Current sheet formation at a magnetic neutral line in Hall magnetohydrodynamics

Abstract

The inner structure of collisionless magnetic reconnection: The electron-frame dissipation measure and Hall fields Phys.The dynamics of a plasma in the vicinity of a neutral line of the magnetic field is considered in the framework of incompressible Hall magnetohydrodynamics ͑MHD͒. A self-similar solution for the collapse to a current sheet is obtained. Numerical and analytical results are presented. In contrast to the standard incompressible MHD model in two dimensions, the Hall effect leads to the formation of finite-time singularities of the velocity derivatives and the electric current density at the neutral line. Analytical expressions are derived for the collapse time and the form of the solution near the singularity, which agree closely with the numerical results. If the ion skin depth is set to zero, the singularity formation time becomes infinite, corresponding to the standard MHD result. Implications of the results for Hall MHD models of fast magnetic reconnection are discussed.

Influence of the Hall effect on the reconnection rate at line-tied magnetic X-points

“…Our results are qualitatively consistent with an exact selfsimilar solution for the current sheet formation in Hall MHD, which predicts the sheet formation time that is an increasing function of a global length scale (Litvinenko 2007). They are also broadly consistent with exact steady incompressible Hall MHD solutions which suggest that Hall effects can both speed up and slow down magnetic reconnection, depending on the structure of the magnetic field (Dorelli 2003;Craig & Watson 2005).…”

Hall MHD and Electron Inertia Effects in Current Sheet Formation at a Magnetic Neutral Line

“…These equations generalise those derived in Ref. [23] for the case γ = const. On differentiating Eq.…”

“…For example, (4.13) is not satisfied for the particular case considered in Ref. [23] where α 0 = β 0 , γ 0 = 0.5 and f 0 = g 0 = 0. We also note that this criterion predicts exponential evolution for the case d i = 0.…”

“…The fundamental equations are presented in Section 2. In Sections 3 and 4, we generalise previous studies [23] by considering a general set of initial conditions and derive a criterion for the formation of a finite-time singularity. The new solution reduces to the exponentially evolving MHD solution upon setting the Hall term to zero.…”

Comment on “Exact solutions and singularities of an X-point collapse in Hall magnetohydrodynamics” [J. Math. Phys. 59, 061509 (2018)]

“…In a recent paper [1], Janda investigated the solutions to a 2D model based on the equations of dissipationless Hall magnetohydrodynamics (MHD). The model (also previously studied by Litvinenko [2] and Litvinenko & McMahon [3]) involves the coupling between the incompressible fluid velocity V = z × ∇φ + V z z, which is expressed in terms of the electrostatic potential φ(x, y, t) = γ(t) xy and the parallel fluid velocity V z (x, y, t) = β 1 (t) x 2 + β 2 (t) y 2 , and the magnetic field B = ∇ψ × z + B z z, which is expressed in terms of the parallel vector potential ψ(x, y, t) = α 1 (t) x 2 − α 2 (t) y 2 and the parallel magnetic field B z (x, y, t) = b(t) x y. Here, the Hall MHD fields (φ, V z , ψ, B z ) are dimensionless and (x, y) have been normalized to a characteristic length scale L. The time-dependent coefficients (α 1 , α 2 , β 1 , β 2 , b) satisfy the coupled ordinary differential equations [1] α 1 − 2γ α 1 = 2b (d i α 1 − d 2 e β 1 ), (1)…”