What drives the solar activity and gives rise to sunspots? This is a key question which is still difficult to answer, although there are some hypotheses about various
mechanisms. The answer is most likely to involve magnetic interaction with turbulent flow and electric conduction.
Petrovay (2000) provides a short review on the matter, basing the discussion around an equation describing magnetic induction in a turbulent conductive medium:
In this equation the term dtB denotes the rate of change in the magnetic field B, and the parameter e describes the flow structure: e = aB — flV x B, where the parameters a and fl are functions describing the nature of the turbulent velocity fields.
According to Petrovay there are three different basic models attempting to explain the solar activity:
a Overshoot models b Interface dynamos c Flux transport circulation conveyor belt
These models must be able to account for the observed solar features: (i) the pole-to-equator diffusion in the convective zone with an 11/22 year periodicity; (ii) the characteristic migration patterns in the butterfly diagram shown in Figure 4.3; (iii) the confinement of large active regions to ±< 35°; (iv) the radial field at low latitudes appears to be ^ radians out of phase with the toroidal field at the same latitude; and (v) the phases of the two branches of the butterfly diagrams tend to have a difference in phase by radians so that the polar field reversal occurs slightly after the sunspot maximum.
Large-scale solar active regions in Figure 4.3, according to Petrovay, can be interpreted as tracers of a subsurface toroidal magnetic field. The latitudinal migration pattern can be partly due to the toroidal and partly due to a poloidal field structure. A toroidal field gives a zonal displacement whereas a poloidal structure can explain meridional displacements.
The mean field dynamo theory assumes that most of the shear (fi) is concentrated in a thin layer near the bottom of the convergence zone also known as the tachocline. The tachocline thickness is « 0.1R0 (R0 is the solar radius). The mean field dynamo theory has been the framework for a number of different models. The most recent models can be classified under four different categories: (a) overshoot layer models (OL dynamos), (b) distributed wave models (IF dynamos), (c) co-spatial transport models (CP dynamos), and (d) distributed transport models (BL dynamos). A short summary of these different model types is given below. A main difference between the various models can be regarded as different ways of interpreting and treating the a-effect (see box): cyclic convection (a is positive in the unstable
layer and negative in the overshoot layer below); magnostrophic waves (a is negative); flux loop a (positive); unstable global-scale Rossby waves on the tacho-cline with non-zero mean helicity that may produce an a-effect.
Models with positive so-called "a-effect"4 (in the solar convective zone) that is consistent with an inwards increasing rotational rate can reproduce the 11-year cycle period. The equatorward migration of the sunspots (butterfly diagram branches) can be explained in terms of a dynamo wave and longitude-latitude phase relationship. The problem is that the radial rotation profile that this model assumes seems to be wrong.
It has been observed that short-lived active regions in the high latitude tend to lie on the backward extension of the low latitude butterfly wings similar to polar faculae. There have also been suggestions that about half of the polar faculae are associated with east-west oriented magnetic dipoles which have been interpreted as parts of the toroidal field.
It was discovered in the 1970s that the magnetic field in the solar photosphere is present in strong intermittent form and concentrated in long flux tubes. This observation is problematic in terms of our physical interpretation. The flux tube theory,
4 Interaction between the flow structure and the magnetic field on which e depends in equation (4.5).
on which the flux emergence models are based, states that the tubes are stored near the convection zone base or lower overshoot layer, but this means that the field strength is difficult to explain (|£| « 105 G).
Overshoot layer models, also known as "co-spatial wave models'', require a < 0 in order to get the right migration direction. These models do a good job predicting the butterfly diagrams, but tend to produce cycle periods that are too short.
Distributed wave models assume an abrupt spatial change in diffusivity that results in the excitation of dynamo waves. Strong toroidal fields can be explained by these models, but evaluation of these are hindered by numerical difficulties.
Co-spatial transport models explain the field migration as a result of density pumping or advection of the magnetic field. These models do not address the question regarding the origin of the deep toroidal field.
Distributed transport models do not interpret the butterfly diagram as a manifestation of a dynamo wave, but a consequence of a conveyor belt action. Low latitudinal confinement of strong activity needs another dynamo mechanism kick and requires unrealistically low turbulent diffusivity. These models account for the confinement of the strong activity at low latitudes and predict migration characteristics similar to those in Figure 4.3.
In summary, none of the models can fully account for the observed features of the solar cycle and all these have to make some assumption about the Sun. There have been some severe difficulties, even with the classical theory of the dynamo wave origin of the butterfly diagram. Thus, the jury is still out, according to Petrovay, on what makes the Sun "tick".
Pores and sunspots are exclusively formed within regions of enhanced magnetic fluxes recently emerged from the solar surface, referred to as bipolar active regions. The emergence time is defined as the time span from the first appearance of the bipolar region5 to maximum development.
The origin of the solar cycle was briefly discussed above, and now attention will be given to the individual sunspots. How do these arise? There are various hypothetical sunspot models that come under two types: (i) convective or hydrodynamical or (ii) magnetic cooling. The first category explains the origin of the sunspots as being due to an initial cooling and subsequent intensification of the magnetic fields.
The second category, which is more generally accepted nowadays, assumes that the sunspots are formed as a result of regional intensification of the magnetic fields. The magnetic fields are thought to suppress the convection in the sunspots, and thereby produce a region of colder and darker material. The presence of a solar magnetic field is explained by a dynamo theory similar to the geomagnetic field model, and will be elaborated in Section 4.5.3.
5 On Kitt Peak magnetogram.
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