Statistical analysis of stochastic magnetic fields
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abstract
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.
