Originally proposed by Parker (1979b) to explain the umbral dots, Spruit and Scharmer (2006) and Scharmer and Spruit (2006) extended the concept of the field-free gaps to explain the bright penumbral filaments and they realized that such a configuration may also produce the dark cores within bright filaments, caused by a subtle radiative effect at the top of the field-free gap. The idea of field-free gaps in the penumbra is that the inclined penumbral magnetic field produces bright elongations instead of dots. The gaps are supposed to be void of magnetic field and to be connected to the surrounding quiet Sun. Within the gaps, overturning convection transports ample amounts of heat, which would account for the brightness of the penumbra. The convective flow field is directed upwards along the central lane of the filament and downward at the edges of the long sides of the filaments. Within the field-free gap there may exist a radial outflow that corresponds to the Evershed flow. The problem with this description is that the Evershed flow, which is observed to be magnetized, needs to be non-magnetized in the field-free gap. This is, however, not an intrinsic limitation of the model since the field-free gap model could be generalized to accommodate a horizontal field.
The field-free gap model shares some similarity with the model of the convective rolls proposed by Danielson (1961) (see also Grosser, 1991, for a numerical investigation on this model). Here convective rolls lie radially aligned next to each other, two such rolls would form one filament as they rotate in opposite direction, producing an upflow in the central lane and a downflow at the lateral lanes. While the resulting flow near the level is very similar to the flow assumed in the gap model and essentially indistinguishable in observations, there is nevertheless a profound difference. The upflow in the gap model is very deep reaching and connects, therefore, to layers with large heat capacity, while shallow convection rolls alone cannot provide the brightness of the penumbra over long time scales.
Danielson assumed that a horizontal magnetic field component would be associated with the rolls. This model has been discarded for two reasons: (1) There was no evidence for the corresponding convective flow field, and (2) a major fraction of the magnetic flux in the penumbra is directed upwards, and not horizontal. However, reason (1) depends on spatial resolution and the issue is not settled yet, as we cannot rule out the existence small amplitude vertical motions of a few hundred m s–1. Reason (2) could be overcome by assuming that the rolls are separated by less inclined (more vertical) magnetic field lines, which constitute a more or less static background magnetic field. And, magnetized rolls interlaced by a static background field that is less inclined relative to the vertical would also meet the observational requirements of two magnetic components in the penumbra. In this respect, at least in principle, it is possible that the horizontal magnetic component carries an Evershed flow.
The problem here is that – up to now – there is only little support for downflows along the edges of bright filaments. Rimmele (2008) finds a convective-roll like flow field in a filament that extends into the umbra for a sunspot close to disk center. However, he uses filtergrams with only two spectral positions at modest spectral resolution. Bellot Rubio et al. (2010), who acquired spectroscopic data at high spatial and spectral resolution, did not find indications for up- and downflows associated with a dark-cored bright filament at disk center. Still, the crucial question of vertical flows in the penumbra is not settled. In order to minimize the effects of the horizontal radial outflow and of possible flows in azimuthal direction, sunspot observations at disk center are needed to learn about the presumably small vertical flow component. See also the discussion in Section 3.7 for further detail.
Another class of simplified models is based on the assumption that penumbral filaments can be identified with primarily horizontal flux tubes near the surface. The properties of flux tubes have been studied in stationary setups as well as dynamic configurations.
In the self-consistent magneto-static tripartite sunspot model of Jahn and Schmidt (1994) the surplus brightness of the penumbra relative to the umbra is produced by a heat transfer through the magnetopause, i.e., through the interface between the quiet Sun and the penumbra. This additional heat is thought to be distributed horizontally by interchange convection of magnetic flux tubes. The idea of dynamic magnetic flux tubes is compatible with the observationally finding of multiple magnetic components in the penumbra.
This motivated the study of the dynamics of a single thin magnetic flux tube as it evolves in a 2D static model background Schlichenmaier et al. (1998a,b). However, these studies did not confirm the concept of interchanging magnetic flux tubes that distribute heat horizontally. Instead, these studies created a new picture: The simulated tube lies along the magnetopause of the tripartite sunspot model and is taken to be a bundle of magnetic field lines with penumbral properties. Initially the tube is in magneto-static equilibrium. However, at a magnetopause that is sufficiently inclined, radiative heat exchange between the tube and the hotter quiet Sun triggers an instability: A thin magnetic flux tube that initially lies along the magnetopause, (a) feels the hotter quiet Sun, (b) heats up by radiation most effectively just beneath the photosphere, (c) expands, (d) rises through the sub-photospheric convectively unstable stratification, and (e) develops an upflow along the tube, which brings hot sub-photospheric plasma into the photosphere. (f) This hot upflow cools radiatively in the photosphere and streams radially outwards with supercritical velocity. The radiative cooling sustains the gas pressure gradient that drives the flow. (g) The outflow intrudes the convectively stable photosphere up to a height of some 50 to 100 km. The equilibrium height is determined by the balance of the diamagnetic force, which pulls the conducting tube upwards toward decreasing magnetic field strength and the downward acting buoyancy, which increases as the tube is being pulled up in a convectively stable stratification.
The moving tube scenario successfully explains a number of observational findings: (i) Penumbral grains are the photospheric footpoints of the tube, where the hot and bright plasma enters the photosphere. (ii) The upflow turns horizontally outwards in the photosphere and cools radiatively until it reaches temperature equilibrium. This determines the length of the penumbral grains (Schlichenmaier et al., 1999). (iii) The footpoints migrate inward, as many observed penumbral grains do (e.g., Sobotka and Sütterlin, 2001). (iv) The horizontal outflow corresponds to the Evershed flow. (v) The tube constitutes a flow channel being embedded in a background magnetic field. This is in agreement with the uncombed penumbra, and produces realistic maps of NCP.
Living Rev. Solar Phys. 8, (2011), 3
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