The third-order law for magnetohydrodynamic turbulence with constant shear

The scaling laws of mixed third‐order structure functions for isotropic, homogeneous, and incompressible magnetohydrodynamic (MHD) turbulence have been recently applied in solar wind studies, even though there is recognition that isotropy is not well satisfied. Other studies have taken account of the anisotropy induced by a constant mean magnetic field. However, large‐scale shear can also cause departures from isotropy. Here we examine shear effects in the simplest case, and derive the third‐order laws for MHD turbulence with constant shear, where homogeneity is still assumed. This generalized scaling law has been checked by data from direct numerical simulations (DNS) of two‐dimensional (2D) MHD and is found to hold across the inertial range. These results suggest that third‐order structure function analysis and interpretation in the solar wind should be undertaken with some caution, since, when present, shear can change the meaning of the third‐order relations

von Kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality

We argue that the hypothesis of preservation of shape of dimensionless second- and third-order correlations during decay of incompressible homogeneous magnetohydrodynamic (MHD) turbulence requires, in general, at least two independent similarity length scales. These are associated with the two Elsässer energies. The existence of similarity solutions for the decay of turbulence with varying cross-helicity implies that these length scales cannot remain in proportion, opening the possibility for a wide variety of decay behaviour, in contrast to the simpler classic hydrodynamics case. Although the evolution equations for the second-order correlations lack explicit dependence on either the mean magnetic field or the magnetic helicity, there is inherent implicit dependence on these (and other) quantities through the third-order correlations. The self-similar inertial range, a subclass of the general similarity case, inherits this complexity so that a single universal energy spectral law cannot be anticipated, even though the same pair of third-order laws holds for arbitrary cross-helicity and magnetic helicity. The straightforward notion of universality associated with Kolmogorov theory in hydrodynamics therefore requires careful generalization and reformulation in MHD

Solar wind fluctuations and the von Kármán–Howarth equations: The role of fourth-order correlations

The von Kármán-Howarth (vKH) hierarchy of equations relate the second-order correlations of the turbulent fluctuations to the third-order ones, the third-order to the fourth-order, and so on. We recently demonstrated [1] that for MHD, self-similar solutions to the vKH equations seem to require at least two independent similarity lengthscales (one for each Elsässer energy), so that compared to hydrodynamics a richer set of behaviors seems likely to ensue. Moreover, despite the well-known anisotropy of MHD turbulence with a mean magnetic field (B₀), the equation for the second-order correlation does not contain explicit dependence on B₀. We show that there is, however, implicit dependence on B₀ via the third-order correlations, which themselves have both explicit B₀-dependence and also their own implicit dependence through fourth-order correlations. Some subtleties and consequences of this implicit-explicit balance are summarized here. In addition, we present an analysis of simulation results showing that the evolution of turbulence can depend strongly on the initial fourth-order correlations of the system. This leads to considerable variation in the energy dissipation rates. Some associated consequences for MHD turbulence are discussed

The third-order law for magnetohydrodynamic turbulence with shear: Numerical investigation

The scaling laws of third-order structure functions for isotropic, homogeneous, and incompressible magnetohydrodynamic (MHD) turbulence relate the observable structure function with the energy dissipation rate. Recently [ Wan et al. Phys. Plasmas 16, 090703 (2009) ], the theory was extended to the case in which a constant velocity shear is present, motivated by the application of the third-order law to the solar wind. We use direct numerical simulations of two-dimensional MHD with shear to confirm this new generalization of the theory. The presence of the shear effect broadens the circumstances in which the law can be applied. Important implications for laboratory and space plasmas are discussed

Generation of X-points and secondary islands in 2D magnetohydrodynamic turbulence

We study the time development of the population of X-type critical points in a two-dimensional magnetohydrodynamic model during the early stages of freely decaying turbulence. At sufficiently high magnetic Reynolds number Rem, we find that the number of neutral points increases as Rem3/2, while the rates of reconnection at the most active sites decrease. The distribution of rates remains approximately exponential. We focus in particular on delicate issues of accuracy, which arise in these numerical experiments, in that the proliferation of X-points is also a feature of under-resolved simulations. The “splitting” of neutral points at high Reynolds number appears to be a fundamental feature of the cascade that has important implications for understanding the relationship between reconnection and turbulence, an issue of considerable importance for the Magnetospheric Multiscale and Solar Probe missions as well as observation of reconnection in the solar wind

Fourier-Hermite decomposition of the collisional Vlasov-Maxwell system: implications for the velocity space cascade

Turbulence at kinetic scales is an unresolved and ubiquitous phenomenon that characterizes both space and laboratory plasmas. Recently, new theories, {\it in-situ} spacecraft observations and numerical simulations suggest a novel scenario for turbulence, characterized by a so-called phase space cascade -- the formation of fine structures, both in physical and velocity space. This new concept is here extended by directly taking into account the role of inter-particle collisions, modeled through the nonlinear Landau operator or the simplified Dougherty operator. The characteristic times, associated with inter-particle correlations, are derived in the above cases. The implications of introducing collisions on the phase space cascade are finally discussed.Comment: Special issue featuring the invited talks from the International Congress on Plasma Physics (ICPP) in Vancouver, Canada 4-8 June 201

Statistical properties of solar wind discontinuities, intermittent turbulence, and rapid emergence of non-Gaussian distributions

Recent studies have compared properties of the magnetic field in simulations of Hall MHD turbulence with spacecraft data, focusing on methods used to identify classical discontinuities and intermittency statistics. Comparison of ACE solar wind data and simulations of 2D and 3D turbulence shows good agreement in waiting‐time analysis of magnetic discontinuities, and in the related distribution of magnetic field increments. This supports the idea that the magnetic structures in the solar wind may emerge fast and locally from nonlinear dynamics that can be understood in the framework of nonlinear MHD theory. The analysis suggests that small scale current sheets form spontaneously and rapidly enough that some of the observed solar wind discontinuities may be locally generated, representing boundaries between interacting flux tubes

Multipoint Turbulence Analysis with Helioswarm

Exploration of plasma dynamics in space, including turbulence, is entering a new era of multi-satellite constellation measurements that will determine fundamental properties with unprecedented precision. Familiar but imprecise approximations will need to be abandoned and replaced with more advanced approaches. We present a preparatory study of the evaluation of second- and third-order statistics, using simultaneous measurements at many points. Here, for specificity, the orbital configuration of the NASA Helioswarm mission is employed in conjunction with three-dimensional magnetohydrodynamics numerical simulations of turbulence. The Helioswarm 9-spacecraft constellation flies virtually through the turbulence to compare results with the exact numerical statistics. We demonstrate novel increment-based techniques for the computation of (1) the multidimensional spectra and (2) the turbulent energy flux. This latter increment-space estimate of the cascade rate, based on the third-order Yaglom-Politano-Pouquet theory, uses numerous increment-space tetrahedra. Our investigation reveals that Helioswarm will provide crucial information on the nature of astrophysical turbulence