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Monthly Notices of the Royal Astronomical Society

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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Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades in Magnetized Weakly Collisional Plasmas

Context. Density and velocity fluctuations on virtually all scales observed with modern telescopes show that molecular clouds (MCs) are turbulent. The forcing and structural characteristics of this turbulence are, however, still poorly understood.Aims. To shed light on this subject, we study two limiting cases of turbulence forcing in numerical experiments: solenoidal (divergence-free) forcing and compressive (curl-free) forcing, and compare our results to observations.Methods. We solve the equations of hydrodynamics on grids with up to 10243  cells for purely solenoidal and purely compressive forcing. Eleven lower-resolution models with different forcing mixtures are also analysed.Results. Using Fourier spectra and Δ -variance, we find velocity dispersion-size relations consistent with observations and independent numerical simulations, irrespective of the type of forcing. However, compressive forcing yields stronger compression at the same rms Mach number than solenoidal forcing, resulting in a three times larger standard deviation of volumetric and column density probability distributions (PDFs). We compare our results to different characterisations of several observed regions, and find evidence of different forcing functions. Column density PDFs in the Perseus MC suggest the presence of a mainly compressive forcing agent within a shell, driven by a massive star. Although the PDFs are close to log-normal, they have non-Gaussian skewness and kurtosis caused by intermittency. Centroid velocity increments measured in the Polaris Flare on intermediate scales agree with solenoidal forcing on that scale. However, Δ -variance analysis of the column density in the Polaris Flare suggests that turbulence is driven on large scales, with a significant compressive component on the forcing scale. This indicates that, although likely driven with mostly compressive modes on large scales, turbulence can behave like solenoidal turbulence on smaller scales. Principal component analysis of G216-2.5 and most of the Rosette MC agree with solenoidal forcing, but the interior of an ionised shell within the Rosette MC displays clear signatures of compressive forcing.Conclusions. The strong dependence of the density PDF on the type of forcing must be taken into account in any theory using the PDF to predict properties of star formation. We supply a quantitative description of this dependence. We find that different observed regions show evidence of different mixtures of compressive and solenoidal forcing, with more compressive forcing occurring primarily in swept-up shells. Finally, we emphasise the role of the sonic scale for protostellar core formation, because core formation close to the sonic scale would naturally explain the observed subsonic velocity dispersions of protostellar cores.

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Comparing the statistics of interstellar turbulence in simulations and observations - Solenoidal versus compressive turbulence forcing

The role of turbulence and magnetic fields is studied for star formation in molecular clouds. We derive and compare six theoretical models for the star formation rate (SFR)—the Krumholz & McKee (KM), Padoan & Nordlund (PN), and Hennebelle & Chabrier (HC) models, and three multi-freefall versions of these, suggested by HC—all based on integrals over the log-normal distribution of turbulent gas. We extend all theories to include magnetic fields and show that the SFR depends on four basic parameters: (1) virial parameter αvir; (2) sonic Mach number ; (3) turbulent forcing parameter b, which is a measure for the fraction of energy driven in compressive modes; and (4) plasma with the Alfven Mach number . We compare all six theories with MHD simulations, covering cloud masses of 300 to 4 × 106 M ☉ and Mach numbers -50 and -∞, with solenoidal (b = 1/3), mixed (b = 0.4), and compressive turbulent (b = 1) forcings. We find that the SFR increases by a factor of four between and 50 for compressive turbulent forcing and αvir ~ 1. Comparing forcing parameters, we see that the SFR is more than 10 times higher with compressive than solenoidal forcing for simulations. The SFR and fragmentation are both reduced by a factor of two in strongly magnetized, trans-Alfvenic turbulence compared to hydrodynamic turbulence. All simulations are fit simultaneously by the multi-freefall KM and multi-freefall PN theories within a factor of two over two orders of magnitude in SFR. The simulated SFRs cover the range and correlation of SFR column density with gas column density observed in Galactic clouds, and agree well for star formation efficiencies SFE = 1%-10% and local efficiencies = 0.3-0.7 due to feedback. We conclude that the SFR is primarily controlled by interstellar turbulence, with a secondary effect coming from magnetic fields.

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The Star Formation Rate of Turbulent Magnetized Clouds: Comparing Theory, Simulations, and Observations

We derive an analytical theory of the prestellar core initial mass function (IMF) based on an extension of the Press-Schechter statistical formalism. Our approach relies on the general concept of the gravothermal and gravoturbulent collapse of a molecular cloud, with a selection criterion based on the thermal or turbulent Jeans mass, which yields the derivation of the mass spectrum of self-gravitating objects in a quiescent or a turbulent environment. The same formalism also yields the mass spectrum of non-self-gravitating clumps produced in supersonic flows. The mass spectrum of the self-gravitating cores reproduces well the observed IMF. The theory predicts that the shape of the IMF results from two competing contributions, namely, a power law at large scales and an exponential cutoff (lognormal form) centered around the characteristic mass for gravitational collapse. The cutoff exists both in the case of thermal or turbulent collapse, provided that the underlying density field has a lognormal distribution. Whereas pure thermal collapse produces a power-law tail steeper than the Salpeter value, -->dN/dlog M M−x with -->x 1.35, the latter is recovered exactly for the (three-dimensional) value of the spectral index of the velocity power spectrum, -->n 3.8, found in observations and in numerical simulations of isothermal supersonic turbulence. Indeed, the theory predicts that -->x = (n + 1)/(2n − 4) for self-gravitating structures and -->x = 2 − n'/3 for non-self-gravitating structures, where -->n' is the power spectrum index of -->log ρ . We show that, whereas supersonic turbulence promotes the formation of both massive stars and brown dwarfs, it has an overall negative impact on star formation, decreasing the star formation efficiency. This theory provides a novel theoretical foundation to understand the origin of the IMF and provides useful guidance to numerical simulations exploring star formation, while making testable predictions.

http://iopscience.iop.org/article/10.1086/589916/pdf

Analytical Theory for the Initial Mass Function: CO Clumps and Prestellar Cores

We perform three-dimensional (3D) compressible MHD simulations over many dynamical times for an extended range of sonic and Alfven Mach numbers and analyze the statistics of 3D density and 2D column density, which include probability distribution functions, spectra, skewness, kurtosis, She-Leveque exponents, and genus. In order to establish the relation between the statistics of the observables, i.e., column densities, and the underlying 3D statistics of density, we analyze the effects of cloud boundaries. We define the parameter space for 3D measures to be recovered from column densities. In addition, we show that for subsonic turbulence the spectra of density fluctuations are consistent with k-7/3 in the case of a strong magnetic field and k-5/3 in the case of a weak magnetic field. For supersonic turbulence we confirm the earlier findings of the shallow spectra of density and Kolmogorov spectra of the logarithm of density. We find that the intermittencies of the density and velocity are very different.

http://iopscience.iop.org/article/10.1086/511515/pdf

Density Fluctuations in MHD Turbulence: Spectra, Intermittency and Topology

We study the statistics of density for supersonic turbulence in a medium with magnetic pressure larger than the gaseous pressure. Our simulations exhibit clumpy density structures, with the contrast increasing with the Mach number. At Mach 10, the densities of some clumps are 3 orders of magnitude higher than the mean density. These clumps give rise to a flat and approximately isotropic density spectrum corresponding to the random distribution of clumps in space. We claim that the clumps originate from our random, isotropic turbulence driving. When the contribution from those clumps is suppressed by studying the logarithm of density, the density statistics exhibit scale-dependent anisotropy consistent with the models in which density structures arise from shearing by Alfven waves. It is noteworthy that originally such models were advocated for the case of low-Mach, nearly incompressible turbulence.

Density Scaling and Anisotropy in Supersonic Magnetohydrodynamic Turbulence

We present numerical simulations of driven magnetohydrodynamic (MHD) turbulence with weak/moderate imposed magnetic fields. The main goal is to clarify dynamics of magnetic field growth. We also investigate the effects of the imposed magnetic fields on the MHD turbulence, including, as a limit, the case of zero external field. Our findings are as follows. First, when we start off simulations with weak mean magnetic field only (or with small scale random field with zero imposed field), we observe that there is a stage at which magnetic energy density grows linearly with time. Runs with different numerical resolutions and/or different simulation parameters show consistent results for the growth rate at the linear stage. Second, we find that, when the strength of the external field increases, the equilibrium kinetic energy density drops by roughly the product of the rms velocity and the strength of the external field. The equilibrium magnetic energy density rises by roughly the same amount. Third, when the external magnetic field is not very strong (say, less than ~0.2 times the rms velocity when measured in the units of Alfven speed), the turbulence at large scales remains statistically isotropic, i.e., there is no apparent global anisotropy of order B 0/v. We discuss implications of our results on astrophysical fluids.

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Growth of magnetic fields induced by turbulent motions

We study the statistics of density for supersonic turbulence in a medium with magnetic pressure larger than the gaseous pressure. This study is motivated by molecular cloud research. Our simulations exhibit clumpy density structures, which contrast increases with the Mach number. At 10 Machs densities of some clumps are three orders of magnitude higher than the mean density. These clumps give rise to flat and approximately isotropic density spectrum corresponding to the random distribution of clumps in space. We claim that the clumps originate from our random, isotropic turbulence driving. When the contribution from those clumps is suppressed by studying logarithm of density, the density statistics exhibit scale-dependent anisotropy consistent with the models where density structures arise from shearing by Alfv\'en waves. It is noteworthy that originally such models were advocated for the case of low-Mach, nearly incompressible turbulence.

https://arxiv.org/pdf/astro-ph/0502547.pdf

Density scaling and anisotropy in supersonic MHD turbulence

We consider stationary, forced, imbalanced, or cross helical MHD Alfvenic turbulence where the waves traveling in one direction have higher amplitudes than the opposite waves. This paper is dedicated to so-called strong turbulence, which cannot be treated perturbatively. Our main result is that the anisotropy of the weak waves is stronger than the anisotropy of strong waves. We propose that critical balance, which was originally conceived as a causality argument, has to be amended by what we call a propagation argument. This revised formulation of critical balance is able to handle the imbalanced case and reduces to the old formulation in the balanced case. We also provide a phenomenological model of energy cascading and discuss the possibility of self-similar solutions in a realistic setup of driven turbulence.

Andrey Beresnyak

Papers

Density Fluctuations in MHD Turbulence: Spectra, Intermittency and Topology

Density Scaling and Anisotropy in Supersonic Magnetohydrodynamic Turbulence

Growth of magnetic fields induced by turbulent motions

Density scaling and anisotropy in supersonic MHD turbulence

Strong Imbalanced Turbulence