G. G. Howes

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

Magnetic Fluctuation Power Near Proton Temperature Anisotropy Instability Thresholds in the Solar Wind

The proton temperature anisotropy in the solar wind is known to be constrained by the theoretical thresholds for pressure-anisotropy-driven instabilities. Here, we use approximately 1x10;{6} independent measurements of gyroscale magnetic fluctuations in the solar wind to show for the first time that these fluctuations are enhanced along the temperature anisotropy thresholds of the mirror, proton oblique firehose, and ion cyclotron instabilities. In addition, the measured magnetic compressibility is enhanced at high plasma beta (beta_{ parallel} greater, similar1) along the mirror instability threshold but small elsewhere, consistent with expectations of the mirror mode. We also show that the short wavelength magnetic fluctuation power is a strong function of collisionality, which relaxes the temperature anisotropy away from the instability conditions and reduces correspondingly the fluctuation power.

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.

IDENTIFICATION OF KINETIC ALFVÉN WAVE TURBULENCE IN THE SOLAR WIND

The nature of small-scale turbulent fluctuations in the solar wind is investigated using a comparison of Cluster magnetic and electric field measurements to predictions arising from models consisting of either kinetic Alfven waves or whistler waves. The electric and magnetic field properties of these waves from linear theory are used to construct spacecraft-frame frequency spectra of (|δE|/|δB|) s/c and (|δB ∥|/|δB|) s/c , allowing for a direct comparison to spacecraft data. The measured properties of the small-scale turbulent fluctuations, found to be inconsistent with the whistler wave model, agree well with the prediction of a spectrum of kinetic Alfven waves with nearly perpendicular wavevectors.

THE SLOW-MODE NATURE OF COMPRESSIBLE WAVE POWER IN SOLAR WIND TURBULENCE

We use a large, statistical set of measurements from the Wind spacecraft at 1 AU, and supporting synthetic spacecraft data based on kinetic plasma theory, to show that the compressible component of inertial range solar wind turbulence is primarily in the kinetic slow mode. The zero-lag cross-correlation C(?n, ?B ?) between proton density fluctuations ?n and the field-aligned (compressible) component of the magnetic field ?B ? is negative and close to ?1. The typical dependence of C(?n, ?B ?) on the ion plasma beta ? i is consistent with a spectrum of compressible wave energy that is almost entirely in the kinetic slow mode. This has important implications for both the nature of the density fluctuation spectrum and for the cascade of kinetic turbulence to short wavelengths, favoring evolution to the kinetic Alfv?n wave mode rather than the (fast) whistler mode.

Nonlinear Phase Mixing and Phase-Space Cascade of Entropy in Gyrokinetic Plasma Turbulence

Electrostatic turbulence in weakly collisional, magnetized plasma can be interpreted as a cascade of entropy in phase space, which is proposed as a universal mechanism for dissipation of energy in magnetized plasma turbulence. When the nonlinear decorrelation time at the scale of the thermal Larmor radius is shorter than the collision time, a broad spectrum of fluctuations at sub-Larmor scales is numerically found in velocity and position space, with theoretically predicted scalings. The results are important because they identify what is probably a universal Kolmogorov-like regime for kinetic turbulence; and because any physical process that produces fluctuations of the gyrophase-independent part of the distribution function may, via the entropy cascade, result in turbulent heating at a rate that increases with the fluctuation amplitude, but is independent of the collision frequency.

Gyrokinetic turbulence: a nonlinear route to dissipation through phase space

This paper describes a conceptual framework for understanding kinetic plasma turbulence as a generalized form of energy cascade in phase space. It is emphasized that conversion of turbulent energy into thermodynamic heat is only achievable in the presence of some (however small) degree of collisionality. The smallness of the collision rate is compensated for by the emergence of a small-scale structure in the velocity space. For gyrokinetic turbulence, a nonlinear perpendicular phase-mixing mechanism is identified and described as a turbulent cascade of entropy fluctuations simultaneously occurring at spatial scales smaller than the ion gyroscale and in velocity space. Scaling relations for the resulting fluctuation spectra are derived. An estimate for the collisional cutoff is provided. The importance of adequately modelling and resolving collisions in gyrokinetic simulations is briefly discussed, as well as the relevance of these results to understanding the dissipation-range turbulence in the solar wind and the electrostatic microturbulence in fusion plasmas.

A weakened cascade model for turbulence in astrophysical plasmas

A refined cascade model for kinetic turbulence in weakly collisional astrophysical plasmas is presented that includes both the transition between weak and strong turbulence and the effect of nonlocal interactions on the nonlinear transfer of energy. The model describes the transition between weak and strong MHD turbulence and the complementary transition from strong kinetic Alfven wave (KAW) turbulence to weak dissipating KAW turbulence, a new regime of weak turbulence in which the effects of shearing by large scale motions and kinetic dissipation play an important role. The inclusion of the effect of nonlocal motions on the nonlinear energy cascade rate in the dissipation range, specifically the shearing by large-scale motions, is proposed to explain the nearly power-law energy spectra observed in the dissipation range of both kinetic numerical simulations and solar wind observations.