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Astron. Astrophys. 319, 699-719 (1997)

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Current sheets in two-dimensional potential magnetic fields

III. Formation in complex topology configurations and application to coronal heating

J.J. Aly 1 and T. Amari 1, 2

1 CEA/DSM/DAPNIA, Service d'Astrophysique (CNRS/URA 2052), C.E. Saclay, F-91191 Gif-sur-Yvette Cedex, France
2 Observatoire de Paris, DASOP (CNRS/URA 326), F-92195 Meudon Principal Cedex, France

Received 25 July 1996 / Accepted 20 August 1996

Abstract

We study the spontaneous formation of a current sheet (CS) in an x-invariant y -symmetric magnetic field [FORMULA] occupying the half-space [FORMULA], and embedded in a pressureless perfectly conducting plasma. At the initial time [FORMULA], [FORMULA] is potential and quadrupolar, and therefore its lines in a poloidal plane have a complex topology: there is either one separatrix, which contains a neutral X-point or is tangent to the y -axis (X- and U-topology, respectively), or two separatrices extending to infinity (I-topology). For [FORMULA], the field is made to evolve quasi-statically by imposing its footpoints on the boundary [FORMULA] to move parallel to the y -axis at the slow velocity [FORMULA]. It thus passes through a sequence of configurations which are either potential equilibria or quasi-potential singular equilibria, the latter containing a CS, assumed a priori to be vertical.

We compute analytically [FORMULA] and its free-energy contents [FORMULA] as functionals of [FORMULA] (this boundary value depending on [FORMULA] and [FORMULA]), and also, when there is a CS, of the unknown heights [FORMULA] and [FORMULA] of its bottom and top, respectively. We derive equations satisfied by the latter quantities, and use them to show that: (i) When the initial field is of the U- or I-type, a CS - and a vertical one indeed - is actually present at time t if and only if the potential field [FORMULA] associated to [FORMULA] has a X-topology. (ii) When the initial field is of the X-type, a CS exists in general at each time [FORMULA], but it is vertical if and only if a quite specific condition is satisfied - which may not be the case for arbitrarily chosen data and puts a limit on the generality of our model. Finally, we derive for [FORMULA], [FORMULA], [FORMULA] and [FORMULA] useful approximate explicit expressions, which are valid just after the CS has started forming at some time [FORMULA].

As an application, we consider a plasma heating process in which a field evolving through a sequence of singular equilibria as described above, relaxes at each time [FORMULA] ([FORMULA]) to a new potential equilibrium, the vertical CS being destroyed by some reconnection process. We present an estimate of the resulting heating rate, which is found to depend on the ratio [FORMULA] (assumed to be [FORMULA]) of a given phenomenological dissipation time [FORMULA] to the ideal evolution time [FORMULA] of the system. The relevance of this process for heating a stellar corona is briefly discussed.

Key words: MHD – plasmas – Sun: coronae – stars: coronae

Send offprint requests to: T. Amari

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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