The next theory, which still plays a role in our present understanding, came from Biermann (1941) and Alfvén (1942): In a highly ionized plasma the electric conductivity can be so large that the magnetic fields are frozen-in to the plasma. Biermann realized that the magnetic field in sunspots itself could be the reason for the spot coolness: Outside a sunspot energy is transported to the surface by overturning convection. In a sunspot the convective flow is inhibited by magnetic tension. Hence, a sunspot is dark because it is cooler, and it is cooler because the magnetic field suppresses the heat transport by convection. Hence, the darkness of a spot is due to a decreased surface brightness. Compared to the sunspot surroundings, which has a heat flux of 6.31 × 107 W/m2, the average penumbra heat flux is reduced by 25% and the umbra heat flux by 77%, respectively (e.g., Jahn and Schmidt, 1994). These values for the heat flux correspond to effective temperatures of 5777 K1, 5275 K, and 4000 K of quiet Sun, penumbra, and umbra, respectively.
Another concept, discussed by Jahn (1992) and relying on an idea from Hoyle (1949), also plays an important role in our understanding: plasma of a monolithic bundle of magnetic field lines increases with depth beneath the photosphere. Hence, to balance the hydrostatic stratifications in and outside the spot and to conserve the magnetic flux, must increase with depth and the spot diameter must decrease. With this geometrical funnel and the assumption that the umbra is thermally isolated from the penumbra and the surroundings, Hoyle constructed the following explanation for the umbral darkness: At a certain depth a given heat flux enters the umbra from below. This entering heat flux is the same as in the sunspot surroundings. Then, even if all that energy is transported (by magneto-convection) to the surface it will dilute because the umbral area at the surface is larger. And as we will discuss further down, this effect is present in the tripartite models of Jahn and Schmidt (1994).
Deinzer (1965)2 pointed out that convection cannot be suppressed completely since sunspots umbrae are still as hot as 4000 K, and radiative heating or heat conduction cannot supply the necessary heat flux. From this theoretical argument it must be concluded that energy transport by convection must exist in sunspots and, in particular, in sunspots umbrae. Meyer et al. (1974) studied the possible modes of magneto-convection and found that for the first 2000 km beneath the surface, the magnetic diffusivity, , is smaller than thermal diffusivity, . In this surface region, convection sets in as overstable oscillations (corresponding to standing Alfvén waves). The latter was also proposed earlier by Savage (1969) and has been found in idealized magneto-convection models, as we describe in Section 3.5.
In deeper layers, down to 20 000 km overturning convection takes place. Hence, heat transport by (magneto-) convection is essential in sunspots, meaning that magnetic fields cannot inhibit convection. It is true, however, that the magnetic field modifies the mode of convection. In Sections 3.5 and 3.6.2, we will describe simulations of magneto-convection in strong magnetic fields, but it should be noted that until today there is no parametrized theory of heat transport in magneto-convection as there is the mixing length theory (or more sophisticated moment approaches) for convection without magnetic field.
Realizing that global sunspot models must rely on energy transport by convection, a mixing length theory with a depth dependent mixing length parameter was developed (Deinzer, 1965; Jahn, 1989). A model that could be compared quantitatively with observed sunspots was presented by Jahn and Schmidt (1994). Most notably it turned out that these models rely on Hoyle’s concept: The ‘tripartite’ model has three stratifications: the thermally isolated umbra, the penumbra, and the quiet Sun. In a typical model, the umbra diameter doubles between 15 Mm depth and the surface, i.e., the area increases by a factor of 4 and the heat flux is reduced by the same amount. In some of these models, the regular solar heat flux enters the umbra at a depth of 15 Mm leading to the same umbral surface brightness as it is observed, i.e., Hoyle’s concept is built-in in the tripartite models. Note that the depth dependence of the mixing length parameter decides about the temperature gradient of the umbral depth stratification.
For a long time another expectation of a funnel-shaped spot was discussed: It was thought that the funnel blocks heat, which accumulates beneath the spot such that a bright ring around the spot should be produced. Several observational investigations have found some evidence for the existence of such bright rings: Fowler et al. (1983) found bright rings in the 0.1 – 0.3% range, the more recent study by Rast et al. (2001) brightness enhancements of 0.5 – 1%. A major challenge in these observational studies is the proper separation from the dominant effect of facular brightening, which is a near surface effect unrelated to the presence of a sunspot. Whether there is observational evidence for a bright ring independent from facular brigthening remains an open issue. It is, however, clear from the present observational constraints that the amplitude of such a bright ring would be insufficient to account for the heat flux blocked by the sunspot. Spruit (1977) resolved this puzzle: The thermal conductivity of the convection zone is so large that thermal disturbances are smoothed out so effectively that the remaining temperature fluctuations are too small to be observed. In a later review paper, Spruit (1992) made this effect “more understandable with a kitchen analogy”: If you put a small piece of a thermally insulating material on top of a electrically heated copper plate, there will not be an enhanced temperature in the vicinity of the insulator, since the blocked heat will be distributed very effectively across the copper plate. The insulator will be fairly cool at the top, but the copper plate, which has a high heat conductivity, will essentially have the same high temperature everywhere. This idea has been applied subsequently in more detailed sunspot models by Spruit (1982a), Spruit (1982b), Foukal et al. (1983), Chiang and Foukal (1985) and was recently reviewed by Spruit (2000). Also MHD models with radiative transfer do not show evidence for bright rings (Rempel, 2011c).
Living Rev. Solar Phys. 8, (2011), 3
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