Scaling theory of three-dimensional magnetic reconnection spreading

Abstract

We develop a first-principles scaling theory of the spreading of three-dimensional (3D) magnetic reconnection of finite extent in the out of plane direction. This theory addresses systems with or without an out of plane (guide) magnetic field, and with or without Hall physics. The theory reproduces known spreading speeds and directions with and without guide fields, unifying previous knowledge in a single theory. New results include the following: (1) reconnection spreads in a particular direction if an x-line is induced at the interface between reconnecting and non-reconnecting regions, which is controlled by the out of plane gradient of the electric field in the outflow direction. (2) The spreading mechanism for anti-parallel collisionless reconnection is convection, as is known, but for guide field reconnection it is magnetic field bending. We confirm the theory using 3D two-fluid and resistive-magnetohydrodynamics simulations. (3) The theory explains why anti-parallel reconnection in resistive-magnetohydrodynamics does not spread. (4) The simulation domain aspect ratio, associated with the free magnetic energy, influences whether reconnection spreads or convects with a fixed x-line length. (5) We perform a simulation initiating anti-parallel collisionless reconnection with a pressure pulse instead of a magnetic perturbation, finding spreading is unchanged rather than spreading at the magnetosonic speed as previously suggested. The results provide a theoretical framework for understanding spreading beyond systems studied here and are important for applications including two-ribbon solar flares and reconnection in Earth's magnetosphere.

Three-Dimensional Magnetic Reconnection Spreading in Current Sheets of Non-Uniform Thickness

“…Most of the previous theoretical and numerical work on the spreading of reconnection has addressed quasi-2D anti-parallel reconnection in uniform current sheets with an initial half-thickness w 0 comparable to the ion inertial scale d i = c/ω pi , where c is the speed of light in vacuum and ω pi is the ion plasma frequency. The consensus is that reconnection spreads orthogonal to the reconnection plane with the velocity of the current carriers (Huba & Rudakov, 2002Shay et al, 2003;Karimabadi et al, 2004;Lapenta et al, 2006;Shepherd & Cassak, 2012; T. K. M. Nakamura et al, 2012;Meyer III, 2013;Jain et al, 2013;Jain & Büchner, 2017;Arencibia et al, 2021). This directionality of the spreading is consistent with observations of reconnection during substorms, which spread in the dawnward direction (McPherron et al, 1973;Nagai, 1982;Nagai et al, 2013).…”

Three‐Dimensional Magnetic Reconnection Spreading in Current Sheets of Non‐Uniform Thickness

“…The strength of the normal magnetic field component B y near the reconnection region is an indicator of the presence of reconnection (Arencibia et al, 2021;Huba & Rudakov, 2002;Jain & Büchner, 2017;Li et al, 2020). The average magnitude of B y at the left and right downstream edges of the electron diffusion region is a proxy for the reconnection rate; we denote this quantity as 𝐴𝐴 B𝐵𝑦𝑦(𝑧𝑧𝑧 𝑧𝑧) , given by…”

“…In a small interval of time Δ t , B y propagates a distance Δ z in the z direction, and the spreading speed is defined as v s = Δ z /Δ t . It is estimated from a scaling analysis of Equation , giving $$${v}_{s}\sim -c\frac{{\Delta }{E}_{x}}{{\Delta }{B}_{y}}.$$$ For anti‐parallel reconnection, it was argued that the main contributor to E x is the Hall electric field − J z B y / nec (Arencibia et al., 2021), where n is the upstream density and e is the elementary charge, so that $$${v}_{s}\sim c\frac{{\Delta }\left({J}_{z}{B}_{y}/nec\right)}{{\Delta }{B}_{y}}.$$$ For a current sheet of uniform thickness, J z and n are independent of z , so Equation becomes $$${v}_{s}\sim \frac{{J}_{z}}{ne}.$$$ This result provided a first‐principles scaling prediction of the previously known result that spreading of anti‐parallel reconnection in a current sheet of uniform thickness occurs at the speed of the current carriers (Huba & Rudakov, 2002, 2003; Jain et al., 2013; Shay et al., 2003). Since J z ∼ cB 0 /4 πw , Equation gives $$${v}_{s}\sim \frac{c{B}_{0}}{4\pi new}\sim \frac{{c}_{A0}{d}_{i}}{w},$$$ where $${d}_{i}=c/{\omega }_{pi}={\left({m}_{i}{c}^{2}/4\pi n{e}^{2}\right)}^{1/2}$$ is th...…”

“…Qiu et al (2017) analyzed six two-ribbon flare events that show ribbon elongation/spreading occurs at speeds typically slower than the coronal Alfvén speed by as much as an order of magnitude. One previously known mechanism that could explain a sub-Alfvénic reconnection spreading speed is that the current sheet could have uniform thickness but be thicker than ion inertial scales (Arencibia et al, 2021;Shay et al, 2003). This may be a potential explanation for unidirectional spreading of ribbons with a uniform speed in flare events with a weak guide field, such as fig.…”

The Spatial Extent of Magnetopause Magnetic Reconnection From In Situ THEMIS Measurements

“…PIC models have shown reconnection to spread at slower rates when the current sheet is thicker (Li et al., 2020). Thick current sheets in night‐side PIC models are used to confine reconnection extent (Liu et al., 2019), although it has been shown that simulation size can affect spreading properties of varying current sheet thickness in a two‐fluid code (Arencibia et al., 2021).…”