Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades in Magnetized Weakly Collisional Plasmas
06 May 2009-Astrophysical Journal Supplement Series (IOP Publishing Ltd.)-Vol. 182, Iss: 1, pp 310-377
TL;DR: In this paper, a theoretical framework for understanding plasma turbulence in astrophysical plasmas is presented, 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.
Abstract: 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.
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TL;DR: The first direct determination of the dissipation range of magnetofluid turbulence in the solar wind at the electron scales is reported and a remarkable agreement with theoretical predictions of a quasi-two-dimensional cascade into Kinetic Alfvén Waves (KAW).
Abstract: We report the first direct determination of the dissipation range of magnetofluid turbulence in the solar wind at the electron scales. Combining high resolution magnetic and electric field data of the Cluster spacecraft, we computed the spectrum of turbulence and found two distinct breakpoints in the magnetic spectrum at 0.4 and 35 Hz, which correspond, respectively, to the Doppler-shifted proton and electron gyroscales, ${f}_{{\ensuremath{\rho}}_{p}}$ and ${f}_{{\ensuremath{\rho}}_{e}}$. Below ${f}_{{\ensuremath{\rho}}_{p}}$, the spectrum follows a Kolmogorov scaling ${f}^{\ensuremath{-}1.62}$, typical of spectra observed at 1 AU. Above ${f}_{{\ensuremath{\rho}}_{p}}$, a second inertial range is formed with a scaling ${f}^{\ensuremath{-}2.3}$ down to ${f}_{{\ensuremath{\rho}}_{e}}$. Above ${f}_{{\ensuremath{\rho}}_{e}}$, the spectrum has a steeper power law $\ensuremath{\sim}{f}^{\ensuremath{-}4.1}$ down to the noise level of the instrument. We interpret this as the dissipation range and show a remarkable agreement with theoretical predictions of a quasi-two-dimensional cascade into Kinetic Alfv\'en Waves (KAW).
580 citations
TL;DR: It is shown that the electron Larmor radius plays the role of a dissipation scale in space plasma turbulence and the spectra form a quasiuniversal spectrum following the Kolmogorov's law at MHD scales.
Abstract: To investigate the universality of magnetic turbulence in space plasmas, we analyze seven time periods in the free solar wind under different plasma conditions. Three instruments on Cluster spacecraft operating in different frequency ranges give us the possibility to resolve spectra up to 300 Hz. We show that the spectra form a quasiuniversal spectrum following the Kolmogorov's law $\ensuremath{\sim}{k}^{\ensuremath{-}5/3}$ at MHD scales, a $\ensuremath{\sim}{k}^{\ensuremath{-}2.8}$ power law at ion scales, and an exponential $\ensuremath{\sim}\mathrm{exp} [\ensuremath{-}\sqrt{k{\ensuremath{\rho}}_{e}}]$ at scales $k{\ensuremath{\rho}}_{e}\ensuremath{\sim}[0.1,1]$, where ${\ensuremath{\rho}}_{e}$ is the electron gyroradius. This is the first observation of an exponential magnetic spectrum in space plasmas that may indicate the onset of dissipation. We distinguish for the first time between the role of different spatial kinetic plasma scales and show that the electron Larmor radius plays the role of a dissipation scale in space plasma turbulence.
437 citations
TL;DR: In this paper, a three-dimensional magnetohydrodynamic (MHD) model for the propagation and dissipation of Alfv?n waves in a coronal loop is developed, which includes the lower atmospheres at the two ends of the loop.
Abstract: A three-dimensional magnetohydrodynamic (MHD) model for the propagation and dissipation of Alfv?n waves in a coronal loop is developed The model includes the lower atmospheres at the two ends of the loop The waves originate on small spatial scales (less than 100?km) inside the kilogauss flux elements in the photosphere The model describes the nonlinear interactions between Alfv?n waves using the reduced MHD approximation The increase of Alfv?n speed with height in the chromosphere and transition region (TR) causes strong wave reflection, which leads to counter-propagating waves and turbulence in the photospheric and chromospheric parts of the flux tube Part of the wave energy is transmitted through the TR and produces turbulence in the corona We find that the hot coronal loops typically found in active regions can be explained in terms of Alfv?n wave turbulence, provided that the small-scale footpoint motions have velocities of 1-2?km?s?1 and timescales of 60-200?s The heating rate per unit volume in the chromosphere is two to three orders of magnitude larger than that in the corona We construct a series of models with different values of the model parameters, and find that the coronal heating rate increases with coronal field strength and decreases with loop length We conclude that coronal loops and the underlying chromosphere may both be heated by Alfv?nic turbulence
424 citations
University of California, Berkeley1, Queen Mary University of London2, Smithsonian Astrophysical Observatory3, University of Minnesota4, University of New Hampshire5, University of Maryland, College Park6, University of Orléans7, Imperial College London8, University of Colorado Boulder9, Goddard Space Flight Center10, University of Maryland, Baltimore County11, University of Iowa12, University of Michigan13, University of Applied Sciences and Arts Northwestern Switzerland FHNW14, University of Paris15, Princeton University16, Johns Hopkins University Applied Physics Laboratory17, University of California, Los Angeles18
TL;DR: Measurements from the Parker Solar Probe show that slow solar wind near the Sun’s equator originates in coronal holes, and plasma-wave measurements suggest the existence of electron and ion velocity-space micro-instabilities that are associated with plasma heating and thermalization processes.
Abstract: During the solar minimum, when the Sun is at its least active, the solar wind1,2 is observed at high latitudes as a predominantly fast (more than 500 kilometres per second), highly Alfvenic rarefied stream of plasma originating from deep within coronal holes. Closer to the ecliptic plane, the solar wind is interspersed with a more variable slow wind3 of less than 500 kilometres per second. The precise origins of the slow wind streams are less certain4; theories and observations suggest that they may originate at the tips of helmet streamers5,6, from interchange reconnection near coronal hole boundaries7,8, or within coronal holes with highly diverging magnetic fields9,10. The heating mechanism required to drive the solar wind is also unresolved, although candidate mechanisms include Alfven-wave turbulence11,12, heating by reconnection in nanoflares13, ion cyclotron wave heating14 and acceleration by thermal gradients1. At a distance of one astronomical unit, the wind is mixed and evolved, and therefore much of the diagnostic structure of these sources and processes has been lost. Here we present observations from the Parker Solar Probe15 at 36 to 54 solar radii that show evidence of slow Alfvenic solar wind emerging from a small equatorial coronal hole. The measured magnetic field exhibits patches of large, intermittent reversals that are associated with jets of plasma and enhanced Poynting flux and that are interspersed in a smoother and less turbulent flow with a near-radial magnetic field. Furthermore, plasma-wave measurements suggest the existence of electron and ion velocity-space micro-instabilities10,16 that are associated with plasma heating and thermalization processes. Our measurements suggest that there is an impulsive mechanism associated with solar-wind energization and that micro-instabilities play a part in heating, and we provide evidence that low-latitude coronal holes are a key source of the slow solar wind. Measurements from the Parker Solar Probe show that slow solar wind near the Sun’s equator originates in coronal holes.
391 citations
TL;DR: In this article, the authors show that turbulent cascade leads to generation of coherent structures in the form of current sheets that steepen to electron scales, triggering strong localized heating of the plasma.
Abstract: An unsolved problem in plasma turbulence is how energy is dissipated at small scales. Particle collisions are too infrequent in hot plasmas to provide the necessary dissipation. Simulations either treat the fluid scales and impose an ad hoc form of dissipation (e.g., resistivity) or consider dissipation arising from resonant damping of small amplitude disturbances where damping rates are found to be comparable to that predicted from linear theory. Here, we report kinetic simulations that span the macroscopic fluid scales down to the motion of electrons. We find that turbulent cascade leads to generation of coherent structures in the form of current sheets that steepen to electron scales, triggering strong localized heating of the plasma. The dominant heating mechanism is due to parallel electric fields associated with the current sheets, leading to anisotropic electron and ion distributions which can be measured with NASA's upcoming Magnetospheric Multiscale mission. The motion of coherent structures also generates waves that are emitted into the ambient plasma in form of highly oblique compressional and shear Alfven modes. In 3D, modes propagating at other angles can also be generated. This indicates that intermittent plasma turbulence will in general consist of both coherent structures and waves. However, the current sheet heating is found to be locally several orders of magnitude more efficient than wave damping and is sufficient to explain the observed heating rates in the solar wind.
379 citations
References
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TL;DR: In this paper, a summary of what is now known of disk turbulence and some knotty outstanding questions (e.g., what is the physics behind nonlinear field saturation?) for which we may soon begin to develop answers.
Abstract: Recent years have witnessed dramatic progress in our understanding of how turbulence arises and transports angular momentum in astrophysical accretion disks. The key conceptual point has its origins in work dating from the 1950s, but its implications have been fully understood only in the last several years: the combination of a subthermal magnetic field (any nonpathological configuration will do) and outwardly decreasing differential rotation rapidly generates magnetohydrodynamic (MHD) turbulence via a remarkably simple linear instability. The result is a greatly enhanced effective viscosity, the origin of which had been a long-standing problem. The MHD nature of disk turbulence has linked two broad domains of magnetized fluid research: accretion theory and dynamos. The understanding that weak magnetic fields are not merely passively acted upon by turbulence, but actively generate it, means that the assumptions of classical dynamo theory break down in disks. Paralleling the new conceptual understanding has been the development of powerful numerical MHD codes. These have taught us that disks truly are turbulent, transporting angular momentum at greatly enhanced rates. We have also learned, however, that not all forms of disk turbulence do this. Purely hydrodynamic turbulence, when it is imposed, simply causes fluctuations without a significant increase in transport. The interplay between numerical simulation and analytic arguments has been particularly fruitful in accretion disk theory and is a major focus of this article. The authors conclude with a summary of what is now known of disk turbulence and mention some knotty outstanding questions (e.g., what is the physics behind nonlinear field saturation?) for which we may soon begin to develop answers.
2,465 citations
TL;DR: In this article, a detailed derivation of the inertial range spectrum for the weak turbulence of shear Alfven waves is presented, where the authors restrict attention to the symmetric case where the oppositely directed waves carry equal energy fluxes and show that as energy cascades to high perpendicular wavenumbers, interactions become so strong that the assumption of weakness is no longer valid.
Abstract: We continue to investigate the possibility that interstellar turbulence is caused by nonlinear interactions
among shear Alfven waves. Here, as in Paper I, we restrict attention to the symmetric case where the oppositely directed waves carry equal energy fluxes. This precludes application to the solar wind in which the outward flux significantly exceeds the ingoing one. All our detailed calculations are carried out for an incompressible
magnetized fluid. In incompressible magnetohydrodynamics (MHD), nonlinear interactions only occur between oppositely direct waves. Paper I contains a detailed derivation of the inertial range spectrum for the weak turbulence of shear Alfven waves. As energy cascades to high perpendicular wavenumbers, interactions become so strong that the assumption of weakness is no longer valid. Here, we present a theory for the strong turbulence of shear Alfven waves. It has the following main characteristics. (1) The inertial-range energy spectrum exhibits a critical balance beween linear wave periods and nonlinear turnover timescales. (2) The "eddies" are elongated in the direction of the field on small spatial scales; the parallel and perpendicular components of the wave vector, k_z and k_⊥, are related by k_z ≈ k^(2/3) _⊥L^(-1/3), where L is the outer scale of the turbulence. (3) The "one-dimensional" energy spectrum is proportional to k^(-5/3) _⊥-an anisotropic Kolmogorov energy spectrum. Shear Alfvenic turbulence mixes specific entropy as a passive contaminant. This gives rise to an electron density power spectrum whose form mimics the energy spectrum of the turbulence. Radio, wave scattering by these electron density fluctuations produces anisotropic scatter-broadened images. Damping by ion-neutral collisions restricts Alfvenic turbulence to highly ionized regions of the interstellar medium. We expect negligible generation of compressive MHD waves by shear Alfven waves belonging to the critically balanced cascade. Viscous and collisionless damping are also unimportant in the interstellar medium (ISM). Our calculations support the general picture of interstellar turbulence advanced by Higdon.
2,248 citations
2,131 citations