Time‐dependent magnetic annihilation at a stagnation point

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Abstract
Magnetic reconnection is a fundamental process that can take place in astrophysical or laboratory plasmas. It occurs within regions of large magnetic gradient where the magnetic field is no longer frozen to the plasma but instead diffuses through it, releasing magnetic energy and causing a change in the connectivity of the field lines. In particular, on the Sun, magnetic reconnection is believed to play an important role in coronal heating, X ray bright points, solar flares and canceling magnetic features. Here, an exact time‐dependent solution of the MHD equations for magnetic annihilation in response to a time‐varying stagnation point flow is presented. The main assumptions in this model are that the magnetic field lines are straight (so that there is no magnetic tension acting on the plasma) and the flow that carries the field lines together is of stagnation point type. This is a reasonable model for the resistive MHD behaviour near the X point of a reconnecting field, especially when the central diffusion region is long, as in the flux pile up regime (Priest and Forbes 1986). The general solution is used to conduct a series of numerical experiments, namely, the evolution of different initial magnetic profiles in a steady flow; the effect of a sudden change in magnetic diffusivity on an initially steady state; the effects of a velocity that either increases linearly in time or ramps up from one steady value to another. The results exhibit the effects of subtle imbalances in diffusion and advection but have the following general features: (1) a diffusion layer is created, the thickness of which is determined by the nature of the plasma flow; (2) the magnetic field outside the diffusion region is determined by advection and will either exponentially increase, exponentially decay or remain in a steady state, depending on whether the initial magnetic profile B (y) ∼ y−n has n < 1, n > 1, or n = 1, respectively; and (3) the magnetic field within the diffusion layer tends to respond to the advected magnetic field at its edge.

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