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Introduction

Magnetic reconnection occurs as a result of non-ideal effects in Ohm's law. Physically, the close encounter of magnetic field lines causes the magnetic field gradients to become locally strong, thus enhancing the formally weak non-ideal process in Ohm's law. Hence, reconnection is a localized process. Reconnection allows the rapid conversion of magnetic energy into kinetic energy [1]. One case of particular interest is magnetic reconnection at X-type neutral points. These points are essentially where hyperbolic magnetic fields meet and create magnetic neutral lines within the plasma flow in the form of an X-point [13].

A thin neutral current sheet is then formed when plasma collapses near the neutral line of the applied magnetic field. In resistive magnetohydrodynamics (MHD) the ion inflow is the only means to transport magnetic flux into the reconnection layer. As the magnetic flux continually accumulates in the region of the neutral sheet, the total current and the sheet width increase until large magnetic pressure gradients develop, which inhibit the ion inflow [15]. The Hall effect [16] can overcome this [4,7], thanks to the decoupling of electrons from ions on length scales below the ion skin depth di. If the reconnection layer width is less than di , the electron inflow can keep transporting the magnetic flux into the reconnection layer and hence reduce the flux pile-up. Previous numerical work [12, 10, 5, 17] indicated that the dissipation in Hall MHD, as di increases, changes from an elongated sheet geometry (Sweet-Parker type [8, 9]) to a more open X-point geometry (Petschek type [3]). However, recent fully kinetic simulations [3, 6] and EMHD-based treatments [2] have shown that elongated current sheets are also possible. To deepen the controversy, more recent particle-in-cell simulations [11] show spatial localization of the out-of-plane current to within a few di 's of the X-line.

In an effort to shed more light on this issue, Shivamoggi [15] investigated whether the Hall effects favor the hyperbolicity of the magnetic field near a two-dimensional X-type magnetic neutral line. In this investigation, asymptotic solutions to a non-linear ordinary differential equation are used. In our investigation we shall determine if those approximations satisfy exact equations in order to give the solutions more credence.

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