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Astron. Astrophys. 339, 610-614 (1998)

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3. The self-excited dynamo

Sunspots exhibit axial magnetic fields and require a local azimuthal current distribution, like a solenoid. Many authors have discussed current distributions in sunspots, but without considering the generation of the currents. See, for example, Osherovich & Garcia (1990), and Molodenskii & Solov'ev (1993).

Sunspots, like magnetic elements, are magnetic flux tubes (Lorrain & Salingaros 1993), and they are associated with downflow. Both are self-excited dynamos of a type that was suggested several years ago by the authors listed in the Introduction.

Fig. 1 shows a plausible set of streamlines in a region of the Sun where there is downflow. The velocity vector of the downwelling plasma has a negative radial component, in cylindrical coordinates. The plasma density increases with depth. Below the region shown in the figure, the plasma diverges and returns to the surface. It is presumably this outflow that was observed under c) and d) above.

[FIGURE] Fig. 1. The self-excited magnetic-flux-tube dynamo of a sunspot is similar to that of a magnetic element. For references, see the Introduction and Sect. 3. Schematic diagram of streamlines for the downflowing plasma in a sunspot: the plasma flows downward and inward, and the density increases with depth. Below the region shown here, the plasma flow diverges and turns upward. We use cylindrical coordinates [FORMULA], with the unit vector [FORMULA] pointing up. The radial component [FORMULA] of the plasma velocity [FORMULA] is negative . Assuming a seed magnetic field [FORMULA] that points up, the [FORMULA] field and the resulting induced current, of density [FORMULA], point in the positive azimuthal direction shown. The magnetic field of this induced current points up, in the same direction as the seed field . The induced current thus amplifies the axial magnetic field. This is positive feedback. If, on the contrary, the seed magnetic field points down, then [FORMULA], [FORMULA], and its magnetic field all change sign. The induced current again amplifies the axial magnetic field. We thus have a self-excited dynamo that amplifies a seed axial magnetic field, whatever its polarity. The asymptotic value of B is a function of the power that drives the convection (Lorrain 1995).

Assume a seed axial magnetic field [FORMULA] that points up, as in the figure. The electric current density induced in a plasma flowing at a velocity [FORMULA] in a field [FORMULA] is given by Ohm's law for moving conductors,

[EQUATION]

where [FORMULA] is the conductivity. The term [FORMULA] comes from electric volume charges, if any, in the convecting plasma, while [FORMULA] is the electric field induced by a time-dependent magnetic field, according to the law of Faraday.

We use cylindrical coordinates [FORMULA], and assume a steady state and axisymmetry:

[EQUATION]

We also set

[EQUATION]

We have set [FORMULA], despite the fact that, in Sect. 5, that condition is not quite satisfied. For a more general discussion, see Lorrain & Salingaros (1993).

The electrostatic space charge density [FORMULA] inside a conductor that moves at a velocity [FORMULA] in a magnetic field [FORMULA] is given by (van Bladel 1984; Lorrain et al. 1988; Lorrain 1990; Lorrain & Koutchmy 1995, 1996)

[EQUATION]

With the above assumptions, [FORMULA] and [FORMULA]. Thus, for a steady state,

[EQUATION]

In Fig. 1, the radial component [FORMULA] of the plasma velocity [FORMULA] is negative, while [FORMULA] is positive, so that [FORMULA] and [FORMULA] are both azimuthal, in the direction shown. The increase of the plasma density with depth maintains the negative radial velocity along the flux tube.

The magnetic field of the induced azimuthal electric current [FORMULA] points up in Fig. 1, in the same direction as the assumed seed magnetic field : the induced azimuthal current amplifies any existing axial magnetic field. This is characteristic of a self-excited dynamo (Lorrain 1996).

If, instead, the seed magnetic field points down then [FORMULA] and [FORMULA] change direction and the magnetic field of the induced azimuthal current points down, again in the same direction as the seed field. So self-excitation applies for either polarit y of [FORMULA]. The axial magnetic field B tends to an asymptotic value that is a function of the power that drives the convection (Lorrain 1996).

In Fig. 1, if [FORMULA] flares slightly, as it undoubtedly does (Sect. 5), the magnetic force density [FORMULA], which is normal to [FORMULA], points somewhat downward and anchors the magnetic flux tube, canceling the upward buoyancy force on the tube.

Now suppose that the magnetic field is established with [FORMULA] pointing up, and that the flow reverses sign: the downwelling becomes upwelling. As the upflowing plasma rises, it expands because of the decreasing ambient pressure, its radial velocity [FORMULA] is positive, and it generates a [FORMULA] field that points in the [FORMULA] direction and a corresponding current whose magnetic field opposes the existing field. Since the flux tube has an inherent time constant (Lorrain & Salingaros 1993), its magnetic field decreases more or less slowly. This is just what Kosovichev (1996) observed: there is an upflow in a decaying active region.

The induction equation for this dynamo reduces to the identity [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: October 21, 1998
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