Finite Electron Mass Research Articles

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.

Hybrid Model of a Stationary Plasma Thruster Taking into Account the Finite Electron Mass

Hybrid Model of a Stationary Plasma Thruster Taking into Account the Finite Electron Mass

An E and B gyrokinetic simulation model for kinetic Alfvén waves in tokamak plasmas

The gyrokinetic particle simulation is a powerful tool for studies of transport, nonlinear phenomenon, and energetic particle physics in tokamak plasmas. While most gyrokinetic simulations make use of the scalar and vector potentials, a new model (GK-E&B) has been developed by using the E and B field in a general form and has been implemented in simulating kinetic Alfvén waves in uniform plasma [Chen et al., Sci. China: Phys., Mech. Astron. 64, 245211 (2021)]. In our work, the Chen et al. GK-E&B model has been expressed, in general, tokamak geometry using the local orthogonal coordinates and general tokamak coordinates. Its reduction for uniform plasma is verified, and the numerical results show good agreement with the original work. The theoretical dispersion relation and numerical results in the local model in screw pinch geometry are also in excellent agreement. Numerical results show excellent performance in a realistic parameter regime of burning plasmas with high values of β/(Mek⊥2ρi2), which is a challenge for traditional methods due to the “cancellation” problem. As one application, the GK-E&B model is implemented with kinetic electrons in the local single flux surface limit. With the matched International Tokamak Physics Activity-Toroidicity-induced Alfvén Eigenmodes parameters adopted, numerical results show the capability of the GK-E&B in treating the parallel electron Landau damping for realistic tokamak plasma parameters. As another application, the global GK-E&B model has been implemented with the dominant electron contribution in the cold electron limit. Its capability in simulating the finite E|| due to the finite electron mass is demonstrated.

On Ohm’s law in reduced plasma fluid models

Drift-reduced MHD models are widely used to study magnetised plasma phenomena, in particular for magnetically confined fusion applications, as well as in solar and astrophysical research. This letter discusses the choice of Ohm’s law in these models, the resulting dispersion relations for the dynamics parallel to the magnetic field, and the implications for numerical simulations. We find that if electron pressure is included in Ohm’s law, then both electromagnetic and finite electron mass effects must also be included in order to obtain physical dispersion relations. A simple modification to the plasma vorticity is also found which improves handling of low density regions, of particular relevance to the simulation of the boundary region of magnetised plasmas.

Equilibration of the chiral asymmetry due to finite electron mass in electron-positron plasma

We calculate the rate of collisional decay of the axial charge in an ultrarelativistic electron-positron plasma, also known as the chirality flipping rate. We find that contrary to the existing estimates, the chirality flipping rate appears already in the first order in the fine-structure constant $\ensuremath{\alpha}$ and is therefore orders of magnitude greater than previously believed. The main channels for the rapid relaxation of the axial charge are the collinear emission of a weakly damped photon and the Compton scattering. The latter contributes to the $\mathcal{O}(\ensuremath{\alpha})$ result because of the infrared divergence in its cross section, which is regularized on the soft scale $\ensuremath{\sim}eT$ due to the thermal corrections. Our results are important for the description of the early Universe processes (such as leptogenesis or magnetogenesis) that affect differently left- and right-chiral fermions of the Standard Model, as discussed in more details in the companion Letter.

Comparative Analysis of the Various Generalized Ohm's Law Terms in Magnetosheath Turbulence as Observed by Magnetospheric Multiscale

AbstractDecomposing the electric field (E) into the contributions from generalized Ohm's law provides key insight into both nonlinear and dissipative dynamics across the full range of scales within a plasma. Using high‐resolution, multispacecraft measurements of three intervals in Earth's magnetosheath from the Magnetospheric Multiscale mission, the influence of the magnetohydrodynamic, Hall, electron pressure, and electron inertia terms from Ohm's law, as well as the impact of a finite electron mass, on the turbulent E spectrum are examined observationally for the first time. The magnetohydrodynamic, Hall, and electron pressure terms are the dominant contributions to E over the accessible length scales, which extend to scales smaller than the electron gyroradius at the greatest extent, with the Hall and electron pressure terms dominating at sub‐ion scales. The strength of the nonideal electron pressure contribution is stronger than expected from linear kinetic Alfvén waves and a partial antialignment with the Hall electric field is present, linked to the relative importance of electron diamagnetic currents in the turbulence. The relative contribution of linear and nonlinear electric fields scale with the turbulent fluctuation amplitude, with nonlinear contributions playing the dominant role in shaping E for the intervals examined in this study. Overall, the sum of the Ohm's law terms and measured E agree to within ∼20% across the observable scales. These results both confirm general expectations about the behavior of E in turbulent plasmas and highlight features that should be explored further theoretically.

Magnetic reconnection in 3D fusion devices: non-linear reduced equations and linear current-driven instabilities

Magnetic reconnection in 3D fusion devices is investigated. With the use of Boozer co-ordinates, we reduce the non-linear resistive magnetohydrodynamic equations in the limit of large aspect ratio and finite pressure fluctuations, to obtain a set of non-linear equations suitable for magnetic reconnection studies in stellarators. Magnetic flux unfreezing due to a finite electron mass is also considered. Equations that govern the linear regime and some of their general properties are given. We emphasise the role of magnetic geometry and identify how some aspects of stellarator optimisation could have an impact on reconnecting instabilities, in particular by exacerbating those enabled by electron inertia. The effect of 3D coupling on the linear reconnection rates and the mode structure is quantitatively addressed in the case in which the equilibrium rotational transform has one specific resonant location for which one mode can reconnect while coupled to an arbitrary number of non-resonant harmonics. The full problem is rigorously reduced to an equivalent cylindrical one, by introducing some geometrically modified plasma inertial and dissipative scales. The 3D scalings for the growth rates of reconnection instabilities and their destabilisation criteria are given.

Electron skin depth in low density plasmas

We rescale the generalized Ohm’s law and consider the limits that imply the electric field which moves with an ideal (inviscid and perfectly conducting) plasma be proportional to the time rate of change of the current density. Therefore, we show that those limits are satisfied by a sufficiently low electron number density, $n_{\text{e}}$ . We also show that the electron–ion collision frequency, $\unicode[STIX]{x1D708}_{\text{ei}}$ , is much smaller than the ion cyclotron frequency, $\unicode[STIX]{x1D714}_{\text{ci}}$ . The combination of that condition with the Lawson criterion for a typical deuterium–tritium fusion in ITER reveals a lower bound for the geometric mean of the confinement time, $\unicode[STIX]{x1D70F}_{\text{C}}$ , and collision interval, $\sqrt{\unicode[STIX]{x1D70F}_{\text{C}}/\unicode[STIX]{x1D708}_{\text{ei}}}\gg 10^{-5}~\text{s}$ . For that reaction, we estimate that $n_{\text{e}}\sim 10^{19}~\text{m}^{-3}$ , and contrast typical parameters of our fully ionized gas with those of warm, hot and thermonuclear plasmas. When, in addition, $n_{\text{e}}$ varies slowly in time and weakly in space, we generalize Alfvén’s theorem, by showing that the frozen-in condition holds true for an effective magnetic field, which depends on a finite electron skin depth. We perform a (divergenceless) helical perturbation on an axisymmetric equilibrium, to derive a dispersion relation in the cylindrical tokamak limit, and, subsequently, apply our analytical formulation to the peaked model, which assumes a logarithmic derivative profile for the poloidal component of the equilibrium magnetic field. In that formulation, the definition of the safety factor in terms of the effective field yields a shift in the magnetic surfaces. We find that the instability peak may triple that predicted on neglect of a finite electron mass. We also find that inertial effects may centuple the radius of the stable cylindrical column.

Variational integrators for inertial magnetohydrodynamics

Recently, an extended version of magnetohydrodynamics that incorporates electron inertia, dubbed inertial magnetohydrodynamics, has been proposed [Lingam et al., Phys. Lett. A, 379, 570–576 (2015)]. This model features a noncanonical Hamiltonian formulation with a number of conserved quantities, including the total energy and modified versions of magnetic and cross helicity. In this work, a variational integrator is presented which preserves these conservation laws to machine accuracy. As long as effects due to finite electron mass are neglected, the scheme preserves the magnetic field line topology so that unphysical reconnection is absent. Only when the effects of finite electron mass are added, magnetic reconnection takes place. The excellent conservation properties of the method are illustrated by numerical examples in 2D.

Insights into neutrino decoupling gleaned from considerations of the role of electron mass

We present calculations showing how electron rest mass influences entropy flow, neutrino decoupling, and Big Bang Nucleosynthesis (BBN) in the early universe. To elucidate this physics and especially the sensitivity of BBN and related epochs to electron mass, we consider a parameter space of rest mass values larger and smaller than the accepted vacuum value. Electromagnetic equilibrium, coupled with the high entropy of the early universe, guarantees that significant numbers of electron–positron pairs are present, and dominate over the number of ionization electrons to temperatures much lower than the vacuum electron rest mass. Scattering between the electrons–positrons and the neutrinos largely controls the flow of entropy from the plasma into the neutrino seas. Moreover, the number density of electron–positron-pair targets can be exponentially sensitive to the effective in-medium electron mass. This entropy flow influences the phasing of scale factor and temperature, the charged current weak-interaction-determined neutron-to-proton ratio, and the spectral distortions in the relic neutrino energy spectra. Our calculations show the sensitivity of the physics of this epoch to three separate effects: finite electron mass, finite-temperature quantum electrodynamic (QED) effects on the plasma equation of state, and Boltzmann neutrino energy transport. The ratio of neutrino to plasma–component energy scales manifests in Cosmic Microwave Background (CMB) observables, namely the baryon density and the radiation energy density, along with the primordial helium and deuterium abundances. Our results demonstrate how the treatment of in-medium electron mass (i.e., QED effects) could translate into an important source of uncertainty in extracting neutrino and beyond-standard-model physics limits from future high-precision CMB data.

Viscous-resistive layer in Rayleigh-Taylor instability

In this work, new scaling laws of the time growth rate γ of the Rayleigh-Taylor instability with the plasma resistivity η, kinematic viscosity ν, and electron number density ne are derived. A viscosity scale is defined in terms of the time decay of the perturbative fluid flow perpendicular to the equilibrium magnetic field, at the quasi-static approximation. Such a scale provides the identification of a viscous layer that can be combined with the resistive layer to produce a viscous-resistive layer. The latter, in turn, is found to satisfy an algebraic biquadratic equation. When viscous effects are negligible, it is shown that the viscous-resistive layer is given by the resistive layer. Somewhat surprisingly, when viscous effects cannot be neglected, it is shown that the viscous-resistive layer is given by the geometric mean of the resistive and viscous layers. A dispersion relation for the time growth rate is derived in terms of the viscous-resistive layer. When viscous effects cannot be neglected, two new scaling laws are found. At the quasi-static approximation, it is shown that γ ∼ (ην)1∕4. However, on account of a finite electron mass, it is shown that γ∼(ν/ne)1/3. Further developments of our formulation are addressed in connection with a finite compressibility in the perturbative flow.

The electric field in capacitively coupled RF discharges: a smooth step model that includes thermal and dynamic effects

The electric field in radio-frequency driven capacitively coupled plasmas (RF-CCP) is studied, taking thermal (finite electron temperature) and dynamic (finite electron mass) effects into account. Two dimensionless numbers are introduced, the ratios of the electron Debye length to the minimum plasma gradient length l (typically the sheath thickness) and of the RF frequency to the electron plasma frequency . Assuming both numbers small but finite, an asymptotic expansion of an electron fluid model is carried out up to quadratic order inclusively. An expression for the electric field is obtained which yields (i) the space charge field in the sheath, (ii) the generalized Ohmic and ambipolar field in the plasma, and (iii) a smooth interpolation for the transition in between. The new expression is a direct generalization of the Advanced Algebraic Approximation (AAA) proposed by the same author (2009 J. Phys. D: Appl. Phys. 42 194009), which can be recovered for , and of the established Step Model (SM) by Godyak (1976 Sov. J. Plasma Phys. 2 78), which corresponds to the simultaneous limits , . A comparison of the hereby proposed Smooth Step Model (SSM) with a numerical solution of the full dynamic problem proves very satisfactory.

Relaxation model for extended magnetohydrodynamics: Comparison to magnetohydrodynamics for dense Z-pinches

It is shown that the two-fluid model under a generalized Ohm’s law formulation and the resistive magnetohydrodynamics (MHD) can both be described as relaxation systems. In the relaxation model, the under-resolved stiff source terms constrain the dynamics of a set of hyperbolic equations to give the correct asymptotic solution. When applied to the collisional two-fluid model, the relaxation of fast time scales associated with displacement current and finite electron mass allows for a natural transition from a system where Ohm’s law determines the current density to a system where Ohm’s law determines the electric field. This result is used to derive novel algorithms, which allow for multiscale simulation of low and high frequency extended-MHD physics. This relaxation formulation offers an efficient way to implicitly advance the Hall term and naturally simulate a plasma-vacuum interface without invoking phenomenological models. The relaxation model is implemented as an extended-MHD code, which is used to analyze pulsed power loads such as wire arrays and ablating foils. Two-dimensional simulations of pulsed power loads are compared for extended-MHD and MHD. For these simulations, it is also shown that the relaxation model properly recovers the resistive-MHD limit.

Generalized theory of annularly bounded helicon waves

The generalized dispersion relationship for annularly bounded helicon plasma is derived. The theory considers the effect of finite electron mass, which results in the plasma field described by superposition of classical helicon solution and the Trivelpiece-Gould (TG) wave. The solution is obtained for an insulating boundary. The solution shows that the wave structure is heavily affected by the presence of the TG wave, even when a strong axial magnetic field is applied. However, the electric field profile shows that a pair of external rf antennas can still be coupled to the strong radial electric field in the plasma to excite the wave. Moreover, the general solution permits an almost continuous spectrum of parallel wave numbers. The study of k-f and n0-B0 diagrams shows that the linear relation obtained using classical theory holds everywhere except at low applied magnetic field strength.

Hybrid simulation of reforming shocks with electron mass and pressure tensor effects

We examine the dynamics of low plasma beta (β = 0.05), high Alfvén Mach number (MA = 4 ∼ 5), quasi‐perpendicular (θbn = 87°) collisionless shocks using two one‐dimensional hybrid simulation codes, a standard hybrid code with massless fluid electrons and a new extended hybrid code with finite electron mass, and the full electron pressure tensor. Both hybrid codes predict similar macroscopic shock structures with well defined foot, ramp, overshoot and downstream regions and a shock that reforms with a quasi‐cyclic change of the shock front structure. The shock reformation period predicted by the standard hybrid code is 1.8Ωci−1 (Ωci is the upstream ion gyrofrequency). In contrast to the standard hybrid simulations, the additional electron dissipation in the new extended hybrid code leads to a strong ion thermalization and phase space mixing between upstream incoming and reflected ions via plasma wave interactions in the foot, and formation of ion phase space holes. Whistler waves with typical wavelength ≃0.3vAΩci−1 and frequency ≃5.5Ωci−1 are observed in the foot, and ramp region which help to stop the shock steepening, and reduce the predicted shock reformation period to 1.6Ωci−1.

Electron inertial effects on rotating magnetic field current drive

The effect of finite electron mass on the formation and sustainment of a field reversed configuration (FRC) by rotating magnetic fields (RMF) is studied. The importance of inertial effects is measured by the ratio between the RMF frequency (ω) and the electron-ion collision frequency (ν). In the limit where this ratio is very small (ω∕ν→0), previous results corresponding to massless electrons are recovered. When ω∕ν increases there are significant changes in the value of the minimum external rotating field needed to sustain the FRC and the time required to reach a steady state. Since ν decreases with increasing temperature and decreasing density, these effects are expected to become more important as fusion relevant temperatures are approached.

Three-Dimensional Full-Wave Propagation Code for Cold Plasma

Electromagnetic wave propagation in the Alfvén frequency range in two-dimensional (2-D) and three-dimensional (3-D) plasma configurations is investigated by using a recently developed full-wave code, LEMan. The code solves the Maxwell equation globally in a fully 3-D numerical equilibrium geometry with nested magnetic surfaces and implements a cold plasma model with finite electron mass. Energy balance is used for the self-consistency and convergence checks. The code is applied for several 2-D systems and a 3-D system, and results are compared with a simple cylindrical configuration.

Two-loop form factors in QED

We evaluate the on shell form factors of the electron for arbitrary momentum transfer and finite electron mass, at two loops in QED, by integrating the corresponding dispersion relations, which involve the imaginary parts known since a long time. The infrared divergences are parameterized in terms of a fictitious small photon mass. The result is expressed in terms of harmonic polylogarithms of maximum weight 4. The expansions for small and large momentum transfer are also given.

Vertex diagrams for the QED form factors at the 2-loop level

We carry out a systematic investigation of all the 2-loop integrals occurring in the electron vertex in QED in the continuous D-dimensional regularization scheme, for on-shell electrons, momentum transfer t=− Q 2 and finite squared electron mass m e 2= a. We identify all the master integrals (MIs) of the problem and write the differential equations in Q 2 which they satisfy. The equations are expanded in powers of ϵ=(4− D)/2 and solved by the Euler's method of the variation of the constants. As a result, we obtain the coefficients of the Laurent expansion in ϵ of the MIs up to zeroth order expressed in close analytic form in terms of harmonic polylogarithms.

Helicon waves in a cylindrical plasma

The dispersion relation for helicon waves in a uniform bounded plasma is derived by including the finite electron mass. The eigenmodes are identified and the coupling mechanism between the Ez and Bz modes is discussed. This is important since an essential part of the physics associated with the application of helicon waves for the generation and heating of plasmas consists in coupling the whistler branch with the Ez mode, which can interact directly with the electrons.PACS No.: 52.35-g