Statistical analysis of stochastic magnetic fields

Previous work has introduced scale-split energy density psi_{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, partial differential_{t}S_{p}>0. An increasing magnetic stochasticity in turn leads to disalignments of the coarse-grained fields B_{d} at smaller scales, d 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}= partial differential_{t}S_{p}=0and partial differential_{t}T_{2} 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}= partial differential_{t}S_{p}, tau_{p}= partial differential_{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.