2.4 Stability of monolithic models

Sunspot models have recently been reviewed in detail by Moradi et al. (2010Jump To The Next Citation Point), but see also Spruit (1981a) and Jahn (1997Jump To The Next Citation Point). Here we consider the issue of sunspot stability in monolithic models: Sunspots are stable relative to the dynamical time, i.e., Alfvén waves are estimated to travel across a spot in about 1 hour, while the lifetime is in order of weeks. How can it be that all this dynamic fine structure constitutes a spot that is globally stable?

The question of stability can be addressed if one assumes a monolithic vertical magneto-hydrostatic magnetic flux tube that fans out with height, being bounded by current sheets (Pizzo, 1986; Jahn, 1989Jump To The Next Citation Point; Pizzo, 1990Jump To The Next Citation Point; Jahn and Schmidt, 1994Jump To The Next Citation Point). In such models it is implicitly assumed that the heat transport is attributed to magneto-convection. Yet, the heat transport is not described dynamically, but parametrized in the context of mixing length theory (e.g., Hansen and Kawaler, 1994; Stix, 2004) with a reduced mixing length parameter (Deinzer, 1965; Jahn, 1989). Thereby the dynamic fine structure is ignored and only their averaged effect on the stratification for umbra and penumbra is accounted for. As a consequence, the magnetic field in the models does not really consist out of simple and straight field lines, but must be represented by a mean field, which result from averaging the small scales. In monolithic models it is then implicitly assumed that this mean field follows the rules of magnetostatics.

View Image

Figure 1: The tripartite model from Jahn and Schmidt (1994Jump To The Next Citation Point) consists of three stratifications as sketched in the left panel: An umbra, a thick penumbra, and the quiet Sun. The umbra is thermally isolated and confined by a current sheet called peripatopause, while the penumbra is confined by a current sheet called magnetopause. For depths smaller than zbp the magnetopause transmits 70% of the quiet Sun heat flux, thereby accounting for the surplus brightness of the penumbra when compared to the umbra. The right panel shows a surface plot of the (logarithmic) magnetic pressure. Credit: Jahn and Schmidt (1994Jump To The Next Citation Point), reproduced with permission of the ESO.

Exemplary for the class of monolithic models, we consider the tripartite sunspot model presented by Jahn and Schmidt (1994Jump To The Next Citation Point) (but see also Pizzo, 1990, for a nicely constructed model). It is configured to be in magneto-static equilibrium with a total pressure balance horizontally and a hydrostatic equilibrium vertically. The three stratification are separated by two current sheets between the umbra, and penumbra, and between the penumbra and the quiet Sun horizontally, as sketched in the left panel of Figure 1View Image. The gas pressure jumps are balanced by magnetic pressure, which is shown logarithmically as a surface plot in the right panel of Figure 1View Image. The tripartite model describes a sunspot down to 15 Mm beneath the surface. This configuration can be stable against the interchange instability (Meyer et al., 1977; Schüssler, 1984; Buente et al., 1993) in the first 5 Mm or so beneath the photosphere (Jahn, 1997). In these upper layers of the convection zone the inclination of the magnetopause, i.e., the interface between spot and surrounding, is so large that buoyancy forces make the spot to float on the granulation. In deeper layers, beyond 5 Mm, the inclination of the outermost magnetic field line, i.e., the magnetopause, is small relative to the vertical. There, interchange (fluting) instability is no longer suppressed by buoyancy effects, and the magnetic configuration of a monolithic sunspot is unstable. In these depths one would expect that strands of field lines separate to form a spaghetti configuration. However, in this depth range η is larger than κ, and magneto-convection is active, such that those spaghetti are far from being al dente: The magnetic and non-magnetic plasma is expected to mix.

Indeed, it has been proposed that the magnetic field strength progressively weakens in these deep layers shortly after the formation of a sunspot. The decreasing field strength, the convective motions, and the interchange instability dynamically disrupt the sunspot magnetic field from the deeper roots (Schüssler and Rempel, 2005Jump To The Next Citation Point). Hence, the magnetic field in the deeper layers may be dispersed, but the floating part of the sunspot is stable.

As mentioned before, it is essential to realize that monolithic models cannot be static. To transport sufficient energy, magneto-convection must also be present in the upper layers where it is stable against fluting. This seems feasible and we see no necessity to discard monolithic models and to prefer jelly fish models, instead. Jelly fish models may be compelling to explain umbral dots and bright filaments, but we are faced with the following problems: (1) How is the separation of plasma into magnetic and non-magnetic components maintained? How could that be achieved in a turbulent and convectively unstable stratification? At least in layers deeper than 2 Mm, magneto-convection is expected to mix the plasma. (2) What makes the sunspot to be stable and to behave like a coherent structure in which the field strength and inclination monotonically decreases outwards from spot center?

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