## 4. A dynamical numerical model for the formation of field-aligned potential drops in the corona of an accretion diskIn this section we study the formation of localized field-aligned electric fields in the corona of an AGN in a idealized dynamical numerical model. For this we use a Cartesian (for simplicity we choose a slab geometry) 3D resistive MHD code that integrates the following balance equations: Here, , , We consider one single sheared coronal loop (see Fig. 1), i.e. a magnetic flux tube with a current flowing due to the magnetic shear. Since we are mainly interested in the region we do not have to take into account coronal current closure. An appropriate idealized initial configuration for the numerical simulation is the following force-free magnetic field, given by (cf. Birk and Otto 1996): where and denote the constant main component and the shear (toroidal) component of the magnetic field, respectively. The choice provides us with a generic magnetic field configuration with a current sheet due to magnetic shear. An alternative choice would be which would require some pressure profile whereas in the present approach we could start with a homogeneous plasma. We note that the actual choice does not alter the results significantly as long as we deal with a current sheet with . We note that we study a single thin acceleration region, as a part of an extended relatively thin current sheet, at about . It is to expected that a number of such regions at different altitudes and latitudes permanently form due to the mechanical shear forces (cf. Sect. 2). The change of through the current sheet , the half-width of the latter and the critical current density , are related by Ampère's law: The threshold current for
microinstabilities can be related to the critical drift velocity
(cf. Papadopoulos 1979) by
. Since the currents are generated by shearing
forces in the disk with and transmitted along
the magnetic field lines into the corona with ,
we expect the critical drift speed
( is the ion thermal velocity). The value of
the scaling factor, In order to model a macroscopic resistive instability at an
altitude As a boundary condition for we use These velocity perturbations with an amplitude chosen as
of the Alfvén velocity mimic
differential or convective plasma motion at one end of the considered
coronal loop that due to the quasi-ideality of the plasma leads to a
further shear of the magnetic field and an increase of the
field-aligned current density. An anomaleous resistivity will be
switched on if this current density exceeds a critical value
(we start with a marginal current density),
and thus, gives rise to the macroscopic resistive instability. We
localize the resistivity in the For the numerical realization we pose line symmetry as boundary
conditions in the During the dynamical evolution of the resistive instability the field-aligned current density is reduced as shown in the snapshot after 60 Alfvénic transit times (Fig. 2) (the initial current density was chosen as 1 in normalized units). This reduction appears due to the fact that magnetic energy stored due to the shear can now be released and is converted in bulk kinetic energy as well as thermal energy during the instability process.
Fig. 3 shows a perspective view of magnetic field lines. One set of field lines traverse the entire acceleration region whereas the other one represents flux that has been reconnected slightly earlier.
The main point is that during the dynamics a relatively strong field-aligned electric field ( evolves very fast (, ) that grows up to a maximum value of for the parameters chosen (Fig. 4).
Since the strength of the electric field depends in particular on the magnitude of the resistivity (or the magnetic Reynoldsnumber) and the critical current density, our quantitative results are to be understood as rough but quite realistic order of magnitude estimations. The evolving field-aligned electric fields can accelerate charged particles as it is the case in so-called "auroral potential structures" observed in the Earth's magnetosphere (cf. Mozer 1981) and consequently may play an important role for the solution of the injection problem. Fig. 5 shows the generalized electric potential
© European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |