A. A. Schekochihin

ASTROPHYSICAL GYROKINETICS: KINETIC AND FLUID TURBULENT CASCADES IN MAGNETIZED WEAKLY COLLISIONAL PLASMAS

This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulent motions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the inertial range above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvenic fluctuations and a passive cascade of density and magnetic-field-strength fluctuations. The former are governed by the reduced magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvenic component (in the collisional limit, these compressive fluctuations become the slow and entropy modes of the conventional MHD). In the dissipation range below ion gyroscale, there are again two cascades: the kinetic-Alfven-wave (KAW) cascade governed by two fluid-like electron reduced magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations both in space and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was Landau-damped at the ion gyroscale to collisional scales in the phase space and leads to ion heating. The KAW energy is similarly damped at the electron gyroscale and converted into electron heat. Kolmogorov-style scaling relations are derived for all of these cascades. The relationship between the theoretical models proposed in this paper and astrophysical applications and observations is discussed in detail.This paper presents a theoretical framework for understand ing plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and densit y fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulent motions at weakly collisional an d collisionless scales. This paper focuses on turbulence in a strong mean magnetic field (the guide field). T he key assumptions behind the theory developed here are that the turbulent fluctuations are anisotropic wit h respect to the mean field and that their frequency is low compared to the ion cyclotron frequency. The turbulen ce is assumed to be stirred (forced) at some system-specific outer scale L. The energy injected at this scale has to be dissipated into h eat, which ultimately cannot be accomplished without collisions. A kinetic cascadedevelops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic c as ade in various scale ranges depends on the physics of plasma fluctuations that can exist there. There ar four special scales that separate physically distinct regimes: the electron gyroscale ρe, the ion gyroscaleρi , the mean free path λmfpi and the electron heat diffusion scale (mi/me)λmfpi (me andmi are electron and ion masses). In each of the scale ranges sepa rated by these scales, a number of physically meaningful and rigorously ju stifiable simplifications of the fully kinetic plasma description are possible. These are derived systematicall y vi a hierarchy of asymptotic expansions. The result is that, in each scale range, the fully kinetic proble m is reduced to a more physically transparent and computationally tractable system of equations, which are d erived in a rigorous way. In the “inertial range” above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvénic fluctuations and a passive cascade of density and magnetic-field-strength fluc tuations. The former are governed by two fluid-like Reduced Magnetohydrodynamic (RMHD) equations at both the c ollisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines as ociated with the Alfvénic component (in the collisional limit, these passive fluctuations become the sl ow and entropy modes of the conventional MHD). In the “dissipation range” between the ion and electron gyroscales, there are again two cascades: the kineticAlfvén-wave (KAW) cascade governed by two fluid-like Electr on Reduced Magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations b oth in space and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was dampe d by collisionless wave-particle interaction at the ion gyroscle to collisional scales and leads to ion heating. The KAW energy is similarly damped at the electron gyroscale and is converted into electron heat. The relation ship between the theoretical models proposed in this paper and astrophysical applications and observations is d i cussed in detail. Subject headings: magnetic fields—methods: analytical—MHD—plasmas—turbul ence

Instability of current sheets and formation of plasmoid chains

Current sheets formed in magnetic reconnection events are found to be unstable to high-wavenumber perturbations. The instability is very fast: its maximum growth rate scales as S1∕4vA∕LCS, where LCS is the length of the sheet, vA the Alfven speed, and S the Lundquist number. As a result, a chain of plasmoids (secondary islands) is formed, whose number scales as S3∕8.

Simulations of the Small-Scale Turbulent Dynamo

We report the results of an extensive numerical study of the small-scale turbulent dynamo. The primary focus is on the case of large magnetic Prandtl numbers Prm, which is relevant for hot low-density astrophysical plasmas. A Prm parameter scan is given for the model case of viscosity-dominated (low Reynolds number) turbulence. We concentrate on three topics: magnetic energy spectra and saturation levels, the structure of the magnetic field lines, and intermittency of the field strength distribution. The main results are as follows: (1) the folded structure of the field (direction reversals at the resistive scale, field lines curved at the scale of the flow) persists from the kinematic to the nonlinear regime; (2) the field distribution is self-similar and appears to be lognormal during the kinematic regime and exponential in the saturated state; and (3) the bulk of the magnetic energy is at the resistive scale in the kinematic regime and remains there after saturation, although the magnetic energy spectrum becomes much shallower. We propose an analytical model of saturation based on the idea of partial two-dimensionalization of the velocity gradients with respect to the local direction of the magnetic folds. The model-predicted saturated spectra are in excellent agreement with numerical results. Comparisons with large-Re, moderate-Prm runs are carried out to confirm the relevance of these results and to test heuristic scenarios of dynamo saturation. New features at large Re are elongation of the folds in the nonlinear regime from the viscous scale to the box scale and the presence of an intermediate nonlinear stage of slower than exponential magnetic energy growth accompanied by an increase of the resistive scale and partial suppression of the kinetic energy spectrum in the inertial range. Numerical results for the saturated state do not support scale-by-scale equipartition between magnetic and kinetic energies, with a definite excess of magnetic energy at small scales. A physical picture of the saturated state is proposed. Subject heading gs: magnetic fields — methods: numerical — MHD — plasmas — turbulence

A Model of Turbulence in Magnetized Plasmas: Implications for the Dissipation Range in the Solar Wind

This paper studies the turbulent cascade of magnetic energy in weakly col- lisional magnetized plasmas. A cascade model is presented, based on the assumptions of local nonlinear energy transfer in wavenumber space, critical balance between linear propagation and nonlinear interaction times, and the applicability of linear dissipation rates for the nonlinearly turbulent plasma. The model follows the nonlinear cascade of energy from the driving scale in the MHD regime, through the transition at the ion Lar- mor radius into the kinetic Alfven wave regime, in which the turbulence is dissipated by kinetic processes. The turbulent fluctuations remain at frequencies below the ion cy- clotron frequency due to the strong anisotropy of the turbulent fluctuations, kk ≪ k⊥ (implied by critical balance). In this limit, the turbulence is optimally described by gy- rokinetics; it is shown that the gyrokinetic approximation is well satisfied for typical slow solar wind parameters. Wave phase velocity measurements are consistent with a kinetic Alfven wave cascade and not the onset of ion cyclotron damping. The conditions under which the gyrokinetic cascade reaches the ion cyclotron frequency are established. Cas- cade model solutions imply that collisionless damping provides a natural explanation for the observed range of spectral indices in the dissipation range of the solar wind. The dis- sipation range spectrum is predicted to be an exponential fall off; the power-law behav- ior apparent in observations may be an artifact of limited instrumental sensitivity. The cascade model is motivated by a programme of gyrokinetic simulations of turbulence and particle heating in the solar wind.

Astrophysical Gyrokinetics: Basic Equations and Linear Theory

Magnetohydrodynamic (MHD) turbulence is encountered in a wide variety of astrophysical plasmas, including accretion disks, the solar wind, and the interstellar and intracluster medium. On small scales, this turbulence is often expected to consist of highly anisotropic fluctuations with frequencies small compared to the ion cyclotron frequency. For a number of applications, the small scales are also collisionless, so a kinetic treatment of the turbulence is necessary. We show that this anisotropic turbulence is well described by a low-frequency expansion of the kinetic theory called gyrokinetics. This paper is the first in a series to examine turbulent astrophysical plasmas in the gyrokinetic limit. We derive and explain the nonlinear gyrokinetic equations and explore the linear properties of gyrokinetics as a prelude to nonlinear simulations. The linear dispersion relation for gyrokinetics is obtained, and its solutions are compared to those of hot-plasma kinetic theory. These results are used to validate the performance of the gyrokinetic simulation code GS2 in the parameter regimes relevant for astrophysical plasmas. New results on global energy conservation in gyrokinetics are also derived. We briefly outline several of the problems to be addressed by future nonlinear simulations, including particle heating by turbulence in hot accretion flows and in the solar wind, the magnetic and electric field power spectra in the solar wind, and the origin of small-scale density fluctuations in the interstellar medium.

Kinetic Simulations of Magnetized Turbulence in Astrophysical Plasmas

This Letter presents the first ab initio, fully electromagnetic, kinetic simulations of magnetized turbulence in a homogeneous, weakly collisional plasma at the scale of the ion Larmor radius (ion gyroscale). Magnetic- and electric-field energy spectra show a break at the ion gyroscale; the spectral slopes are consistent with scaling predictions for critically balanced turbulence of Alfvén waves above the ion gyroscale (spectral index -5/3) and of kinetic Alfvén waves below the ion gyroscale (spectral indices of -7/3 for magnetic and -1/3 for electric fluctuations). This behavior is also qualitatively consistent with in situ measurements of turbulence in the solar wind. Our findings support the hypothesis that the frequencies of turbulent fluctuations in the solar wind remain well below the ion cyclotron frequency both above and below the ion gyroscale.

Fast magnetic reconnection in the plasmoid-dominated regime.

A conceptual model of resistive magnetic reconnection via a stochastic plasmoid chain is proposed. The global reconnection rate is shown to be independent of the Lundquist number. The distribution of fluxes in the plasmoids is shown to be an inverse-square law. It is argued that there is a finite probability of emergence of abnormally large plasmoids, which can disrupt the chain (and may be responsible for observable large abrupt events in solar flares and sawtooth crashes). A criterion for the transition from the resistive magnetohydrodynamic to the collisionless regime is provided.

Formation of Plasmoid Chains in Magnetic Reconnection

A detailed numerical study of magnetic reconnection in resistive MHD for very large, previously inaccessible, Lundquist numbers (10(4) <or= S <or= 10(8)) is reported. Large-aspect-ratio Sweet-Parker current sheets are shown to be unstable to super-Alfvénically fast formation of plasmoid (magnetic-island) chains. The plasmoid number scales as S(3/8) and the instability growth rate in the linear stage as S(1/4), in agreement with the theory by Loureiro et al. [Phys. Plasmas 14, 100703 (2007)]. In the nonlinear regime, plasmoids continue to grow faster than they are ejected and completely disrupt the reconnection layer. These results suggest that high-Lundquist-number reconnection is inherently time-dependent and hence call for a substantial revision of the standard Sweet-Parker quasistationary picture for S>10(4).

Plasma Instabilities and Magnetic Field Growth in Clusters of Galaxies

We show that under very general conditions, cluster plasmas threaded by weak magnetic fields are subject to very fast growing plasma instabilities driven by the anisotropy of the plasma pressure (viscous stress) with respect to the local direction of the magnetic field. Such an anisotropy will naturally arise in any weakly magnetized plasma that has low collisionality and is subject to stirring. The magnetic field must be sufficiently weak for the instabilities to occur, viz., beta>Re^{1/2}. The instabilities are captured by the extended MHD model with Braginskii viscosity. However, their growth rates are proportional to the wavenumber down to the ion gyroscale, so MHD equations with Braginskii viscosity are not well posed and a fully kinetic treatment is necessary. The instabilities can lead to magnetic fields in clusters being amplified from seed strength of ~10^{-18} G to dynamically important strengths of ~10 microG on cosmologically trivial time scales (~10^8 yr). The fields produced during the amplification stage are at scales much smaller than observed. Predicting the saturated field scale and structure will require a kinetic theory of magnetized cluster turbulence.We show that under very general conditions, cluster plasmas threaded by weak magnetic fields are subject to very fast growing plasma instabilities driven by the anisotropy of the plasma pressure (viscous stress) with respect to the local direction of the magnetic field. Such an anisotropy will naturally arise in any weakly magnetized plasma that has low collisionality and is subject to stirring. The magnetic field must be sufficiently weak for the instabilities to occur, viz., β Re1/2. The instabilities are captured by the extended MHD model with Braginskii viscosity. However, their growth rates are proportional to the wavenumber down to the ion gyroscale, so MHD equations with Braginskii viscosity are not well posed and a fully kinetic treatment is necessary. The instabilities can lead to magnetic fields in clusters being amplified from seed strength of ~10-18 G to dynamically important strengths of ~10 μG on cosmologically trivial timescales (~108 yr). The fields produced during the amplification stage are at scales much smaller than observed. Predicting the saturated field scale and structure will require a kinetic theory of magnetized cluster turbulence.

Anisotropy of solar wind turbulence between ion and electron scales.

The anisotropy of turbulence in the fast solar wind, between the ion and electron gyroscales, is directly observed using a multispacecraft analysis technique. Second order structure functions are calculated at different angles to the local magnetic field, for magnetic fluctuations both perpendicular and parallel to the mean field. In both components, the structure function value at large angles to the field S{⊥} is greater than at small angles S{∥}: in the perpendicular component S{⊥}/S{∥}=5±1 and in the parallel component S{⊥}/S{∥}>3, implying spatially anisotropic fluctuations, k{⊥}>k{∥}. The spectral index of the perpendicular component is -2.6 at large angles and -3 at small angles, in broad agreement with critically balanced whistler and kinetic Alfvén wave predictions. For the parallel component, however, it is shallower than -1.9, which is considerably less steep than predicted for a kinetic Alfvén wave cascade.