University of Central Florida Undergraduate Research Journal - Numerical Solutions to Hall Magnetohydrodynamic Equations near an X-type Magnetic Neutral Line
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The Hall Magnetohydrodynamic (MHD) model is a new paradigm for describing fast magnetic reconnection processes in space and laboratory plasmas. Current sheets form and store enormous amounts of magnetic energy at X-type magnetic neutral points, which is released as magnetic storms when the sheets break up. The fast magnetic reconnection process impacts solar flares and Earth's geomagnetic sub-storms, which affect global weather. The fast magnetic reconnection process also influences fusion reactors, which may be used as a future energy source. Numerical analysis of approximate solutions to the Hall MHD equations at X-type magnetic neutral points offer these solutions further credence and enhance our understanding of the aforementioned physical phenomena. The solutions to the Hall MHD equations must obey compatibility conditions like incompressibility and an exact invariant. The result, achieved through asymptotic analysis, correctly fits physical laws up to O(t). On the other hand, the exact invariant can be used to refine the asymptotics to any order of accuracy. The asymptotic solutions fully satisfy the plasma incompressibility condition. The most notable of changes between the MHD solutions and the Hall MHD solutions is the ability of the Hall term to prevent the finite-time singularity that appears in MHD. This leads to the prevention of the current density blow-up at long times in Hall MHD.

KEYWORDS: plasma physics, magnetic reconnection, nonlinear differential equations, Taylor's theorem

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