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Astron. Astrophys. 318, 947-956 (1997)

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5. Effects of the currents at chromospheric heights

The horizontal force balance equation can be written as

[EQUATION]

The third term on the LHS of Eq. (19) gives the [FORMULA] 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 [FORMULA] between the two current systems is approximatly equal to

[EQUATION]

where [FORMULA] and [FORMULA] are the external pressure and the maximum value of the total vertical current [FORMULA]. Consequently the external cylindrical current system must open and increase in diameter in order to reduce the amplitude of the outward electromagnetic [FORMULA] force.

In the internal current shell, the [FORMULA] 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 [FORMULA] is maximum leads to

[EQUATION]

Here [FORMULA] is the radial distance at which [FORMULA] is maximum, [FORMULA] is the external pressure and [FORMULA]. 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 [FORMULA]. Then, for a constant maximum current [FORMULA], 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 [FORMULA]:

[EQUATION]

With the boundaries conditions used here (i.e. [FORMULA]  G and [FORMULA]  km s-1), the radial dimension  [FORMULA] is less than 100 km. Therefore, as illustrated in Fig. 3a, for these boundary conditions a significant pressure must be present in the tube to enlarge it to a radial size of 100 km.

[FIGURE] Fig. 3. Radial dependence at three heights in the atmosphere, respectively 655, 705, and 755 km, for a flux tube of radius 100 km with an azimuthal velocity of neutrals at its boundary of 300 m s-1 of respectively: a - [FORMULA] the ratio of the local gas pressure to the external gas pressure, b - [FORMULA] the radial velocity of neutrals. For each height, the two scale factor values give the inverse of the required increase of the radius of the internal and external cylindrical currents required for having a positive gas pressure at center and between the two currents shells.

Then the high pressure gradient required to balance the [FORMULA] 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.

[FIGURE] Fig. 4. Vertical dependence of the maxima of the radial velocities of neutrals (dotted line) and ions (dashed line).

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 [FORMULA] is such that [FORMULA] ; where [FORMULA] is the radial extension of the photospheric pressure increase and [FORMULA] is the vertical extension of the chromospheric layer from which neutrals escape with a velocity  [FORMULA], and [FORMULA] is the ratio of chromospheric to photospheric densities of neutrals. An outflow at a velocity of 10 km s-1 over a depth of 50 km at a height of 800 km implies a moderate upflow [FORMULA]  m s-1 compared to the upflow velocity found in Sect. 4.1. The consequences of the possible upflows and outflows of neutrals across lines of forces found in this Section and in Sect. 4 are examined below.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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