3.4 Description of idealized models

3.4.1 Convective models: field-free gap and convective rolls

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 τ = 1 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 (2008Jump To The Next Citation Point) 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. (2010Jump To The Next Citation Point), 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.

3.4.2 Flux tube models: stationary and dynamic

Another class of simplified models is based on the assumption that penumbral filaments can be identified with primarily horizontal flux tubes near the τ = 1 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.

Weak magnetic field at footpoint:
The gas pressure gradient that drives the flow is caused by a surplus gas pressure building up inside the part of the tube that rises through the sub-photospheric stratification. At the footpoint, i.e., the intersection of the tube with the transition layer from convectively unstable to stable, the gas pressure is high, and in order to balance the total pressure with the surroundings, the magnetic field strength is strongly decreased relative to the surroundings. In this sense the upflow footpoints can be considered as regions of weak magnetic field strength. In other words, the moving tube model is a magneto-convective mode, which consists of a region of weak-field plasma that harbors hot upflows and that travels inwards. In principle, the effect leading to the up- and outflow works like an inverse convective collapse: In the classical convective collapse the plasma in the tube is cooled and a downflow occurs. Here, the heating of the plasma results in an upflow, and consequently the magnetic field strength in the tube decreases as the flow continues. In the photosphere, the gas pressure gradient is sustained by radiative cooling.

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.

Magneto-convective overshoot:
An interesting effect that can be studied with the idealized moving tube model, is related to overshooting (Schlichenmaier, 2002, 2003Jump To The Next Citation Point). The upflow shoots into the convectively stable photosphere, and is turned horizontally by the magnetic curvature forces along the tube. The dominating forces here are the centrifugal force of the flow, κρv2 and the magnetic curvature force, 2 κB ∕(4π), with κ being the curvature. In equilibrium v equals vA, with vA being the Alfvén velocity. During the evolution of the tube the velocity is roughly constant, but the magnetic field strength and, hence, vA decreases, leading to an overshoot into the photosphere that is convectively stable. As the plasma overshoots the density increases and buoyancy forces pull the flow back down. This results in an oscillation of the outflow around its equilibrium position such that the tube adopts a wave-like shape, i.e., the plasma first shoots up and then down, again passing the equilibrium position. Such a wave can be considered quasi-stationary, and the crest of such a wave can be compared with the properties of a siphon flow (see below). Hence, the flow yields a serpentine shape, looking like a sea serpent, and evidence for such radially aligned up and downflows has been presented by Sainz Dalda and Bellot Rubio (2008). The amplitude of this wave increases as the magnetic field strength decreases, and eventually the downflow part dives in the sub-photosphere. There the stratification is convectively unstable and the magnetic flux tube experiences a dynamic evolution, that produces outward propagating waves. This scenario produces downflows and makes the tube to disappear within the penumbra. Thereby it would solve a problem of the moving tube model: the out-flow would not extend into the surrounding canopy, but would disappear within the penumbra, as it is observed.

Serpentine flow:
Such a two-dimensional serpentine solution was criticized to be unstable in three dimensions (Thomas, 2005Jump To The Next Citation Point), arguing that buoyancy forces make the wavy tube to fall over sideways. But this argument is not valid, since the influence of the upflow at the footpoint of the tube is not taken into account. At the footpoint the plasma is ejected upwards into the photosphere and due to conservation of momentum, the plasma overshoots and follows an up- and down wavy behavior. The fact that the density at the upper crest is larger than in the surroundings does not make the tube to fall over. As an analogy, one may think of a jet of water directed upwards with a garden hose. As long as the jet is pointing upwards with the hose (at the footpoint), the jet of water will not fall over. The jet of water will not fall over, even though the density of water is larger than that of the surrounding air. Since the footpoint of the upflowing flux tube and its inclination is constrained, the boundary condition circumvents the wavy flow to fall over. Therefore, the argument of Thomas (2005) is only true for a serpentine flow without a footpoint, and is not applicable here.

Siphon flows:
Siphon flow arches are stationary magnetic flux tube models, which were proposed to explain the Evershed flow (e.g., Meyer and Schmidt, 1968; Thomas, 1988; Degenhardt, 1991; Thomas and Montesinos, 1991). This class of models makes the ad hoc assumption of different magnetic field strengths at the two footpoints of a magnetic arch, which is responsible for a gas pressure gradient along the tube driving the flow. In the dynamic sea-serpent solutions (see above) a quasi-stationary solution exists (Schlichenmaier, 2003). This solution corresponds to one (out of four) particular siphon flow solution: a flow with a supercritical flow speed along the arch. Since the pressure difference at the footpoints has to be assumed a-priori, this is not a fully self-consistent explanation of the Evershed flow, but these models allow to describe properties of the flow in the visible layers of the sun.

Heat transport:
Temporal measurements of the intensity evolution rule out the existence of interchange convection (Solanki and Rüedi, 2003) and, also, the numerical work of the moving tube model did not confirm the concept of interchange convection of magnetic flux tubes as the heating mechanism for the surplus brightness of the penumbra: A crucial result of the numerical investigation is that a tube rises and develops an upflow, but the upflow does not stop nor does the tube sink back down to the magnetopause. Hence, instead of interchange convection the moving tube simulations suggests that the heating occurs in form of upflow channels along magnetic field lines. Ruiz Cobo and Bellot Rubio (2008) demonstrate that such an upflow is capable to account for the brightness of the penumbra and that such upflows can produce dark-cored bright filaments with a length of up to 3 Mm. Yet, even if such upflows can transport enough heat to account for the brightness of the penumbra, Schlichenmaier and Solanki (2003) have shown that downflows within the penumbra are obligatory: If each hot upflow along a tube turns horizontal and stays horizontal, there is not enough space for all the horizontal tubes. Hence, the horizontal magnetic tubes must turn downwards after a few Mms to make room for more tubes with more upflows. The downward directed part of the tube is then associated with a downflow. In this respect, the overshoot scenario (serpentine flow) may help: the hot upflow cools and the cool downflow heats up in the hot sub-photosphere, and re-enters the photosphere as a hot upflow. Hence, the moving tube scenario encounters problems in accounting for sufficient heat transport, but there are ways to solve the heat transport problem with channeled flows. And these channeled flows are driven by radiative cooling. In present-day MHD simulations the energy transport in umbral dots, light bridges, and filaments in the inner penumbra is accomplished by a magneto-convective mode, which may be characterized as convective elongated cells, very similar to the energy transport in granulation. The following section will summarize models of sunspot fine structure based on MHD simulations with radiative transfer.


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