theory, constructed on the basis of two special theories [Zhang et al., Phys. Fluids B 4, 2729 (1992); Y. Z. Zhang and T. Xie, Nucl. Fusion Plasma Phys. 33, 193 (2013)], shows that a weak up-down asymmetric mode structure is normally formed in an up-down symmetric equilibrium; the weak up-down asymmetry in mode structure is the manifestation of non-trivial higher order effects beyond the standard ballooning equation. It is shown that the asymmetric mode may have even higher growth rate than symmetric modes. The salient features of the theory are illustrated by investigating a fluid model for the ion temperature gradient (ITG) mode. The two dimensional (2D) analytical form of the ITG mode, solved in ballooning representation, is then converted into the radial-poloidal space to provide the natural boundary condition for solving the 2D mathematical local eigenmode problem. We find that the analytical expression of the mode structure is in a good agreement with finite difference solution. This sets a reliable framework for quasi-linear computation.
Dynamics of ion beam charge neutralization by ferroelectric plasma sources
Ferroelectric Plasma Sources (FEPSs) can generate plasma that provides effective space-charge neutralization of intense high-perveance ion beams, as has been demonstrated on the Neutralized Drift Compression Experiment NDCX-I and NDCX-II. This article presents experimental results on charge neutralization of a high-perveance 38 keV Ar+ beam by a plasma produced in a FEPS discharge. By comparing the measured beam radius with the envelope model for space-charge expansion, it is shown that a charge neutralization fraction of 98% is attainable with sufficiently dense FEPS plasma. The transverse electrostatic potential of the ion beam is reduced from 15 V before neutralization to 0.3 V, implying that the energy of the neutralizing electrons is below 0.3 eV. Measurements of the time-evolution of beam radius show that near-complete charge neutralization is established ∼5 μs after the driving pulse is applied to the FEPS and can last for 35 μs. It is argued that the duration of neutralization is much longer than a reasonable lifetime of the plasma produced in the sub-μs surface discharge. Measurements of current flow in the driving circuit of the FEPS show the existence of electron emission into vacuum, which lasts for tens of μs after the high voltage pulse is applied. It is argued that the beam is neutralized by the plasma produced by this process and not by a surface discharge plasma that is produced at the instant the high-voltage pulse is applied.
Suppression of thermal conduction in a mirror-unstable plasma
The plasma of galaxy clusters is subject to firehose and mirror instabilities at scales of order the ion Larmor radius. The mirror instability generates fluctuations of magnetic-field strength δB/B ∼ 1. These fluctuations act as magnetic traps for the heat-conducting electrons, suppressing their transport. We calculate the effective parallel thermal conductivity in the ICM in the presence of the mirror fluctuations for different stages of the evolution of the instability. The mirror fluctuations are limited in amplitude by the maximum and minimum values of the field strength, with no large deviations from the mean value. This key property leads to a finite suppression of thermal conduction at large scales. We find suppression down to ≈0.2 of the Spitzer value for the secular phase of the perturbations’ growth, and ≈0.3 for their saturated phase. The effect operates in addition to other suppression mechanisms and independently of them. Globally, fluctuations δB/B ∼ 1 can be present on much larger scales, of the order of the scale of turbulent motions. However, we do not expect large suppression of thermal conduction by these, because their scale is considerably larger than the collisional mean free path of the ICM electrons. The obtained suppression of thermal conduction by a factor of ∼5 appears to be characteristic and potentially universal for a weakly collisional mirror-unstable plasma.
Gyrokinetic neoclassical study of the bootstrap current in the tokamak edge pedestal with fully nonlinear Coulomb collisions
As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons.
Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal.
An inaccuracy in Koh et al.'s result is found in the steep edge pedestal that originated from a small error in the collisional momentum conservation.
The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. A new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.
Equilibrium drives of the low and high field side n = 2 plasma response and impact on global confinement
The nature of the multi-modal $n = 2$ plasma response and its impact on global confinement
is studied as a function of the axisymmetric equilibrium pressure, edge safety factor,
collisionality, and L-versus H-mode conditions.
Varying the relative phase ($\Delta\phi_{UL}$)
between upper and lower in-vessel coils demonstrates that different $n = 2$ poloidal spectra
preferentially excite different plasma responses. These different plasma response modes
are preferentially detected on the tokamak high-field side (HFS) or low-field side (LFS)
midplanes, have different radial extents, couple differently to the resonant surfaces, and have
variable impacts on edge stability and global confinement. In all equilibrium conditions
studied, the observed confinement degradation shares the same $\Delta\phi_{UL}$ dependence as the
coupling to the resonant surfaces given by both ideal (IPEC) and resistive (MARS-F) MHD
computation. Varying the edge safety factor shifts the equilibrium field-line pitch and thus
the $\Delta\phi_{UL}$ dependence of both the global confinement and the $n = 2$ magnetic response.
As edge safety factor is varied, modeling finds that the HFS response (but not the LFS
response), the resonant surface coupling, and the edge displacements near the X-point all
share the same $\Delta\phi_{UL}$ dependence. The LFS response magnitude is strongly sensitive to the
core pressure and is insensitive to the collisionality and edge safety factor. This indicates that
the LFS measurements are primarily sensitive to a pressure-driven kink-ballooning mode that
couples to the core plasma. MHD modeling accurately reproduces these (and indeed all) LFS
experimental trends and supports this interpretation. In contrast to the LFS, the HFS magnetic
response and correlated global confinement impact is unchanged with plasma pressure, but is
strongly reduced in high collisionality conditions in both H- and L-mode. This experimentally
suggests the bootstrap current drives the HFS response through the kink-peeling mode drive,
though surprisingly weak or no dependence on the bootstrap current is seen in modeling.
Instead, modeling is revealed to be very sensitive to the details of the edge current profile
and equilibrium truncation. Holding truncation fixed, most HFS experimental trends are not
captured, thus demonstrating a stark contrast between the robustness of the HFS experimental
results and the sensitivity of its computation.
On the inward drift of runaway electrons in plateau regime
The well observed inward drift of current carrying runaway electrons during runaway plateau phase after disruption is studied by considering the phase space dynamic of runaways in a large aspect ratio toroidal system. We consider the case where the toroidal field is unperturbed and the toroidal symmetry of the system is preserved. The balance between the change in canonical angular momentum and the input of mechanical angular momentum in such a system requires runaways to drift horizontally in configuration space for any given change in momentum space. The dynamic of this drift can be obtained by integrating the modified Euler-Lagrange equation over one bounce time. It is then found that runaway electrons will always drift inward as long as they are decelerating. This drift motion is essentially non-linear, since the current is carried by runaways themselves, and any runaway drift relative to the magnetic axis will cause further displacement of the axis itself. A simplified analytical model is constructed to describe such inward drift both in the ideal wall case and no wall case, and the runaway current center displacement as a function of parallel momentum variation is obtained. The time scale of such displacement is estimated by considering effective radiation drag, which shows reasonable agreement with the observed displacement time scale. This indicates that the phase space dynamic studied here plays a major role in the horizontal displacement of runaway electrons during plateau phase.
Radially dependent large-scale dynamos in global cylindrical shear flows and the local cartesian limit
For cylindrical differentially rotating plasmas, we study large-scale magnetic field generation from finite amplitude non-axisymmetric perturbations by comparing numerical simulations with quasi-linear analytic theory. When initiated with a vertical magnetic field of either zero or finite net flux, our global cylindrical simulations exhibit the magnetorotational instability (MRI) and large-scale dynamo growth of radially alternating mean fields, averaged over height and azimuth. This dynamo growth is explained by our analytic calculations of a non-axisymmetric fluctuation-induced electromotive force that is sustained by azimuthal shear of the fluctuating fields. The standard ‘Ω effect’ (shear of the mean field by differential rotation) is unimportant. For the MRI case, we express the large-scale dynamo field as a function of differential rotation. The resulting radially alternating large-scale fields may have implications for angular momentum transport in discs and corona. To connect with previous work on large-scale dynamos with local linear shear and identify the minimum conditions needed for large-scale field growth, we also solve our equations in local Cartesian coordinates. We find that large-scale dynamo growth in a linear shear flow without rotation can be sustained by shear plus non-axisymmetric fluctuations – even if not helical, a seemingly previously unidentified distinction. The linear shear flow dynamo emerges as a more restricted version of our more general new global cylindrical calculations.
Pressure-driven amplification and penetration of resonant magnetic perturbations
We show that a resonant magnetic perturbation applied to the boundary of an ideal plasma screw-pinch equilibrium with nested surfaces can penetrate inside the resonant surface and into the core. The response is significantly amplified with increasing plasma pressure. We present a rigorous verification of nonlinear equilibrium codes against linear theory, showing excellent agreement.
Verification of the ideal magnetohydrodynamic response at rational surfaces in the VMEC code
The VMEC nonlinear ideal MHD equilibrium code [S. P. Hirshman and J. C. Whitson, Phys. Fluids 26, 3553 (1983)] is compared against analytic linear ideal MHD theory in a screw-pinch-like configuration. The focus of such analysis is to verify the ideal MHD response at magnetic surfaces which possess magnetic transform (ι) which is resonant with spectral values of the perturbed boundary harmonics. A large aspect ratio circular cross section zero-beta equilibrium is considered. This equilibrium possess a rational surface with safety factor q = 2 at a normalized flux value of 0.5. A small resonant boundary perturbation is introduced, exciting a response at the resonant rational surface. The code is found to capture the plasma response as predicted by a newly developed analytic theory that ensures the existence of nested flux surfaces by allowing for a jump in rotational transform (ι=1/q). The VMEC code satisfactorily reproduces these theoretical results without the necessity of an explicit transform discontinuity (Δι) at the rational surface. It is found that the response across the rational surfaces depends upon both radial grid resolution and local shear (dι/dΦ), where ι is the rotational transform and Φ the enclosed toroidal flux). Calculations of an implicit Δι suggest that it does not arise due to numerical artifacts (attributed to radial finite differences in VMEC) or existence conditions for flux surfaces as predicted by linear theory (minimum values of Δι). Scans of the rotational transform profile indicate that for experimentally relevant levels of transform shear the response becomes increasing localised. Careful examination of a large experimental tokamak equilibrium, with applied resonant fields, indicates that this shielding response is present, suggesting the phenomena is not limited to this verification exercise.
Large-volume flux closure during plasmoid-mediated reconnection in Coaxial Helicity Injection
A large-volume flux closure during transient coaxial helicity injection (CHI) in NSTX-U is demonstrated through resistive magnetohydrodynamics (MHD) simulations. Several major improvements, including the improved positioning of the divertor poloidal field coils, are projected to improve the CHI start-up phase in NSTX-U. Simulations in the NSTX-U configuration with constant in time coil currents show that with strong flux shaping the injected open field lines (injector flux) rapidly reconnect and form large volume of closed flux surfaces. This is achieved by driving parallel current in the injector flux coil and oppositely directed currents in the flux shaping coils to form a narrow injector flux footprint and push the injector flux into the vessel. As the helicity and plasma are injected into the device, the oppositely directed field lines in the injector region are forced to reconnect through a local Sweet–Parker type reconnection, or to spontaneously reconnect when the elongated current sheet becomes MHD unstable to form plasmoids. In these simulations for the first time, it is found that the closed flux is over 70% of the initial injector flux used to initiate the discharge. These results could work well for the application of transient CHI in devices that employ super conducting coils to generate and sustain the plasma equilibrium.
Blob Structure and Motion in the Edge and SOL of NSTX
The structure and motion of discrete plasma blobs (a.k.a. filaments) in the edge and scrape-off layer of NSTX is studied for representative Ohmic and H-mode discharges. Individual blobs were tracked in the 2D radial versus poloidal plane using data from the gas puff imaging diagnostic taken at 400 000 frames $s^{−1}$. A database of blob amplitude, size, ellipticity, tilt, and velocity was obtained for ~45 000 individual blobs. Empirical relationships between various properties are described, e.g. blob speed versus amplitude and blob tilt versus ellipticity. The blob velocities are also compared with analytic models.
Post calibration of the two-dimensional electron cyclotron emission imaging instrument with electron temperature characteristics of the magnetohydrodynamic instabilities
The electron cyclotron emission imaging (ECEI) instrument is widely used to study the local electron temperature $(T_e)$ fluctuations by measuring the ECE intensity $I_{ECE} \propto T_e$ in tokamak plasmas.
The ECEI measurement is often processed in a normalized fluctuation quantity against the time averaged value due to complication in absolute calibration. In this paper, the ECEI channels are relatively calibrated using the flat $T_e$ assumption of the sawtooth crash or the tearing mode island and a proper extrapolation.
The 2-D relatively calibrated electron temperature $(T_{e,rel})$ images are reconstructed and the displacement amplitude of the magnetohydrodynamic modes can be measured for the accurate quantitative growth analysis.
Higher order volume-preserving schemes for charged particle dynamics
A class of higher order numerical methods for advancing the charged particles in a general electromagnetic field is developed based on processing technique. By taking the volume-preserving methods as the kernel, the processed methods are still volume-preserving, and preserve the conservative quantities for the Lorenz force system. Moreover, this class of numerical methods are explicit and are more efficient compared with other higher order composition methods. Linear stability analysis is given by applying the numerical methods to the test equation. It is shown that the newly constructed higher order methods have the better stability property. This allows the use of larger step sizes in their implementation.
Suppressed gross erosion of high-temperature lithium via rapid deuterium implantation
Lithium-coated high-Z substrates are planned for use in the NSTX-U divertor and are a candidate plasma facing component (PFC) for reactors, but it remains necessary to characterize the gross Li erosion rate under high plasma fluxes ($>10^{23} m^{−2} s^{−1}$), typical for the divertor region. In this work, a realistic model for the compositional evolution of a Li/D layer is developed that incorporates first principles molecular dynamics (MD) simulations of D diffusion in liquid Li. Predictions of Li erosion from a mixed Li/D material are also developed that include formation of lithium deuteride (LiD). The erosion rate of Li from LiD is predicted to be significantly lower than from pure Li. This prediction is tested in the Magnum-PSI linear plasma device at ion fluxes of $10^{23}–10^{24} m^{−2} s^{−1}$ and Li surface temperatures ≤800 °C. Li/LiD coatings ranging in thickness from 0.2 to 500 μm are studied. The dynamic D/Li concentrations are inferred via diffusion simulations. The pure Li erosion rate remains greater than Langmuir Law evaporation, as expected. For mixed-material Li/LiD surfaces, the erosion rates are reduced, in good agreement with modelling in almost all cases. These results imply that the temperature limit for a Li-coated PFC may be significantly higher than previously imagined.
Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov–Maxwell equations
Particle-in-cell (PIC) simulation is the most important numerical tool in plasma physics. However, its long-term accuracy has not been established. To overcome this difficulty, we developed a canonical symplectic PIC method for the Vlasov–Maxwell system by discretising its canonical Poisson bracket. A fast local algorithm to solve the symplectic implicit time advance is discovered without root searching or global matrix inversion, enabling applications of the proposed method to very large-scale plasma simulations with many, e.g. $10^9$, degrees of freedom. The long-term accuracy and fidelity of the algorithm enables us to numerically confirm Mouhot and Villani's theory and conjecture on nonlinear Landau damping over several orders of magnitude using the PIC method, and to calculate the nonlinear evolution of the reflectivity during the mode conversion process from extraordinary waves to Bernstein waves.
We demonstrate that in a 3D resistive magnetohydrodynamic simulation, for some parameters it is possible to form a stationary state in a tokamak where a saturated interchange mode in the center of the discharge drives a near helical flow pattern that acts to nonlinearly sustain the configuration by adjusting the central loop voltage through a dynamo action.
This could explain the physical mechanism for maintaining stationary nonsawtoothing “hybrid” discharges, often referred to as “flux pumping.”
Mesh generation for confined fusion plasma simulation
XGC1 and M3D-$C^1$ are two fusion plasma simulation codes being developed at Princeton Plasma Physics Laboratory.
XGC1 uses the particle-in-cell method to simulate gyrokinetic neoclassical physics and turbulence (Chang et al. Phys Plasmas 16(5):056108, 2009; Ku et al. Nucl Fusion 49:115021, 2009; Admas et al. J Phys 180(1):012036, 2009). M3D-$C^1$ solves the two-fluid resistive magnetohydrodynamic equations with the $C^1$ finite elements (Jardin J comput phys 200(1):133-152, 2004; Jardin et al. J comput Phys 226(2):2146-2174, 2007; Ferraro and Jardin J comput Phys 228(20):7742-7770, 2009; Jardin J comput Phys 231(3):832-838, 2012; Jardin et al. Comput Sci Discov 5(1):014002, 2012; Ferraro et al. Sci Discov Adv Comput, 2012; Ferraro et al. International sherwood fusion theory conference, 2014). This paper presents the software tools and libraries that were combined to form the geometry and automatic meshing procedures for these codes. Specific consideration has been given to satisfy the mesh configuration and element shape quality constraints of XGC1 and M3D-$C^1$.