## 3. Currents generated in a partially ionized medium## 3.1. Basic equationsThe solution of the equation of dynamics for the horizontal motions of a partially ionized plasma (Paper 1, see also Hénoux and Somov, 1993 and 1994) gives a set of four 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 circuitEvery layer 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 The resistance of the part of the circuit respectively above and
below a layer where the integration is made from the layer where the integration is made from the layer The resulting current density in the layer 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 systemThe 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 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
© European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |