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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
Received 25 July 1996 / Accepted 20 August 1996
We study the spontaneous formation of a current sheet (CS) in an x-invariant y -symmetric magnetic field occupying the half-space , and embedded in a pressureless perfectly conducting plasma. At the initial time , 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 , the field is made to evolve quasi-statically by imposing its footpoints on the boundary to move parallel to the y -axis at the slow velocity . 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 and its free-energy contents as functionals of (this boundary value depending on and ), and also, when there is a CS, of the unknown heights and 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 associated to has a X-topology. (ii) When the initial field is of the X-type, a CS exists in general at each time , 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 , , and useful approximate explicit expressions, which are valid just after the CS has started forming at some time .
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 () 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 (assumed to be ) of a given phenomenological dissipation time to the ideal evolution time 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
Online publication: July 3, 1998