bi-Maxwellian Distribution Function Research Articles

Detection of Electron Temperature Anisotropy by an X-ray Crystal Spectrometer in the Large Helical Device

Abstract We examine Ar XVI spectra measured by an X-ray crystal spectrometer to estimate electron temperature anisotropy in high-temperature electron cyclotron heated plasmas in the Large Helical Device (LHD). We calculate the atomic structure and electron impact excitation cross section between magnetic sublevels of Ar XVI. By assuming a bi-Maxwellian electron distribution function and considering the mount parameters of the X-ray spectrometer in LHD, we model the expected intensity ratio q/r as a function of Te⊥B and Te||B, where ⊥ B and ||B denote perpendicular and parallel components to the magnetic axis of the toroidal plasma, respectively. The calculation results show that the intensity ratio of q and r, which are formed by inner-shell excitation from the ground state of Ar15+, is sensitive to electron temperature anisotropy. We apply the calculation results to the LHD experiments. The ratio of q/r changes with variations in electron density and collision frequency. In the core and low νe region, Te⊥B is predominant, and the electron temperature becomes isotropic above νe > 104 Hz. By combining electron temperature measurements from Thomson scattering and radial profile of Ar15+ ions estimated using the extreme ultraviolet spectrometer, local values of Te⊥B/Te||B are quantitatively estimated using the q/r ratio. The derived Te⊥B/Te||B is compared with collision frequency, radial electric field, and effective helical ripple, and the experimental results are explained qualitatively.

Plasma properties in the vicinity of the last closed flux surface in hydrogen and helium fusion plasma discharges

The origin of the bi-Maxwellian electron energy distribution function (EEDF) observed in the scrape-off layer (SOL) of tokamak plasmas by means of Langmuir probes is still under discussion. It has been assumed that the ionization of hydrogen and deuterium neutrals by thermal electrons penetrating the SOL from the bulk plasma is the main reason for the appearance of a second Maxwellian. To validate this assumption, radial measurements of the electron temperatures and densities, or the plasma properties in helium plasmas in the GOLEM tokamak and the TJ-II stellarator were performed. The radial profiles of the low-temperature electron group densities follow the trend of the calculated radial profiles of the electron sources arising from the ionization of neutrals in both deuterium and helium plasmas in TJ-II. The difference in the radial location where the bi-Maxwellian EEDF appears can be explained by the difference in the rate coefficients for ionization of deuterium and helium. The results of probe measurements in GOLEM and the WEST tokamak divertor, at one radial location in the SOL, are compatible with the hypothesis concerning the ionization of neutral atoms and the type of the EEDF.

Statistical Study of Anisotropic Proton Heating in Interplanetary Magnetic Switchbacks Measured by Parker Solar Probe

Magnetic switchbacks, which are large angular deflections of the interplanetary magnetic field, are frequently observed by Parker Solar Probe (PSP) in the inner heliosphere. Magnetic switchbacks are believed to play an important role in the heating of the solar corona and the solar wind as well as the acceleration of the solar wind in the inner heliosphere. Here, we analyze magnetic field data and plasma data measured by PSP during its second and fourth encounters, and select 71 switchback events with reversals of the radial component of the magnetic field at times of unchanged electron-strahl pitch angles. We investigate the anisotropic thermal kinetic properties of plasma during switchbacks in a statistical study of the measured proton temperatures in the parallel and perpendicular directions as well as proton density and specific proton fluid entropy. We apply the “genetic algorithm” method to directly fit the measured velocity distribution functions in field-aligned coordinates using a two-component bi-Maxwellian distribution function. We find that the protons in most switchback events are hotter than the ambient plasma outside the switchbacks, with characteristics of parallel and perpendicular heating. Specifically, significant parallel and perpendicular temperature increases are seen for 45 and 62 of the 71 events, respectively. We find that the density of most switchback events decreases rather than increases, which indicates that proton heating inside the switchbacks is not caused by adiabatic compression, but is probably generated by nonadiabatic heating caused by field–particle interactions. Accordingly, the proton fluid entropy is greater inside the switchbacks than in the ambient solar wind.

Influence of electron energy distribution on fluid models of a low-pressure inductively coupled plasma discharge

The aim of the present paper is to examine the influence of assumption on the electron energy distribution function on the relation between the plasma potential and the electron temperature for both electropositive (argon) and electronegative (chlorine) plasmas. A one-dimensional fluid model is used for simplicity although similar results were obtained using a self-consistent two-dimensional fluid model coupled with the Maxwell's equations for inductively coupled plasmas. We find that for electropositive plasma only a bi-Maxwellian electron energy distribution function provides reasonable results compared to measurements in low-pressure inductively coupled plasmas, namely, the increasing plasma potential for increasing electron temperature. For electronegative plasma, the plasma potential is an increasing function of the electron temperature for all electron distributions considered in the model. However, the scaling factors do not agree with the conventional plasma theory. We explain these results by the deviation of electrons from a Boltzmann distribution, which is due to non-equilibrium and non-local nature of plasma at the low-pressure conditions.

Experimental observations of local plasma parameters in the COMPASS divertor in NBI-assisted L-mode plasmas

This paper presents an experimental characterization of the local plasma parameters in the COMPASS open divertor in deuterium, L-mode plasmas by means of Langmuir probes during neutral beam injection (NBI) heating. The effect of NBI heating of the core plasma on the local divertor plasma parameters is investigated under different plasma configurations and different directions of the magnetic field and plasma current for different line-averaged electron densities. At low line-averaged densities below 6 × 1019 m-3, a low NBI heating power level does not lead to any significant changes in the plasma parameters in the divertor region. It is also found that for a reversed magnetic field the influence of NBI in the divertor is stronger at higher line-averaged electron density and plasma current. In the divertor, the impact of the NBI heating is more pronounced at high delivered powers (above P NBI = 320 kW) and line-averaged densities (above 7 × 1019 m-3). The electron temperature and the floating potential at the outer target increases respectively with 10–50% and 100%, while the plasma potential in the inner divertor increases twofold. At a higher line-averaged density, the NBI heating affects the edge plasma, namely, the electron temperature in the midplane scrape-off layer and the heating moves toward the divertor. Thus, a twice as high electron temperature is observed in the divertor and a bi-Maxwellian electron energy distribution function arises.

Impact of non-thermal electrons on spatial damping: a kinetic model for the parallel propagating modes

Abstract A sequence of in situ measurements points the presence of non-thermal species in the profile of particle distributions. This study highlights the role of such energetic electrons on the wave-spectrum. Using Vlasov–Maxwell’s model, the dispersion relations of the parallel propagating modes along with the space scale of damping are discussed using non-relativistic bi-Maxwellian and bi-Kappa distribution functions under the weak field approximation, i.e., ω − k . v > Ω 0 $\left\vert \omega -\mathbf{k}.\mathbf{v}\right\vert { >}{{\Omega}}_{0}$ . Power series and asymptotic expansions of plasma dispersion functions are performed to derive the modes and spatial damping of waves, respectively. The role of these highly energetic electrons is illustrated on real frequency and anomalous damping of R and L-modes which is in fact controlled by the parameter κ in the dispersion. Further, we uncovered the effect of external magnetic field and thermal anisotropy on such spatial attenuation. In global perspective of the kinetic model, it may be another step.

Analysis of influences of pressure anisotropies on the 3D MHD equilibrium in LHD

3D equilibria with an anisotropic pressure component in the large helical device are analyzed with respect to their magnetic axis locations. The anisotropic extension of the 3D equilibrium solver variational moments equilibrium code, anisotropic neumann inverse moments equilibrium code, is used to compute fixed-boundary plasma equilibria based on a bi-Maxwellian distribution function describing the anisotropic particles. Different heating scenarios are assessed by means of parallel and perpendicular pressure anisotropies with different radial anisotropic pressure profiles imposed. A theoretical predicted scaling of the magnetic axis location with the auxiliary parameter βeq as predicted for classical stellarators and heliotrons by Hitchon [Nucl. Fusion 23, 383 (1983)] is found to be applicable to the large helical device in the case of a flat hot-particle profile for parallel or weak perpendicular dominated anisotropies with β⊥/β∥≤2. For strong perpendicular or non-flat hot-particle profiles, a deviation from the predicted scaling of the magnetic axis location is found. Whereas center-peaked profiles show a stronger shift of the magnetic axis, edge-peaked profiles show no significant change of its radial location. High critical magnetic fields are identified as a necessary condition for strong perpendicular anisotropies. The observed deviations are ascribed to the magnetic field structure and negative pressure gradients. The invalidity of the theoretical predictions in the case of certain configurations is found to be caused by higher-order terms in the pressure components, which are not accounted for by the ordering on which the theory is based.

Application of subtracted bi-Maxwellian distribution function in propagation of electron cyclotron waves in magnetospheric plasma of an outer planet

For the study of propagation of electron cyclotron waves in magnetospheric plasma, besides the two distribution functions: (i) bi-Maxwellian distribution function, and (ii) generalized Lorentzian (kappa) distribution function, subtracted bi-Maxwellian distribution function is also used in some investigations. In all these investigations, temperature anisotropy is considered, besides a distribution function. In the investigations, for the subtracted bi-Maxwellian distribution function, plasma is considered either relativistic or non-relativistic. We have looked into some investigations where subtracted bi-Maxwellian distribution function is used and we have analyzed them. We have also considered AC electric field perpendicular to the magnetic field in the magnetospheric plasma, and have found that their claim of relativistic treatment of plasma is not valid. Further, we have found that the dispersion relation used for propagation of waves parallel to the magnetic field is erroneous. Further, the dispersion relation for oblique propagation of waves also is erroneous. Therefore, the results of those investigations are wrong. We have also found that for the same set of input parameters, different results are found, though all of them are wrong.

A simple method for theoretical determination of the radius- and time-dependent electron temperatures in nanosecond pulsed longitudinal discharges in helium and neon assuming a bi-Maxwellian electron energy distribution function

Assuming a bi-Maxwellian electron energy distribution function, the temporal and radial distribution of the electron temperature is determined in nanosecond pulsed longitudinal discharges used for excitation of two prospective high-power gas-discharge lasers, namely, a deep ultraviolet Cu+ Ne-H2-CuBr laser and a He-Sr+ recombination laser. For this purpose, the parameters of the set of nonstationary heat-conduction equations for the two groups of electrons, namely, the electrical power density and the specific heat capacity, are determined for each group. A 2D(r,t) numerical model is also developed in order to solve the set of the equations.

Electron cyclotron waves in plasma in magnetosphere of a planet having perpendicular AC electric field

In a series of papers, scientists have discussed about electron cyclotron waves in plasma having AC electric field perpendicular to ambient magnetic field. In some investigations, scientists have claimed to consider relativistic plasma. We have reinvestigated the work and found that their claim of relativistic treatment of plasma is not valid. We have further found that the dispersion relation used by them is erroneous and therefore their results are not reliable. Using, individually, two distribution functions: (i) bi-Maxwellian distribution function, and (ii) Lorentzian (kappa) distribution function, for propagation of waves parallel to magnetic field, for both distribution functions, we have found that for known value of electron cyclotron frequency ωc, the real oscillation frequency ωr depends on temperature anisotropy only. We have also found that the growth rate γ increases continuously with the increase of wavevector k. These results are very different from earlier investigations. We have also found that the study of oblique propagation, carried out in a number of investigations, is not correct.

Electron cyclotron waves in plasma in magnetosphere of a planet having perpendicular DC electric field

Scientists have always been interested in the study of electron cyclotron waves in plasma in magnetosphere of a planet. Series of papers are published where AC electric field is taken perpendicular to magnetic field in the magnetosphere. Some scientists have claimed to consider DC electric field, but they are found not to have any electric field. We have discussed propagation of electron cyclotron waves when constant electric field is perpendicular to magnetic field in megnetosphere of a planet. Using bi-Maxwellian distribution function, we have found that for the known value of electron cyclotron frequency ωc, the real oscillation frequency ωr depends only on the temperature anisotropy. We have also found that the growth rate γ increases continuously with the increase of the wavevector k. It is interesting to note that the dispersion relation used in series of papers is erroneous, and it has been taken as universal relation.

Study of relativistic beam of electron on whistler mode waves for subtracted distribution in Saturnian magnetosphere

Very low frequency electromagnetic waves in the magnetosphere of Saturn are the focus of the present study. Hot injected beam effect on these waves like whistler mode waves for relativistic and non-relativistic subtracted bi-Maxwellian distribution function have been studied in the presence of perpendicular A.C electric field. By using the method of characteristics solutions and kinetic approach, expression for dispersion relation and growth rate has been derived via detailed calculations and derivations. By changing parameters of plasma like: ac frequency, thermal anisotropy, number density, parametric analysis has been done. The influence of AC frequency on Doppler shift is analyzed, and the comparative study of the effects of oblique and parallel propagating waves on the growth rate has been done. Some different results were found using the subtracted bi-Maxwellian distribution function discussed with the results of bi-Maxwellian distribution function. It can be seen that the effective parameters for generating whistler mode waves are not only temperature anisotropy, but also relativistic factor, alternating field frequency, subtracted distribution amplitude and the width of the loss cone distribution function, which have been discussed in the results and discussion section.

Exploring the effect of various plasma parameters on whistler mode growth rates in the Jovian magnetosphere

In 1979, after plasma envelope exploration, Voyager 1 and 2 revealed that Jovian magnetosphere consists of an unusual mixture of ions like hydrogen, sulphur, oxygen etc., in similar proportions. The present study observed that the waves in Jovian magnetosphere propagate in whistler-mode, with some similarities to whistler-mode auroral hiss in the Earth’s magnetosphere. The dispersion relation has been deduced and calculated in detail for oblique propagating waves in presence of parallel AC electric field for bi-Maxwellian distribution function. Magnetic field model for different values of latitude at radial distance $17 R_{J}$ has been reported. By using the method of characteristic solution, relativistic growth rate has been calculated. Data provided by spacecrafts like Pioneer 10 and 11, Voyager 1 and 2, while exploring the magnetosphere of Jupiter, has been used to plot graphs of variation of growth rate for different values of various plasma parameters like temperature anisotropy, angle of wave propagation, AC frequency etc. The effect on growth rate by these plasma parameters is shown by graphs.

Effects of Non-Maxwellian Electron Distribution Function to the Propagation Coefficients of Electromagnetic Waves in Plasma

The values of the propagation constants of electromagnetic (EM) waves through the plasma using as the example of a homogeneous plasma layer for different electron distribution functions (EDFs) are the subject of this paper. The Maxwellian, Druyvesteyn, Pipe-line, and Bi-Maxwellian distribution functions were considered in this paper. Using values of the complex dielectric permittivity obtained from using different EDFs, the results for corresponding reflection, transmission, and absorption coefficients can be significantly different. The EDFs sometimes can have a considerable effect on the dielectric properties of plasma. Hence, comparing the results obtained, it is helpful to analyze the interaction between EM waves and plasma and provide a theoretical basis for the experiments.

A New Inner Heliosphere Proton Parameter Dataset from the Helios Mission

In the near future, Parker Solar Probe and Solar Orbiter will provide the first comprehensive in-situ measurements of the solar wind in the inner heliosphere since the Helios mission in the 1970s. We describe a reprocessing of the original Helios ion distribution functions to provide reliable and reproducible data to characterise the proton core population of the solar wind in the inner heliosphere. A systematic fitting of bi-Maxwellian distribution functions was performed to the raw Helios ion distribution function data to extract the proton core number density, velocity, and temperatures parallel and perpendicular to the magnetic field. We present radial trends of these derived proton parameters, forming a benchmark to which new measurements in the inner heliosphere will be compared. The new dataset has been made openly available for other researchers to use, along with the source code used to generate it.

New Closures for More Precise Modeling of Landau Damping in the Fluid Framework.

Incorporation of kinetic effects such as Landau damping into a fluid framework was pioneered by Hammett and Perkins, by obtaining closures of the fluid hierarchy, where the gyrotropic heat flux fluctuations or the deviation of the fourth-order gyrotropic fluid moment are expressed through lower-order fluid moments. To obtain a closure of a fluid model expanded around a bi-Maxwellian distribution function, the usual plasma dispersion function Z(ζ) that appears in kinetic theory or the associated plasma response function R(ζ)=1+ζZ(ζ) has to be approximated with a suitable Padé approximant in such a way that the closure is valid for all ζ values. Such closures are rare, and the original closures of Hammett and Perkins are often employed. Here we present a complete mapping of all plausible Landau fluid closures that can be constructed at the level of fourth-order moments in the gyrotropic limit and we identify the most precise closures. Furthermore, by considering 1D closures at higher-order moments, we show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing convergence of the fluid and collisionless kinetic descriptions.

Deuterium temperature, drift velocity, and density measurements in non-Maxwellian plasmas at ASDEX Upgrade

We measure the deuterium density, the parallel drift velocity, and parallel and perpendicular temperatures (, ) in non-Maxwellian plasmas at ASDEX Upgrade. This is done by taking moments of the ion velocity distribution function measured by tomographic inversion of five simultaneously acquired spectra of -light. Alternatively, we fit the spectra using a bi-Maxwellian distribution function. The measured kinetic temperatures ( keV, keV) reveal the anisotropy of the plasma and are substantially higher than the measured boron temperature (7 keV). The Maxwellian deuterium temperature computed with TRANSP (6 keV) is not uniquely measurable due to the fast ions. Nevertheless, simulated kinetic temperatures accounting for fast ions based on TRANSP ( keV, keV) are in excellent agreement with the measurements. Similarly, the Maxwellian deuterium drift velocity computed with TRANSP (300 km s−1) is not uniquely measurable, but the simulated kinetic drift velocity accounting for fast ions agrees with the measurements (400 km s−1) and is substantially larger than the measured boron drift velocity (270 km s−1). We further find that ion cyclotron resonance heating elevates and each by 2 keV without evidence for preferential heating in the spectra. Lastly, we derive an expression for the 1D projection of an arbitrarily drifting bi-Maxwellian onto a diagnostic line-of-sight.

Plasma potential and electron temperature evaluated by ball-pen and Langmuir probes in the COMPASS tokamak

The radial distributions of the main plasma parameters in the scrape-off-layer of the COMPASS tokamak are measured during L-mode and H-mode regimes by using both Langmuir and ball-pen probes mounted on a horizontal reciprocating manipulator. The radial profile of the plasma potential derived previously from Langmuir probes data by using the first derivative probe technique is compared with data derived using ball-pen probes. A good agreement can be seen between the data acquired by the two techniques during the L-mode discharge and during the H-mode regime within the inter-ELM periods. In contrast with the first derivative probe technique, the ball-pen probe technique does not require a swept voltage and, therefore, the temporal resolution is only limited by the data acquisition system. In the electron temperature evaluation, in the far scrape-off layer and in the limiter shadow, where the electron energy distribution is Maxwellian, the results from both techniques match well. In the vicinity of the last closed flux surface, where the electron energy distribution function is bi-Maxwellian, the ball-pen probe technique results are in agreement with the high-temperature components of the electron distribution only. We also discuss the application of relatively large Langmuir probes placed in parallel and perpendicularly to the magnetic field lines to studying the main plasma parameters. The results obtained by the two types of the large probes agree well. They are compared with Thomson scattering data for electron temperatures and densities. The results for the electron densities are compared also with the results from ASTRA code calculation of the electron source due to the ionization of the neutrals by fast electrons and the origin of the bi-Maxwellian electron energy distribution function is briefly discussed.

Cyclotron instabilities driven by temperature anisotropy in the solar wind

Kinetic plasma instabilities are important for regulating the temperature anisotropies of electrons and ions in solar wind. For the low beta regime, it is known that electromagnetic ion/electron cyclotron instabilities are important, but in the literature these unstable modes are discussed under the assumption of parallel propagation. The present paper extends the analysis to two (or with cylindrical symmetry, three) dimensions. The analysis is further extended to include quasilinear description with the assumption of the bi-Maxwellian velocity distribution function. Such an analysis lays the foundation for an eventual study in which cyclotron instabilities as well as obliquely propagating unstable modes such as the mirror instability are simultaneously taken into account. The present paper first lays down the basis for such future efforts in which the two- or three dimensional linear and quasilinear theories of cyclotron instabilities in the low beta regime are formulated.

Kinetic instabilities in the solar wind driven by temperature anisotropies

The present paper comprises a review of kinetic instabilities that may be operative in the solar wind, and how they influence the dynamics thereof. The review is limited to collective plasma instabilities driven by the temperature anisotropies. To limit the scope even further, the discussion is restricted to the temperature anisotropy-driven instabilities within the model of bi-Maxwellian plasma velocity distribution function. The effects of multiple particle species or the influence of field-aligned drift will not be included. The field-aligned drift or beam is particularly prominent for the solar wind electrons, and thus ignoring its effect leaves out a vast portion of important physics. Nevertheless, for the sake of limiting the scope, this effect will not be discussed. The exposition is within the context of linear and quasilinear Vlasov kinetic theories. The discussion does not cover either computer simulations or data analyses of observations, in any systematic manner, although references will be made to published works pertaining to these methods. The scientific rationale for the present analysis is that the anisotropic temperatures associated with charged particles are pervasively detected in the solar wind, and it is one of the key contemporary scientific research topics to correctly characterize how such anisotropies are generated, maintained, and regulated in the solar wind. The present article aims to provide an up-to-date theoretical development on this research topic, largely based on the author’s own work.