3. Currents generated in a partially ionized medium
3.1. Basic equations
Here is the electric conductivity and where and are respectively the neutral-ion and neutral-electron collisional frequencies taken from Kubát and Karlický (1986). The equations (6) and (7) are derived in Appendix (Eqs. B.3 and C.1). The radial and vertical current densities are related by the particle conservation law
and we have the additional relations between variables
The radial variation of the vertical component of the field was taken to be identical to the one that corresponds to null azimuthal velocities and to a linear dependence on radial distance of the neutral radial velocity. Then the set of Eqs. (4) to (7) was solved iteratively. Since , the vertical electric current intensity, appears in Eq. (6), the current densities and cannot be derived locally, i.e. independently of the contribution of the other atmospheric layers. A circuit model is necessary to relate the total current to the current densities.
3.2. Electric current circuit
Every layer l acts as a current generator in a circuit that extends above and below this layer. Two main circuit models are possible. One where the flux tube is at the foot of a magnetic loop, and the circuit extends to the other foot of this loop. Another model where the flux tube opens and keeps its axial symmetry. In this case, currents transverse to the magnetic field are required in the upper coronal and lower convective part of the flux tube to close the circuit.
In all cases the contributions of every layer to the circuit
regions placed above and below it are proportional to the inverse
ratio of the resistances of these parts of the circuit. The use of the
parallel conductivity to estimate the resistance is valid for a loop
type circuit and, as shown below, is still a good approximation in the
case of an open flux-tube:
where L and are the length of the flux tube and the radial distance between the two opposite cylindrical current channels. The transverse Hall conductivity is equal to , where and are respectively the electron gyrofrequency and the ion-electron collisional time, and is the conductivity parallel to field lines. Consequently in the low density corona the ratio is close to . For an electron density in the coronal part of the flux tube of 1010 cm-3 and a magnetic field of 100 G, the product is . Therefore, for a radial distance between the two cylindrical shells of current of about 10 km, and over a length L of 104 km, the transverse and longitudinal resistances are equal. On the other hand, in the convective zone for a magnetic field of 100 G, at densities higher than about 1018 cm-3, the conductivity is isotropic. Consequently, we estimated the resistance of the coronal and photospheric parts of an open flux tube by using for the conductivity.
The resistance of the part of the circuit respectively above and below a layer l are respectively
where the integration is made from the layer l to the uppest layer N of the VAL C atmospheric model (Vernazza et al. 1981) used and
where the integration is made from the layer l to the lowest layer of the VAL C atmospheric model. Consequently the contributions and to the current flowing above or below every layer l are given by
The resulting current density in the layer k is
and the total current . Iterations are made between the two systems of Eqs (5) to (8) and (10) to (12). It is worth noticing that equation (5) shows that the lower the higher the radial current density must be, in order to generate a given breaking force. Consequently, the radial currents are generated in low vertical magnetic fields.
As shown in subsequent sections, pressure enhancements are generated in the flux tube that either slow down the inward radial velocity of neutrals or generate outward radial motions. The current generating layers are restrained to the ones where the radial velocity of neutrals is negative, corresponding to an inflow of angular momentum. This condition limits the vertical extension of the DC current generator.
3.3. Characteristics of the current system
The upper part of the ensemble of current generating layers will send (receive) currents predominantly into (from) the section of the circuit above it, since its resistance is lower than the resistance of the section below it. This conclusion reverses for the lower layers of this ensemble. Therefore, we expect the sign of the field aligned currents to change at some depth in the solar atmosphere. Similar conclusion holds in the model of current generation in a twisted magnetic field frozen in a plasma. In this case the maximum of the magnetic twist is usually assumed to be located at photospheric level, and currents above and below this level flow in opposite directions.
When the partial ionization of the plasma is taken into account, the height at which the vertical currents change of sign depends on the height dependence of the azimuthal velocity. In the numerical application presented here, where the azimuthal velocity at the periphery of the flux tube is constant and equal to 0.3 km s-1, the change of sign occurs at a height of 200 km above the level where the continuum optical depth is unity.
The radial currents are generated in low vertical magnetic fields. The radial current density amplitude shows a maximum and then decreases inwards. Accordingly the vertical currents must flow to neutralize the radial current. Two systems of vertical currents are then generated that flow in opposite directions. These currents flow in two cylindrical shells near the boundary of the flux tube.
The characteristics of the system of currents have been computed for a flux tube of radius 100 km with a vertical magnetic field on the vertical axis of the tube equal to 1000 Gauss and an azimuthal velocity of 0.3 km s-1 at the boundary. The radial variation of the vertical component of the magnetic field is plotted in Fig. 1. Figs. 2a,b show the radial dependence of the radial and vertical current densities and at the altitudes of -25 and 50 km above , and Fig. 2c gives at the altitudes of 150 and 350 km.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998