Scaling of Anisotropy in Hydromagnetic Turbulence

Abstract

We present evidence that anisotropy of low frequency plasma turbulence scales linearly with the ratio of fluctuating to total magnetic field strength for a useful range of parameters, for incompressible, weakly compressible, and driven magnetohydrodynamic turbulence. [S0031-9007(98) PACS numbers: 47.65. + a, 52.35.Ra, 95.30.Qd Evidence accumulated over the past several decades indicates that a large-scale applied (dc) magnetic field imposes a preferred direction on turbulence, and thus plays an important role in plasma diffusion [1], energetic particle scattering [2], and plasma heating [3][4][5]. Each of these in turn may significantly influence large-scale flows and structure [6][7][8]. The interplay between turbulence and large-scale magnetic field suggests a crucial role of rotational symmetry or "geometry" of the fluctuations in many astrophysical plasma settings. There has been considerable recent interest in detection and understanding of anisotropy of fluctuations in solar, interplanetary, and galactic plasmas, and thus it would appear to be of importance to understand mechanisms that can produce and regulate anisotropy in fluid-scale plasma turbulence. In this Letter we show, using numerical solutions of magnetohydrodynamics (MHD), that anisotropy produced by spectral transfer scales in a systematic way with applied field strength. In particular, an angular measure of the anisotropy of the spectrum varies linearly with field strength over a useful range of applied field magnitudes. A simple argument, based upon the physics of reduced MHD [9][10][11], explains this scaling property as well as its saturation.Within the MHD framework, anisotropy associated with a (uniform) dc magnetic field ͑B 0 ͒ may take a number of forms [12][13][14]. Here we are concerned specifically with dynamical development of spectral anisotropy due to asymmetry of nonlinear spectral transfer relative to the mean field direction [14,15]. This anisotropy is characterized by gradients across the mean magnetic field that are relatively larger than gradients along the field. Such features can be readily observed in fluctuations of plasma fluid velocity, magnetic field, and density, and have been observed in the solar wind [16][17][18], the solar corona [19], the interstellar medium [20,21], and in various laboratory plasma devices [22,23]. The limiting case, when all variations are perpendicular to the mean field, and the parallel coordinate is ignorable, is known as two-dimensional (2D) turbulence. The opposite limit, with perpendicular coordinates ignorable, often called "slab" symmetry, is traditionally employed in linear wave theory [2,16]. Turbulence that is "quasi-2D" is described by "reduced" MHD equations that emerge naturally in the theory of nearly incompressible MHD [24] for low plasma b.It is well known that anisotropy can be generated robustly through rapid turbulent wave-vector-space spectral transfer in the directions transverse to the mean field [14]. Parallel spectral transfer is relatively suppressed, so the spe...

Reduced magnetohydrodynamics and parallel spectral transfer

“…The ratio 19 of these two timescales yields a parameter central to RMHD-and indeed MHD turbulence in general: 3,9,12,13,[20][21][22][23] …”

On the dissipation of magnetic fluctuations in the solar wind

“…In this picture, no dissipation is needed to explain the steepening. Furthermore, it is now believed that the MHD turbulence cascade is highly anisotropic, with a significant fraction of turbulent energy cascades mostly in a quasi-2D fashion, perpendicular to the background magnetic field Bo [Shebalin et al, 1983;Matthaeus et al, 1998]. …”

Kinetic helicity and MHD turbulence

“…Observational, simulation, and modelling support for the quality of this approximation is well documented (see e.g. Tu and Marsch 1995;Matthaeus et al 1996Matthaeus et al , 1998Matthaeus et al , 1999. Thus, the incompressible results presented below are also likely to be of relevance to various compressible space physics and astrophysical systems.…”

“…In physical space, this manifests itself as flows that develop fine-scale structure perpendicular to the mean field, while maintaining essentially unchanged characteristic parallel lengthscales. Important refinements to this picture are discussed elsewhere (Grappin 1986;Kinney and McWilliams 1998; Matthaeus et al 1998;Oughton et al 1998). The net result is that after a few eddy-turnover times, the turbulence is in a state which is well characterized by descriptions based on reduced MHD (RMHD), even when the initial states are emphatically not RMHD-like (Strauss 1976;Montgomery 1982;Zank and Matthaeus 1992).…”

“…Furthermore, the usual ordering θ ψ,a < θ v,b < θ ω,j is seen, suggesting that the smaller scales are more anisotropic. Discussions on the scaling of anisotropy angles with the ratio of fluctuating to total magnetic field strength can be found in Matthaeus et al (1998) and Oughton et al (1998).…”