## 5. Effects of the currents at chromospheric heightsThe horizontal force balance equation can be written as The third term on the LHS of Eq. (19) gives the force. The direction of this force changes between the two vertical current systems. At the boundary of the flux tube in the external DC current shell, the radial component of the electromagnetic force is directed outwards and can be compensated only by a pressure gradient. The resulting minimum of pressure between the two current systems is approximatly equal to where and are the external pressure and the maximum value of the total vertical current . Consequently the external cylindrical current system must open and increase in diameter in order to reduce the amplitude of the outward electromagnetic force. In the internal current shell, the force is directed inwards. By pinch effect, this force can maintain a strong vertical magnetic field at the center of the flux tube, evoiding its opening with height and allowing a thin, nearly constant cross section, flux tube to extend into the chromosphere. Integrating Eq. (19) from the axis of the flux tube to the location where the current is maximum leads to Here is the radial distance at which is maximum, is the external pressure and . The magnetic field at the boundary of the flux tube is assumed to be negligible. For low pressure forces, the magnetic field structure is force-free, corresponding to a balance between the energy stored in the vertical and azimuthal components of the field. The first term on the RHS of Eq. (21) represents the energy of the vertical component and, since the magnetic flux is constant, is proportional to . Then, for a constant maximum current , if plasma is injected from the photosphere into the chromospheric part of a magnetic flux tube increasing the gas pressure, the tube must widen in order to reduce this first term. The equilibrium size of the flux tube will depend on the gas pressure, which is dependent on the flux and ionization state of the gas injected in the tube since, as discussed below, the non-ionized fraction of the plasma can escape across magnetic field lines. For a prescribed radial dimension of the internal current shell, the radial dependence of pressure necessary to satisfy the force balance equation is shown in Fig. 3a at three heights in the chromosphere. This figure corresponds to a flux tube of 100 km radius with an azimuthal velocity of neutrals high enough to generate a vertical current that could confine a 1000 G field in a force-free situation. The radius corresponding to the force-free solution is proportional to the current intensity : With the boundaries conditions used here (i.e.
G and
km s
Then the high pressure gradient required to balance the force leads to a high speed radial outflow of neutrals as shown in Fig. 3b. This flow of neutrals across lines of force is possible since the densities are small enough for ions and neutrals not being coupled by collisions. The height dependence of the maxima of outward radial velocities of neutrals, when present, and of the inward velocities of ions and neutrals at the same radial distance are represented in Fig. 4. In the photosphere, ions and neutrals are collisionaly coupled. Higher in the atmosphere, neutrals can move across the field lines. The reversal in the sign of the neutral radial velocity takes place around the temperature minimum region.
The situation described in the preceding section requires that a
significant upflow of gas balances the escape of neutrals at
chromospheric level. The upward velocity is
such that ; where is the
radial extension of the photospheric pressure increase and
is the vertical extension of the chromospheric
layer from which neutrals escape with a velocity
, and is the ratio of
chromospheric to photospheric densities of neutrals. An outflow at a
velocity of 10 km s © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |