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 ${\mathcal{L}}_p$ norm $S_p\left(t\right)=\frac{1}{2}{\parallel 1-{\widehat{\mathbf{B}}}_l·{\widehat{\mathbf{B}}}_L\parallel }_p, p\mathrm{th}$-order magnetic stochasticity of the stochastic field $\mathbf{B}(\mathbf{x},t)$, based on the coarse-grained fields ${\mathbf{B}}_l$ and ${\mathbf{B}}_L$ at different scales $l\ne L$. 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 $S_p\left(t\right)$ 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.

Figure 1

Parallel, ${\lambda }_{\parallel }$, and perpendicular, ${\lambda }_{\perp }$, wavelengths with respect to the large-scale magnetic field ${\mathbf{B}}_L$ (coarse-grained $\mathbf{B}$ on scale $L$), in a region of scale $l<L$ with local field ${\mathbf{B}}_l$ (coarse-grained $\mathbf{B}$ on scale $l<L$). The angle $\theta$ is a stochastic variable because of the randomness inherent in $\mathbf{B}$, hence it can be used to quantify magnetic stochasticity level. Its local relationship with ${\lambda }_{\parallel }$ and ${\lambda }_{\perp }$, on the other hand, provides a means to relate the stochasticity level $S_2\left(t\right)$, defined by Eq. (10), to magnetic diffusion, Eq. (18). This relationship is quantified by Eq. (21).

Figure 2

Plots of $f\left(t\right)$ (curves) defined by Eq. (22) with respect to time and its time average ${\langle f\left(t\right)\rangle }_T$ (horizontal lines) for $l=3,L=11$ (dotted, cyan), $l=5,L=11$ (dashed, red), and $l=7,L=11$ (solid, yellow). For the correct value of $\beta$, the time average of this function, ${\langle f\left(t\right)\rangle }_T$, should be almost independent of scale and approximately equal to the half of the diffusion coefficient $\alpha$, defined by Eq. (18). For different scales, the standard deviation and relative standard deviation corresponding to $\beta =1$ (a) are much larger that their counterparts for $\beta =3$ (b). The numerical values of $\alpha$, in the same subvolume, are shown in Table 1. This result holds in different subvolumes of the simulation box.

Figure 3

Same as Fig. 2 but for $\beta =5$ (a) and $\beta =7$ (b) (see also Table 1). The value of $f\left(t\right)$ becomes smaller by increasing $\beta$; however, its relative standard deviation ramps up indicating that $f\left(t\right)$ is not a constant for $\beta >3$ and thus no constant diffusion coefficient can be defined. For $\beta \ne 3$, in general, a slight change in scale leads to a large relative change in diffusion coefficient.