Electron Magnetohydrodynamics Research Articles

Unsteady Magnetohydrodynamics PDE of Monge–Ampère Type: Symmetries, Closed-Form Solutions, and Reductions

The paper studies an unsteady equation with quadratic nonlinearity in second derivatives, that occurs in electron magnetohydrodynamics. In mathematics, such PDEs are referred to as parabolic Monge–Ampère equations. An overview of the Monge–Ampère type equations is given, in which their unusual qualitative features are noted. For the first time, the Lie group analysis of the considered highly nonlinear PDE with three independent variables is carried out. An eleven-parameter transformation is found that preserves the form of the equation. Some one-dimensional reductions allowing to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. A large number of new additive, multiplicative, generalized, and functional separable solutions are obtained. Special attention is paid to the construction of exact closed-form solutions, including solutions in elementary functions (in total, more than 30 solutions in elementary functions were obtained). Two-dimensional symmetry and non-symmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations, linear and nonlinear heat type equations, and nonlinear filtration equations). The obtained results and exact solutions can be used to evaluate the accuracy and analyze the adequacy of numerical methods for solving initial boundary value problems described by highly nonlinear partial differential equations.

Magneto-inertial range dominated by magnetic helicity in space plasmas

Magneto-inertial range dominated by magnetic helicity has been studied using results of the numerical simulations, laboratory measurements, solar, solar wind, and the Earth’s and planets’ magnetosphere observations (spacecraft measurements). The spectral data have been compared with the theoretical results based on the distributed chaos notion in the frames of the Kolmogorov–Iroshnikov phenomenology. The transition from magnetohydrodynamics to kinetics in the electron and Hall magnetohydrodynamics, and in a fully kinetic 3D approach, as well as in the solar wind, solar photosphere, and at the special events (reconnections, Kelvin–Helmholtz instability, isolated flux tube interchanges, etc.) in the magnetosphere of Earth, Saturn, Jupiter, and Mercury has been also discussed.

Numerical study of the influence of electron inertial effects and electron dynamics on tearing mode instability

The tearing mode instabilities were numerically studied in two distinct models: the finite electron inertial magnetohydrodynamics (MHD) and the electron MHD (EMHD). The finite electron inertial MHD model employed a modified Hall-MHD model that incorporated the electron inertial effects in the generalized Ohm’s Law. On the other hand, the electron dynamics were described by the EMHD model. It is found that both electron inertial effects and electron dynamics significantly influence the linear and nonlinear growth of tearing mode instabilities, with electron dynamics playing a more dominant role. The dependence of the linear growth rate of tearing modes on the electron inertial length de was investigated. The results show that electron inertial effects enhance the growth rate but resemble the behavior of resistivity η. Whereas, in the EMHD model, electron inertia plays a dominant role in tearing mode instabilities. Additionally, a study on the nonlinear saturation of (2,1) tearing modes was conducted, demonstrating consistency with relevant analytical theories. The study indicates that, in both models, the magnetic island exhibits faster growth and achieves a larger saturated island width as de increases.

A numerical study of electron-magnetohydrodynamics tearing modes in parameter ranges of experimental interest

We perform a numerical study of the linear dynamics of tearing modes in slab incompressible electron-magnetohydrodynamics (EMHD) by considering some parameter ranges, which can be of interest for laboratory plasmas (e.g., helicon devices) or for astrophysics (e.g., solar-wind turbulence). To this purpose, several non-ideal effects are simultaneously retained (finite electron inertia, resistivity, and electron viscosity), and we make distinction between the dissipation coefficients in the direction parallel and perpendicular to the guide field. We thus identify some new reconnection regimes, characterized by a departure from the customary monotonic power-law scalings of the growth rates with respect to the non-ideal parameters. The results here presented can provide a useful indication for future studies of EMHD regimes relevant to experiments and for extensions of the EMHD tearing mode modeling to more complete regimes including kinetic effects (e.g., “electron-only” reconnection in kinetic regimes).

Charged particle transport coefficient challenges in high energy density plasmas

High energy density physics (HEDP) and inertial confinement fusion (ICF) research typically relies on computational modeling using radiation-hydrodynamics codes in order to design experiments and understand their results. These tools, in turn, rely on numerous charged particle transport and relaxation coefficients to account for laser energy absorption, viscous dissipation, mass transport, thermal conduction, electrical conduction, non-local ion (including charged fusion product) transport, non-local electron transport, magnetohydrodynamics, multi-ion-species thermalization, and electron-ion equilibration. In many situations, these coefficients couple to other physics, such as imposed or self-generated magnetic fields. Furthermore, how these coefficients combine are sensitive to plasma conditions as well as how materials are distributed within a computational cell. Uncertainties in these coefficients and how they couple to other physics could explain many of the discrepancies between simulation predictions and experimental results that persist in even the most detailed calculations. This paper reviews the challenges faced by radiation-hydrodynamics in predicting the results of HEDP and ICF experiments with regard to these and other physics models typically included in simulation codes.

The Okubo–Weiss-type topological criteria in two-dimensional magnetohydrodynamic flows

The Okubo–Weiss (Okubo, Deep-Sea Res., vol. 17, issue 3, 1970, pp. 445–454; Weiss, Physica D, vol. 48, issue 2, 1991, pp. 273–294) criterion has been widely used as a diagnostic tool to divide a two-dimensional (2-D) hydrodynamical flow field into hyperbolic and elliptic regions. This paper considers extension of these ideas to 2-D magnetohydrodynamic (MHD) flows, and presents an Okubo–Weiss-type criterion to parameterize the magnetic field topology in 2-D MHD flows. This ensues via its topological connections with the intrinsic metric properties of the underlying magnetic flux manifold, and is illustrated by recasting the Okubo–Weiss-type criterion via the 2-D MHD stationary generalized Alfvénic state condition to approximate the slow-flow-variation ansatz imposed in its derivation. The Okubo–Weiss-type parameter then turns out to be related to the sign definiteness of the Gaussian curvature of the magnetic flux manifold. A similar formulation becomes possible for 2-D electron MHD flows, by using the generalized magnetic flux framework to incorporate the electron-inertia effects. Numerical simulations of quasi-stationary vortices in 2-D MHD flows in the decaying turbulence regime are then given to demonstrate that the Okubo–Weiss-type criterion is able to separate the MHD flow field into elliptic and hyperbolic field configurations very well.

Influence of ion-to-electron temperature ratio on tearing instability and resulting subion-scale turbulence in a low- β e collisionless plasma

A two-field gyrofluid model including ion finite Larmor radius (FLR) corrections, magnetic fluctuations along the ambient field, and electron inertia is used to study two-dimensional reconnection in a low βe collisionless plasma, in a plane perpendicular to the ambient field. Both moderate and large values of the ion-to-electron temperature ratio τ are considered. The linear growth rate of the tearing instability is computed for various values of τ, confirming the convergence to reduced electron magnetohydrodynamics predictions in the large τ limit. Comparisons with analytical estimates in several limit cases are also presented. The nonlinear dynamics leads to a fully developed turbulent regime that appears to be sensitive to the value of the parameter τ. For τ = 100, strong large-scale velocity shears trigger Kelvin–Helmholtz instability, leading to the propagation of the turbulence through the separatrices, together with the formation of eddies of size of the order of the electron skin depth. In the τ = 1 regime, the vortices are significantly smaller and their accurate description requires that electron FLR effects be taken into account.

Almost sure existence of global weak solutions for supercritical electron MHD

Almost sure existence of global weak solutions for supercritical electron MHD

Topology of turbulence within collisionless plasma reconnection

In near-collisionless plasmas, which are ubiquitous in astrophysics, entropy production relies on fully-nonlinear processes such as turbulence and reconnection, which lead to particle acceleration. Mechanisms for turbulent reconnection include multiple magnetic flux ropes interacting to generate thin current sheets which undergo reconnection, leading to mixing and magnetic merging and growth of coherent structures, unstable reconnection current layers that fragment and turbulent reconnection outflows. All of these processes act across, and encompass, multiple reconnection sites. We use Magnetospheric Multi Scale four-point satellite observations to characterize the magnetic field line topology within a single reconnection current layer. We examine magnetopause reconnection where the spacecraft encounter the Electron Diffusion Region (EDR). We find fluctuating magnetic field with topology identical to that found for dynamically evolving vortices in hydrodynamic turbulence. The turbulence is supported by an electron-magnetohydrodynamic (EMHD) flow in which the magnetic field is effectively frozen into the electron fluid. Accelerated electrons are found in the EDR edge where we identify a departure from this turbulent topology, towards two-dimensional sheet-like structures. This is consistent with a scenario in which sub-ion scale turbulence can suppress electron acceleration within the EDR which would otherwise be possible in the electric field at the X-line.

On the Propagation of Whistler‐Mode Waves in the Magnetic Ducts

AbstractThis paper studies extremely low frequency (ELF) whistler‐mode waves' behavior within small‐scale magnetic field irregularities in the Earth's magnetosphere, known as magnetic ducts. Based on the magnetic fields' magnitude inside and outside these ducts, they are categorized as high‐magnetic ducts and low‐magnetic ducts (LBD). Using the whistler‐mode dispersion relation analysis, our primary focus is to show that LBDs are prone to leak electromagnetic energy outside the duct. We further investigate the hypothesis that whistlers can propagate within LBDs without any signal loss when the width of the duct corresponds to an integer multiple of the perpendicular wavelengths of the waves inside it. This condition offers a straightforward and effective method for identifying non‐leaking eigenmodes of LBDs. Our analysis of this non‐leaking condition reveals that every LBD possesses a finite number of non‐leaking eigenmodes directly proportional to the duct's width and the magnitude of the ambient magnetic field within it. The analytical results are then validated using two‐dimensional, time‐dependent simulations of the electron‐Magnetohydrodynamics model. Also, we model the non‐leaking propagation of an ELF whistler‐mode wave observed inside the LBD by the NASA Magnetospheric Multiscale mission satellites.

Dissipation wavenumber and regularity for electron magnetohydrodynamics

We consider the electron magnetohydrodynamics (MHD) with static background ion flow. A special situation of B(x,y,t)=∇×(ae→z)+be→z with scalar-valued functions a(x,y,t) and b(x,y,t) was studied numerically in the physics paper [8]. The authors concluded from numerical simulations that there is no evidence of dissipation cutoff for the electron MHD. In this paper we show the existence of determining wavenumber for the electron MHD, and establish a regularity condition only on the low modes of the solution. Our results suggest that the conclusion of the physics paper on the dissipation cutoff for the electron MHD is debatable.

Axi-symmetric solutions for active vector models generalizing 3D Euler and electron–MHD equations

We study systems interpolating between the 3D incompressible Euler and electron–MHD equations, given by ∂tB+V⋅∇B=B⋅∇V,V=−∇×(−Δ)−aB,∇⋅B=0,where B is a time-dependent vector field in R3. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range 1/2<a<1. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions.

Asymptotic scalings of fluid, incompressible “electron-only” reconnection instabilities: Electron-magnetohydrodynamics tearing modes

We perform a numerical study of the scaling laws of tearing modes in different parameter regimes of incompressible fluid electron magnetohydrodynamics, both in the small and large wavelength limits, as well as for the fastest growing mode that can be destabilized in a large aspect ratio current sheet. We discuss the relevance of these results, also for the interpretation of the “electron-only reconnection regime,” recently identified in spacecraft measures and in numerical simulations of solar wind turbulence. We restrict here to a single parameter study, in which we selectively consider only one non-ideal effect among electron inertia, perpendicular resistivity, and perpendicular electron viscosity, and we also consider the cases in which a proportionality exists between the parallel and the perpendicular dissipative coefficients. While some known theoretical results are thus confirmed, in other regimes and/or wavelength limits, corrections are proposed with respect to some theoretical estimates already available in the literature. In other cases, the scalings are provided for the first time. All numerical results are justified in terms of heuristic arguments based on the measurement of the scaling laws of some new microscopic scales associated with the gradients of the eigenfunctions. The alternative scalings we have found are consistent with this interpretation.

Electron magnetohydrodynamics Grad–Shafranov reconstruction of the magnetic reconnection electron diffusion region

We present a study of the electron magnetohydrodynamics Grad–Shafranov (GS) reconstruction of the electron diffusion region (EDR) of magnetic reconnection. Two-dimensionality of the magnetoplasma configuration and steady state are the two basic assumptions of the GS reconstruction technique, which represent the method’s fundamental limitations. The present study demonstrates that the GS reconstruction can provide physically meaningful results even when these two assumptions, which are hardly fulfilled in spacecraft observations, are violated. This conclusion is supported by the reconstruction of magnetic configurations of two EDRs, encountered by the Magnetospheric Multiscale (MMS) Mission on July 11, 2017 and September 8, 2018. Here, the former event exhibited a violation of two-dimensionality, and the latter event exhibited a violation of steady state. In both cases, despite the deviations from the ideal model configuration, reasonable reconstruction results are obtained by implementing the herein introduced compressible GS reconstruction model. In addition to the discussed fundamental limitations, all existing versions of the GS reconstruction technique rely on a number of minor simplifying assumptions, which restrict the model scope and efficiency. We study the prospects for further model improvement and generalization analytically. Our analysis reveals that nearly all these minor limitations can be overcome by using a polynomial MMS-tailored reconstruction technique in the space of rotationally invariant variables instead of Cartesian coordinates.

Active vector models generalising 3D Euler and electron–MHD equations

We introduce an active vector system, which generalises both the 3D Euler equations and the electron–magnetohydrodynamic equations (E–MHD). We may as well view the system as singularised systems for the 3D Euler equations, in which case the equations of (E–MHD) correspond to the order two more singular one than the 3D Euler equations. The generalised surface quasi-geostrophic equation (gSQG) can be also embedded into a special case of our system when the unknown functions are constant in one coordinate direction. We investigate some basic properties of this system as well as the conservation laws. In the case when the system corresponds up to order one more singular than the 3D Euler equations, we prove local well-posedness in the standard Sobolev spaces. The proof crucially depends on a sharp commutator estimate similar to the one used for (gSQG) in Chae et al (2012 Commun. Pure Appl. Math. 65 1037–66). Since the system covers many areas of both physically and mathematically interesting cases, one can expect that there are various related problems to be investigated, parts of which are discussed here.

PLASMA AND MAGNETIC FIELD DYNAMICS IN POS: PIC SIMULATIONS

2D3V particle in cell computer simulations for plasma, magnetic field and current density spatio-temporal dynamics in plasma opening switch (POS) are presented for various initial plasma densities. The current channel formation in the POS’s plasma, its passage through the plasma bridge, and its destruction at the boundary of the plasma jumper are studied. It is shown, that during the penetration of the magnetic field into the POS's plasma and current channel formation, electron vortices are excited. The passage speed of the POS's plasma by vortices decreases with increasing plasma density, which is qualitatively consistent with estimates based on classical electron magnetohydrodynamics (EMHD).

Cascade conditions in electron magneto-hydrodynamic turbulence

The basic features of electron whistler waves can be described by electron magneto-hydrodynamics. In terms of this model, we discuss criteria and relevant parameter ranges for dual and forward cascades in electron whistler turbulence. It is an advantage to use the set of eigenfunctions for the “curl” operator in the analysis. It is demonstrated that three wavemodes with same helicity support a dual cascade as know also from spatially two dimensional turbulence, while modes with mixed helicities include the classical cascade of energy from long to shorter wavelengths. The analysis is based on the conservation of energy and magnetic helicity during the wave interactions. For stationary conditions in large closed systems, a detailed balance can be expected. It is, however, possible for large-scale self-organized structures to develop as transient phenomena. These can be found when solving initial value problem with a strong imbalance of positive and negative helicities. For plasmas in nature or laboratory, similar conditions can occur when the system has boundary conditions with preference for emission of oscillations with one sign of helicity.

Wave turbulence in inertial electron magnetohydrodynamics

A wave turbulence theory is developed for inertial electron magnetohydrodynamics (IEMHD) in the presence of a relatively strong and uniform external magnetic field $\boldsymbol {B_0} = B_0 \hat {\boldsymbol {e}}_\|$ . This regime is relevant for scales smaller than the electron inertial length $d_e$ . We derive the kinetic equations that describe the three-wave interactions between inertial whistler or kinetic Alfvén waves. We show that for both invariants, energy and momentum, the transfer is anisotropic (axisymmetric) with a direct cascade mainly in the direction perpendicular ( $\perp$ ) to $\boldsymbol {B_0}$ . The exact stationary solutions (Kolmogorov–Zakharov spectra) are obtained for which we prove the locality. We also found the Kolmogorov constant $C_K \simeq 8.474$ . In the simplest case, the study reveals an energy spectrum in $k_\perp ^{-5/2} k_\|^{-1/2}$ (with k the wavenumber) and a momentum spectrum enslaved to the energy dynamics in $k_\perp ^{-3/2} k_\|^{-1/2}$ . These solutions correspond to a magnetic energy spectrum ${\sim }k_\perp ^{-9/2}$ , which is steeper than the EMHD prediction made for scales larger than $d_e$ . We conclude with a discussion on the application of the theory to space plasmas.

On the Cauchy Problem for the Hall and Electron Magnetohydrodynamic Equations Without Resistivity I: Illposedness Near Degenerate Stationary Solutions

In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev space $$H^{s}$$ and even in any Gevrey spaces. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which the $$L^{2}$$ norm (energy) of the solution stays constant in time. The core illposedness (or instability) mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols. The method developed in this work is sharp and robust, in that we also prove nonlinear $$H^{s}$$ -illposedness (for s arbitrarily high) in the presence of fractional dissipation of any order less than 1, matching the previously known wellposedness results. The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problems for the incompressible Hall and electron-MHD equations. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field.

Lagrangian regularity of the electron magnetohydrodynamics flow on a bounded domain

In this paper we investigate the regularity in time of the Lagrangian flow associated with the electron magnetohydrodynamics (e-MHD) equations on a bounded domain with a smooth (ultradifferentiable) boundary. This model is widely used in controlled magnetic fusion, in space and astrophysics plasmas and also in physics of solids. We show that initial data with limited smoothness in Sobolev spaces induce a Lagrangian flow-map X and a Lagrangian magnetic vector potential A (viz. the magnetic vector potential evaluated at the Lagrangian spatial point X), which are ultradifferentiable in time, with the two particular cases of real analytic and Gevrey time regularity. It turns out that the Lagrangian canonical momentum P, the Lagrangian magnetic field B, and the Lagrangian electric field E inherit this Lagrangian regularity property. Among others, the proof makes crucial use of a novel Lagrangian formulation of the e-MHD in terms of the Lagrangian fields (X,A,P,B,E). A by-product of this Lagrangian and constructive proof is the design of arbitrary high-order semi-Lagrangian schemes to solve the e-MHD equations on a bounded domain.