Electron Magnetohydrodynamic Turbulence Research Articles

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.

Two- and Three-dimensional Nonlinear Instabilities of Whistler Waves

Abstract Recently, two-dimensional (2D) nonlinear instabilities of whistler waves from resonant wave–wave interaction have gained much attention in numerical simulations as well as observations in space plasmas. In this paper, both 2D and three-dimensional (3D) nonlinear instabilities of whistler waves are investigated using electron magnetohydrodynamics (EMHD). It is found that decay instabilities can excite waves with a broadband wavenumber spectrum, including highly oblique propagating whistler waves. Whistler waves with λ e k = 1 would excite counter-propagating whistler waves, and the wave with can produce both co- and counter-propagating whistler waves, where λ e is the electron inertial length and k is the wavenumber. Moreover, it is shown that 3D instabilities have similar nonlinear growth rate distributions as in 2D decay for the azimuthal wavelength much larger than λ e . These results suggest that nonlinear wave–wave interaction can play an important role in the scattering of whistler waves in the solar wind and the Earth’s magnetosphere, and are also helpful for understanding nonlinear wave–wave interaction in the formation and development of EMHD turbulence.

Spectra of Turbulence during the Dipolarization of the Magnetic Field

The presence of heterogeneity in turbulent processes has been analyzed, and the spectra of turbulence have been obtained for the regions before and during the dipolarization of the magnetic field in the Earth’s magnetospheric tail from the measurements of four space vehicles of the Cluster-2 mission (the event of September 21, 2005). The spectral and wavelet analysis was supplemented by the investigations of the fluctuation kurtosis for the magnetic field absolute value. In the region of the magnetic field dipolarization in the magnetospheric tail, a decreasing horizontal component of the magnetic field in parts of the tail and an increasing vertical component, kurtosis variations, the presence of strong Pc5 and Pc4 pulsations as well as direct and inverse cascades, a break in the spectra at the frequencies below the proton gyrofrequency, and a change in the character of turbulent motions at different time scales (at large time scales, the turbulent flow corresponds to the homogeneous models of Kolmogorov and Iroshnikov–Kraichnan; at smaller time scales, the turbulent flow is described by the electron magnetohydrodynamic turbulence model) have been detected. Using the measurements from different space vehicles, it was possible to estimate the velocity of the plasma flow in the tail direction.

Spectral evolution of helical electron magnetohydrodynamic turbulence

AbstractProperties of turbulence in space plasmas above and below the proton gyroscale are very different and different descriptions should be used for the large and the small scales. On scales larger than the proton gyroscale, we can treat the plasmas as conducting fluids and use magnetohydrodynamics (MHD). However, on scales smaller than the proton gyroscale, we cannot use MHD due to various kinetic effects. Electron magnetohydrodynamics (EMHD) is a simple fluid‐like model for small‐scale plasmas. In this paper, we discuss properties of EMHD waves and turbulence. In particular, we focus on inverse cascade in EMHD turbulence. Since EMHD waves are helical and magnetic helicity is a conserved quantity, inverse cascade is expected to be a common phenomenon in EMHD. We first focus on inverse cascade and spectral evolution of completely imbalanced EMHD wave packets, in which all waves are moving in the same direction. Then, we discuss inverse cascade and spectral evolution of driven imbalanced EMHD turbulence. We also discuss “pinning” of energy spectra in imbalanced EMHD turbulence.

Forward and Inverse Cascades in EMHD Turbulence

Electron magnetohydrodynamics (EMHD) provides a simple fluid-like description of physics below the proton gyro-scale in collisionless plasmas, such as the solar wind. In this paper, we discuss forward and inverse cascades in EMHD turbulence in the presence of a strong mean magnetic field. Similar to Alfvén waves, EMHD waves, or EMHD perturbations, propagate along magnetic field lines. Therefore, two types of EMHD waves can exist: waves moving parallel to and waves moving anti-parallel to the the magnetic field lines. For energy cascade in EMHD turbulence, the relative amplitudes of opposite-traveling waves are important. When the amplitudes are balanced, we will see fully-developed forward cascade with a k-7/3 energy spectrum and a scale-dependent anisotropy. On the other hand, when the amplitudes are imbalanced, we will see inverse cascade, as well as (presumably not fully developed) forward cascade. The underlying physics for the inverse cascade is magnetic helicity conservation.

INVERSE CASCADE IN IMBALANCED ELECTRON MAGNETOHYDRODYNAMIC TURBULENCE

Electron magnetohydrodynamics (EMHD) provides a fluid-like description of small-scale magnetized plasmas. Balanced EMHD turbulence has been studied for a long time. However, driven imbalanced EMHD turbulence, in which waves moving in one direction (dominant waves) have higher amplitudes than waves moving in the other direction (sub-dominant waves), has not been well studied. In this paper, we numerically study driven three-dimensional imbalanced weak EMHD turbulence. We find the following results. First, in driven imbalanced EMHD turbulence, we clearly observe inverse cascade of magnetic helicity, as well as magnetic energy. This is because magnetic helicity is a conserved quantity and non-zero magnetic helicity is injected into the system in driven imbalanced EMHD turbulence. Second, the magnetic energy spectrum of the dominant waves on scales larger than the energy injection scale does not show a single power-law spectrum, which indicates that the inverse cascade is not a self-similar process. The peak of the spectrum roughly follows a k –3/2 spectrum, which can be explained by a Kolmogorov-type argument for weak turbulence. Third, a small amount of sub-dominant waves is induced by the dominant waves on large scales and the ratio of helicity densities of the dominant and the sub-dominant waves on large scales seems to converge to a certain value.

Electron magnetohydrodynamic turbulence: universal features

The energy cascade of electron magnetohydrodynamic (EMHD) turbulence is considered. Fractal and multi-fractal models for the energy dissipation field are used to determine the spatial intermittency corrections to the scaling behavior in the high-wavenumber (electron hydrodynamic limit) and low-wavenumber (magnetization limit) asymptotic regimes of the inertial range. Extrapolation of the multi-fractal scaling down to the dissipative microscales confirms in these asymptotic regimes a dissipative anomaly previously indicated by the numerical simulations of EMHD turbulence. Several basic features of the EMHD turbulent system are found to be universal which seem to transcend the existence of the characteristic length scale d e (which is the electron skin depth) in the EMHD problem: equipartition spectrum; Reynolds-number scaling of the dissipative microscales; scaling of the probability distribution function (PDF) of the electron-flow velocity (or magnetic field) gradient (even with intermittency corrections); dissipative anomaly; and critical exponent scaling.

Anomalousk⊥−8/3Spectrum in Electron Magnetohydrodynamic Turbulence

Electron magnetohydrodynamic turbulence is investigated under the presence of a relatively strong external magnetic field b0e∥ and through three-dimensional direct numerical simulations. Our study reveals the emergence of a k⊥(-8/3) scaling for the magnetic energy spectrum at scales k∥(D)≤k⊥≤k⊥(D), where k∥(D) and k⊥(D) are, respectively, the typical largest dissipative scales along and transverse to the b0 direction. Unlike standard magnetohydrodynamic, this turbulence regime is characterized by filaments of electric currents parallel to b0. The anomalous scaling is in agreement with a heuristic model in which the transfer in the parallel direction is negligible. Implications for solar wind turbulence are discussed.

Electron magnetohydrodynamics: Dynamics and turbulence

We consider dynamics and turbulent interaction of whistler modes within the framework of inertialess electron magnetohydrodynamics (EMHD). We argue that there is no energy principle in EMHD: any stationary closed configuration is neutrally stable. On the other hand, the relaxation principle, the long term evolution of a weakly dissipative system towards Taylor-Beltrami state, remains valid in EMHD. We consider the turbulent cascade of whistler modes. We show that (i) harmonic whistlers are exact nonlinear solutions; (ii) collinear whistlers do not interact (including counterpropagating); (iii) waves with the same value of the wave vector k(1)=k(2) do not interact; (iv) whistler modes have a dispersion that allows a three-wave decay, including into a zero frequency mode; (v) the three-wave interaction effectively couples modes with highly different wave numbers and propagation angles. In addition, linear interaction of a whistler with a single zero mode can lead to spatially divergent structures via parametric instability. All these properties are drastically different from MHD, so that the qualitative properties of the Alfvén turbulence can not be transferred to the EMHD turbulence. We derive the Hamiltonian formulation of EMHD, and using Bogoliubov transformation reduce it to the canonical form; we calculate the matrix elements for the three-wave interaction of whistlers. We solve numerically the kinetic equation and show that, generally, the EMHD cascade develops within a broad range of angles, while transiently it may show anisotropic, nearly two-dimensional structures. Development of a cascade depends on the forcing (nonuniversal) and often fails to reach a steady state. Analytical estimates predict the spectrum of magnetic fluctuations for the quasi-isotropic cascade [proportionality]k(-2). The cascade remains weak (not critically balanced). The cascade is UV local, while the infrared locality is weakly (logarithmically) violated.

Magnetic Helicity Conservation and Inverse Energy Cascade in Electron Magnetohydrodynamic Wave Packets

Electron magnetohydrodynamics (EMHD) provides a fluidlike description of small-scale magnetized plasmas. An EMHD wave propagates along magnetic field lines. The direction of propagation can be either parallel or antiparallel to the magnetic field lines. We numerically study propagation of three-dimensional (3D) EMHD wave packets moving in one direction. We obtain two major results. (1) Unlike its magnetohydrodynamic (MHD) counterpart, an EMHD wave packet is dispersive. Because of this, EMHD wave packets traveling in one direction create opposite-traveling wave packets via self-interaction and cascade energy to smaller scales. (2) EMHD wave packets traveling in one direction clearly exhibit inverse energy cascade. We find that the latter is due to conservation of magnetic helicity. We compare inverse energy cascade in 3D EMHD turbulence and two-dimensional (2D) hydrodynamic turbulence.

Consequence of space correlation foliation for electron magnetohydrodynamic turbulence

Homogeneous electron magnetohydrodynamic turbulence is investigated under the presence of an imposed magnetic field. It is shown that this situation is well described in physical space by a divergence relation, which expresses the statistical conservation of magnetic energy flux through the inertial range. The ansatz is made that the development of anisotropy implies a foliation of space correlation with an algebraic relation between the axial and radial components of the separation vector r. A direct consequence is the possibility to derive a new universal law for the third-order moments, which is parametrized by the intensity of anisotropy. The so-called critical balance condition is used to fix this parameter and find a unique vectorial expression.

SIMULATIONS OF ELECTRON MAGNETOHYDRODYNAMIC TURBULENCE

We present numerical simulations of electron magnetohydrodynamic (EMHD) and electron reduced MHD (ERMHD) turbulence. Comparing scaling relations, we find that both EMHD and ERMHD turbulence show similar spectra and anisotropy. We develop new techniques to study anisotropy of EMHD turbulence. Our detailed study of anisotropy of EMHD turbulence supports our earlier result of k || k 1/3 ⊥ scaling, where k || and k ⊥ are wavenumbers parallel and perpendicular to local direction of magnetic field, respectively. We find that the high-order statistics show a scaling that is similar to the She-Leveque scaling for incompressible hydrodynamic turbulence and different from that of incompressible MHD turbulence. We observe that the bispectra, which characterize the interaction of different scales within the turbulence cascade, are very different for EMHD and MHD turbulence. We show that both decaying and driven EMHD turbulence have the same statistical properties. We calculate the probability distribution functions (PDFs) of MHD and EMHD turbulence and compare them with those of interplanetary turbulence. We find that, as in the case of the solar wind, the PDFs of the increments of magnetic field strength in MHD and EMHD turbulence are well described by the Tsallis distribution. We discuss implications of our results for astrophysical situations, including the advection-dominated accretion flows and magnetic reconnection.

Cascades in decaying three-dimensional electron magnetohydrodynamic turbulence

AbstractDecaying electron magnetohydrodynamic (EMHD) turbulence in three dimensions is studied via high-resolution numerical simulations. The resulting energy spectra asymptotically approach a k−2 law with increasing RB, the ratio of the nonlinear to linear time scales in the governing equation, consistent with theoretical predictions. No evidence is found of a dissipative cutoff, consistent with non-local spectral energy transfer and recent studies of 2D EMHD turbulence. Dissipative cutoffs found in previous studies are explained as artificial effects of hyperdiffusivity. In another similarity to 2D EMHD turbulence, relatively stationary structures are found to develop in time, rather than the variability found in ordinary or MHD turbulence. Further, cascades of energy in 3D EMHD turbulence are found to be suppressed in all directions under the influence of a uniform background field. Energy transfer is further reduced in the direction parallel to the field, displaying scale-dependent anisotropy. Finally, the governing equation is found to yield a weak inverse cascade, at least partially transferring magnetic energy from small to large scales.

Forward and inverse cascades in decaying two-dimensional electron magnetohydrodynamic turbulence

Electron magnetohydrodynamic (EMHD) turbulence in two dimensions is studied via high-resolution numerical simulations with a normal diffusivity. The resulting energy spectra asymptotically approach a k−5/2 law with increasing RB, the ratio of the nonlinear to linear time scales in the governing equation. No evidence is found of a dissipative cutoff, consistent with nonlocal spectral energy transfer. Dissipative cutoffs found in previous studies are explained as artificial effects of hyperdiffusivity. Relatively stationary structures are found to develop in time, rather than the variability found in ordinary or MHD turbulence. Further, EMHD turbulence displays scale-dependent anisotropy with reduced energy transfer in the direction parallel to the uniform background field, consistent with previous studies. Finally, the governing equation is found to yield an inverse cascade, at least partially transferring magnetic energy from small to large scales.

Driven dissipative whistler wave turbulence

Driven dissipative whistler wave turbulence in two-dimensional electron magnetohydrodynamics is investigated using very high resolution nonlinear fluid simulations. It is shown that a dual cascade phenomenon of mean magnetic potential and energy invariants is in agreement with predictions based on statistical ensemble and Kolmogorov’s theories. Turbulent length scales larger than the electron skin depth (whistler wave regime) exhibit a spectral break in the vicinity of the forcing wave number that separates the inverse and forward cascade regimes. On the other hand, length scales smaller than the electron skin depth behave like hydrodynamic eddies in which both small and large scale regimes exhibit identical turbulent spectra. In both cases, however, turbulent fluctuations follow an exact Kolmogorov-type spectra. While wave effects are strong in the whistler wave regime, they are absent entirely in the hydrodynamics regime of the driven electron magnetohydrodynamic turbulence.

The Anisotropy of Electron Magnetohydrodynamic Turbulence

We present numerical studies of three-dimensional electron magnetohydrodynamic (EMHD) turbulence. We investigate the cascade timescale and anisotropy of freely decaying, strong EMHD turbulence with zero electron skin depth. The cascade time scales with k-4/3. Our numerical results clearly show scale-dependent anisotropy. We find that the observed anisotropy is consistent with k∥ ∝ k, where k∥ and k⊥ are wavenumbers parallel and perpendicular to (local) mean magnetic field, respectively.

On the generation of mean fields by small-scale electron magnetohydrodynamic turbulence

The problem of the generation of mean magnetic fields by small-scale turbulence within the framework of electron magnetohydrodynamics (EMHD) is considered. Two EMHD models are investigated, a two and one-half dimensional (212D) model in which the magnetic field has all three spatial components but, due to a strong external field, depends only on two coordinates, and a fully three-dimensional (3D) model with an imposed stationary and homogeneous magnetic field. It is shown that in the case of 212D turbulence two possible mechanisms are responsible for the generation of mean magnetic fields. The first one is similar to the α-effect in the MHD dynamo problem and is due to a nonzero helicity of the turbulence. The second one is related to the anisotropy of the turbulence, which can give rise to negative dissipation (resistivity, viscosity) of the mean field. The influence of electron inertia on the above effects is analyzed. Inertia results in a qualitative modification of the helicity effects and may lead to a change in sign of the turbulent viscosity. The criteria for the generation of mean magnetic fields are obtained. In the case of the 3D model, the generation of large-scale helicons by the small-scale helicon turbulence is studied within the framework of the adiabatic approximation. A closed set of equations for the evolution of both the magnetic field of the large-scale helicon and of the generalized action of the small-scale turbulence is obtained. The criterion for the resonant instability of a large-scale helicon due to its interaction with small-scale helicon turbulence is obtained.

Anisotropic Turbulence in Two‐dimensional Electron Magnetohydrodynamics

Spectral anisotropy within decaying incompressible electron magnetohydrodynamics turbulence is investigated by two-dimensional fluid simulations. An initially isotropic turbulent spectrum exhibits anisotropy in the spectral cascade because of the presence of a self-consistent large-scale or an externally imposed magnetic field. Our simulations show that the anisotropy scales linearly with the strength of the ambient magnetic field. The linear scaling of the anisotropic spectral cascades appears to be rather generic as turbulent excitation scales larger than kde 1, the collisionless electron inertial or skin depth length scale (de), follow a similar trend.

Scaling of Spectral Anisotropy in Two-Dimensional Electron Magneto-Hydrodynamic Turbulence

It is demonstrated using numerical simulation that spectral anisotropy in two-dimensional decaying electron magneto-hydrodynamic turbulence varies linearly with the ratio of fluctuating to mean magnetic field. Turbulent excitation scales larger ( k λ e <1) as well as smaller ( k λ e >1) than the electron skin depth scale length (λ e ) are shown to exhibit such linear scaling behavior.

Electron magnetohydrodynamic turbulence in a high-beta plasma. I. Plasma parameters and instability conditions

The interaction of a dense discharge plasma with a weak external magnetic field has been studied experimentally. The electron pressure exceeds the field pressure and forms a magnetic hole in the plasma interior. The ions are unmagnetized, while the electrons are in a transition regime from none to full magnetization. The electron confinement changes from Boltzmann equilibrium to magnetic confinement. The pressure balance equation does not describe the diamagnetism because ambipolar E×B drifts oppose the diamagnetic drift. The net drift exceeds the sound speed by an order of magnitude and produces a strong two-stream cross-field instability. Although its spectrum is close to the lower hybrid instability, there are significant differences from the classical lower hybrid instability, e.g., the presence of strong magnetic fluctuations. These fall into the regime of electron magnetohydrodynamics (EMHD) with unmagnetized but mobile ions. While the EMHD turbulence is the main focus of the two following companion papers, this first paper describes the plasma diamagnetism and basic parameters that lead to the instability.