Parallel and perpendicular cascades in solar wind turbulence

MHD-scale fluctuations in the velocity, magnetic, and density fields of the solar wind are routinely observed. The evolution of these fluctuations, as they are transported radially outwards by the solar wind, is believed to involve both wave and turbulence processes. The presence of an average magnetic field has important implications for the anisotropy of the fluctuations and the nature of the turbulent wavenumber cascades in the directions parallel and perpendicular to this field. In particular, if the ratio of the rms magnetic fluctuation strength to the mean field is small, then the parallel wavenumber cascade is expected to be weak and there are difficulties in obtaining a cascade in frequency. The latter has been invoked in order to explain the heating of solar wind fluctuations (above adiabatic levels) via energy transfer to scales where ion-cyclotron damping can occur. Following a brief review of classical hydrodynamic and magnetohydrodynamic (MHD) cascade theories, we discuss the distinct nature of parallel and perpendicular cascades and their roles in the evolution of solar wind fluctuations

Reduced magnetohydrodynamics and parallel spectral transfer

The self-consistency of the reduced magnetohydrodynamics (RMHD) model is explored by examining whether (parallel) spectral transfer might invalidate the assumptions employed in deriving it. Using direct numerical simulations we find that transfer of energy to structures with high parallel wavenumber is in fact limited by ongoing perpendicular transfer. Thus, the dynamics associated with RMHD models remains consistent with the underlying assumptions of RMHD. In particular, in well-resolved simulations it is neither necessary nor correct to introduce additional dissipation terms that (artificially) damp spectral transfer parallel to the mean magnetic field B0

A two-component phenomenology for homogeneous magnetohydrodynamic turbulence

A one-point closure model for energy decay in three-dimensional magnetohydrodynamic (MHD) turbulence is developed. The model allows for influence of a large-scale magnetic field that may be of strength sufficient to induce Alfvén wave propagation effects, and takes into account components of turbulence in which either the wave-like character is negligible or is dominant. This two-component model evolves energy and characteristic length scales, and may be useful as a simple description of homogeneous MHD turbulent decay. In concert with spatial transport models, it can form the basis for approximate treatment of low-frequency plasma turbulence in a variety of solar, space, and astrophysical contexts

Current singularities at finitely compressible three-dimensional magnetic null points

The formation of current singularities at line-tied two- and three-dimensional (2D and 3D, respectively) magnetic null points in a nonresistive magnetohydrodynamic environment is explored. It is shown that, despite the different separatrix structures of 2D and 3D null points, current singularities may be initiated in a formally equivalent manner. This is true no matter whether the collapse is triggered by flux imbalance within closed, line-tied null points or driven by externally imposed velocity fields in open, incompressible geometries. A Lagrangian numerical code is used to investigate the finite amplitude perturbations that lead to singular current sheets in collapsing 2D and 3D null points. The form of the singular current distribution is analyzed as a function of the spatial anisotropy of the null point, and the effects of finite gas pressure are quantified. It is pointed out that the pressure force, while never stopping the formation of the singularity, significantly alters the morphology of the current distribution as well as dramatically weakening its strength. The impact of these findings on 2D and 3D magnetic reconnection models is discussed

Direct comparisons of compressible magnetohydrodynamics and reduced magnetohydrodynamics turbulence

Direct numerical simulations of low Mach number compressible three-dimensional magnetohydrodynamic (CMHD3D) turbulence in the presence of a strong mean magnetic field are compared with simulations of reduced magnetohydrodynamics (RMHD). Periodic boundary conditions in the three spatial coordinates are considered. Different sets of initial conditions are chosen to explore the applicability of RMHD and to study how close the solution remains to the full compressible MHD solution as both freely evolve in time. In a first set, the initial state is prepared to satisfy the conditions assumed in the derivation of RMHD, namely, a strong mean magnetic field and plane-polarized fluctuations, varying weakly along the mean magnetic field. In those circumstances, simulations show that RMHD and CMHD3D evolve almost indistinguishably from one another. When some of the conditions are relaxed the agreement worsens but RMHD remains fairly close to CMHD3D, especially when the mean magnetic field is large enough. Moreover, the well-known spectral anisotropy effect promotes the dynamical attainment of the conditions for RMHD applicability. Global quantities (mean energies, mean-square current, and vorticity) and energy spectra from the two solutions are compared and point-to-point separation estimations are computed. The specific results shown here give support to the use of RMHD as a valid approximation of compressible MHD with a mean magnetic field under certain but quite practical conditions

On the accuracy of simulations of turbulence

The widely recognized issue of adequate spatial resolution in numerical simulations of turbulence is studied in the context of two-dimensional magnetohydrodynamics. The familiar criterion that the dissipation scale should be resolved enables accurate computation of the spectrum, but fails for precise determination of higher-order statistical quantities. Examination of two straightforward diagnostics, the maximum of the kurtosis and the scale-dependent kurtosis, enables the development of simple tests for assessing adequacy of spatial resolution. The efficacy of the tests is confirmed by examining a sample problem, the distribution of magnetic reconnection rates in turbulence. Oversampling the Kolmogorov dissipation scale by a factor of 3 allows accurate computation of the kurtosis, the scale-dependent kurtosis, and the reconnection rates. These tests may provide useful guidance for resolution requirements in many plasma computations involving turbulence and reconnection

A two-component phenomenology for the evolution of MHD turbulence

Incompressible MHD turbulence with a mean magnetic field B₀ develops anisotropic spectral structure and can be simply described only by including at least two distinct fluctuation components. These are conveniently referred to as “waves,” for which propagation effects are important, and “quasi-2D” turbulence, for which nonlinear effects dominate over propagation ones. The quasi-2D component has wavevectors approximately perpendicular to B₀. These two idealized ingredients capture the essential physics of propagation (high frequency fluctuations) and strong turbulence (low frequency fluctuations.) Here we present a two-component energy-containing range phenomenology for the evolution of homogeneous MHD turbulence

General second-rank correlation tensors for homogeneous magnetohydrodynamic turbulence

The properties and structure of second-order (Cartesian) correlation tensors are derived for the general case of two solenoidal random vector fields. The theory is intended to describe homogeneous magnetohydrodynamic turbulence, with no assumed rotational or reflectional symmetries. Each correlation tensor can be written in terms of four scalar generating functions and the relationship of these functions to the potentials that generate the poloidal and toroidal components of the underlying vector fields is derived. The physical nature of the scalar functions is investigated and their true or pseudoscalar character is ascertained. In our general discussion we clarify several misleading statements dating back to Robertson’s original paper in the field [Proc. Camb. Philos. Soc. 36, 209 (1940)]. It is also shown that using the one-dimensional correlation function, it is possible to obtain spectral information on the induced electric field in directions perpendicular to the measurement direction

Scaling of spectral anisotropy with magnetic field strength in decaying magnetohydrodynamic turbulence

Space plasmameasurements,laboratory experiments, and simulations have shown that magnetohydrodynamic(MHD)turbulence exhibits a dynamical tendency towards spectral anisotropy given a sufficiently strong background magnetic field. Here the undriven decaying initial-value problem for homogeneous MHDturbulence 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 (RMHD) 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 (isotropic) initial conditions