Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence

We present the results of our detailed pseudospectral direct numerical simulation (DNS) studies, with up to 1024(3) collocation points, of incompressible, magnetohydrodynamic (MHD) turbulence in three dimensions, without a mean magnetic field. Our study concentrates on the dependence of various statistical properties of both decaying and statistically steady MHD turbulence on the magnetic Prandtl number Pr-M over a large range, namely 0.01 <= Pr-M <= 10. We obtain data for a wide variety of statistical measures, such as probability distribution functions (PDFs) of the moduli of the vorticity and current density, the energy dissipation rates, and velocity and magnetic-field increments, energy and other spectra, velocity and magnetic-field structure functions, which we use to characterize intermittency, isosurfaces of quantities, such as the moduli of the vorticity and current density, and joint PDFs, such as those of fluid and magnetic dissipation rates. Our systematic study uncovers interesting results that have not been noted hitherto. In particular, we find a crossover from a larger intermittency in the magnetic field than in the velocity field, at large Pr-M, to a smaller intermittency in the magnetic field than in the velocity field, at low Pr-M. Furthermore, a comparison of our results for decaying MHD turbulence and its forced, statistically steady analogue suggests that we have strong universality in the sense that, for a fixed value of Pr-M, multiscaling exponent ratios agree, at least within our error bars, for both decaying and statistically steady homogeneous, isotropic MHD turbulence.

Magnetohydrodynamic turbulence

We simulate incompressible, MHD turbulence using a pseudo-spectral code. Our major conclusions are as follows.

1) MHD turbulence is most conveniently described in terms of counter propagating shear Alfvén and slow waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves.

2) MHD turbulence is anisotropic with energy cascading more rapidly along k_⊥ than along k_∥, where k_⊥ and k_∥ refer to wavevector components perpendicular and parallel to the local magnetic field. Anisotropy increases with increasing k_⊥ such that excited modes are confined inside a cone bounded by k_∥ ∝ k^γ_⊥ where γ less than 1. The opening angle of the cone, θ(k_⊥) ∝ k^-(1-γ)_⊥, defines the scale dependent anisotropy.

3) MHD turbulence is generically strong in the sense that the waves which comprise it suffer order unity distortions on timescales comparable to their periods. Nevertheless, turbulent fluctuations are small deep inside the inertial range. Their energy density is less than that of the background field by a factor θ² (k_⊥)≪1.

4) MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counter propagating waves. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfvén waves are responsible for the mapping's shear and slow waves for its dilatation. The amplitude of the former exceeds that of the latter by 1/θ(k_⊥) which accounts for dominance of the shear Alfvén waves in controlling the cascade dynamics.

5) Passive scalars mixed by MHD turbulence adopt the same power spectrum as the velocity and magnetic field perturbations.

6) Decaying MHD turbulence is unstable to an increase of the imbalance between the flux of waves propagating in opposite directions along the magnetic field. Forced MHD turbulence displays order unity fluctuations with respect to the balanced state if excited at low k by δ(t) correlated forcing. It appears to be statistically stable to the unlimited growth of imbalance.

7) Gradients of the dynamic variables are focused into sheets aligned with the magnetic field whose thickness is comparable to the dissipation scale. Sheets formed by oppositely directed waves are uncorrelated. We suspect that these are vortex sheets which the mean magnetic field prevents from rolling up.

8) Items (1)-(5) lend support to the model of strong MHD turbulence put forth by Goldreich and Sridhar (1995, 1997). Results from our simulations are also consistent with the GS prediction γ = 2/3. The sole not able discrepancy is that the 1D power law spectra, E(k_⊥) ∝ k^-∝_⊥, determined from our simulations exhibit ∝ ≈ 3/2, whereas the GS model predicts ∝ = 5/3.

The decay of Saffman and batchelor turbulence subject to rotation, stratification or an imposed magnetic field

We consider unforced, statistically-axisymmetric turbulence evolving in the presence of a background rotation, an imposed stratification, or a uniform magnetic field. We focus on two canonical cases: Saffman turbulence, in which E(κ → 0) ∼ κ 2, and Batchelor turbulence, in which E(κ → 0) ∼ κ 4. It has recently been shown that, provided the large scales evolve in a self-similar manner, then u ⊥ 2ℓ ⊥ 2ℓ // = constant in Saffman turbulence and u ⊥ 2ℓ ⊥ 4ℓ // = constant in Batchelor turbulence (Davidson, 2009, 2010). Here the subscripts ⊥ and // indicate directions perpendicular and parallel to the axis of symmetry, and ℓ ⊥, ℓ //, and u ⊥ are suitably defined integral scales. These constraints on the integral scales allow us to make simple, testable predictions for the temporal evolution of ℓ ⊥, ℓ //, and u ⊥ in rotating, stratified and MHD turbulence.

VELOCITY FIELD OF COMPRESSIBLE MAGNETOHYDRODYNAMIC TURBULENCE: WAVELET DECOMPOSITION AND MODE SCALINGS

We study compressible magnetohydrodynamic turbulence, which holds the key to many astrophysical processes, including star formation and cosmic-ray propagation. To account for the variations of the magnetic field in the strongly turbulent fluid, we use wavelet decomposition of the turbulent velocity field into Alfven, slow, and fast modes, which presents an extension of the Cho & Lazarian decomposition approach based on Fourier transforms. The wavelets allow us to follow the variations of the local direction of the magnetic field and therefore improve the quality of the decomposition compared to the Fourier transforms, which are done in the mean field reference frame. For each resulting component, we calculate the spectra and two-point statistics such as longitudinal and transverse structure functions as well as higher order intermittency statistics. In addition, we perform a Helmholtz-Hodge decomposition of the velocity field into incompressible and compressible parts and analyze these components. We find that the turbulence intermittency is different for different components, and we show that the intermittency statistics depend on whether the phenomenon was studied in the global reference frame related to the mean magnetic field or in the frame defined by the local magnetic field. The dependencies of the measures we obtained are different for different components of the velocity; for instance, we show that while the Alfven mode intermittency changes marginally with the Mach number, the intermittency of the fast mode is substantially affected by the change.

Particle Acceleration in Turbulence and Weakly Stochastic Reconnection

In this Letter we analyze the energy distribution evolution of test particles injected in three dimensional (3D) magnetohydrodynamic (MHD) simulations of different magnetic reconnection configurations. When considering a single Sweet-Parker topology, the particles accelerate predominantly through a first-order Fermi process, as predicted in [3] and demonstrated numerically in [8]. When turbulence is included within the current sheet, the acceleration rate is highly enhanced, because reconnection becomes fast and independent of resistivity [4,11] and allows the formation of a thick volume filled with multiple simultaneously reconnecting magnetic fluxes. Charged particles trapped within this volume suffer several head-on scatterings with the contracting magnetic fluctuations, which significantly increase the acceleration rate and results in a first-order Fermi process. For comparison, we also tested acceleration in MHD turbulence, where particles suffer collisions with approaching and receding magnetic irregularities, resulting in a reduced acceleration rate. We argue that the dominant acceleration mechanism approaches a second order Fermi process in this case.

Dynamo onset as a first-order transition: Lessons from a shell model for magnetohydrodynamics

We carry out systematic and high-resolution studies of dynamo action in a shell model for magnetohydro-dynamic (MHD) turbulence over wide ranges of the magnetic Prandtl number Pr-M and the magnetic Reynolds number Re-M. Our study suggests that it is natural to think of dynamo onset as a nonequilibrium first-order phase transition between two different turbulent, but statistically steady, states. The ratio of the magnetic and kinetic energies is a convenient order parameter for this transition. By using this order parameter, we obtain the stability diagram (or nonequilibrium phase diagram) for dynamo formation in our MHD shell model in the (Pr-M(-1), Re-M) plane. The dynamo boundary, which separates dynamo and no-dynamo regions, appears to have a fractal character. We obtain a hysteretic behavior of the order parameter across this boundary and suggestions of nucleation-type phenomena.

Intermittent and Scale-Invariant Intensity Fluctuations in Hot Coronal Loops

ABSTRACT BODY: To resolve outstanding questions on heating of coronal loops, we study intensity fluctuations in inter-moss portions of active region core loops as observed with AIA/SDO. The 94Å fluctuations (Figure 1) have structure on timescales shorter than radiative and conductive cooling times. Each of several strong 94Å brightenings is followed after ~8 min by a broader peak in the cooler 335Å emission. This indicates that we see emission from the hot component of the 94Å contribution function. No hotter contributions appear, and we conclude that the 94Å intensity can be used as a proxy for energy injection into the loop plasma. The probability density function of the observed 94Å intensity has 'heavy tails' that approach zero more slowly than the tails of a normal distribution. Hence, large fluctuations dominate the behavior of the system. The resulting 'intermittence' is associated with power-law or exponential scaling of the related variables, and these in turn are associated with turbulent phenomena. The intensity plots in Figure 1 resemble multifractal time series, which are common to various forms of turbulent energy dissipation. In these systems a single fractal dimension is insufficient to describe the dynamics and instead there is a spectrum of fractal dimensions that quantify the self-similar properties. Figure 2 shows the multifractal spectrum from our data to be invariant over timescales from 24 s to 6.4 min. We compare these results to outputs from theoretical energy dissipation models based on MHD turbulence, and in some cases we find substantial agreement, in terms of intermittence, multifractality and scale invariance. Figure 1. Time traces of 94A intensity in the inter-moss portions of four AR core loops. Figure 2. Multifractal spectra showing timescale invariance. The four cases of Figure 1 are included.

The generation and amplification of intergalactic magnetic fields in analogue laboratory experiments with high power lasers

The advent of high-power laser facilities has, in the past two decades, opened a new field of research where astrophysical environments can be scaled down to laboratory dimensions, while preserving the essential physics. This is due to the invariance of the equations of magneto-hydrodynamics to a class of similarity transformations. Here we review the relevant scaling relations and their application in laboratory astrophysics experiments with a focus on the generation and amplification of magnetic fields in cosmic environment. The standard model for the origin of magnetic fields is a multi stage process whereby a vanishing magnetic seed is first generated by a rotational electric field and is then amplified by turbulent dynamo action to the characteristic values observed in astronomical bodies. We thus discuss the relevant seed generation mechanisms in cosmic environment including resistive mechanism, collision-less and fluid instabilities, as well as novel laboratory experiments using high power laser systems aimed at investigating the amplification of magnetic energy by magneto-hydrodynamic (MHD) turbulence. Future directions, including efforts to model in the laboratory the process of diffusive shock acceleration are also discussed, with an emphasis on the potential of laboratory experiments to further our understanding of plasma physics on cosmic scales.

DIFFUSION OF MAGNETIC FIELD AND REMOVAL OF MAGNETIC FLUX FROM CLOUDS VIA TURBULENT RECONNECTION

The diffusion of astrophysical magnetic fields in conducting fluids in the presence of turbulence depends on whether magnetic fields can change their topology via reconnection in highly conducting media. Recent progress in understanding fast magnetic reconnection in the presence of turbulence reassures that the magnetic field behavior in computer simulations and turbulent astrophysical environments is similar, as far as magnetic reconnection is concerned. This makes it meaningful to perform MHD simulations of turbulent flows in order to understand the diffusion of magnetic field in astrophysical environments. Our studies of magnetic field diffusion in turbulent medium reveal interesting new phenomena. First of all, our three-dimensional MHD simulations initiated with anti-correlating magnetic field and gaseous density exhibit at later times a de-correlation of the magnetic field and density, which corresponds well to the observations of the interstellar media. While earlier studies stressed the role of either ambipolar diffusion or time-dependent turbulent fluctuations for de-correlating magnetic field and density, we get the effect of permanent de-correlation with one fluid code, i.e., without invoking ambipolar diffusion. In addition, in the presence of gravity and turbulence, our three-dimensional simulations show the decrease of the magnetic flux-to-mass ratio as the gaseous density at the center of the gravitational potential increases. We observe this effect both in the situations when we start with equilibrium distributions of gas and magnetic field and when we follow the evolution of collapsing dynamically unstable configurations. Thus, the process of turbulent magnetic field removal should be applicable both to quasi-static subcritical molecular clouds and cores and violently collapsing supercritical entities. The increase of the gravitational potential as well as the magnetization of the gas increases the segregation of the mass and magnetic flux in the saturated final state of the simulations, supporting the notion that the reconnection-enabled diffusivity relaxes the magnetic field + gas system in the gravitational field to its minimal energy state. This effect is expected to play an important role in star formation, from its initial stages of concentrating interstellar gas to the final stages of the accretion to the forming protostar. In addition, we benchmark our codes by studying the heat transfer in magnetized compressible fluids and confirm the high rates of turbulent advection of heat obtained in an earlier study.

Reconnection current sheet structure in a turbulent medium

In the presence of turbulence, magnetic field lines lose their dynamical identity and particles entrained on field lines diffuse through space at a rate determined by the amplitude of the turbulence. In previous work (Lazarian and Vishniac, 1999; Kowal et al., 2009; Eyink et al., 2011) we showed that this leads to reconnection speeds which are independent of resistivity. In particular, in Kowal et al. (2009) we showed that numerical simulations were consistent with the predictions of this model. Here we examine the structure of the current sheet in simulations of turbulent reconnection. Laminar flows consistent with the Sweet-Parker reconnection model produce very thin and well ordered currents sheets. On the other hand, the simulations of Kowal et al. (2009) show a strongly disordered state even for relatively low levels of turbulence. Comparing data cubes with and without reconnection, we find that large scale field reversals are the cumulative effect of many individual eddies, each of which has magnetic properties which are not very different from turbulent eddies in a homogeneous background. This implies that the properties of stationary and homogeneous MHD turbulence are a reasonable guide to understanding turbulence during large scale magnetic reconnection events. In addition, dissipation and high energy particle acceleration during reconnection events take place over a macroscopic volume, rather than being confined to a narrow zone whose properties depend on microscopic transport coefficients.

Magnetic flux transport by turbulent reconnection in astrophysical flows

The role of magnetohydrodynamics (MHD) turbulence in astrophysical environments is still highly debated. An important question that permeates this debate is the transport of magnetic flux. This is particularly important, for instance, in the context of star formation. When clouds collapse gravitationally to form stars, there must be some magnetic flux transport. Otherwise, the newborn stars would have magnetic fields several orders of magnitude larger than the observed ones. Also, the magnetic flux that is dragged in the late stages of the formation of a star can remove all the rotational support from the accretion disc that grows around the protostar. The efficiency of the mechanism that is often invoked to allow transport of magnetic fields at different stages of star formation, namely ambipolar diffusion, has recently been put in check. We discuss here an alternative mechanism for magnetic flux transport which is based on turbulent fast magnetic reconnection. We review recent results from three-dimensional MHD numerical simulations that indicate that this mechanism is very efficient in decoupling and transporting magnetic flux from the inner denser regions to the outskirts of collapsing clouds at different stages of star formation. We discuss this mechanism also in the context of dynamo processes and speculate that it can play a role both in solar dynamo and in accretion disc dynamo processes.