Simulations of Incompressible Magnetohydrodynamic Turbulence

Abstract

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 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, respectively. Anisotropy increases with increasing k_⊥ such that excited modes are confined inside a cone bounded by k_∥ ∝ k^y_⊥, where γ < 1. The opening angle of the cone, Θ(k_⊥) ∝ k^⊥^(-(1-y)), defines the scale-dependent anisotropy. (3) The one-dimensional inertial range energy spectrum is well fitted by a power law, E(k_⊥) ∝ k_⊥^(-ɑ), with α > 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 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. (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.

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