CURRENT SHEETS IN 2-DIMENSIONAL POTENTIAL MAGNETIC-FIELDS .3. FORMATION IN COMPLEX TOPOLOGY CONFIGURATIONS AND APPLICATION TO CORONAL HEATING

We study the spontaneous formation of a current sheet (CS) in an x-inv ariant y-symmetric magnetic field B(y, z, t) occupying the half-space {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 X-point or is tang ent to the y-axis (X- and U-topology, respectively), or two separatric es extending to infinity (I-topology). For t greater than or equal to 0, the held is made to evolve quasi-statically by imposing its footpoi nts on the boundary {z = 0} to move parallel to the y-axis at the slow velocity v(y, t). It thus passes through a sequence of configurations which are either potential equilibria or quasi-potential singular equ ilibria, the latter containing a CS, assumed a priori to be vertical. We compute analytically B(y, z, t) and its free-energy contents delta 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 unknow n 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 s how that: (i) When the initial field is of the U- or I-type, a CS - an d 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 X-t opology. (ii) When the initial field is of the X-type, 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 arbitrar ily chosen data and puts a limit on the generality of our model. Final ly, we derive for z(1)(t), z(2)(t), B(y, z, t) and delta W(t) useful a pproximate explicit expressions, which are valid just after the CS has started forming at some time t(c) greater than or equal to 0. As an a pplication, we consider a plasma heating process in which a field evol ving through a sequence of singular equilibria as described above, rel axes at each time t(k) = k tau(D) (k = 1, 2,..., N) to a new potential equilibrium, the vertical CS being destroyed by some reconnection pro cess. We present an estimate of the resulting heating rate, which is f ound to depend on the ratio tau(D)/tau(ev), (assumed to be << 1) of a given phenomenological dissipation time tau(D) to the ideal evolution time tau(ev) of the system. The relevance of this process for heating a stellar corona is briefly discussed.