Topological skeleton of the 2-D slightly non-ideal MHD system close to X-type magnetic null points – an analysis of the general solution for the generic case

Abstract

Abstract. The appearance of eruptive space plasma processes, e.g. in eruptive flares as observed in the solar atmosphere, is usually assumed to be caused by magnetic reconnection, often connected with singular points of the magnetic field.We are interested in the general relation between the eigenvalues of the Jacobians of the plasma velocity and the magnetic field and their relation to the shape of a spatially varying, localized non-idealness or resistivity, i.e. we are searching for the general solution. We perform a local analysis of almost all regular, generic, structurally stable non-ideal or resistive MHD solutions. Therefore we use Taylor expansions of the magnetic field, the velocity field and all other physical quantities, including the non-idealness, and with the method of comparison of coefficients, the non-linear resistive MHD system is solved analytically, locally in a close vicinity of the null point.We get a parameterised general solution that provides us with the topological and geometrical skeleton of the resistive MHD fields. These local solutions provide us with the "roots" of field and streamlines around the null points of basically all possible 2-D reconnection solutions. We prove mathematically that necessarily, the flow close to the magnetic X-point must also be of X-point type to guarantee positive dissipation of energy and annihilation of magnetic flux. We also prove that, if the non-idealness has only a onedimensional, sheet-like structure, only one separatrix line can be crossed by the plasma flow, similar to known reconnective annihilation solutions.

Topological Structures of Velocity and Electric Field in the Vicinity of a Cusp-type Magnetic Null Point

“…Our topological considerations revealed that the resistivity surface is a hyperbolic paraboloid, meaning that the resistivity has a saddle-point structure in the vicinity of the magnetic cusp rather than a maximum of the resistivity like in the case of a regular structurally stable magnetic X-point (Nickeler et al 2012).…”

“…Magnetic reconnection is a key process for understanding magnetic structures and their topological change in astrophysical plasmas. Many investigations analyzing especially topological or geometrical properties of physical fields, focusing either on some or involving all of them, have been done in various contexts and approaches: for example resistive magnetohydrodynamics (MHD) in 2 dimensions (2D) (Priest & Cowley 1975), kinematic ideal MHD in 3D (Lau & Finn 1990), purely magnetically in 2D and 3D (Parnell et al 1996), resistive MHD in 3D with constant resistivity (Titov & Hornig 2000), resistive MHD in 3D with locally varying but prescribed resistivity (Hornig & Priest 2003;Priest & Pontin 2009), Hall-MHD in 2.5D (Litvinenko 2009), resistive, kinematic MHD in 3D with varying resistivity (Wyper & Jain 2011), and resistive MHD in 2D with localized but consistent resistivity (Nickeler et al 2012).…”

“…In 2D, the search of the Poincaré class of the total electric field is linked to the Poincaré class of the resistivity. It is well known that for exact and analytical reconnection solutions, it is not sufficient to assume any non-ideal term or non-idealness or for example a constant resistivity (Priest et al 1994;Craig & Henton 1995;Watson & Craig 1998;Throumoulopoulos & Tasso 2000;Titov & Hornig 2000;Nickeler et al 2012Nickeler et al , 2014. For the classical role of reconnection in 2D it is inevitable that the plasma flow can cross both magnetic separatrix branches, as this scenario constitutes the reconnection solution.…”

“…This characteristic does often not exist if a constant resistivity is assumed (Craig & Henton 1995), or if the non-idealness shows a one-dimensional character, like in the case of current (sheet) depending resistivities. Very often such resistive or non-ideal terms only allow for crossing one of the two branches of the separatrix (Nickeler et al 2012), which are the so-called reconnective annihilation solutions.…”