Current sheets in twodimensional potential magnetic fields. III. Formation in complex topology configurations and application to coronal heating.
Abstract
We study the spontaneous formation of a current sheet (CS) in an xinvariant ysymmetric magnetic field B(y,z,t) occupying the halfspace {z>0}, and embedded in a pressureless perfectly conducting plasma. At the initial time t=0, B(y,z,0) 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 Xpoint or is tangent to the yaxis (X and Utopology, respectively), or two separatrices extending to infinity (Itopology). For t>=0, the field is made to evolve quasistatically by imposing its footpoints on the boundary {z=0} to move parallel to the yaxis at the slow velocity v(y,t). It thus passes through a sequence of configurations which are either potential equilibria or quasipotential singular equilibria, the latter containing a CS, assumed a priori to be vertical. We compute analytically B(y,z,t) and its freeenergy contents δ W (t) as functionals of B_z_(y,0,t) (this boundary value depending on B_z_(y,0,0) and v(y,t)), and also, when there is a CS, of the unknown heights z_1_(t) and z_2_(t) 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 Itype, a CS  and a vertical one indeed  is actually present at time t if and only if the potential field B^p^(y,z,t) associated to B_z_(y,0,t) has a Xtopology. (ii) When the initial field is of the Xtype, a CS exists in general at each time t>0, 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 z_1_(t), z_2_(t), B(y,z,t) and δW(t) useful approximate explicit expressions, which are valid just after the CS has started forming at some time t_c_>=0. 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 t_k_ = k τ_D_ (k=1,2, ...,N) 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 τ_D_/τ_ev_ (assumed to be <<1) of a given phenomenological dissipation time τ_D_ to the ideal evolution time τ_ev_ of the system. The relevance of this process for heating a stellar corona is briefly discussed.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 March 1997
 Bibcode:
 1997A&A...319..699A
 Keywords:

 MHD;
 PLASMAS;
 SUN: CORONAE;
 STARS: CORONAE