Finite Electron Inertia Research Articles

Fast, purely growing collisionless reconnection as an eigenfunction problem related to but not involving linear whistler waves

If either finite electron inertia or finite resistivity is included in 2D magnetic reconnection, the two-fluid equations become a pair of second-order differential equations coupling the out-of-plane magnetic field and vector potential to each other to form a fourth-order system. The coupling at an X-point is such that out-of-plane even-parity electric and odd-parity magnetic fields feed off each other to produce instability if the scale length on which the equilibrium magnetic field changes is less than the ion skin depth. The instability growth rate is given by an eigenvalue of the fourth-order system determined by boundary and symmetry conditions. The instability is a purely growing mode, not a wave, and has growth rate of the order of the whistler frequency. The spatial profile of both the out-of-plane electric and magnetic eigenfunctions consists of an inner concave region having extent of the order of the electron skin depth, an intermediate convex region having extent of the order of the equilibrium magnetic field scale length, and a concave outer exponentially decaying region. If finite electron inertia and resistivity are not included, the inner concave region does not exist and the coupled pair of equations reduces to a second-order differential equation having non-physical solutions at an X-point.

A robust method for handling low density regions in hybrid simulations for collisionless plasmas

A robust method to handle vacuum and near vacuum regions in hybrid simulations for space and astrophysical plasmas is presented. The conventional hybrid simulation model dealing with kinetic ions and a massless charge-neutralizing electron fluid is known to be susceptible to numerical instability due to divergence of the whistler-mode wave dispersion, as well as division-by-density operation in regions of low density. Consequently, a pure vacuum region is not allowed to exist in the simulation domain unless some ad hoc technique is used. To resolve this difficulty, an alternative way to introduce finite electron inertia effect is proposed. Contrary to the conventional method, the proposed one introduces a correction to the electric field rather than the magnetic field. It is shown that the generalized Ohm's law correctly reduces to Laplace's equation in a vacuum which therefore does not involve any numerical problems. In addition, a variable ion-to-electron mass ratio is introduced to reduce the phase velocity of high frequency whistler waves at low density regions so that the stability condition is always satisfied. It is demonstrated that the proposed model is able to handle near vacuum regions generated as a result of nonlinear self-consistent development of the system, as well as pure vacuum regions set up at the initial condition, without losing the advantages of the standard hybrid code.

Seed island formation by forced magnetic reconnection

Neoclassical tearing modes observed in experiments often grow from seed magnetic islands induced by triggers like sawteeth. The formation of seed islands is studied in this paper using both the reduced MHD and two-fluid equations, with the trigger being modelled by externally applied resonant magnetic perturbations. In the linear phase the growth rate of the driven mode is found to be the same as that of the trigger. A slowly growing trigger drives a tearing mode, while a fast one drives a kink-like mode, which becomes a tearing mode later when the trigger's growth slows down. A finite ion sound Larmor radius (ion Larmor radius by using electron temperature) and electron inertia are found to lead to a larger seed island for a given external perturbation. The electron diamagnetic drift and plasma rotation, if increasing the relative rotation between the trigger and the driven mode, decrease the seed island width.

On the question of hysteresis in Hall magnetohydrodynamic reconnection

Controversy has been raised regarding the cause of hysteresis, or bistability, of solutions to the equations that govern the geometry of the reconnection region in Hall magnetohydrodynamic (MHD) systems. This brief communication presents a comparison of the frameworks within which this controversy has arisen and illustrates that the Hall MHD hysteresis originally discovered numerically by Cassak et al. [Phys. Rev. Lett. 95, 235002 (2005)] is a different phenomenon from that recently reported by Zocco et al. [Phys. Plasmas 16, 110703 (2009)] on the basis of analysis and simulations in electron MHD with finite electron inertia. We demonstrate that the analytic prediction of hysteresis in EMHD does not describe or explain the hysteresis originally reported in Hall MHD, which is shown to persist even in the absence of electron inertia.

Effect of electron inertial delay on Debye sheath formation

Present contribution deals with the role of weak but finite electron inertia on the sheath formation condition. As reported earlier this becomes effective when the ions' drift velocity exceeds the phase velocity of the acoustic wave fluctuations. Such situation has natural existence near the sheath edge. Keeping this in mind we have revisited the problem of usual Bohm sheath condition. Analytical and numerical analysis have been performed to re- derive the local condition for plasma sheath formation. It is found that the weak but finite electron inertia reduces the threshold value of ion Mach number that may be, at least in principle, of qualitative value to define the sheath edge boundary. Consideration of finite but weak equilibrium electron flow at the defined sheath edge shrinks the width of non- neutral space charge layer over which major potential drop and charge imbalance occurs. Detailed numerical analysis and results of quantitative and qualitative importance are included in the text.

Self-gravitational instability of rotating viscous Hall plasma with arbitrary radiative heat-loss functions and electron inertia

The effects of arbitrary radiative heat-loss functions and Hall current on the self-gravitational instability of a homogeneous, viscous, rotating plasma has been investigated incorporating the effects of finite electrical resistivity, finite electron inertia and thermal conductivity. A general dispersion relation is obtained using the normal mode analysis with the help of relevant linearized perturbation equations of the problem, and a modified Jeans criterion of instability is obtained. The conditions of modified Jeans instabilities and stabilities are discussed in the different cases of our interest. We find that the presence of arbitrary radiative heat-loss functions and thermal conductivity modifies the fundamental Jeans criterion of gravitational instability into a radiative instability criterion. The Hall parameter affects only the longitudinal mode of propagation and it has no effect on the transverse mode of propagation. For longitudinal propagation, it is found that the condition of radiative instability is independent of the magnetic field, Hall parameter, finite electron inertia, finite electrical resistivity, viscosity and rotation; but for the transverse mode of propagation it depends on the finite electrical resistivity, the strength of the magnetic field, and it is independent of rotation, electron inertia and viscosity. From the curves we find that the presence of thermal conductivity, finite electrical resistivity and density-dependent heat-loss function has a destabilizing influence, while viscosity and magnetic field have a stabilizing effect on the growth rate of an instability. The effect of arbitrary heat-loss functions is also studied on the growth rate of a radiative instability.

The scaling of forced collisionless reconnection

This paper presents two-fluid simulations of forced magnetic reconnection with finite electron inertia in a collisionless three-dimensional (3D) cube with periodic boundaries in all three directions. Comparisons are made to analogous two-dimensional (2D) simulations. Reconnection in this system is driven by a spatially localized forcing function that is added to the ion momentum equation inside the computational domain. Consistent with previous results in similar, but larger forced 2D simulations [B. Sullivan, B. N. Rogers, and M. A. Shay, Phys. Plasmas 12, 122312 (2005)], for sufficiently strong forcing the reconnection process is found to become Alfvénic in both 2D and 3D, i.e., the inflow velocity scales roughly like some fraction of the Alfvén speed based on the upstream reconnecting magnetic field, and the system takes on a stable configuration with a dissipation region aspect ratio on the order of 0.15.

Self-gravitating rotating anisotropic pressure plasma in presence of Hall current and electrical resistivity using generalized polytrope laws

The effects of uniform rotation, finite electrical resistivity, electron inertia, and Hall current on the self-gravitational instability of anisotropic pressure plasma with generalized polytrope laws have been studied. A general dispersion relation is obtained with the help of the relevant linearized perturbed magnetohydrodynamic (MHD) equations incorporating the relevant contributions of various effects of the problem using the method of normal mode analysis. The general dispersion relation is further reduced for the special cases of rotation; i.e., parallel and perpendicular to the direction of the magnetic field. The longitudinal and transverse modes of propagation are discussed separately for investigation of condition of instability. The effects of rotation, Hall current, finite electron inertia, and polytropic indices are discussed on the gravitational, “firehose,” and “mirror” instabilities. The numerical calculations have been performed to obtain the dependence of the growth rate of the gravitational unstable mode on the various physical parameters involved. The finite electrical resistivity, rotation, and Hall current have a stabilizing influence on the growth rate of the unstable mode of wave propagation. The finite electrical resistivity removes the effect of magnetic field and polytropic index from the condition of instability in the transverse mode of propagation for both the cases of rotation. It is also found that the Jeans criterion of gravitational instability depends upon rotation, electron inertia, and polytropic indices. In the case of transverse mode of propagation with the axis of rotation parallel to the magnetic field, it is observed that the region of instability and the value of the critical Jeans wavenumber are larger for the Chew–Goldberger–Low set of equations in comparison with the MHD set of equations. The stability of the system is discussed by applying Routh–Hurwitz criterion. The inclusion of rotation or Hall current or both together depresses the growth rate of mirror instability. We also note that the condition of mirror instability depends upon polytropic indices.

Wave dynamics of an electrojet: generalized Farley–Buneman instability

AbstractIn this paper we generalize the classical Farley–Buneman (FB) instability to include space-charge effects and finite electron inertia. The former effect makes the ion-acoustic wave dispersive with the usual resonance appearing at the ion plasma frequency, but other than that the structure of the FB instability remains intact. However, the inclusion of the latter, finite electron inertia, gives rise to the propagating electron-cyclotron mode, albeit modified by collisions. In the presence of differential electron streaming relative to the ions, the interaction between this mode, attempting to propagate against the stream, but convected forward by the stream, and a forward propagating ion-acoustic mode, gives rise to a new instability distinct from the FB instability. The process may be thought of in terms of the coupling between negative energy waves (electron-cyclotron waves attempting to propagate against the stream) and positive energy waves (forward propagating ion-acoustic waves). In principle, the instability simply requires super-ion acoustic streaming electrons and the corresponding growth rates are of the order of one half of the lower hybrid frequency, which are faster than the corresponding FB growth rates. For conditions appropriate to the middle day-side E-region this instability excites a narrow band of frequencies just below the ion plasma frequency. Its role in the generation of electrojet irregularities may be as important as the classical FB instability.

Cross-Scale Coupling Within Rolled-Up MHD-Scale Vortices and Its Effect on Large Scale Plasma Mixing Across the Magnetospheric Boundary

Kelvin-Helmholtz Instability (KHI) is an MHD-scale instability that grows in a velocity shear layer such as the low-latitude boundary layer of the magnetosphere. KHI is driven unstable when a velocity shear is strong enough to overcome the stabilization effect of magnetic field. When the shear is significantly strong, vortices in the nonlinear stage of KHI is so rolled-up as to situate magnetospheric plasma outward of the magnetosheath plasma and vice versa. The big question is if such highly rolled-up vortices contribute significantly to the plasma transport across the boundary and to the filling of the plasma sheet by cool magnetosheath component, which is observed under northward Interplanetary Magnetic Field (IMF) condition. Here we review our recent results from two-fluid simulations of MHD-scale KHI with finite electron inertia taken into account. The results indicate that there is coupling between the MHD-scale dynamics and electron-scale dynamics in the rolled-up stage of the vortices. While the details differ depending on the initial magnetic geometry, the general conclusion is that there is significant modification of the MHD-scale vortex flow pattern via coupling to the micro-physics. The kick-back from the parasitic micro-physics enhances highly the potential for large-scale plasma mixing of the parent MHD-scale vortices, which is prohibited by definition in ideal-MHD. We also review our recent 3-D MHD simulation results indicating that KHI vortex can indeed roll-up in the magnetotail-flank situation despite the strong stabilization by the lobe magnetic field. These results encouraged us to search for evidence of rolled-up vortices in the Cluster formation flying observations. As reviewed in this paper, a nice event was found during northward IMF interval. This interval is when the plasma transport via large scale reconnection becomes less efficient. The finding supports the argument that KHI is playing some role in transporting solar wind into the magnetosphere when the normal mode of transport cannot dominate.

The scaling of forced collisionless reconnection

We present two-fluid simulations of forced magnetic reconnection with finite electron inertia in a collisionless two-dimensional slab geometry. Reconnection in this system is driven by a spatially localized forcing function that is added to the ion momentum equation inside the computational domain. The resulting forced reconnection process is studied as a function of the temporal and spatial structure of the forcing function, the plasma β, and strength of the out-of-plane guide magnetic field component, and the electron to ion mass ratio. Consistent with previous results found in unforced, large Δ′ systems, for sufficiently strong forcing the reconnection process is found to become Alfvénic, i.e., the inflow velocity scales roughly like some small fraction of the Alfvén speed based on the reconnecting component of the magnetic field just upstream of the dissipation region. The magnitude of this field and thus the rate of reconnection is controlled by the behavior of the forcing function. When the forcing strength is below a certain level, fast reconnection is not observed.

Turbulence in low-β reconnection

Using a two-dimensional two-fluid numerical model with finite electron inertia, we have observed turbulence to occur in simulations of plasmas undergoing magnetic reconnection when the plasma β is sufficiently small. This turbulence is caused by secondary instabilities, which are excited by large gradients in the out-of-plane current. These gradients spontaneously form due to the dynamics of ion force balance across the layer, and first arise in the outflow regions downstream from the magnetic x point. This turbulence is not observed to have a substantial effect on the rate of reconnection.

Effects of the seasonal asymmetry in ionospheric Pedersen conductance on the appearance of discrete aurora

A two‐dimensional numerical two‐fluid MHD model of the coupled magnetosphere‐ionosphere system has been used to examine the effects of seasonal asymmetry in ionospheric conductance on the development of feedback instability and associated energetic electron precipitation. Effects of finite electron inertia and anomalous resistivity in the magnetospheric MHD model lead to the formation of parallel electric fields above the ionosphere as the feedback instability develops. The Fridman‐Lemaire model of energetic electron precipitation has been used to calculate the energy flux carried into the auroral ionosphere by accelerated electrons. The electron energy flux into the lower conductivity winter ionosphere is significantly greater than that into the summer ionosphere due to the asymmetry in ionospheric conductance. The higher energy flux into the winter ionosphere may increase the occurrence of discrete aurora in the winter hemisphere as detected by satellite auroral imaging.

Effect of cross-field flow on inertial Alfvén waves of small transverse scale

This analytic study examines the effect of cross-field flow on a microscopic current channel. To illustrate the subtle interplay between finite electron inertia and flows, a simple model of excitation of a microscopic current channel is considered. The model consists of a slab antenna exciter with a transverse width on the order of the electron skin-depth driven at a single frequency. The antenna is fixed in the laboratory frame and embedded within a plasma that has a uniform drift across the confining magnetic field. The combined effects of the plasma flow and the intrinsic, collisionless, cross-field expansion of the current channel lead to standing wave structures across the confining magnetic field. The resulting parallel electric fields generate an array of current filaments of alternating polarity which individually have transverse width smaller than the original channel. These results may help interpret laboratory and spacecraft measurements of Alfvénic turbulence and could lead to the development of a diagnostic tool to map plasma flows.

Splitting of the spectra of MHD waves and the structure of a satellite alfvén resonance in a cold plasma in a strong axial magnetic field and a weak helical field

The possibility is demonstrated of splitting the eigenfrequencies of MHD plasma waves in a stellarator with a weakly rippled helical confining magnetic field. The distribution of the fields of an Alfven wave in the satellite Alfven resonance region is investigated when the influence of the helical ripple in a confining magnetic field on the resonance structure is comparable with the effects of the finite ion Larmor radius, electron inertia, and collisions between plasma particles.

Dispersive width of the Alfvénic field line resonance

The results from a numerical investigation of the dependence of the eigenfrequency and structure of Alfvénic field line resonance (FLR) layers on the dispersion of the resonant Alfvén waves are presented. The investigation is based on the linear, reduced, two‐fluid MHD‐kinetic model considered in the dipole magnetic field geometry with a realistic distribution of the background plasma parameters along the entire auroral flux tube. It is shown that (1) dispersion of the small‐scale Alfvén wave due to the finite electron inertia (inertial dispersions) decreases the eigenfrequency of the Alfvénic FLR and broadens the FLR layer in the direction along the transverse gradient in the background Alfvén speed; (2) dispersion of the wave due to the finite plasma temperature (kinetic dispersion) increases the FLR eigenfrequency and broadens the resonance layer in the opposite direction; and (3) on the field line where the effect of the kinetic dispersion balances the effect of the inertial dispersion, the eigenfrequency becomes independent of the transverse wavelength of the resonant wave. As a result, a FLR occurring on such a particular field line develops fields and currents with smaller transverse scale sizes and larger amplitudes than FLRs on field lines where inertial and kinetic dispersions are not in a balance.

Small‐scale, dispersive field line resonances in the hot magnetospheric plasma

The formation, temporal behavior, and spatial structure of field line resonance (FLR) layers formed by shear Alfvén waves standing along auroral magnetic field lines between the ionospheres are investigated when the layer develops transverse structure on the scale of the ion Larmour radius. Using a new numerical model including full ion Larmour radius correction in dipole magnetic geometry with realistic distributions of background plasma temperature and density, the following is shown: (1) Hot magnetospheric ions significantly retard the development of a parallel electric field in ion gyroscale dispersive Alfvén waves. (2) A fundamental FLR forming near L = 7.5 can contract to a transverse scale size of several hundred of meters in the direction perpendicular to the geomagnetic field at ionospheric altitudes, with a parallel electric field sufficient to produce a k V potential drop along the resonance field line from the ionosphere up to ∼ 4 RE altitude, in the region where the wave dispersion is due to the finite electron inertia. (3) A plasma density depletion in the lower auroral magnetosphere (≈ 2–5 RE geocentric distance) enables the formation of a nonradiative fundamental FLR. (4) Dispersive FLRs for the higher harmonics are more radiative at the equatorial magnetosphere than the fundamental mode.

Generation of kinetic Alfvén waves by upper-hybrid pump waves

The nonlinear mechanism for kinetic-Alfvén-wave (KAW) excitation by upper-hybrid waves (UHWs) is discussed. Taking into account perpendicular dispersion of KAWs, caused by effects of finite ion Larmor radius and electron inertia, we examine a new channel for UHW decay, in which a pump UHW decays into another UHW and an ultralow-frequency wave, KAW: UHW→KAW+UHW. A nonlinear dispersion relation is derived, and the growth rate of the parametric decay instability is calculated for a pump UHW propagating at an arbitrary angle to the background magnetic field. We find that the resulting KAWs often have a two-peaked spectrum with different perpendicular dispersions. Using satellite observations, the analytical results are applied to show that the considered process represents an effective mechanism for KAW generation and the consequent spreading of the UHW spectrum in the Earth's magnetosphere and solar corona.

Numerical simulations and simplified models of nonlinear electron inertial Alfvén waves

We describe nonlinear resonance absorption of compressional Alfvén waves in a model magnetosphere. It is shown that the ponderomotive force of excited standing shear Alfvén waves can lead to nonlinear saturation and spatial structuring of field line resonances and that in low‐beta plasmas ponderomotive saturation may occur before other nonlinear saturation processes such as the Kelvin‐Helmholtz instability. The effects of finite electron inertia are also considered, and it is shown that spatial structuring of field line resonances may occur with scale sizes compatible with those found in discrete auroral arcs. Results are discussed in the context of three‐dimensional numerical solutions to the resistive magnetohydrodynamic equations and based on a simplified analytical model of nonlinear resonance absorption.

Effect of Boundary, Electron Inertia and Streaming on LargeAmplitude Waves in a Relativistic Plasma

Nonlineaer wave propagation in a relativistic bounded plasma isanalysed through the pseudopotential approach by taking into accountthe streaming of electrons, and electron inertia. The finite geometryis seen to effect both the forward moving and reflected modes ofpropagation. Its dependence on the electron–ion mass ratio isalso critically analysed. In the second part of the paper we study theformation of solitons through the pseudopotential approach. Thevariation of the form of the pseudopotential is studied with respectto the radius of the bounding system, the electron–ion massratio, and the streaming velocity u0. It isobserved that the finite geometry and electron inertia significantlyaffect the solitary wave formation.