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.

[1] An analytical model of steady-state magnetic reconnection in a collisionless incompressible plasma is developed using the electron Hall MHD approximation. It is shown that the initial complicated system of equations may be split into a system of independent equations, and the solution of the problem is based on the solution of the Grad-Shafranov equation for a magnetic potential. This equation is found to be fundamental for the whole problem analysis. An electric field potential jump across the electron diffusion region and the separatrices is proved to be the necessary condition for steady-state reconnection. Besides of this fact, it is found that the protons in-plane motion obeys to Bernoulli law. The solution obtained demonstrates all essential Hall reconnection features, namely proton acceleration up to Alfvén velocities and the formation of Hall current systems and a magnetic field structure as expected.

[1] The Sweet-Parker analysis of the inner electron diffusion region of collisionless magnetic reconnection is presented. The study includes charged particles motion near the X-line and an appropriate approximation of the off-diagonal term for the electron pressure tensor. The obtained scaling shows that the width of the inner electron diffusion region is equal to the electron inertial length, and that electrons are accelerated up to the electron Alfvén velocity in X-line direction. The estimated effective plasma conductivity is based on the electron gyrofrequency rather than the binary collision frequency, and gives the extreme (minimal) value of the plasma conductivity similar to Bohm diffusion. The scaling properties are verified by means of Particle-in-Cell simulations. An ad hoc parameter needs to be introduced to the scaling relations in order to better match the theory and simulations.

Fast magnetic reconnection is an explosive plasma process, bringing the topological reconfiguration of magnetic fields, plasma heating, and acceleration in laboratory and space plasmas (e.g., Gonzalez & Parker, 2016;Yamada et al., 2010). In general, this is a time-dependent multi-scale three-dimensional process (e.g., Bhattacharjee, 2004;Dorfman et al., 2013;Frank, 1999;Xiao et al., 2006), but sometimes reconnection may demonstrate a symmetric configuration and be quasi-stationary. Particularly, at the day side of the Earth's magnetopause, quasi-stationary reconnection has been detected in-situ on several occasions (e.g., Gosling et al., 1982;Phan et al., 2004;Retinò et al., 2005) as well as anti-parallel reconnection (Cassak & Fuselier, 2016, and references therein). The latter is more common in the Earth's magnetotail (Paschmann et al., 2013). It is also important that in many cases reconnection can be studied analytically in the frame of two-dimensional models for the considerable length of the reconnection X-line. Configurations with short X-lines demonstrate spreading in the X-line direction (see, e.g., Li et al., 2020, and references therein) in course of time. At last, both at the dayside magnetopause and in the magnetotail (e.g., Cassak &

Highlights► We study the Kelvin–Helmholtz instability at boundary layers around of Venus. ► The stability of the induced magnetopause and the ionopause is examined. ► The ionopause seems to be stable due to a large density jump across this boundary. ► The instability evolves into its nonlinear phase on the magnetopause at solar maximum. ► Loss rates are therefore lower than previously assumed.

[1] The paper presents the detailed numerical investigation of the "double-gradient mode," which is believed to be responsible for the magnetotail flapping oscillations-the fast vertical (normal to the layer) oscillations of the Earth's magnetotail plasma sheet with a quasiperiod 100-200 s. The instability is studied using the magnetotail near-equilibrium configuration. For the first time, linear three-dimensional numerical analysis is complemented with full 3-D MHD simulations. It is known that the "double-gradient mode" has unstable solutions in the region of the tailward growth of the magnetic field component, normal to the current sheet. The unstable kink branch of the mode is the focus of our study. Linear MHD code results agree with the theory, and the growth rate is found to be close to the peak value, provided by the analytical estimates. Full 3-D simulations are initialized with the numerically relaxed magnetotail equilibrium, similar to the linear code initial condition. The calculations show that current layer with tailward gradient of the normal component of the magnetic field is unstable to wavelengths longer than the curvature radius of the field line. The segment of the current sheet with the earthward gradient of the normal component makes some stabilizing effect (the same effect is registered in the linearized MHD simulations) due to the minimum of the total pressure localized in the center of the sheet. The overall growth rate is close to the theoretical double-gradient estimate averaged over the computational domain.

[1] A 2.5-D analytical electron Hall magnetohydrodynamic model of steady state magnetic reconnection in a collisionless compressible plasma with a constant electron temperature is developed. It is shown that as in the incompressible case, the solution of the Grad-Shafranov equation for the magnetic potential is a basis for the problem analysis. The formation of the double electric layers and layers of low-density plasma, mapping the magnetic separatrices, are investigated. It is found that the formation of depletion layers should not be governed by the out-of-plane magnetic field, but rather, the origin of these layers lies inside the electron diffusion region. The double electric layers are found to be thin separatrices-elongated sheets, whose cross sections are of the order of the electron diffusion region half width. These charged layers provide the presence of the strong electric field orthogonal to the in-plane projection of the magnetic field, which forces electrons to accelerate into the out-of-plane direction. Outside of the double electric layers, the condition of quasi-neutrality of the plasma is found to be fulfilled to high accuracy.

Abstract. Magnetic reconnection is believed to be responsible for various explosive processes in the space plasma including magnetospheric substorms. The Hall effect is proved to play a key role in the reconnection process. An analytical model of steady-state magnetic reconnection in a collisionless incompressible plasma is developed using the electron Hall MHD approximation. It is shown that the initial complicated system of equations may split into a system of independent equations, and the solution of the problem is based on the Grad-Shafranov equation for the magnetic potential. The results of the analytical study are further compared with a two-dimensional particle-in-cell simulation of reconnection. It is shown that both methods demonstrate a close agreement in the electron current and the magnetic and electric field structures obtained. The spatial scales of the acceleration region in the simulation and the analytical study are of the same order. Such features like particles trajectories and the in-plane electric field structure appear essentially similar in both models.