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_par ~ k_perp^(1/3) scaling. We find that the high-order statistics show a scaling that is similar to the She-Leveque scaling. 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 ADAFs and magnetic reconnection.
Deep Dive into 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_par ~ k_perp^(1/3) scaling. We find that the high-order statistics show a scaling that is similar to the She-Leveque scaling. 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 st
Turbulence at scales below the proton gyroradius is of great importance in many astrophysical applications. Such turbulence involving motions of electrons is essential for understanding the small-scale magnetic field dynamics of plasmas. It is also important for understanding of magnetic fields in the crust of a neutron star (Goldreich & Reisenegger 1992). This turbulence has been measured at by solar wind probes (Leamon et al. 1998).
The origin of the small-scale turbulence in a magnetized plasma is easy to understand if we think of what is happening with turbulent motions at small scales. When turbulence is driven on large scales, turbulence energy cascades down to smaller scales. The nature of magnetized turbulence from the outer scale to the proton gyroradius scale is relatively well known. Magnetized turbulence above the proton gyroradius can be described by the standard magnetohydrodynamics (MHD). As the name implies, the standard MHD treats the plasma as a single fluid. MHD turbulence can be decomposed into cascades of Alfven, fast and slow modes (Goldreich & Sridhar 1995;Lithwick & Goldreich 2001;Cho & Lazarian 2002, 2003). While fast and slow modes get damped at larger scales, Alfvenic modes can cascade down to the proton gyroradius scale. Near and below the proton gyroradius scale, single-fluid description fails and we should take into account kinetic effects. Then what will happen to the Alfven modes that reach the proton gyroradius scale?
Recent years, the nature of small-scale MHD turbulence in the solar wind has drawn interest from researchers (Howes et al. 2008a,b;Matthaeus, Servidio, & Dmitruk 2008;Saito et al. 2008;Gary, Saito, & Li 2008;Schekochihin et al. 2009;Dmitruk & Matthaeus 2006). In situ measurements of the solar wind show magnetic fluctuations over a broad range of frequencies. In the rest frame of the spacecrafts, the magnetic fluctuations show a broken power-law spectrum. For example, Leamon et al. (1999) reported that, at ∼ 0.2Hz, the spectrum breaks from a ν -1.67 power-law to a ν -2.91 power-law. In general, the spectral break-point lies in the range 0.2Hz ν 0.5Hz (see Saito et al. 2008). The ν -1.67 power-law at ν 0.2Hz seems to be relatively robust and represents inertial range of Alfvenic MHD turbulence. However, the power index for ν 0.5Hz seems to vary between -2 and -4 (Leamon et al. 1998;Smith et al. 2006). This range, characterized by a steeper power-law index, is termed “dispersion range” (Stawicki, Gary, & Li 2001), which is different from the dissipation range. The true dissipation scale of turbulence may lie at the end of the dispersion range. When we convert frequency to length scale, the broken powerlaw implies that the magnetic energy spectrum changes from a k -1.67 inertial range spectrum to a steeper dispersion range spectrum, as we move from large scales to small scales. The transition from the inertial range to the dispersion range occurs near the proton gyro-scale ρ i . (However, it is also possible that it occurs at the ion inertial scale d i = ρ i / √ β i , where β i is the ion plasma β. See discussion in Schekochihin et al. 2009). The identity of the dispersion range turbulence is still under debate.
Small-scale magnetized turbulence also plays important roles in other astrophysical objects. Among them, the crust of neutron stars gives us useful insights on the electron MHD (EMHD) model of small-scale magnetized tur-bulence. In the crust of neutron stars, ions are virtually immobile and, thus, the ion gyroradius scale can be regarded as infinite. Therefore, turbulence in the crust should be similar to small-scale turbulence. Since ions are immobile, they provide a smooth charge background and electrons carry all the current, so that
where v e is the electron velocity, J is electric current density, B is magnetic field, c is the speed of light, n e is the electron number density, and e is the absolute value of the electric charge. Inserting this relation into the magnetic induction equation, we can obtain the EMHD equation
where η is magnetic diffusivity (see Kingsep, Chukbar, & Yankov 1990 for details about EMHD). Goldreich & Reisenegger (1992) first showed that magnetized turbulence in the crust of neutron stars can be described by the EMHD equation. They discussed the properties of EMHD turbulence in neutron stars and argued that EMHD turbulence can enhance ohmic dissipation of magnetic field in isolated neutron stars (see also Cumming, Arras, & Zweibel 2004).
In this paper, we will focus on spectrum and anisotropy of EMHD turbulence. Earlier researchers convincingly showed that energy spectrum of EMHD turbulence is steeper than Kolmogorov’s k -5/3 spectrum (Biskamp, Schwarz, & Drake 1996;Biskamp et al. 1999;Ng et al.2003). They found that the energy spectrum follows E(k) ∝ k -7/3 .
(3)
The steep energy spectrum can be explained by the following Kolmogorov-type argument (Biskamp et al. 1996). Suppose that the eddy interaction time for eddies o
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
