Statistical analysis of stochastic magnetic fields.

Previous work has introduced scale-split energy density ψ_{l,L}(x,t)=1/2B_{l}·B_{L} for vector field B(x,t) coarse grained at scales l and L, in order to quantify the field stochasticity or spatial complexity. In this formalism, the L_{p} norms S_{p}(t)=1/2||1-B[over ̂]_{l}·B[over ̂]_{L}||_{p}, pth-order stochasticity level, and E_{p}(t)=1/2||B_{l}B_{L}||_{p}, pth order mean cross energy density, are used to analyze the evolution of the stochastic field B(x,t). Application to turbulent magnetic fields leads to the prediction that turbulence in general tends to tangle an initially smooth magnetic field increasing the magnetic stochasticity level, ∂_{t}S_{p}>0. An increasing magnetic stochasticity in turn leads to disalignments of the coarse-grained fields B_{d} at smaller scales, d≪L, thus they average to weaker fields B_{L} at larger scales upon coarse graining, i.e., ∂_{t}E_{p}<0. Magnetic field resists the tangling effect of the turbulence by means of magnetic tension force. This can lead at some point to a sudden slippage between the field and fluid, decreasing the stochasticity ∂_{t}S_{p}<0 and increasing the energy ∂_{t}E_{p}>0 by aligning small-scale fields B_{d}. Thus the maxima (minima) of magnetic stochasticity are expected to approximately coincide with the minima (maxima) of cross energy density, occurrence of which corresponds to slippage of the magnetic field through the fluid. In this formalism, magnetic reconnection and field-fluid slippage both correspond to T_{p}=∂_{t}S_{p}=0and∂_{t}T_{2}<0. Previous work has also linked field-fluid slippage to magnetic reconnection invoking totally different approaches. In this paper, (a) we test these theoretical predictions numerically using a homogeneous, incompressible magnetohydrodynamic (MHD) simulation. Apart from expected small-scale deviations, possibly due to, e.g., intermittency and strong field annihilation, the theoretically predicted global relationship between stochasticity and cross energy is observed in different subvolumes of the simulation box. This indicate ubiquitous local field-fluid slippage and reconnection events in MHD turbulence. In addition, (b) we show that the conditions T_{p}=∂_{t}S_{p}=0and∂_{t}T_{p}<0 lead to sudden increases in kinetic stochasticity level, i.e., τ_{p}=∂_{t}s_{p}(t)>0 with s_{p}(t)=1/2||1-u[over ̂]_{l}.u[over ̂]_{L}||_{p}, which may correspond to fluid jets spontaneously driven by sudden field-fluid slippage-magnetic reconnection. Otherwise, they may correspond only to field-fluid slippage without energy dissipation. This picture, therefore, suggests defining reconnection as field-fluid slippage (changes in S_{p}) accompanied with magnetic energy dissipation (changes in E_{p}). All in all, these provide a statistical approach to the reconnection in terms of the time evolution of magnetic and kinetic stochasticities, S_{p} and s_{p}, their time derivatives, T_{p}=∂_{t}S_{p}, τ_{p}=∂_{t}s_{p}, and corresponding cross energies, E_{p}, e_{p}(t)=1/2||u_{l}u_{L}||_{p}. Furthermore, (c) we introduce the scale-split magnetic helicity based on which we discuss the energy or stochasticity relaxation of turbulent magnetic fields-a generalized Taylor relaxation. Finally, (d) we construct and numerically test a toy model, which resembles a classical version of quantum mean field Ising model for magnetized fluids, in order to illustrate how turbulent energy can affect magnetic stochasticity in the weak field regime.

Magnetic stochasticity and diffusion.

We develop a quantitative relationship between magnetic diffusion and the level of randomness, or stochasticity, of the diffusing magnetic field in a magnetized medium. A general mathematical formulation of magnetic stochasticity in turbulence has been developed in previous work in terms of the L_{p} norm S_{p}(t)=1/2∥1-B[over ̂]_{l}·B[over ̂]_{L}∥_{p}, pth-order magnetic stochasticity of the stochastic field B(x,t), based on the coarse-grained fields B_{l} and B_{L} at different scales l≠L. For laminar flows, the stochasticity level becomes the level of field self-entanglement or spatial complexity. In this paper, we establish a connection between magnetic stochasticity S_{p}(t) and magnetic diffusion in magnetohydrodynamic (MHD) turbulence and use a homogeneous, incompressible MHD simulation to test this prediction. Our results agree with the well-known fact that magnetic diffusion in turbulent media follows the superlinear Richardson dispersion scheme. This is intimately related to stochastic magnetic reconnection in which superlinear Richardson diffusion broadens the matter outflow width and accelerates the reconnection process.

Topology and stochasticity of turbulent magnetic fields.

We present a mathematical formalism for the topology and stochasticity of vector fields based on renormalization group methodology. The concept of a scale-split energy density, ψ_{l,L}=B_{l}·B_{L}/2 for vector field B(x,t) renormalized at scales l and L, is introduced in order to quantify the notion of the field topological deformation, topology change, and stochasticity level. In particular, for magnetic fields, it is shown that the evolution of the field topology is directly related to the field-fluid slippage, which has already been linked to magnetic reconnection in previous work. The magnitude and direction of stochastic magnetic fields, shown to be governed, respectively, by the parallel and vertical components of the renormalized induction equation with respect to the magnetic field, can be studied separately by dividing ψ_{l,L} into two (3+1)-dimensional scalar fields. The velocity field can be approached in a similar way. Magnetic reconnection can then be defined in terms of the extrema of the L_{p} norms of these scalar fields. This formulation in fact clarifies different definitions of magnetic reconnection, which vaguely rely on the magnetic field topology, stochasticity, and energy conversion. Our results support the well-founded yet partly overlooked picture in which magnetic reconnection in turbulent fluids occurs on a wide range of scales as a result of nonlinearities at large scales (turbulence inertial range) and nonidealities at small scales (dissipative range). Lagrangian particle trajectories, as well as magnetic field lines, are stochastic in turbulent magnetized media in the limit of small resistivity and viscosity. The magnetic field tends to reduce its stochasticity induced by the turbulent flow by slipping through the fluid, which may accelerate fluid particles. This suggests that reconnection is a relaxation process by which the magnetic field lowers both its topological entanglements induced by turbulence and its energy level.

Inertial-Range Reconnection in Magnetohydrodynamic Turbulence and in the Solar Wind.

In situ spacecraft data on the solar wind show events identified as magnetic reconnection with wide outflows and extended "X lines," 10(3)-10(4) times ion scales. To understand the role of turbulence at these scales, we make a case study of an inertial-range reconnection event in a magnetohydrodynamic simulation. We observe stochastic wandering of field lines in space, breakdown of standard magnetic flux freezing due to Richardson dispersion, and a broadened reconnection zone containing many current sheets. The coarse-grain magnetic geometry is like large-scale reconnection in the solar wind, however, with a hyperbolic flux tube or apparent X line extending over integral length scales.

Flux-freezing breakdown in high-conductivity magnetohydrodynamic turbulence.

The idea of 'frozen-in' magnetic field lines for ideal plasmas is useful to explain diverse astrophysical phenomena, for example the shedding of excess angular momentum from protostars by twisting of field lines frozen into the interstellar medium. Frozen-in field lines, however, preclude the rapid changes in magnetic topology observed at high conductivities, as in solar flares. Microphysical plasma processes are a proposed explanation of the observed high rates, but it is an open question whether such processes can rapidly reconnect astrophysical flux structures much greater in extent than several thousand ion gyroradii. An alternative explanation is that turbulent Richardson advection brings field lines implosively together from distances far apart to separations of the order of gyroradii. Here we report an analysis of a simulation of magnetohydrodynamic turbulence at high conductivity that exhibits Richardson dispersion. This effect of advection in rough velocity fields, which appear non-differentiable in space, leads to line motions that are completely indeterministic or 'spontaneously stochastic', as predicted in analytical studies. The turbulent breakdown of standard flux freezing at scales greater than the ion gyroradius can explain fast reconnection of very large-scale flux structures, both observed (solar flares and coronal mass ejections) and predicted (the inner heliosheath, accretion disks, γ-ray bursts and so on). For laminar plasma flows with smooth velocity fields or for low turbulence intensity, stochastic flux freezing reduces to the usual frozen-in condition.