## 3. The self-excited dynamo
Sunspots exhibit axial magnetic fields and require a 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.
Assume a seed axial magnetic field that points up, as in the figure. The electric current density induced in a plasma flowing at a velocity in a field is given by Ohm's law for moving conductors, where is the conductivity. The term comes from electric volume charges, if any, in the convecting plasma, while is the electric field induced by a time-dependent magnetic field, according to the law of Faraday. We use cylindrical coordinates , and assume a steady state and axisymmetry: We also set We have set , 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 inside a conductor that moves at a velocity in a magnetic field is given by (van Bladel 1984; Lorrain et al. 1988; Lorrain 1990; Lorrain & Koutchmy 1995, 1996) With the above assumptions, and . Thus, for a steady state, In Fig. 1, the radial component of the plasma velocity is negative, while is positive, so that and 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
points up in Fig. 1, If, instead, the seed magnetic field points down then
and 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 . The axial magnetic
field In Fig. 1, if flares slightly, as it undoubtedly does (Sect. 5), the magnetic force density , which is normal to , 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 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 is positive, and it generates a field that points in the 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 . © European Southern Observatory (ESO) 1998 Online publication: October 21, 1998 |