Abstract. Direct spectral method simulation of the three-dimensional magnetohydrodynamics (MHD) equations is used to explore anisotropy that develops from initially isotropic fluctuations as a consequence of a uniform applied magnetic field. Spectral and variance anisotropies are investigated in both compressible and incompressible MHD. The nature of the spectral anisotropy is consistent with the model of Shebalin et al. [1983] in which the spectrum broadens in the perpendicular wavenumber direction, the anisotropy being greater for smaller wavenumbers. Here this effect is seen for both incompressible and polytropic compressible MHD. In contrast, the longitudinal (compressive) velocity fluctuations remain isotropic.Variance anisotropy is observed for low plasma beta compressible MHD but not for incompressible MHD. Solar wind observations are qualitatively consistent with both variance and spectral anisotropies of the type discussed here.

We use a two‐dimensional, incompressible MHD spectral code to establish that shear‐driven turbulence is a possible means for producing many observed properties of the evolution of the magnetic and velocity fluctuations in the solar wind and, in particular, the evolution of the cross helicity (“Alfvénicity”) at small scales. We find that large‐scale shear can nonlinearly produce a cascade to smaller scale fluctuations even when the linear Kelvin‐Helmholtz mode is stable and that a roughly power law inertial range is established by this process. While the fluctuations thus produced are not Alfvénic, they are nearly equipartitioned between magnetic and kinetic energy. We report simulations with Alfvénic fluctuations at high wave numbers, both with and without shear layers and find that it is the low cross helicity at low wave numbers that is critical to the cross helicity evolution, rather than the geometry of the flow or the dominance of kinetic energy at large scales. The fluctuations produced by shear effects are shown to evolve similarly but more slowly in the presence of a larger mean field and to be anisotropic with a preferred direction of spectral transfer perpendicular to the mean field. The evolution found is similar to that seen in some other simulations of MHD turbulence, and thus seems in many respects to be a instance of a more generic turbulent evolution rather than due to specific conditions in the solar wind.

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...

Power spectra of solar wind magnetic field and velocity fluctuations more closely resemble those of turbulent fluids (spectral index of −5/3) than they do predictions for magnetofluid turbulence (a −3/2 index). Furthermore, the amount the solar wind is heated by turbulence is uncertain. To aid in the study of both of these issues, we report numerically derived energy cascade rates in magnetohydrodynamic (MHD) turbulence and compare them with predictions of MHD turbulence phenomenologies. Either of the commonly predicted spectral indices of 5/3 and 3/2 are consistent with the simulations. Explicit calculation of inertial range energy cascade rates in the simulations show that for unequal levels of fluctuations propagating parallel and antiparallel to the magnetic field, the majority species always cascades faster than does the minority species, and the cascade rates are in better agreement with a Kolmogoroff‐like MHD turbulence phenomenology than with a generalized Kraichnan phenomenology even in situations where the fluctuations are much smaller than the mean magnetic field. The “Kolmogoroff constant” for MHD turbulence for small normalized cross helicity is roughly 6.7 in two dimensions and 3.6 for one calculation in three dimensions. For large normalized cross helicity, however, none of the existing models can account for the numerical results, although the Kolmogoroff‐like case still works somewhat better than the Kraichnan‐like. In particular, the applied magnetic field has much less influence than expected, and Alfvénicity is more important than predicted. These results imply the need for better phenomenological models to make clear predictions about the solar wind.

Solar wind frequency spectra show a distinct steepening of the ƒ−5/3 power law inertial range spectrum at frequencies above the Doppler‐shifted ion cyclotron frequency. This is commonly attributed to dissipation due to wave‐particle interactions. We consider the extent to which this steepening can be described, using a magnetohydrodynamic formulation that includes the Hall term. An important characteristic of Hall MHD is that although the ion cyclotron resonance is included, there is no wave‐particle dissipation of energy. In this study we use a compressible Hall MHD code with a constant magnetic field and a polytropic equation of state. Artificial dissipation in the form of a bi‐Laplacian operator is used to suppress numerical instabilities, allowing for a clear separation of the dissipative scales from the ion cyclotron scales. A distinct steepening appears in the simulation power spectra near the cyclotron resonance for certain types of initial conditions. This steepening is associated with the appearance of right circularly polarized fluctuations at frequencies above the ion cyclotron resonance. Similar steepenings and polarization enhancements are observed in solar wind magnetic field data.

Recent theoretical studies have led to an improvement of our understanding of the behavior of a compressible magnetofluid with an adiabatic equation of state, in the limit of low plasma frame Mach number. Under certain assumptions the lowest‐order behavior is that of incompressible magnetohydrodynamics (MHD), associated with small nonpropagating “pseudosound” density fluctuations. Departures from incompressibility include magnetoacoustic fluctuations, appearing at the same order as the pseudosound. In the present paper the simplest nearly incompressible MHD theory, with a polytropic equation of state, is reviewed, with an emphasis on observable consequences, particularly for solar wind turbulence. A central feature of the theory is the development of a relationship between the spectra of density fluctuations and of magnetic and velocity fluctuations. Here this relationship is extended to include uniform magnetic field effects, the possibility of anisotropic turbulence, and the influences of magnetic and cross helicities. Consequences of the theory, including Mach number scalings of density fluctuations and their wave number dependence, the anticorrelation of mechanical and magnetic pressure, and the reduction of density fluctuations in Alfvénic periods, are discussed in terms of Voyager solar wind observations. Finally, we present results of a simulation using a two‐dimensional compressible MHD code that illustrate the appearance of similar anticorrelations.

[1] We examine the influence of Hall effect and MHD turbulence on the reconnection rate of a large scale periodic sheet pinch in 2.5 dimensional compressible Hall MHD. Simulations indicate that the reconnection rate is enhanced both by increasing Hall parameter and by increasing turbulence amplitude.

Space plasma measurements, laboratory experiments, and simulations have shown that magnetohydrodynamic ͑MHD͒ turbulence exhibits a dynamical tendency towards spectral anisotropy given a sufficiently strong background magnetic field. Here the undriven decaying initial-value problem for homogeneous MHD turbulence is examined with the purpose of characterizing the variation of spectral anisotropy of the turbulent fluctuations with magnetic field strength. Numerical results for both incompressible and compressible MHD are presented. A simple model for the scaling of this spectral anisotropy as a function of the fluctuating magnetic field over total magnetic field is offered. The arguments are based on ideas from reduced MHD ͑RMHD͒ dynamics and resonant driving of certain non-RMHD modes. The results suggest physical bases for explaining variations of the anisotropy with compressibility, Reynolds numbers, and spectral width of the ͑isotropic͒ initial conditions.