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

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3. A kinematic description of field-aligned electric potential drops in the corona of an accretion disk

Magnetic fields have long been considered as an important element in the dynamics of accretion disks, primarily as a mechanism for supplying internal stresses required for efficient angular momentum transfer (Eardley and Lightman 1975; Ichimaru 1977; Hawley and Balbus 1991, 1992). In these models magnetic fields can be generated within the inner portion of an accretion disk by the joint action of thermal convection and differential rotation along Keplerian orbits. Field amplification will then be limited by nonlinear effects; as a consequence of buoyancy, magnetic flux will be expelled from the disk, leading to an accretion disk corona consisting of many magnetic loops where the energy is stored and probably transferred via magnetic reconnection into heat and particle acceleration.

The buildup of magnetic fields within the disk is limited by nonlinear effects related to convection. Since convection takes place primarily perpendicular to the plane of the disk, we shall assume differential rotation to remain the dominant mechanism for the toroidal magnetic field [FORMULA] generation; the generation of the poloidal field [FORMULA] will then be dominated by convection-mediated effects. For convection cells whose aspect ratio is [FORMULA], the poloidal magnetic field spatial scale will then be of the order of the convective cell size H, equal to the half-thickness of the disk. To describe the generation of [FORMULA] we invoke magnetic flux conservation:

[EQUATION]

The maximal toroidal field strength is given by the energy density of the turbulence in the disk (Galeev et al. 1979)

[EQUATION]

where [FORMULA] denotes the sound velocity.

The temperature [FORMULA] of the disk is given by (Straumann 1986)

[EQUATION]

with the temperature at the inner edge [FORMULA] of the disk

[EQUATION]

Here we use the accretion rate [FORMULA]. This temperature corresponds to a sound velocity of [FORMULA]

With the same parameters for the particle density we obtain a central value of about [FORMULA]. We get the maximum toroidal field strength of about [FORMULA] or a poloidal field of about [FORMULA], respectively.

Field-aligned electric fields can be discussed either in terms of generalized Ohm's law

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA] denote the electric field, the plasma velocity and some unspecified nonidealness [FORMULA]. One either has to specify for the parallel component of the nonidealness [FORMULA] or one can explore definite properties of the magnetic field and the plasma flow that necessarily imply a significant field-aligned electric field on the grounds of a kinematic approach (Schindler 1991; Schindler et. al 1991). The formation of such parallel electric fields can be understood as a consequence of some driving voltage [FORMULA] (where the integration is carried out along a magnetic line element [FORMULA]) and a local violation of ideal Ohm's law (cf. Schindler et al., 1991; Kuijpers, 1996).

It is a matter of question which processes lead to current concentrations and thereby to [FORMULA]. Possible mechanisms are superheating-instabilities (Coppi and Friedland 1971; Coppi 1975; Spicer 1976) radiation induced instabilities (Schmutzler and Lesch 1989) or various kinds of current-driven microinstabilities (Lesch 1991), in particular the lower-hybrid-drift and the ion-cyclotron drift instabilities that are distinguished from others by their rather non-stringent onset criteria (cf. Papadopoulos 1979; Huba 1985).

Obviously, the kinematic approach in terms of magnetic field and the flow patterns bypasses the physical foundation of a specified violation of ideal Ohm's law, e.g. the microscopic mechanism by which field-aligned currents become unstable and give rise to localized regions of anomaleous momentum transfer. Making use of this approach one has to discuss the relevant quantity (cf. Hesse and Schindler 1988)

[EQUATION]

where the integral is extended over the arc length of a given magnetic field line ds. The quantity U is to be understood as a "generalized" field-aligned electric potential that for a steady state configuration is identical to the electrostatic potential. Schindler et al. (1991) show that U has to satisfy the following equations

[EQUATION]

and

[EQUATION]

Here, [FORMULA] and [FORMULA] denote the Euler potentials ([FORMULA]) and the brackets denote differences in the quantities on either sides of the nonidealness along magnetic field lines. Thus, Eqs. (10) and (11) show that the difference in the magnetic field and the plasma flow on either side outside the nonideal region is a measure for the generalized potential U. This measure does not depend explicitly on the rather involved physics that support the anomaleous dissipation required for [FORMULA].

We define appropriate Euler potentials by assuming an almost bipolar magnetic field, which implies [FORMULA] where [FORMULA] is the azimuthal angle in spherical coordinates and [FORMULA] and [FORMULA] denote the dipole moment and the radius of the central compact object. If one maps the values for [FORMULA] and the poloidal angle [FORMULA] into the corona of the AGN by keeping them constant, Euler potentials are defined as [FORMULA]. From Eq. (10) we get an order of magnitude estimation for U:

[EQUATION]

The typical magnetic flux in Quasar jets is found to be [FORMULA] (Standke et al. 1996). In order to accelerate electrons up to [FORMULA] we need a generalized potential drop [FORMULA]. Thus, if we think about intraday variability (cf. Standke et al.1996), which gives an upper limit of [FORMULA], and assume a latitudinal width of the acceleration region of the order of [FORMULA] the poloidal magnetic flux difference has to be [FORMULA], i.e. only a small amount of the total magnetic flux is associated with the acceleration process. If we assume the acceleration region to be located in a distance of about [FORMULA] ([FORMULA] is the Schwarzschild radius of the central object), which implies [FORMULA] for a magnetic field profile [FORMULA] and [FORMULA], the region of [FORMULA] covers an area of about [FORMULA]. For any effective particle acceleration the length of the acceleration region [FORMULA] must not exceed the loss length due to synchrotron radiation

[EQUATION]

In the considered physical situation this implies current sheets with widths of the order of [FORMULA] with [FORMULA] (cf. Sect. 5). However, we note that variability on shorter time scales allows for thinner current sheets.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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