Jason L. Maron

Simulations of Incompressible Magnetohydrodynamic Turbulence

We simulate incompressible MHD turbulence using a pseudospectral code. Our major conclusions are: (1) MHD turbulence is most conveniently described in terms of counterpropagating shear Alfven and slow waves. Shear Alfven waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfven waves. Cascades composed entirely of shear Alfven 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, respectively. Anisotropy increases with increasing k_⊥ such that excited modes are confined inside a cone bounded by k_∥ ∝ k^y_⊥, where γ 1. (4) MHD turbulence is generically strong in the sense that the waves that 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 of Θ^((α-1)/(1-γ)) « 1. (5) MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counterpropagating waves. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfven waves are responsible for the mappings 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 Alfven waves in controlling the cascade dynamics. (6) Passive scalars mixed by MHD turbulence adopt the same power spectrum as the velocity and magnetic field perturbations. (7) Decaying MHD turbulence is unstable to an increase of the imbalance between the fluxes 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. (8) 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. (9) Items 1-6 lend support to the model of strong MHD turbulence put forth by Goldreich & Sridhar (GS). Results from our simulations are also consistent with the GS prediction γ = 2/3, as are those obtained previously by Cho & Vishniac. The sole notable discrepancy is that one-dimensional energy spectra determined from our simulations exhibit α ≈ 3/2, whereas the GS model predicts α = 5/3. Further investigation is needed to resolve this issue.

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

The Onset of a Small-Scale Turbulent Dynamo at Low Magnetic Prandtl Numbers

We study numerically the dependence of the critical magnetic Reynolds number Rmc for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number Pm = Rm/Re < 1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rmc → const as Re → ∞ (a small-scale dynamo exists at low Pm) or Rmc/Re = Pmc → const as Re → ∞ (no small-scale dynamo exists at low Pm). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, Rmc grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rmc ~ 500 for Re ~ 3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that Rmc is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve Rmc(Re).

A model of nonlinear evolution and saturation of the turbulent MHD dynamo

The growth and saturation of magnetic field in conducting turbulent media with large magnetic Prandtl numbers are investigated. This regime is very common in low-density hot astrophysical plasmas. During the early (kinematic) stage, weak magnetic fluctuations grow exponentially and concentrate at the resistive scale, which lies far below the hydrodynamic viscous scale. The evolution becomes nonlinear when the magnetic energy is comparable to the kinetic energy of the viscous-scale eddies. A physical picture of the ensuing nonlinear evolution of the MHD dynamo is proposed. Phenomenological considerations are supplemented with a simple Fokker-Planck model of the nonlinear evolution of the magnetic-energy spectrum. It is found that, while the shift of the bulk of the magnetic energy from the subviscous scales to the velocity scales may be possible, it occurs very slowly - at the resistive, rather than dynamical, timescale (for galaxies, this means that the generation of large-scale magnetic fields cannot be explained by this mechanism). The role of Alfvenic motions and the implications for the fully developed isotropic MHD turbulence are discussed.

Critical magnetic Prandtl number for small-scale dynamo.

We report a series of numerical simulations showing that the critical magnetic Reynolds number Rm_c for the nonhelical small-scale dynamo depends on the Reynolds number Re. Namely, the dynamo is shut down if the magnetic Prandtl number Pr=Rm/Re is less than some critical value Pr_c<1 even for Rm for which dynamo exists at Pr>=1. We argue that, in the limit of Re->infinity, a finite Pr_c may exist. The second possibility is that Pr_c->0 as Re->infinity, while Rm_c tends to a very large constant value inaccessible at current resolutions. If there is a finite Pr_c, the dynamo is sustainable only if magnetic fields can exist at scales smaller than the flow scale, i.e., it is always effectively a large-Pr dynamo. If there is a finite Rm_c, our results provide a lower bound: Rm_c<220 for Pr<=1/8. This is larger than Rm in many planets and in all liquid-metal experiments.

The Small-Scale Structure of Magnetohydrodynamic Turbulence with Large Magnetic Prandtl Numbers

We study the intermittency and field-line structure of the MHD turbulence in plasmas with very large magnetic Prandtl numbers. In this regime, which is realized in the interstellar medium, some accretion disks, protogalaxies, galaxy-cluster gas, the early universe, etc., magnetic fluctuations can be excited at scales below the viscous cutoff. The salient feature of the resulting small-scale magnetic turbulence is the folded structure of the fields. It is characterized by very rapid transverse spatial oscillation of the field direction, while the field lines remain largely unbent up to the scale of the flow. Quantitatively, the fluctuation level and the field-line geometry can be studied in terms of the statistics of the field strength and of the field-line curvature. In the kinematic limit, the distribution of the field strength is an expanding lognormal, while that of the field-line curvature K is stationary and has a power tail ~K-13/7. The field strength and curvature are anticorrelated, i.e., the growing fields are mostly flat, while the sharply curved fields remain relatively weak. The field, therefore, settles into a reduced-tension state. Numerical simulations demonstrate three essential features of the nonlinear regime. First, the total magnetic energy is equal to the total kinetic energy. Second, the intermittency is partially suppressed compared to the kinematic case, as the fields become more volume-filling and their distribution develops an exponential tail. Third, the folding structure of the field is unchanged from the kinematic case: the anticorrelation between the field strength and the curvature persists, and the distribution of the latter retains the same power tail. We propose a model of back-reaction based on the folding picture that reproduces all of the above numerical results.

Structure of Small-Scale Magnetic Fields in the Kinematic Dynamo Theory

A weak fluctuating magnetic field embedded into a a turbulent conducting medium grows exponentially while its characteristic scale decays. In the interstellar medium and protogalactic plasmas, the magnetic Prandtl number is very large, so a broad spectrum of growing magnetic fluctuations is excited at small (subviscous) scales. The condition for the onset of nonlinear back reaction depends on the structure of the field lines. We study the statistical correlations that are set up in the field pattern and show that the magnetic-field lines possess a folding structure, where most of the scale decrease is due to the field variation across itself (rapid transverse direction reversals), while the scale of the field variation along itself stays approximately constant. Specifically, we find that, though both the magnetic energy and the mean-square curvature of the field lines grow exponentially, the field strength and the field-line curvature are anticorrelated, i.e., the curved field is relatively weak, while the growing field is relatively flat. The detailed analysis of the statistics of the curvature shows that it possesses a stationary limiting distribution with the bulk located at the values of curvature comparable to the characteristic wave number of the velocity field and a power tail extending to large values of curvature where it is eventually cut off by the resistive regularization. The regions of large curvature, therefore, occupy only a small fraction of the total volume of the system. Our theoretical results are corroborated by direct numerical simulations. The implication of the folding effect is that the advent of the Lorentz back reaction occurs when the magnetic energy approaches that of the smallest turbulent eddies. Our results also directly apply to the problem of statistical geometry of the material lines in a random flow.

Effect of fractional kinetic helicity on turbulent magnetic dynamo spectra

Magnetic field amplification in astrophysics ultimately requires an understanding of MHD turbulence. Kinetic helicity has long been known to be important for large-scale field growth in forced MHD turbulence and has been recently demonstrated numerically to be asymptotically consistent with slow mean field dynamo action in a periodic box. Here we show numerically that the magnetic spectrum at and below the forcing scale is also strongly influenced by kinetic helicity. We identify a critical value, fh, crit, above which the magnetic spectrum develops maxima at a wavenumber of 1 scale and at the forcing scale. For f < fh, crit. the field peaks only at the resistive scale. Kinetic helicity may thus be important not only for generating a large-scale field, but also for establishing observed peaks in magnetic spectra at the forcing scale. The turbulent Galactic disk provides an example where both large-scale (greater than the supernova forcing scale) fields and small-scale (less than or equal to forcing scale, with peak at forcing scale) fields are observed. We discuss this and the potential application to the protogalaxy, but we also emphasize the limitations in applying our results to these systems.

The Nonlinear Magnetic Cascade

The Galactic magnetic field has an energy density comparable to that of the interstellar medium turbulence and a coherence spanning the Galaxy. It is not known if this field was formed before, during, or after the Galaxy. However, it is often assumed to originate from a turbulent dynamo process. We investigate the early stages of a Galactic dynamo when the dynamics is well approximated by homogeneous turbulence. Our simulations show that homogeneous magnetized turbulence with large Prandtl number yields magnetic energy at the small, resistive scale rather than at the Galactic scale. Thus, additional phenomena—perhaps helicity generated from Galactic rotation, stratification, and the differential rotation of the disk—are needed to explain the observed field. We simulate the growth of magnetic energy in forced nonhelical turbulence from an initially weak value until it saturates with the same energy density as the turbulence. When the field is dynamically weak, the simulations agree with the kinematic theory. In the long-term saturated state, the magnetic field is strong enough to modify the turbulence. This is the magnetohydrodynamic (MHD) analog of the Kolmogorov problem for hydrodynamic turbulence. The nature of the back-reaction is to neutralize the net shear (stretching) in small-scale eddies that are less energetic than the magnetic field. Only the forcing-scale eddies remain energetic enough to shear and cascade the magnetic field. The magnetic field at all scales therefore forward-cascades at the forcing timescale through spectrally nonlocal interactions with the forcing-scale eddies. Furthermore, the magnetic field folds into a reduced-tension state where field-line curvature anticorrelates with intensity. Direct consequences of these statements are that the magnetic spectrum is largely independent of viscosity and that the magnetic energy is located at the small, resistive scale.

Divergence of neighboring magnetic-field lines and fast-particle diffusion in strong magnetohydrodynamic turbulence, with application to thermal conduction in galaxy clusters.

Using direct numerical simulations, we calculate the rate of divergence of neighboring magnetic-field lines in different types of strong magnetohydrodynamic turbulence. In the static-magnetic-field approximation, our results imply that tangled magnetic fields in galaxy clusters reduce the electron diffusion coefficient and thermal conductivity by a factor of approximately 5-10, relative to their values in a nonmagnetized plasma.