Bi-Maxwellian, slowing-down, and ring velocity distributions of fast ions in magnetized plasmas

Abstract

We discuss analytic fast-ion velocity distribution functions which are useful for basic plasma modelling as illustrated for typical parameters of the future fusion plasma in the tokamak ITER. The Maxwellian is by far the most widespread model for ions and electrons in tokamaks and stellarators. The bi-Maxwellian and the drifting (bi-)Maxwellian are extensions allowing for anisotropy and bulk plasma flow, respectively. For example, fast ions generated by wave heating in the ion cyclotron range of frequencies are often described by bi-Maxwellians or so-called tail temperatures. The ring distribution can serve as a basic building block for arbitrary distributions or as bump-on-tail in stability studies. The isotropic slowing-down distribution is a good model for fusion α-particles. The anisotropic slowing-down distribution occurs for anisotropic particle sources as is typical for neutral beam injection. We physically motivate these distribution functions and present analytic models in various coordinate systems commonly used by theorists and experimentalists. We further calculate 1D projections of the distribution functions onto a diagnostic line-of-sight to gain insight into measurements relying on the Doppler shift.

Fast-ion velocity-space tomography using slowing-down regularization in EAST plasmas with co- and counter-current neutral beam injection

“…Here m i is the fast-ion mass, v = v 2 ⊥ + v 2 is the fast-ion speed and the subscripts of the velocity components v ⊥ and v indicate the directions relative to the magnetic field. Drifting Maxwellian distributions, as described in [39], with estimated local bulk ion temperatures, densities, and drift velocities: T ∼ 1.2 − 1.5 keV, n ∼ 2.5 − 2.8 × 10 19 m −3 and v d ∼ 2.5 − 8 km/s, have been added to the TRANSP distributions in order to emphasize the much higher ion densities at energies below 10 keV compared to the dilute populations at higher energies.…”

“…At high energies, the energy is transferred predominantly to electrons causing little pitch-angle scattering. Collisions with ions dominate below the crossover speed [39]. For the EAST discharges, this is evaluated to be ∼ 26 keV for deuterium.…”

Bayesian inference of ion velocity distribution function from laser-induced fluorescence spectra

“…There are also many other models of ion velocity distribution function in magnetized plasma, as reviewed in Ref. [2]. Depending on the setup, they can also be candidates, not only f (v; n i , T i ) and g(v; n f , v b , v c ), to be evaluated in terms of the statistical adequateness for the observed spectrum.…”

“…The bi-Maxwellian distribution represents the anisotropy of flow and temperature in parallel and perpendicular directions to the external magnetic field [1]. The ring velocity distribution is a model of the energetic ion population that is formed immediately after neutral beam injection [2]. The velocity distribution function of α-particles has a fat tail originating from collisions of these particles with the thermal ions and electrons in the background plasma [1,3,4].…”

Formation of non-thermal electron velocity distribution functions in kinetic magnetic reconnection

“…Analytical studies and kinetic PIC simulations of the ECMI usually have been carried out with ring-beam EVDF 26,[42][43][44][45][46] . For this sake the authors concentrated on EVDFs resembling a ring in the 2D velocity space perpendicular to the local magnetic field.…”