Grad–Shafranov reconstruction of the magnetic configuration in the reconnection X-point vicinity in compressible plasma

Abstract

The reconstruction problem for steady symmetrical two-dimensional magnetic reconnection is addressed in the frame of a two-fluid approximation with neglected ion current. This approach yields Poisson's equation for the magnetic potential of the in-plane magnetic field, where the right-hand side contains the out-of-plane electron current density with the reversed sign. In the simplest case of uniform electron temperature and number density and neglecting the electron inertia, Poisson's equation turns to the Grad-Shafranov one. With boundary conditions fixed at any unclosed curve (the satellite trajectory), both equations result in an ill-posed problem. Since the magnetic configuration in the reconnection region is highly stretched, one can make use of the boundary layer approximation; hence, the problem becomes well-posed. The described approach is generalized for the case of nonuniform electron temperature and number density. The benchmark reconstruction of the PIC simulations data has shown that the main contribution for inaccuracy arises from replacing Poisson's equation by the equation of Grad-Shafranov. Under this substitution, the reachable cross-size of the reconstructed region is shrinking down to fractions of the proton inertial length. Artificial smoothing, demanded by solving the ill-posed problem, and boundary layer approximation represent two alternative methods of problem regularization. In terms of the reconstruction error, they perform nearly the same; the second method benefits from the comparative simplicity and less restrictions imposed on the boundary shape.

The Inertia‐Based Model for Reconstruction of the Electron Diffusion Region

“…Contrary to Korovinskiy et al (2020), where two-fluid MHD formalism is used, in this paper we use a single-fluid model. Hence, all equations in Section II of Korovinskiy et al (2020) remain the same, except for the equations of the proton motion and state, which are not considered here. The set of normalization constants is also changed from the proton scale to the electron one:  e, m e , d e , t*, B 0 ,…”

“…, where e is the elementary charge, m e is the electron mass; d e = c/ω e is the electron inertial length; c is the speed of light;   Omitting all details, addressed in Section II of Korovinskiy et al (2020), where the transition to the electron scale units is performed by the substitution μ = 1, the problem is formulated as follows.…”

“…As for the solution of Equation 17 with omitted term ∂ 2 A/∂x 2 (i.e., solution in BLA), BLA is the regularization procedure itself. The detailed comparison of these two regularization techniques is provided in Korovinskiy et al (2020). Two solutions of Equation 17 are compared to each other and to the in-situ data in terms of the B x value in Figure 6.…”

“…In the interior of this Hall MHD (HMHD) domain, the relation between amplitudes of the ion and electron current densities, | j i | ≪ | j e |, allows the neglect of the ion current, yielding the commonly used electron MHD (EMHD) approximation (e.g., Biskamp, 2000; Bulanov et al., 1992; Ji et al., 2014). Approved to be appropriate for plasmas with high and moderate values of the β parameter (Biskamp et al., 1997), where β is the ratio of the gas and magnetic pressures, the condition | j i | ≪ | j e | is found to fulfill not in immediate vicinity of the X‐point only, but at bigger distances also, where β drops down to ∼0.1 (Korovinskiy et al., 2020). At last, in the very central part of the reconnection region, the EDR, electrons are also demagnetized due to the electron pressure anisotropy and electron inertia at the scale of the electron inertial length, d e (e.g., Hesse et al., 2011; Paschmann et al., 2013).…”

“…At the same time, the calculation of B y from the Ohm's law, implying the term   (ˆ) e y P , stays a single option only until electron inertia is neglected. Keeping electron inertia and assuming n e = const, one arrives at the system of two coupled Poisson's equations for A and B y , which do not contain the term   (ˆ) e y P (see Equation 12, 13 in Sonnerup et al (2016) or Equation 51, 52 in Korovinskiy et al (2020)). In the present paper, we examine the new reconstruction technique, based on the solution of this system, by reconstructing the configuration of EDR, encountered by MMS on July 11, 2017 -an event, reported by Torbert et al (2018) and considered in a number of studies (e.g., Denton et al, 2021;Egedal et al, 2019;Genestreti et al, 2018;Hasegawa et al, 2019;Nakamura et al, 2019).…”