2.3 Subsurface morphology

In the preceding Section 2.2, we implicitly assumed that a sunspot is a funnel-shaped object consisting of magnetized plasma. This class of models is termed monolithic. One can also envisage that bundles of field lines separate beneath the photosphere such that gaps/columns of field-free plasma exist just below the surface. The latter type has been referred to as the jelly fish or cluster or spaghetti models (Parker, 1979bJump To The Next Citation Point,c; Spruit, 1981b; Choudhuri, 1992). The subsurface morphology relates also to the question of what determines the apparent stability of sunspots over long time scales despite more rapid evolution on smaller scales. We will come back to this aspect in Section 4.4.

Spaghetti model and field-free gaps:
The field-free columns close at the photosphere, but broaden with depth, since plasma 2 β = 8πp∕B is about unity in the photosphere and increases with depth: At the surface the magnetic pressure is strong enough to squeeze the field-free gaps such that only small umbral dots remain at the surface as the imprint of the field-free convective columns. An umbral dot would correspond to the peak of a field-free gap. More recently, Spruit and Scharmer (2006Jump To The Next Citation Point) suggested that such field-free gaps in the inclined magnetic field of the penumbra may result in elongated bright filaments, instead of in point-like dots, thereby proposing an explanation for the brightness of the penumbra. The surplus brightness of the penumbra relative to the umbra would then be due to the fact that the convective cell can become larger in the more inclined and weaker magnetic field as in the less inclined (more vertical) and stronger field of the umbra.

Are spots monolithic or spaghetti-like?
At present, we do not have the means to give a rigorous answer. As discussed in Jahn (1992), fine structure phenomena like umbral dots can be explained in both frameworks. The same is true for the umbral heat flux: it is either done by field-free convection in gaps or by a magneto-convective process. The spaghetti model is compelling, since it explains naturally the existence of umbral dots and bright filaments, but it is unknown (a) what forces would keep the magnetic field lines together in the photosphere so that they can form stable long-lived sunspots, and (b) how it happens that the spot has a magnetic field strength that decreases systematically from spot center. In the spaghetti model, magnetic and non-magnetic convecting plasma are distinct from each other, while a monolithic model forms an entity in which magneto-convection takes place. Theoretical considerations like the one mentioned in Section 2.2.3 (Meyer et al., 1974Jump To The Next Citation Point) predict the existence of overturning magneto-convective processes for depths larger than 2000 km in which magnetized and non-magnetized plasma gets mixed, and hence, favor the monolithic approach. These monolithic models have the advantage that they can be investigated in a quantitative manner, at least if one assumes that the magnetic field can be described by a mean magnetic field, which finds itself in a global magnetostatic equilibrium. Obviously, the sunspot brightness and its subsurface structure are intimately connected. In summary, it can be said that the coolness of sunspots relative to the surrounding quiet Sun is explained by the tension of the magnetic field, which tends to suppress convective motions and/or the channeling of heat in an expanding fan, thereby diluting the heat flux. It is more difficult to understand why sunspots are as hot as they are. Since neither radiative transport nor heat conduction can account for the surface brightness of sunspots, the energy must be transported by (magneto-)convective flows. Indeed, the fine structure manifests the inhomogeneities of the magnetic and velocity field and testifies that the energy transport in sunspots happens on small spatial scales by the motion of plasma. The essential question is how magnetized these convective flows are. Observations point toward reduced magnetic field strength in umbral dots (see Section 3.1), which agrees also with recent results from 3D radiative MHD simulations (see Section 3.6.2). Overall, magneto-convective models point toward the presence of overturning convection in both umbra and penumbra that takes place in regions with reduced, albeit not zero field strength. Most importantly, it is found that this mode of convective energy transport can originate in initially monolithic magnetic field, i.e., the presence of sunspot fine structure cannot be taken as support for the “spaghetti-like” subsurface structure.

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