Logo Logo
Switch Language to German
Jafari, Amir; Vishniac, Ethan; Vaikundaraman, Vignesh (2020): Statistical analysis of stochastic magnetic fields. In: Physical Review E, Vol. 101, No. 2, 022122
Full text not available from 'Open Access LMU'.


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 parallel to 1 - (B) over cap (l).(B) over cap (L)parallel to(p), pth-order stochasticity level, and E-p(t)= 1/2 parallel to BlBL parallel to(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 derivative S-t(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., partial derivative E-t(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 partial derivative S-t(p) < 0 and increasing the energy partial derivative E-t(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 = partial derivative S-t(p) = 0 and partial derivative 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 = partial derivative S-t(p) = 0 and partial derivative T-t(p) < 0 lead to sudden increases in kinetic stochasticity level, i.e., tau(p) = partial derivative S-t(p)(t) > 0 with S-p(t) = 1/2 parallel to 1 - (u) over cap (l).(u) over cap (L)parallel to(p) which may correspond to fluid jets spontaneously driven by sudden field-fluid slippagemagnetic 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 derivative S-t(p), tau(p) = partial derivative S-t(p), and corresponding cross energies, E-p, e(p)(t) = 1/2 parallel to u(l)u(L)parallel to(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.