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Register a new userGyrokinetic simulations of fusion plasmas using a spectral velocity space representation
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Joseph Thomas Parker
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Magnetic confinement fusion reactors suffer severely from heat and particle losses through turbulent transport, which has inspired the construction of ever larger and more expensive reactors. Numerical simulations are vital to their design and operation, but particle collisions are too infrequent for fluid descriptions to be valid. Instead, strongly magnetised fusion plasmas are described by the gyrokinetic equations, a nonlinear integro-differential system for evolving the particle distribution functions in a five-dimensional position and velocity space, and the consequent electromagnetic field. Due to the high dimensionality, simulations of small reactor sections require hundreds of thousands of CPU hours on High Performance Computing platforms. We develop a Hankel-Hermite spectral representation for velocity space that exploits structural features of the gyrokinetic system. The representation exactly conserves discrete free energy in the absence of explicit dissipation, while our Hermite hypercollision operator captures Landau damping with few variables. Calculation of the electromagnetic fields becomes purely local, eliminating inter-processor communication in, and vastly accelerating, searches for linear instabilities. We implement these ideas in SpectroGK, an efficient parallel code. Turbulent fusion plasmas may dissipate free energy through linear phase mixing to fine scales in velocity space, as in Landau damping, or through a nonlinear cascade to fine scales in physical space, as in hydrodynamic turbulence. Using SpectroGK to study saturated electrostatic drift-kinetic turbulence, we find that the nonlinear cascade suppresses linear phase mixing at energetically-dominant scales, so the turbulence is fluid-like. We use this observation to derive Fourier-Hermite spectra for the electrostatic potential and distribution function, and confirm these spectra with simulations.
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Plasma turbulence is studied via direct numerical simulations in a two-dimensional spatial geometry. Using a hybrid Vlasov-Maxwell model, we investigate the possibility of a velocity-space cascade. A novel theory of space plasma turbulence has been recently proposed by Servidio {it et al.} [PRL, {bf 119}, 205101 (2017)], supported by a three-dimensional Hermite decomposition applied to spacecraft measurements, showing that velocity space fluctuations of the ion velocity distribution follow a broad-band, power-law Hermite spectrum $P(m)$, where $m$ is the Hermite index. We numerically explore these mechanisms in a more magnetized regime. We find that (1) the plasma reveals spectral anisotropy in velocity space, due to the presence of an external magnetic field (analogous to spatial anisotropy of fluid and plasma turbulence); (2) the distribution of energy follows the prediction $P(m)sim m^{-2}$, proposed in the above theoretical-observational work; and (3) the velocity-space activity is intermittent in space, being enhanced close to coherent structures such as the reconnecting current sheets produced by turbulence. These results may be relevant to the nonlinear dynamics weakly-collisional plasma in a wide variety of circumstances.
In this work, we compare gyrokinetic simulations in stellarators using different computational domains, namely, flux tube, full-flux-surface, and radially global domains. Two problems are studied: the linear relaxation of zonal flows and the linear stability of ion temperature gradient (ITG) modes. Simulations are carried out with the codes EUTERPE, GENE, GENE-3D, and stella in magnetic configurations of LHD and W7-X using adiabatic electrons. The zonal flow relaxation properties obtained in different flux tubes are found to differ with each other and with the radially global result, except for sufficiently long flux tubes, in general. The flux tube length required for convergence is configuration-dependent. Similarly, for ITG instabilities, different flux tubes provide different results, but the discrepancy between them diminishes with increasing flux tube length. Full-flux-surface and flux tube simulations show good agreement in the calculation of the growth rate and frequency of the most unstable modes in LHD, while for W7-X differences in the growth rates are found between the flux tube and the full-flux-surface domains. Radially global simulations provide results close to the full-flux-surface ones. The radial scale of unstable ITG modes is studied in global and flux tube simulations finding that in W7-X, the radial scale of the most unstable modes depends on the binormal wavenumber, while in LHD no clear dependency is found.
The linear gyrokinetic stability properties of magnetically confined electron-positron plasmas are investigated in the parameter regime most likely to be relevant for the first laboratory experiments involving such plasmas, where the density is small enough that collisions can be ignored and the Debye length substantially exceeds the gyroradius. Although the plasma beta is very small, electromagnetic effects are retained, but magnetic compressibility can be neglected. The work of a previous publication (Helander, 2014) is thus extended to include electromagnetic instabilities, which are of importance in closed-field-line configurations, where such instabilities can occur at arbitrarily low pressure. It is found that gyrokinetic instabilities are completely absent if the magnetic field is homogeneous: any instability must involve magnetic curvature or shear. Furthermore, in dipole magnetic fields, the stability threshold for interchange modes with wavelengths exceeding the Debye radius coincides with that in ideal MHD. Above this threshold, the quasilinear particle flux is directed inward if the temperature gradient is sufficiently large, leading to spontaneous peaking of the density profile.
A massively parallel simulation code, called textit{dHybrid}, has been developed to perform global scale studies of space plasma interactions. This code is based on an explicit hybrid model; the numerical stability and parallel scalability of the code are studied. A stabilization method for the explicit algorithm, for regions of near zero density, is proposed. Three-dimensional hybrid simulations of the interaction of the solar wind with unmagnetized artificial objects are presented, with a focus on the expansion of a plasma cloud into the solar wind, which creates a diamagnetic cavity and drives the Interplanetary Magnetic Field out of the expansion region. The dynamics of this system can provide insights into other similar scenarios, such as the interaction of the solar wind with unmagnetized planets.
Observations of ion-scale (k_y*rho_i <= 1) density turbulence of relative amplitude dn_e/n_e <= 0.2% are available on the Mega Amp Spherical Tokamak (MAST) using a 2D (8 radial x 4 poloidal channel) imaging Beam Emission Spectroscopy (BES) diagnostic. Spatial and temporal characteristics of this turbulence, i.e., amplitudes, correlation times, radial and perpendicular correlation lengths and apparent phase velocities of the density contours, are determined by means of correlation analysis. For a low-density, L-mode discharge with strong equilibrium flow shear exhibiting an internal transport barrier (ITB) in the ion channel, the observed turbulence characteristics are compared with synthetic density turbulence data generated from global, non-linear, gyro-kinetic simulations using the particle-in-cell (PIC) code NEMORB. This validation exercise highlights the need to include increasingly sophisticated physics, e.g., kinetic treatment of trapped electrons, equilibrium flow shear and collisions, to reproduce most of the characteristics of the observed turbulence. Even so, significant discrepancies remain: an underprediction by the simulations of the turbulence amplituide and heat flux at plasma periphery and the finding that the correlation times of the numerically simulated turbulence are typically two orders of magnitude longer than those measured in MAST. Comparison of these correlation times with various linear timescales suggests that, while the measured turbulence is strong and may be `critically balanced, the simulated turbulence is weak.
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