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Power and spatial complexity in stochastic reconnection

The level of spatial complexity associated with a given vector field on an arbitrary range of scales \iffalse ${\bf F(x}, t)$ can be quantified by a simple, time-dependent function $S(t)={1\over 2}(1-\hat{\bf F}_l.\hat{\bf F}_L)_{rms}$ with ${\bf F}_l$ (${\bf F}_L$) being the average field in a volume of scale $l$ ($L>l$) and the unit vector defined as $\hat{\bf F}={\bf F}/|{\bf F}|$. Thus,\fi can be quantified by a simple, scale-dependent function of time; $0\leq S(t)\leq 1$. Previous work has invoked kinetic and magnetic complexities, associated with velocity and magnetic fields ${\bf u(x}, t)$ and ${\bf B(x}, t)$, to study magnetic reconnection and diffusion in turbulent and magnetized fluids. In this paper, using the coarse-grained momentum equation, we argue that the fluid jets associated with magnetic reconnection events at an arbitrary scale $l$ in the turbulence inertial range are predominantly driven by the Lorentz force ${\bf{N}}_l=({\bf j\times B})_l-{\bf j}_l\times {\bf B}_l$. This force, is induced by the subscale currents and is analogous to the turbulent electromotive force ${\cal E}_l=({\bf u\times B})_l-{\bf u}_l\times {\bf B}_l$ in dynamo theories. Typically, high (low) magnetic complexities during reconnection imply large (small) spatial gradients for the magnetic field, i.e., strong (weak) Lorentz forces ${\bf N}_l$. Reconnection launches jets of fluid, hence the rate of change of kinetic complexity is expected to strongly correlate with the power injected by the Lorentz force ${\bf N}_l$. We test this prediction using an incompressible, homogeneous magnetohydrodynamic (MHD) simulation and associate it with previous results. It follows that the stronger (weaker) the turbulence, the more (less) complex the magnetic field and the stronger (weaker) the reconnection field and thus the ensuing reconnection.