Electron Thermal Speed Research Articles

Numerical study of turbulent eddy self-interaction in tokamaks with low magnetic shear. Part II: Nonlinear simulations.

Abstract Using local nonlinear gyrokinetic simulations and building up on linear results from a companion paper (Vol\v{c}okas, 2024), we demonstrate that turbulent eddies can extend along magnetic field lines for hundreds of poloidal turns in tokamaks with weak or zero magnetic shear $\hat{s}$. We observe that this parallel eddy length scales inversely with magnetic shear and at $\hat{s}=0$ is limited by the thermal speed of electrons $v_{th,e}$. We examine the consequences of these ``ultra long" eddies on turbulent transport, in particular, how field line topology mediates strong parallel self-interaction. Our investigation reveals that, through this process, field line topology can strongly affect transport. It can cause transitions between different turbulent instabilities and in some cases triple the logarithmic gradient needed to drive a given amount of heat flux. We also extensively study a novel ``eddy squeezing" effect introduced in (Vol\v{c}okas, 2023), which reduces the perpendicular size of eddies and their ability to transport energy, thus representing a novel approach to improve confinement. Finally, we investigate the triggering mechanism of Internal Transport Barriers (ITBs) using low magnetic shear simulations, shedding light on why ITBs are often easier to trigger where the safety factor has a low-order rational value.

Presheath-like structures and effusive particle losses for biased probes at and near I–V electron saturation

A theory for presheath-like structures near probes biased at and above the plasma potential is developed for collisionless plasmas with an electron-neutral mean free path on the order of the chamber scale. The theory predicts presheath-like perturbations to the plasma that result from the free streaming of electrons and an effusion loss process from the chamber at the electrode. For these situations, a loss-cone-like velocity distribution function for electrons is predicted where the loss angle of the depletion region corresponds to the angular size of the electrode at a specified distance. The angle of the loss cone becomes 180° at the sheath edge. In comparison with a previous collisional electron presheath model that required electrons satisfy a Bohm criterion at the sheath edge [Scheiner et al., Phys. Plasmas 22, 123520 (2015)], the present work suggests that no such condition is needed for collisionless low pressure plasmas in the ≲10 mTorr range. The theory predicts the generation of a density depletion of roughly 0.5ne and an electron velocity moment of tens of percent of the electron thermal speed by the sheath edge in a presheath with a potential drop of less than Ti/e. The range of this presheath perturbation is determined by the electrode geometry instead of the collisional mean free path. These predictions are tested against previously published particle in cell simulations and are found to be in good agreement.

On the Plasma Quasi-thermal Noise in the Outer Heliosphere

The recent paper by Li et al. on electron quasi-thermal noise (QTN) in the outer heliosphere is flawed. It assumes the plasma drift speed to be much smaller than the electron thermal speed, even though both quantities are of the same order of magnitude in the outer heliosphere inward of the termination shock, because of the low plasma temperature. In this case, the Langmuir wave dispersion equation and the QTN in the antenna frame are completely changed. Furthermore, these calculations neglect the shot noise, which should produce a large contribution below the plasma frequency with the Voyager antennas in the outer heliosphere.

Slow Electron Holes in the Earth's Magnetosheath

AbstractWe present a statistical analysis of electrostatic solitary waves observed aboard Magnetospheric Multiscale spacecraft in the Earth's magnetosheath. Applying single‐spacecraft interferometry to several hundred solitary waves collected in about 2‐minute interval, we show that almost all of them have the electrostatic potential of positive polarity and propagate quasi‐parallel to the local magnetic field with plasma frame velocities of the order of 100 km/s. The solitary waves have typical parallel half‐widths from 10 to 100 m that is between 1 and 10 Debye lengths and typical amplitudes of the electrostatic potential from 10 to 200 mV that is between 0.01% and 1% of local electron temperature. The solitary waves are associated with quasi‐Maxwellian ion velocity distribution functions, and their plasma frame velocities are comparable with ion thermal speed and well below electron thermal speed. We argue that the solitary waves of positive polarity are slow electron holes and estimate the time scale of their acceleration, which occurs due to interaction with ions, to be of the order of one second. The observation of slow electron holes indicates that their lifetime was shorter than the acceleration time scale. We argue that multi‐spacecraft interferometry applied previously to these solitary waves is not applicable because of their too‐short spatial scales. The source of the slow electron holes and the role in electron‐ion energy exchange remain to be established.

Introducing electromagnetic effects in Soledge3X

AbstractIn the pedestal region, electromagnetic effects affect the evolution of micro‐instabilities and plasma turbulence. The transport code Soledge3X developed by the CEA offers an efficient framework for turbulent 3D simulation on an electrostatic model with a fixed magnetic field. The physical accuracy of the model is improved with electromagnetic induction, driven by the local value of the parallel component of the electromagnetic vector potential , known from Ampère's law. It is solved implicitly in a coupled system with the vorticity equation on the electric potential . The consequence is a basic electromagnetic behavior in the form of shear Alfvén waves. A finite electron mass prevents unphysical speeds but requires solving for the time evolution of the parallel current density in the generalized Ohm's law. This term can be analytically included with little computational overhead in the system on and and improves its numerical condition, facilitating the iterative solving procedure. Simulations on a periodic slab case let us observe the predicted bifurcation of the wave propagation speed between the Alfvén wave and the electron thermal wave speeds for varying perpendicular wavenumbers. The first results on a circular geometry with a limiter attest to the feasibility of turbulent electromagnetic scenarios.

Theory of gradient drift instabilities in low-temperature, partially magnetised plasmas

A fluid dispersion theory in partially magnetised plasmas is analysed to examine the conditions under which large-wavelength modes develop in Penning-type configurations, that is, where an electric field is imposed perpendicular to a homogeneous magnetic field. The fluid dispersion relation assuming a slab geometry shows that two types of low-frequency, gradient drift instabilities occur in the direction of the $\boldsymbol {E} \times \boldsymbol {B}$ and diamagnetic drifts. One type of instability, observed when the equilibrium electric field and plasma density gradient are in the same direction, is similar to the classic modified Simon–Hoh instability. A second instability is found for conditions in which (i) the diamagnetic drift is in the direction opposite to the $\boldsymbol {E} \times \boldsymbol {B}$ drift and (ii) the magnitude of the diamagnetic drift is sufficiently larger than the electron thermal speed. The present fluid dispersion theory suggests that the rotating spokes driven by such fluid instabilities propagate in the same direction as the diamagnetic drift, which can be in the same direction as or opposite to the $\boldsymbol {E} \times \boldsymbol {B}$ drift, depending on the plasma conditions. This finding may account for the observation, in some plasma devices, of the rotation of large-scale structures in both the $\boldsymbol {E} \times \boldsymbol {B}$ and $-\boldsymbol {E} \times \boldsymbol {B}$ directions.

Whistler-regulated Magnetohydrodynamics: Transport Equations for Electron Thermal Conduction in the High-β Intracluster Medium of Galaxy Clusters

Abstract Transport equations for electron thermal energy in the high-β e intracluster medium (ICM) are developed that include scattering from both classical collisions and self-generated whistler waves. The calculation employs an expansion of the kinetic electron equation along the ambient magnetic field in the limit of strong scattering and assumes whistler waves with low phase speeds V w ∼ v te /β e ≪ v te dominate the turbulent spectrum, with v te the electron thermal speed and β e ≫ 1 the ratio of electron thermal to magnetic pressure. We find: (1) temperature-gradient-driven whistlers dominate classical scattering when L c > L/β e , with L c the classical electron mean free path and L the electron temperature scale length, and (2) in the whistler-dominated regime the electron thermal flux is controlled by both advection at V w and a comparable diffusive term. The findings suggest whistlers limit electron heat flux over large regions of the ICM, including locations unstable to isobaric condensation. Consequences include: (1) the Field length decreases, extending the domain of thermal instability to smaller length scales, (2) the heat flux temperature dependence changes from T e 7 / 2 / L to V w nT e ∼ T e 1 / 2 , (3) the magneto-thermal- and heat-flux-driven buoyancy instabilities are impaired or completely inhibited, and (4) sound waves in the ICM propagate greater distances, as inferred from observations. This description of thermal transport can be used in macroscale ICM models.

Ultrafast electron holes in plasma phase space dynamics

Electron holes (EH) are localized modes in plasma kinetic theory which appear as vortices in phase space. Earlier research on EH is based on the Schamel distribution function (df). A novel df is proposed here, generalizing the original Schamel df in a recursive manner. Nonlinear solutions obtained by kinetic simulations are presented, with velocities twice the electron thermal speed. Using 1D-1V kinetic simulations, their propagation characteristics are traced and their stability is established by studying their long-time evolution and their behavior through mutual collisions.

Oblate electron holes are not attributable to anisotropic shielding

The influence of shielding mechanisms on the ratio of perpendicular to parallel scale lengths of multidimensional plasma electron hole equilibria is analyzed theoretically and computationally. It is shown that the “gyrokinetic” model, invoking perpendicular polarization, is based on a misunderstanding and cannot explain the observational trend that greater transverse extent accompanies a lower magnetic field. Instead, the potential in the wings of the hole, outside the region of trapped-electron depletion, has isotropic shielding giving ϕ∝e−r/L/r, with the shielding length L equal to the Debye length for holes much slower than the electron thermal speed. Particle in cell simulations confirm the analysis. Trapped electron charge distribution anisotropy must, therefore, instead underlie the oblate shape of electron holes.

An asymptotic preserving well-balanced scheme for the isothermal fluid equations in low-temperature plasmas at low-pressure

We present a novel numerical scheme for the efficient and accurate solution of the isothermal two-fluid (electron + ion) equations coupled to Poisson's equation for low-temperature plasmas at low-pressure. The model considers electrons and ions as separate fluids, comprising the electron inertia and space charge regions. The discretization of this system with standard explicit schemes is constrained by very restrictive time steps and cell sizes related to the resolution of the Debye length, electron plasma frequency, and electron sound waves. Both sheath and electron inertia are fundamental to fully explain the physics in low-pressure and low-temperature plasmas. However, most of the phenomena of interest for fluid models occur at speeds much slower than the electron thermal speed and are quasi-neutral, except in small charged regions. A numerical method that is able to simulate efficiently and accurately all these regimes is a challenge due to the multiscale character of the problem. In this work, we present a scheme based on the Lagrange-projection operator splitting that preserves the asymptotic regime where the plasma is quasi-neutral with massless electrons. As a result, the quasi-neutral regime is treated without the need of an implicit solver nor the resolution of the Debye length and electron plasma frequency. Additionally, the scheme proves to accurately represent the dynamics of the electrons both at low speeds and when the electron speed is comparable to the thermal speed. In addition, a well-balanced treatment of the ion source terms is proposed in order to tackle problems where the ion temperature is very low compared to the electron temperature. The scheme significantly improves the accuracy both in the quasi-neutral limit and in the presence of plasma sheaths when the Debye length is resolved. In order to assess the performance of the scheme in low-temperature plasmas conditions, we propose two specifically designed test-cases: a quasi-neutral two-stream periodic perturbation with analytical solution and a low-temperature discharge that includes sheaths. The numerical strategy, its accuracy, and computational efficiency are assessed on these two discriminating configurations.

Instabilities and turbulence in low-β guide field reconnection exhausts with kinetic Riemann simulations

The role of turbulence in low-β, guide-field reconnection exhausts is explored in 2D reconnection and 2D and 3D Riemann simulations. The structure of the exhaust and associated turbulence is controlled by a pair of rotational discontinuities (RDs) at the exhaust boundary and a pair of slow shocks (SSs) that are generated by counterstreaming ions beams. In 2D, the exhaust develops large-amplitude striations at the ion Larmor radius scale that are produced by electron-beam-driven ion cyclotron waves. The electron beams driving the instability are injected into the exhaust from one of the RDs. However, in 3D Riemann simulations, the additional dimension (in the out-of-plane direction) results in strong Buneman and electron-electron streaming instabilities at the RD which suppress the electron beam formation and therefore the striations in the exhaust. The strength of the streaming instabilities at the RD is controlled by the ratio of the electron thermal speed to Alfvén speed, with the lower thermal speed being more unstable. In the 3D simulations, an ion-ion streaming instability acts to partially thermalize the counterstreaming ion beams at the SSs. This instability is controlled by the ratio of the sound speed to Alfvén speed and is expected to be stable in the low β solar corona. The results suggest that in a guide field reconnection exhaust with 1≫β>me/mi, the kinetic-scale turbulence that develops will be too weak to play a significant role in energy conversion and particle acceleration. Therefore, the energy conversion will be mostly controlled by laminar physics or multi-x-line reconnection.

A scale-separated approach for studying coupled ion and electron scale turbulence

Multiple space and time scales arise in plasma turbulence in magnetic confinement fusion devices because of the smallness of the square root of the electron-to-ion mass ratio and the consequent disparity of the ion and electron thermal gyroradii and thermal speeds. Direct simulations of this turbulence that include both ion and electron space–time scales indicate that there can be significant interactions between the two scales. The extreme computational expense and complexity of these direct simulations motivates the desire for reduced treatment. By exploiting the scale-separation between ion scales (IS) and electron scales (ES), and expanding the gyrokinetic equations for the turbulence in , we derive such a reduced system of gyrokinetic equations that describes cross-scale interactions. The coupled gyrokinetic equations contain novel terms which provide candidate mechanisms for the observed cross-scale interaction. The ES turbulence experiences a modified drive due to gradients in the IS distribution function, and is advected by the IS drift, which varies in the direction parallel to the magnetic field line. The largest possible cross-scale term in the IS equations is sub-dominant in our expansion. Hence, in our model the IS turbulence evolves independently of the ES turbulence. To complete the scale-separated approach, we provide and justify a parallel boundary condition for the coupled gyrokinetic equations in axisymmetric equilibria based on the standard ‘twist-and-shift’ boundary condition. This approach allows one to simulate multi-scale turbulence using ES flux tubes nested within an IS flux tube.

Kinematics of electrostatic 3-wave decay of generalized Langmuir waves in magnetized plasmas

The decay of generalised Langmuir waves L into backscattered (generalised) Langmuir waves L and ion acoustic waves S or ion cyclotron waves IC, represented by L→L′+S and L→L′+IC, is a fundamental nonlinear process relevant to beam-plasma instabilities in space and laboratory plasmas and to multiple solar system radio emissions. Both magnetization and arbitrary wavevector directions are included for the generalised Langmuir waves, thereby naturally encompassing both conventional Langmuir waves and upper hybrid waves. A recent 1D analysis for L waves with wavevectors closely parallel to the ambient magnetic field B0 in weakly magnetized plasma (angular electron cyclotron frequency Ωe much less than the angular electron plasma frequency ωp) showed that the electrostatic (ES) decay L→L′+S persists for kL<k0, reversing the old prediction based on the unmagnetized dispersion relation. Here, the kinematics for the processes L→L′+S and L→L′+IC are derived in 2 dimensions for approximately electrostatic waves in arbitrary magnetized plasmas and for all wavevector orientations relative to B0. ES decay processes are shown to exist in both weakly and strongly magnetized plasmas and, under most circumstances, for arbitrary L-wavevector directions, including close to perpendicular to B0, and wavenumbers. For L-wavenumbers kL≳2k0, the decay process is very similar to the standard unmagnetized decay for kL close to parallel with B0, proceeding primarily as a backscatter to kL′≈(kL−k0)kL/kL and a trivial forward-scatter solution with kL≈kL. (Here, k0=2ωpvS/3Ve2, VS is the ion acoustic speed, and Ve is the electron thermal speed.) In addition, the decay persists for kL<k0 to very small kL′≈k*=(ωp/c)(1+fp/fce)−1/2 for arbitrary magnetizations and orientations of kL relative to B0, at least for sufficiently large ion-to-electron temperature ratios Ti/Te. Thus, once magnetization effects are included, the decay is kinematically allowed for essentially all initial wavevectors and can proceed for the very fast beams (with kL<k0) for which modulational instability and not ES decay was previously expected to dominate the nonlinear evolution.

Turbulent heating in an inhomogeneous magnetized plasma slab

Observational evidence in space and astrophysical plasmas with a long collisional mean free path suggests that more massive charged particles may be preferentially heated. One possible mechanism for this is the turbulent cascade of energy from injection to dissipation scales, where the energy is converted to heat. Here we consider a simple system consisting of a magnetized plasma slab of electrons and a single ion species with a cross-field density gradient. We show that such a system is subject to an electron drift wave instability, known as the universal instability, which is stabilized only when the electron and ion thermal speeds are equal. For unequal thermal speeds, we find from quasilinear analysis and nonlinear simulations that the instability gives rise to turbulent energy exchange between ions and electrons that acts to equalize the thermal speeds. Consequently, this turbulent heating tends to equalize the component temperatures of pair plasmas and to heat ions to much higher temperatures than electrons for conventional mass-ratio plasmas.

Electron Bulk Acceleration and Thermalization at Earth's Quasiperpendicular Bow Shock.

Electron heating at Earth's quasiperpendicular bow shock has been surmised to be due to the combined effects of a quasistatic electric potential and scattering through wave-particle interaction. Here we report the observation of electron distribution functions indicating a new electron heating process occurring at the leading edge of the shock front. Incident solar wind electrons are accelerated parallel to the magnetic field toward downstream, reaching an electron-ion relative drift speed exceeding the electron thermal speed. The bulk acceleration is associated with an electric field pulse embedded in a whistler-mode wave. The high electron-ion relative drift is relaxed primarily through a nonlinear current-driven instability. The relaxed distributions contain a beam traveling toward the shock as a remnant of the accelerated electrons. Similar distribution functions prevail throughout the shock transition layer, suggesting that the observed acceleration and thermalization is essential to the cross-shock electron heating.

Multiprobe characterization of plasma flows for space propulsion

Plasma engines for space propulsion generate plasma jets (also denominated plasma plumes) having supersonic ion groups with typical speeds in the order of tens of kilometers per second, which lies between electron and ion thermal speeds. Studies of the stationary plasma expansion process using a four-grid retarding field energy analyzer (RFEA), an emissive probe (EP) and a Langmuir probe (LP), all mounted on a three dimensionally (3D) displaced multiprobe structure are discussed. Specifically, the determination of plasma beam properties from the RFEA current-voltage (IV) characteristic curves is presented. The experimental results show the ion energy spectra to be essentially unchanged over 300 mm along the plasma-jet expansion axis of symmetry. The measured ion velocity distribution function (IVDF) results from the superposition of different ion groups and has two dominant populations: A low-energy group constituted of ions from the background plasma is produced by the interaction of the plasma jet with the walls of the vacuum chamber. The fast-ion population is composed of ions from the plasma beam moving at supersonic speeds with respect to the low-energy ions. The decreasing spatial profiles of the plasma-jet current density are compared with those of the low-energy ion group, which are not uniform along the axis of symmetry because of the small contributions from other ion populations with intermediate speeds.

Electron presheaths: the outsized influence of positive boundaries on plasmas

Electron sheaths form near the surface of objects biased more positive than the plasma potential, such as a Langmuir probe collecting electron saturation current. Generally, the formation of electron sheaths requires that the electron-collecting area be sufficiently smaller ( times) than the ion-collecting area. They are commonly thought to be local phenomena that collect the random thermal electron current, but do not otherwise perturb a plasma. Here, using experiments on an electrode embedded in a wall in a helium discharge, particle-in-cell simulations, and theory it is shown that under low temperature plasma conditions () electron sheaths are far from local. Instead, a long presheath region (27 mm, approximately an electron’s mean free path) extends into the plasma where electrons are accelerated via a pressure gradient to a flow speed exceeding the electron thermal speed at the sheath edge. This fast flow is found to excite instabilities, causing strong fluctuations near the sheath edge.

Slow electrostatic fluctuations generated by beam-plasma interaction

Eulerian simulations of the Vlasov-Poisson equations have been employed to analyze the excitation of slow electrostatic fluctuations (with phase speed close to the electron thermal speed), due to a beam-plasma interaction, and their propagation in linear and nonlinear regimes. In 1968, O'Neil and Malmberg [Phys. Fluids 11, 1754 (1968)] dubbed these waves “beam modes.” In the present paper, previous analytical results on the beam modes in both linear and nonlinear regimes have been revisited numerically, pointing out that, when an electron beam is launched in a plasma of Maxwellian electrons and motionless protons and this initial equilibrium is perturbed by a monochromatic density disturbance, the electric field amplitude grows exponentially in time and then undergoes nonlinear saturation, associated with the kinetic effects of particle trapping and phase space vortex generation. Moreover, new numerical results give evidence that, when the initial density perturbation is setup in the form of a low amplitude random phase noise, the whole Fourier spectrum of wavenumbers is excited. As a result, the electric field profile appears as a train of isolated pulses, each of them being associated with a phase space vortex in the electron distribution function. At later times, these vortical structures tend to merge and, correspondingly, the electric pulses collapse, showing the tendency towards a time asymptotic configuration characterized by the appearance of electric soliton-like pulses. This dynamical evolution is driven by purely kinetic processes, possibly at work in many space and laboratory plasma environments.

Interactions of electrons with two lower hybrid waves

The effects of perturbed orbits on the interactions of electrons with two lower hybrid waves, one of which is resonant with electrons at a low phase velocity (vp1 = 3.8Vthe, where vp1 is the wave phase velocity and Vthe is the electron thermal speed) while the other is off-resonant at a high phase velocity (vp2 = 5.5Vthe), have been studied by using the particle simulation code based on the gyro-kinetic electron and fully-kinetic ion (GeFi) model [Lin et al., Plasma Phys. Controlled Fusion 47, 657 (2005)]. When the amplitude of the off-resonant wave is sufficiently small so that the resonances of these two waves do not overlap, the variation of the resonant wave amplitude is similar to that predicted by O'Neil's theory [O'Neil, Plasma Fluid 8, 12 (1965)]. With the increasing amplitude, the two resonances overlap and large scale chaos emerges. As a result, the damping of the resonant wave can be enhanced, which is due to that the trapped electron orbits are significantly perturbed by the off-resonant wave. The diffusion process gives rise to the enhanced damping. When the overlap is sufficiently large, the damping of the off-resonant wave and the oscillatory behavior of the wave amplitude are observed. In addition, the resonant plateau in the distribution function can be broadened due to the change in the chaotic region boundaries as the electron perturbed orbits are taken into account.

On the generation of double layers from ion- and electron-acoustic instabilities

A plasma double layer (DL) is a nonlinear electrostatic structure that carries a uni-polar electric field parallel to the background magnetic field due to local charge separation. Past studies showed that DLs observed in space plasmas are mostly associated with the ion acoustic instability. Recent Van Allen Probes observations of parallel electric field structures traveling much faster than the ion acoustic speed have motivated a computational study to test the hypothesis that a new type of DLs—electron acoustic DLs—generated from the electron acoustic instability are responsible for these electric fields. Nonlinear particle-in-cell simulations yield negative results, i.e., the hypothetical electron acoustic DLs cannot be formed in a way similar to ion acoustic DLs. Linear theory analysis and the simulations show that the frequencies of electron acoustic waves are too high for ions to respond and maintain charge separation required by DLs. However, our results do show that local density perturbations in a two-electron-component plasma can result in unipolar-like electric field structures that propagate at the electron thermal speed, suggesting another potential explanation for the observations.