A novel approach to radially global gyrokinetic simulation using the flux-tube code stella
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70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
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Comparison of local and global gyrokinetic calculations of collisionless zonal flow damping in quasi-symmetric stellarators
The linear collisionless damping of zonal flows is calculated for quasi-symmetric stellarator equilibria in flux-tube, flux-surface, and full-volume geometry. Equilibria are studied from the quasi-helical symmetry configuration of the Helically Symmetric eXperiment (HSX), a broken symmetry configuration of HSX, and the quasi-axial symmetry geometry of the National Compact Stellarator eXperiment (NCSX). Zonal flow oscillations and long-time damping affect the zonal flow evolution, and the zonal flow residual goes to zero for small radial wavenumber. The oscillation frequency and damping rate depend on the bounce-averaged radial particle drift in accordance with theory. While each flux tube on a flux surface is unique, several different flux tubes in HSX or NCSX can reproduce the zonal flow damping from a flux-surface calculation given an adequate parallel extent. The flux-surface or flux-tube calculations can accurately reproduce the full-volume long-time residual for moderate kx, but the oscillation and damping time scales are longer in local representations, particularly for small kx approaching the system size.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A subgrid model for electron-scale turbulent transport in global ion-scale gyrokinetic simulations of tokamak plasmas
A subgrid electron–temperature–gradient (ETG) model is demonstrated here which averages local electron-scale turbulence over intermediate scales in space and time to include in global ion–temperature–gradient (ITG) simulations. This approach results in ion-scale equations, which incorporate the electron heat transport from ETG turbulence and the effects of electron-scale turbulence on the ion scale. Flux-tube ETG Cyclone Base Case simulations are performed at different radial locations and a kinetic form of the flux is added to global ion-scale (IS) simulations as a source term. Analytic radial profiles of ETG heat flux are constructed and compared to flux-tube results at multiple radial locations. Different ratios of ITG to ETG heat flux levels are considered and the results of capturing ETG heat transport in global ion-scale simulations are discussed. The potential coupling of the ETG streamer potential and intermediate-scale zonal flows to the ion scale is further addressed.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Investigating profile stiffness and critical gradients in shaped TCV discharges using local gyrokinetic simulations of turbulent transport
The experimental observation made on the TCV tokamak of a significant confinement improvement in plasmas with negative triangularity ( δ < 0) compared to those with standard positive triangularity has been interpreted in terms of different degrees of profile stiffness (Sauter et al. Phys. Plasmas 21 055906, 2014) and/or different critical gradients. Employing the Eulerian gyrokinetic code GENE (Jenko et al. Phys. Plasmas 7 1904, 2000), profile stiffness and critical gradients are studied under TCV relevant conditions. For the considered experimental discharges, trapped electron modes (TEMs) and electron temperature gradient (ETG) modes are the dominant microinstabilities, with the latter providing a significant contribution to the non-linear electron heat fluxes near the plasma edge. Two series of simulations with different levels of realism are performed, addressing the question of profile stiffness at various radial locations. Retaining finite collisionality, impurities and electromagnetic effects, as well as the physical electron-to-ion mass ratio are all necessary in order to approach the experimental flux measurements. However, flux-tube simulations are unable to fully reproduce the TCV results, pointing towards the need to carry out radially nonlocal (global) simulations, i.e. retaining finite machine size effects, in a future study. Some conclusions about the effect of triangularity can nevertheless be drawn based on the flux-tube results. In particular, the importance of considering the sensitivity to both temperature and density gradient is shown. The flux tube results show an increase of the critical gradients towards the edge, further enhanced when δ < 0, and they also appear to indicate a reduction of profile stiffness towards plasma edge.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
A computational model for nanosecond pulse laser-plasma interactions
A multi-physics numerical model for laser-induced optical breakdown and laser-plasma interaction in a non-equilibrium gas is presented, accounting for: production of priming electrons via multi-photon ionization, energy absorption, cascade ionization, induced hydrodynamic response, and shock formation and propagation. The gas is governed by the Navier-Stokes equations, with non-equilibrium effects taken into account by means of a two-temperature model. The space-time dependence of the laser beam is modeled with a flux-tube formulation for the Radiative Transfer Equation. The flow governing equations are discretized in space using a second-order finite volume method. The semi-discrete equations are marched in time using an implicit-explicit (IMEX) dual time-stepping strategy, where diffusion and chemistry are solved implicitly, whereas convection is explicit. Application to a 20 ns long 50 mJ pulse laser-induced breakdown in quiescent O 2 shows the advantages of this temporal discretization during and just after the laser pulse, while a less-expensive symmetric Strang splitting (with implicit chemistry) is sufficient for the post-breakdown gas dynamics after ≃ 0.1 μs. The integrated model is shown to reproduce key features of corresponding experiments.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Multi-scale dynamics of magnetic flux tubes and inverse magnetic energy transfer
Here, we report on an analytical and numerical study of the dynamics of a three-dimensional array of identical magnetic flux tubes in the reduced-magnetohydrodynamic description of the plasma. We propose that the long-time evolution of this system is dictated by flux-tube mergers, and that such mergers are dynamically constrained by the conservation of the pertinent (ideal) invariants, viz. the magnetic potential and axial fluxes of each tube. We also propose that in the direction perpendicular to the merging plane, flux tubes evolve in a critically balanced fashion. These notions allow us to construct an analytical model for how quantities such as the magnetic energy and the energy-containing scale evolve as functions of time. Of particular importance is the conclusion that, like its two-dimensional counterpart, this system exhibits an inverse transfer of magnetic energy that terminates only at the system scale. We perform direct numerical simulations that confirm these predictions and reveal other interesting aspects of the evolution of the system. We find, for example, that the early time evolution is characterized by a sharp decay of the initial magnetic energy, which we attribute to the ubiquitous formation of current sheets. We also show that a quantitatively similar inverse transfer of magnetic energy is observed when the initial condition is a random, small-scale magnetic seed field.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Kinetic-ballooning-mode turbulence in low-average-magnetic-shear equilibria
In this work, kinetic-ballooning-mode (KBM) turbulence is studied via gyrokinetic flux-tube simulations in three magnetic equilibria that exhibit small average magnetic shear: the Helically Symmetric eXperiment (HSX), the helical-axis Heliotron-J and a circular tokamak geometry. For HSX, the onset of KBM being the dominant instability at low wavenumber occurs at a critical value of normalized plasma pressure $\beta ^{\rm KBM}_{\rm crit}$ that is an order of magnitude smaller than the magnetohydrodynamic (MHD) ballooning limit $\beta ^{\rm MHD}_{\rm crit}$ when a strong ion temperature gradient (ITG) is present. However, $\beta ^{\rm KBM}_{\rm crit}$ increases and approaches the MHD ballooning limit as the ITG tends to zero. For these configurations, $\beta ^{\rm KBM}_{\rm crit}$ also increases as the magnitude of the average magnetic shear increases, regardless of the sign of the normalized magnetic shear. Simulations of Heliotron-J and a circular axisymmetric geometry display behaviour similar to HSX with respect to $\beta ^{\rm KBM}_{\rm crit}$ . Despite large KBM growth rates at long wavelengths in HSX, saturation of KBM turbulence with $\beta > \beta _{\rm crit}^{\rm KBM}$ is achievable in HSX and results in lower heat transport relative to the electrostatic limit by a factor of roughly five. Nonlinear simulations also show that KBM transport dominates the dynamics when KBMs are destabilized linearly, even if KBM growth rates are subdominant to ITG growth rates.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
GX: a GPU-native gyrokinetic turbulence code for tokamak and stellarator design
GX is a code designed to solve the nonlinear gyrokinetic system for low-frequency turbulence in magnetized plasmas, particularly tokamaks and stellarators. In GX, our primary motivation and target is a fast gyrokinetic solver that can be used for fusion reactor design and optimization along with wide-ranging physics exploration. Here, this has led to several code and algorithm design decisions, specifically chosen to prioritize time to solution. First, we have used a discretization algorithm that is pseudospectral in the entire phase space, including a Laguerre–Hermite pseudospectral formulation of velocity space, which allows for smooth interpolation between coarse gyrofluid-like resolutions and finer conventional gyrokinetic resolutions and efficient evaluation of a model collision operator. Additionally, we have built GX to natively target graphics processors (GPUs), which are among the fastest computational platforms available today. Finally, we have taken advantage of the reactor-relevant limit of small $\rho _*$ by using the radially local flux-tube approach. In this paper we present details about the gyrokinetic system and the numerical algorithms used in GX to solve the system. We then present several numerical benchmarks against established gyrokinetic codes in both tokamak and stellarator magnetic geometries to verify that GX correctly simulates gyrokinetic turbulence in the small $\rho _*$. Moreover, we show that the convergence properties of the Laguerre–Hermite spectral velocity formulation are quite favourable for nonlinear problems of interest. Coupled with GPU acceleration, which we also investigate with scaling studies, this enables GX to be able to produce useful turbulence simulations in minutes on one (or a few) GPUs and higher fidelity results in a few hours using several GPUs. GX is open-source software that is ready for fusion reactor design studies.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Self-organized confinement in whole-device modeling of laboratory magnetospheres
Turbulent self-organization driven by global ideal interchange modes in a dipole-confined plasma is explored with self-consistent, whole-plasma simulations using a flux-tube averaged magnetohydrodynamic model in dipole magnetic geometry. We show the existence of robust particle pinch driven by ideal interchange-mode fluctuations, in which the particles are transported up the density gradient. It is found that the plasma profiles in a dipole field spontaneously relax to a marginally stable state as centrally peaked pressure and density are created by global interchange-mode transport.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Global gyrokinetic study of shaping effects on electromagnetic modes at NSTX aspect ratio with ad hoc parallel magnetic perturbation effects
Plasma shaping may have a stronger effect on global turbulence in tight-aspect-ratio tokamaks than in conventional-aspect-ratio tokamaks due to the higher toroidicity and more acute poloidal asymmetry in the magnetic field. In addition, previous local gyrokinetic studies have shown that it is necessary to include parallel magnetic field perturbations in order to accurately compute growth rates of electromagnetic modes in tight-aspect-ratio tokamaks. In this work, the effects of elongation and triangularity on global, ion-scale, linear electromagnetic modes are studied at National Spherical Torus Experiment (NSTX) aspect ratio and high plasma β using the global gyrokinetic particle-in-cell code XGC. The effects of compressional magnetic perturbations are approximated via a well-known modification to the particle drifts that was developed for flux-tube simulations [Joiner et al., Phys. Plasmas 17, 072104 (2010)], without proof of its validity in a global simulation, with the gyrokinetic codes GENE and GEM being used for local verification and global cross-verification. Magnetic equilibria are re-constructed for each distinct plasma profile that is used. Coulomb collision effects are not considered. Within the limitations imposed by the present study, it is found that linear growth rates of electromagnetic modes (collisionless microtearing modes and kinetic ballooning modes) are significantly reduced in a high-elongation and high-triangularity NSTX-like geometry compared to a circular NSTX-like geometry. For example, growth rates of kinetic ballooning modes at high- β are reduced to the level of that of collisionless trapped electron modes.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Global gyrokinetic simulations of kinetic-ballooning-mode turbulence in Wendelstein 7-X
We apply the global gyrokinetic code GENE-3D to perform kinetic-ballooning-mode (KBM) turbulence simulations in the Wendelstein 7-X (W7-X) stellarator using experimentally relevant parameters. In contrast to the flux-tube approach, we observe saturation of the turbulent fluctuations at finite amplitude. Analyzing the turbulent characteristics, we identify the presence of weakly driven KBM turbulence hybridizing with ion-temperature-gradient and trapped-electron-mode turbulence in the scenario under consideration, showing that it is possible that KBM activity can be driven in W7-X, opening up the path for studying this type of turbulence, which is highly relevant for future fusion power plants.
Electrostatics↗
Validation of a comprehensive first-principles-based framework for predicting the performance of future stellarators
This paper presents the validation of the GENE-KNOSOS-Tango framework for recovering both the steady-state plasma profiles in the considered radial domain and selected turbulence trends in a stellarator. This framework couples the gyrokinetic turbulence code GENE, the neoclassical transport code KNOSOS, and the transport solver Tango in a multi-timescale simulation feedback loop. Ion-scale kinetic-electron and electron-scale adiabatic-ion flux-tube simulations were performed to evolve the density and temperature profiles for four OP1.2b W7-X scenarios. The simulated density and temperature profiles showed good agreement with the experimental data using a reasonable set of boundary conditions. Equally important was the reproduction of observed trends for several turbulence properties, such as density fluctuations and turbulent heat diffusivities. Key effects were also touched upon, such as electron-scale turbulence and the neoclassical radial electric field shear. The validation of the GENE-KNOSOS-Tango framework enables credible predictions of physical phenomena in stellarators and reactor performance based on a given set of edge parameters.
Gyrokinetic simulations↗
A phase-shift-periodic parallel boundary condition for low-magnetic-shear scenarios
Abstract We formulate a generalized periodic boundary condition as a limit of the standard twist-and-shift parallel boundary condition that is suitable for simulations of plasmas with low magnetic shear. This is done by applying a phase shift in the binormal direction when crossing the parallel boundary. While this phase shift can be set to zero without loss of generality in the local flux-tube limit when employing the twist-and-shift boundary condition, we show that this is not the most general case when employing periodic parallel boundaries, and may not even be the most desirable. A non-zero phase shift can be used to avoid the convective cells that plague simulations of the three-dimensional Hasegawa–Wakatani system, and is shown to have measurable effects in periodic low-magnetic-shear gyrokinetic simulations. We propose a numerical program where a sampling of periodic simulations at random pseudo-irrational flux surfaces are used to determine physical observables in a statistical sense. This approach can serve as an alternative to applying the twist-and-shift boundary condition to low-magnetic-shear scenarios, which, while more straightforward, can be computationally demanding.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Color halo scenario of charmonium-like hybrids
The internal structures of J PC =1 -- ,(o,1,2) -+ charmonium-like hybrids are investigated under lattice QCD in the quenched approximation. We define the Bethe-Salpeter wave function (Φn(r)) in the Coulomb gauge as the matrix element of a spatially extended hybrid-like operator ($\bar{c}cg$) between the vacuum and n-th state for J PC , with r being the spatial separation between a localized $\bar{c}c$ component and the chromomagnetic strength tensor. These wave functions exhibit some similarities for states with the aforementioned different quantum numbers, and their r-behaviors (no node for the ground states and one node for the first excited states) imply that r can be a meaningful dynamical variable for these states. Additionally, the mass splittings of the ground states and first excited states of charmonium-like hybrids in these channels are obtained for the first time to be approximately 1.2-1.4 GeV. These results do not support the flux-tube description of heavy-quarkonium-like hybrids in the Born-Oppenheimer approximation. In contrast, a charmonium-like hybrid can be viewed as a "color halo" charmonium for which a relatively localized color octet $\bar{c}c$ is surrounded by gluonic degrees of freedom, which can readily decay into a charmonium state along with one or more light hadrons. The color halo picture is compatible with the decay properties of Y (42600) and suggests LHCb and BelleII to search for (0,1,2) -+ charmonium-like hybrids in Xc0,1,2N and J/ψω(φ) final states.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Nonlinear gyrokinetic predictions of SPARC burning plasma profiles enabled by surrogate modeling
Multi-channel, nonlinear predictions of core temperature and density profiles are performed for the SPARC tokamak accounting for both kinetic neoclassical and fully nonlinear gyro-kinetic turbulent fluxes. A series of flux-tube, nonlinear, electromagnetic simulations using the CGYRO code with six gyrokinetic species are coupled to a nonlinear optimizer using Gaussian process regression techniques. The simultaneous evolution of energy sources, including alpha heat, radiation, and energy exchange, coupled with these high fidelity models and techniques, leads to a converged solution in electron temperature, ion temperature and electron density channels with a minimal number of expensive gyrokinetic simulations without compromising accuracy.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
A reduced model for the ITER divertor based on SOLPS solutions for ITER Q = 10 baseline conditions: A. identifying options for the control parameters*
Edge codes such as SOLPS coupled to neutral codes such as EIRENE have become so comprehensive and sophisticated that they now constitute, in effect, 'code-experiments' that, as for actual experiments, can benefit from interpretation using simple models and conceptual frameworks, i.e. reduced models. The first task is the identification of options for the reduced model control parameters that are best suited for control of the action of the divertor, i.e. for control of target power loading and sputter-erosion, primarily. A strong correlation between the electron temperature at the divertor target, T e,t , and the neutral deuterium D 2 density at the target, n D2,t , flux-tube resolved, has recently been reported for a number of code studies including SOLPS-4.3 modeling of a set of ~50 ITER baseline cases: Q DT = 10, q 95 = 3, P SOL = 100 MW, metallic walls, and Ne seeding. This part A of the present study reports new results for largely the same ITER cases, confirming the strong correlations reported earlier between local values of T e,t , and (i) n D2,t , and (ii) normalized volumetric losses of power and pressure in the divertor. Strong correlations have now also been found, and are reported here for the first time, between T e,t and all of the divertor target quantities of practical interest. A physical explanation for this surprising result has not been fully identified; nevertheless it has encouraging implications for reduced modeling of the ITER divertor. For such ITER conditions, (i) the global Ne injection rate, Inj Ne [Ne/s], and (ii) the electron temperature at the location on the target where the peak power deposition occurs, T e,t @q ⊥,pk (eV), are found to be promising reduced model control parameters. In the companion report, part B, a reduced model for the ITER divertor is developed and described in detail, based on reversed-direction 2 point modelling, Rev2PM. The input to the reduced model is a value of the variable pair (T et @q ⟂,pk ,Inj Ne ) and the output are values of the various target as well as divertor-entrance quantities of practical interest, e.g. q ⊥,pk , n e,Xpt (the electron density at the poloidal location of the X-point), etc. In part B the reduced model is quantitatively characterized using one half of the code cases; it is then used to successfully predict (replicate) the code values of e.g. n e,Xpt for the other half of the cases.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
A reduced model for the ITER divertor based on SOLPS solutions for ITER Q = 10 baseline conditions: B. A reduced model based on reversed-direction two point modeling
Edge codes such as SOLPS coupled to neutral codes such as EIRENE have become so comprehensive and sophisticated that they now constitute, in effect, 'code-experiments' that, as for actual experiments, can benefit from interpretation using simple models and conceptual frameworks, i.e. reduced models. The first task is the identification of options for the reduced model control parameters that are best suited for control of the action of the divertor, i.e. for control of target power loading and sputter–erosion, primarily. A strong correlation between the electron temperature at the divertor target, T e,t , and the neutral deuterium D 2 density at the target, n D2,t , flux-tube resolved, has recently been reported for a number of code studies including SOLPS-4.3 modeling of a set of ~50 ITER baseline cases: Q DT = 10, q 95 = 3, P SOL = 100 MW, metallic walls, and Ne seeding (Pitts et al 2019). Part A of the present study reports new results for largely the same ITER cases, confirming the strong correlation reported earlier between local values of T e,t , and (i) n D2,t , and (ii) normalized volumetric losses of power and pressure in the divertor. Strong correlations have now also been found, and are reported here for the first time, between T e,t and all of the divertor target quantities of practical interest. A physical explanation for this surprising result has not as yet been fully identified; nevertheless it has encouraging implications for reduced modeling of the ITER divertor. For such ITER conditions, (i) the global Ne injection rate, Inj Ne (Ne s –1 ), and (ii) the electron temperature at the location on the target where the peak power deposition occurs, T e,t @q ⊥,pk (eV), are found to be promising reduced model control parameters. In this part B, a reduced model for the ITER divertor is developed and described in detail, based on reversed-direction two point modeling, Rev2PM. The input to the reduced model is a value of the variable pair (T e,t @q ⊥,pk , Inj Ne )for a chosen case and the output are values of the various target as well as divertor-entrance quantities of practical interest, e.g. q ⊥,pk , the electron density at the X-point, n e,Xpt , etc. The reduced model was quantitatively characterized using one half of the code cases; it was then used to successfully predict (replicate) the code values of e.g. n e,Xpt for the other half.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Safety First: Stability and Dissipation of Line-tied Force-free Flux Tubes in Magnetized Coronae
Abstract Magnetized plasma columns and extended magnetic structures with both footpoints anchored to a surface layer are an important building block of astrophysical dissipation models. Current loops shining in X-rays during the growth of plasma instabilities are observed in the corona of the Sun and are expected to exist in highly magnetized neutron star magnetospheres and accretion disk coronae. For varying twist and system sizes, we investigate the stability of line-tied force-free flux tubes and the dissipation of twist energy during instabilities using linear analysis and time-dependent force-free electrodynamics simulations. Kink modes ( m = 1) and efficient magnetic energy dissipation develop for plasma safety factors q ≲ 1, where q is the inverse of the number of magnetic field line windings per column length. Higher-order fluting modes ( m > 1) can distort equilibrium flux tubes for q > 1 but induce significantly less dissipation. In our analysis, the characteristic pitch μ ˜ 0 of flux-tube field lines determines the growth rate ( ∝ μ ˜ 0 3 ) and minimum wavelength of the kink instability ( ∝ μ ˜ 0 − 1 ). We use these scalings to determine a minimum flux tube length for the growth of the kink instability for any given μ ˜ 0 . By drawing analogies to idealized magnetar magnetospheres with varying regimes of boundary shearing rates, we discuss the expected impact of the pitch-dependent growth rates for magnetospheric dissipation in magnetar conditions.
79 ASTRONOMY AND ASTROPHYSICS↗