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Astron. Astrophys. 324, 461-470 (1997)

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4. A dynamical numerical model for the formation of field-aligned potential drops in the corona of an accretion disk

In 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, [FORMULA], [FORMULA], p and [FORMULA] denote the plasma density, velocity and pressure and the (anomaleous) resistivity. The code makes use of an explicit difference scheme based on the Leapfrog algorithm that is second order in space and time (cf. Otto 1993). The principle train of thought is the following (cf. Otto and Birk 1993; Birk and Otto 1996): The different convective plasma motion at different regions of the coronae of AGN or differential shear motion of the disk itself results in a magnetic shear and thereby the origin of field-aligned electric currents. When this convective shear motion is strong enough the current density exceeds some critical value and current-driven microinstabilities are excited which in course of their nonlinear saturation lead to an anomaleous resistivity. The excitation of microinstabilities can be regarded as a special case of [FORMULA] in Ohm's law as discussed in the previous section. We note that the necessary onset condition for the scenario we have in mind is any violation of ideal Ohm's law that allows for [FORMULA] ; we just concentrate on [FORMULA] - nonidealities for illustration purposes. The formation of localized regions of anomaleous dissipation gives rise to the onset of magnetic reconnection. During the nonlinear dynamical evolution very localized acceleration regions with fairly high field-aligned electric fields form. In these potential structures electrons can in principle be accelerated up to energies of [FORMULA]. In this section we show results for an exemplary chosen set of parameters. Parts of a realistic parameter space and, in particular, consequences of different altitudes and extensions of the acceleration region for the physical situation under consideration are discussed in the following section.

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 [FORMULA] 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 [FORMULA] and [FORMULA] 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 [FORMULA] which would require some pressure profile [FORMULA] 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 [FORMULA].

We note that we study a single thin acceleration region, as a part of an extended relatively thin current sheet, at about [FORMULA]. 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 [FORMULA] through the current sheet [FORMULA], the half-width of the latter [FORMULA] and the critical current density [FORMULA], are related by Ampère's law:


The threshold current [FORMULA] for microinstabilities can be related to the critical drift velocity [FORMULA] (cf. Papadopoulos 1979) by [FORMULA]. Since the currents are generated by shearing forces in the disk with [FORMULA] and transmitted along the magnetic field lines into the corona with [FORMULA], we expect the critical drift speed [FORMULA] ([FORMULA] is the ion thermal velocity). The value of the scaling factor, f, depends on the prevailing kind of microinstability, which in turn, depends on the local plasma parameters (cf. Papadopoulos 1979; Huba 1985). For our order of magnitude estimations we choose f of the order of unity. Notwithstanding the uncertainty of the current concentration process in most cosmical applications we obtain for the half-width of the current generated by the mechanical shear forces [FORMULA], if we assume a shear magnetic field of [FORMULA] and a critical current density of [FORMULA] ([FORMULA] is the ion thermal velocity) with [FORMULA] (as a density profile we assume [FORMULA]) and [FORMULA] (cf. Ulrich 1991; Nandra and Pounds 1994), With these values the electric field is normalized to [FORMULA] ([FORMULA] is the shear Alfvén velocity [FORMULA]). The magnetic Reynoldsnumber depends on the actual kind of microinstability that is excited. If we assume the excitation of the ion-cyclotron-drift turbulence the resulting anomaleous resistivity is [FORMULA] maximum (cf. Papadopoulos 1979; Huba, 1985), where [FORMULA] is the ion-cyclotron frequency, and thus, the magnetic Reynoldsnumber reads [FORMULA]. We note that the lower-hybrid-drift instability would result in a slightly lower resistivity. For numerical reason we use a somewhat lower magnetic Reynoldsnumber [FORMULA] and rescale our quantitative findings accordingly.

In order to model a macroscopic resistive instability at an altitude H in the corona of an AGN (caused by supercritical currents) we apply the following velocity perturbation as an initial condition:


As a boundary condition for [FORMULA] we use


These velocity perturbations with an amplitude chosen as [FORMULA] 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 [FORMULA] (we start with a marginal current density), and thus, gives rise to the macroscopic resistive instability. We localize the resistivity in the z -direction in order to model an acceleration region of the length of [FORMULA] extended along the poloidal magnetic field.

For the numerical realization we pose line symmetry as boundary conditions in the y -direction and carry out the simulations with 49 grid points in x -direction, 39 grid points in y -direction, and 105 grid points in z -direction, where we use a non-uniform numerical grid with a maximum resolution of 0.05 in the x - and z -direction, 0.4 in the y -direction, and 0.2 in the z -direction. The dimensions of our numerical box are given by [FORMULA], [FORMULA], and [FORMULA] in normalized units with a scaling length of [FORMULA], the half-width of the field-aligned current layer. Once again it should be noted that we mainly model the dissipative part of the (in the z -direction, i.e. parallel to the main component of the magnetic field) extended current sheet which at significant higher and lower altitudes is assumed to be ideal.

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 [FORMULA] (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.

[FIGURE] Fig. 2. Surface plot of the field-aligned current density at [FORMULA] after [FORMULA]. The reduction of the current density (up to [FORMULA] of the initial value chosen as 1 in normalized units) caused by the resistive instability is evident.

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.

[FIGURE] Fig. 3. Perspective view of magnetic field lines having a very central (field lines on the left hand side) or more remote trajectory through the acceleration region

The main point is that during the dynamics a relatively strong field-aligned electric field ([FORMULA] evolves very fast [FORMULA] ([FORMULA], [FORMULA]) that grows up to a maximum value of [FORMULA] for the parameters chosen (Fig. 4).

[FIGURE] Fig. 4. Surface plot of the magnetic field-aligned component of the electric field at [FORMULA] after [FORMULA]. The field-aligned electric field may result in coherent acceleration of charged particles.

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 U (obtained by integrating the field-aligned electric field along magnetic field lines that penetrate the [FORMULA] -region) evolving during the nonlinear instability dynamics. Similar to the Earth's auroral acceleration regions thin elongated regions "potential" structures form. Assuming a central acceleration region localized at [FORMULA] (again we note that for [FORMULA] -nonidealities the actual location of the acceleration region depends on the local plasma parameters allowing for localized dissipation) electrons can in principle be accelerated along the poloidal magnetic field up to a Lorentz factor of [FORMULA]. However, this quantitative result obtained by the simulations is to be understood as an upper limit. In order to arrive at a fully consistent description particle simulations has to be performed with the input parameters for the electric and magnetic field given by the MHD simulations. The resulting particle distribution function has to be analyzed carefully in order to evaluate the effectiveness of the reconnection process in some detail. This task is beyond the scope of this contribution, in which we are restricted to a more qualitative level showing that reconnection is important from order of magnitude estimations, and will be dedicated to future investigations.

[FIGURE] Fig. 5. The generalized electric potential U after [FORMULA]. Similar to the Earth's auroral acceleration regions a thin elongated structure evolves with a maximum of [FORMULA]. In such regions charged particles can in principle be accelerated up to [FORMULA].
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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998