Colloquium: Magnetohydrodynamic turbulence and time scales in astrophysical and space plasmas

Magnetohydrodynamic (MHD) turbulence has been employed as a physical model for a wide range of applications in astrophysical and space plasma physics. This Colloquium reviews fundamental aspects of MHD turbulence, including spectral energy transfer, nonlocality, and anisotropy, each of which is related to the multiplicity of dynamical time scales that may be present. These basic issues are discussed based on the concepts of sweeping of the small scales by a large-scale field, which in MHD occurs due to effects of counterpropagating waves, as well as the local straining processes that occur due to nonlinear couplings. These considerations give rise to various expected energy spectra, which are compared to both simulation results and relevant observations from space and astrophysical plasmas.

Figure 1

Comparison of hydrodynamics and magnetohydrodynamics: (a) In hydrodynamics a mean or large-scale flow sweeps the small-scale eddies without affecting the energy transfer between length scales; (b) in magnetohydrodynamics a mean or large-scale magnetic field $\mathbf{B}$ sweeps oppositely propagating fluctuations ${\mathbf{z}}_{-}$ and ${\mathbf{z}}_{+}$, which affects the energy transfer (illustrated as distortions after the two types of fluctuation have passed through each other).

Figure 2

The trace of the power spectral matrix of the magnetic field fluctuations $(\mathrm{nT}=\text{nanotesla})$ from a $74\text{−}\min$ sample of data values, each of which is an average value taken over $0.12\sec$ of data from the Mariner 10 magnetometer on March 20, 1974. The solar wind data indicate a $-5∕3$ Kolmogorov scaling for the slope of the spectrum. Adapted from 36.

Figure 3

Twenty-eight hours of magnetic-field and plasma data demonstrating the presence of nearly pure Alfvén waves. The upper six curves are $5.04\text{−}\min$ bulk velocity components and magnetic-field components averaged over the plasma probe sampling period. The vertical axes on the right are for the Cartesian velocity components ($R=\text{radial}$, $T=\text{tangential}$, $N=\text{normal}$, relative to ecliptic plane) in $\mathrm{km}∕\sec$; the vertical axes on the left are for the magnetic-field components in nT. The lower two curves are magnetic-field strength and proton number density (in ${\mathrm{cm}}^{-3}$). From 5. Reprinted with permission from Journal of Geophyscal Research, American Geophysical Union.

Figure 4

Plot of the normalized angle-integrated energy spectrum of MHD turbulence vs $\widehat{k}\equiv k{\ell }_d$ (where ${\ell }_d$ is the Kolmogorov dissipation length). All curves are multiplied by $k^{5∕3}$. The Kolmogorov scaling is strongly suggested by this direct numerical simulation (solid curve), from a run that does not have magnetic helicity. The dashed line indicates the Iroshnikov-Kraichnan spectrum $k^{-3∕2}$ (see Sec. 3C2), and the dotted line the Kolmogorov spectrum [see. Eq. (4)] with $C_K=2.3$. From 11.

Figure 5

Magnetic-field lines from a spectral-method simulation of 2D MHD turbulence, and associated randomly occurring electric current sheets and filaments that are oriented so that the current is perpendicular to the magnetic-field activity. The magnitudes of the currents are indicated by gray shading, with the most intense levels being white for out-of-the-plane currents and black for into-the-plane currents.

Figure 6

(Color) Color map of the spatial distribution of magnetic-field intensity in a spectral-method simulation of 3D MHD turbulence. Spatial resolution is $256\times 256\times 256$ rectangular grid points. The mechanical and magnetic Reynolds numbers are 1000.

Figure 7

Omnidirectional energy spectrum vs wave number, computed from a generalized energy spectrum (73). Three values of the magnetic-field strength $B_0=0$, 1, 10, and 100 are shown. The slope flattens and the spectral level increases as the field strength is increased, indicating a smooth transition from the Kolmogorov $-5∕3$ scaling $(B_0=0)$ to a spectrum $B_0=100$ that approaches the Iroshnikov-Kraichnan $-3∕2$ scaling. From 73.

Figure 8

(Color) Color maps of the current density $j_z$ in cross sections $x$, $y$ and $x$, $z$ from 3D simulations of MHD turbulence with an applied dc magnetic field $B_0\widehat{\mathbf{z}}$: top, $B_0=0$; middle, $B_0∕\delta B=1$; bottom, $B_0∕\delta B=8$).

Figure 9

Observations in the inner heliosphere by the Helios mission in time intervals associated with high-speed solar wind: solid lines, one-dimensional spectra $E_{+}\left(k\right)$; dobted lines, $E_{-}\left(k\right)$. While the larger Elsässer field has approximately a power-law spectrum in both examples, the spectrum of the smaller field does not run parallel to it (as would be expected in the strain-dominated case) nor does it have a pure power-law spectrum that extends into the dissipation range where it intersects the spectrum of the larger field (as would be expected in the sweep-dominated case). Adapted from 66.