Dynamos and the Solar Dynamo

A dynamo is a system where a flow field v can sustain by induction a field B against ohmic dissipation and where B is amplified by stretching the field lines [114]. The inductive term in the induction equation (13.5) can be written as (see also 13.46),

at where account has been given to V ■ B = 0. Depending on the velocity gradient, the first term on the right can produce an exponential growth of the field through the induction equation and build a dynamo. The velocity field at work in the solar dynamo is the solar differential rotation, both radially and latitudinally. A stellar dynamo requires that both the poloidal and toroidal components are sustained against ohmic dissipation. It requires the two following effects to generate each of the field components from the other one; if not, the field progressively vanishes by ohmic dissipation:

- The a effect: let us first consider a toroidal field, with field lines parallel to the equator. A small loop of a field line may appear and be essentially contained in a plane orthogonal to the equator. This loop generates a horizontal electrical current, which by Ampere's law creates a field with a poloidal component (a loop could also appear on a poloidal field and create a toroidal field).

- The co effect: a poloidal field in a differentially rotating star is stretched horizontally, creating toroidal field components.

13.1.4.1 The Solar Dynamo

The action of the solar dynamo can be decomposed in the following steps where the a and co effects intervene:

1. Near the minimum of solar activity, there is a weak axisymmetric dipole field

1 G) emerging at high latitude and extending far out in the corona, the field lines cross the equatorial plane at a distance of a few solar radii and connect back to the opposite polar region.

2. The polar regions rotate every 34 days, while the equatorial regions do it in 25.5 days. The inner frozen-in field lines are stretched out in the east-west direction.

The winding-up of the field lines generates (a effect) a strong toroidal field component (several 102 G at low latitudes), converting some of the kinetic energy of differential rotation into magnetic energy. These toroidal components have opposite directions in the two hemispheres.

3. The vertical convective motions create upward magnetic loops, with their planes still oriented in the toroidal direction. This further intensifies the field which locally reaches an intensity of a few kG. The loops of the field lines are twisted in the north-south direction by the Coriolis force, generating a poloidal field component (a effect). When the bunches of field lines erupt at the solar surface, as a result of magnetic buoyancy (Sect. 13.7.2), they produce bipolar active regions. The ordered toroidal field at the base of the convective zone may reach intensities of ~ 104-105 G before erupting.

4. The feet of the emerging loops migrate toward lower latitudes, from about 30-40° to 15°, maybe due to meridional circulation and magnetic diffusion. Magnetic reconnection of northern and southern loops, i.e., a sudden change of the field geometry closing the loops in a different way, is able to rebuild the poloidal field across the equator with a polarity opposite to the initial one. The reconnection liberates large closed loops feeding energy into the corona. The new poloidal field then retracts below the surface in equatorial regions reproducing again the geometry of the first step in the cycle.

The dynamo in a convective region derives its energy mostly from the energy flux of the star, thus there is plenty of energy to feed the dynamo. The main remaining questions about the solar cycle [114] regard the generation of the poloidal field component, the roles of meridional circulation and of small scale turbulence, etc.

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