Kink-like mode of a double gradient instability in a compressible plasma current sheet

A linear MHD instability of the electric current sheet, characterized by a small normal magnetic field component, varying along the sheet, is investigated. The tangential magnetic field component is modeled by a hyperbolic function, describing Harris-like variations of the field across the sheet. For this problem, which is formulated in a 3D domain, the conventional compressible ideal MHD equations are applied. By assuming Fourier harmonics along the electric current, the linearized 3D equations are reduced to 2D ones. A finite difference numerical scheme is applied to examine the time evolution of small initial perturbations of the plasma parameters. This work is an extended numerical study of the so called “double gradient instability”, – a possible candidate for the explanation of flapping oscillations in the magnetotail current sheet, which has been analyzed previously in the framework of a simplified analytical approach for an incompressible plasma. The dispersion curve is obtained for the kink-like mode of the instability. It is shown that this curve demonstrates a quantitative agreement with the previous analytical result. The development of the instability is investigated also for various enhanced values of the normal magnetic field component. It is found that the characteristic values of the growth rate of the instability shows a linear dependence on the square root of the parameter, which scales uniformly the normal component of the magnetic field in the current sheet.     

Theoretical model of steady-state magnetic reconnection in collisionless incompressible plasma based on the Grad–Shafranov equation solution

The problem of steady-state magnetic reconnection in an infinite current layer in collisionless, incompressible, nonresistive plasma, except of the electron diffusion region, is examined analytically using the electron Hall magnetohydrodynamics approach. It is found that this approach allows reducing the problem to the magnetic field potential finding, while last one has to satisfy the Grad–Shafranov equation. The obtained solution demonstrates all essential Hall reconnection features, namely proton acceleration up to Alfvén velocities, the forming of Hall current systems and the magnetic field structure expected. It turns out that the necessary condition of steady-state reconnection to exist is an electric field potential jump across the electron diffusion region and the separatrices. Besides, the powerful mechanism of electron acceleration in X-line direction is required. It must accelerate electrons up to the electron Alfvén velocity inside the diffusion region and on the separatrixes. This is a necessary condition for steady-state reconnection as well.