3.2 Sunspot penumbra and penumbral filaments

3.2.1 Spines and intraspines

The filamentary structure of sunspot penumbra was recognised early in the 19th century in visual observations (see review by Thomas and Weiss, 2008). The progress of observational techniques to attain higher spatial resolution revealed that the sunspot penumbra consists of radially elongated filaments with a width of 0.2 – 0.3” as seen in continuum images (e.g., Danielson, 1961a; Muller, 1976). Resolving the structure of the magnetic field with such high resolution is much more difficult because polarimetric measurements require multiple images taken in different polarization states, and a longer exposure time in a narrow wavelength band to isolate the Zeeman signal in a spectral line. For this reason, until recently many of the investigations of the magnetic field in the penumbra have reported contradictory results.

A hint of fluctuation in the magnetic field, in association with the penumbral filamentary structure, was first reported by Beckers and Schröter (1969a), who reported that the magnetic field was stronger and more horizontal in dark regions of the penumbra. Wiehr and Stellmacher (1989Jump To The Next Citation Point), however, found no general relationship between brightness and the strength of the magnetic field.

Advancement of large solar telescopes at locations with a good seeing conditions made it possible to better resolve the penumbral filamentary structure in spectroscopic and polarimetric observations, and a number of papers on the small-scale magnetic field structures in sunspot penumbra were published in early 1990s. Lites et al. (1990Jump To The Next Citation Point), using the Swedish Vacuum Solar Telescope (SVST) in La Palma, found a rapid change in the inclination of the magnetic field between some dark and light filaments near the edge of the penumbra, while the field strength showed only a gradual variation across the filaments. Degenhardt and Wiehr (1991), using the Gregory Coudé Telescope in Tenerife, found fluctuations in the inclination of the magnetic field vector in the penumbra by 7 – 14°, with steeper (more vertical) regions having a stronger magnetic field.

Schmidt et al. (1992), using the German Vacuum Tower Telescope (VTT) in Tenerife, found more horizontal field lines in dark filaments, while the strength of the magnetic field did not differ between bright and dark penumbral filaments. Title et al. (1993Jump To The Next Citation Point), using a series of Dopplergrams and line-of-sight magnetograms taken by a tunable narrowband filter equipped on SVST, found variations in the inclination of the magnetic field of about ± 18° across penumbral filaments. Lites et al. (1993), using the Advanced Stokes Polarimeter (ASP) on the Dunn Solar Telescope at Sacramento Peak, identified radial narrow lanes in the penumbra where the magnetic field is more vertical and stronger, thereby naming such regions as spines. Their results indicated that spines feature an azimuthal expansion of the magnetic field towards the sunspot’s border. In addition, they found no clear evidence for a spatial correlation between spines and brightness. The correlation between the magnetic field strength and field inclination (i.e., stronger field in spines) was confirmed by Stanchfield II et al. (1997Jump To The Next Citation Point) using ASP data.

With a highly resolved spectrum in the Fe i 684.3 nm spectral line, which is formed in the deep photosphere, Wiehr (2000) found that darker penumbral lanes correlate with a stronger and more horizontal magnetic field, though the slit of the spectrograph sampled only a portion of the penumbrae. Better defined polarization maps of spines were taken with the Swedish 1-m Solar Telescope employing adaptive optics (Langhans et al., 2005), demonstrating that spines are regions with stronger and more vertical magnetic field, and that they are associated with bright penumbral filaments.

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Figure 24: Sunspot AR 10933 observed at Θ = 2.9° on January 5, 2007 with the spectropolarimeter SOT/SP on-board Hinode. Displayed are: a) continuum intensity Ic, b) magnetic field inclination γ, c) divergence of the horizontal component of the magnetic field vector ∇ ⋅ Bh, and d) total field strength B. All parameters were obtained from a Milne–Eddington inversion of the recorded Stokes spectra. The green arrow in panel a indicates the direction of the center of the solar disk. The yellow box surrounds the sunspot region displayed in Figure 34View Image.

High quality vector magnetograms with a high spatial resolution are now routinely obtained by the spectropolarimeter (SP) on-board Hinode. Figure 24View Image (panels a and b) show the continuum intensity and the inclination of the magnetic field for a sunspot observed on January 5, 2007 (AR 10933), located very close to the center of the solar disk (∘ Θ ≈ 2.9). The field inclination was derived by a Milne–Eddington inversion (Section 1.3) as the angle between the magnetic field vector and the line-of-sight, γ (see Equation 27View Equation), but because of the proximity of the sunspot to the disk center, the inclination can be regarded as the inclination of the magnetic field, ζ (Equation 10View Equation), with respect to the local normal to the solar surface: eρ (see Figure 42View Image). It is obvious in the inclination map that the penumbrae consists of radial channels that have alternative larger and smaller field inclination. A close comparison with the continuum image shows that more horizontal field channels in panel b (also called intraspines) tend to be bright filaments in inner penumbra but to be dark filaments in outer penumbra. Panels c and d in Figure 24View Image show the divergence of transverse component of the magnetic field vector (∇ ⋅ Bh ) and the total field strength B, respectively, obtained from the Milne–Eddington inversion. It is confirmed that spines have stronger field than intraspines, as well as a positive field divergence. Also noticeable is the presence of a number of patches that have opposite polarity to the sunspot around the outer border of the penumbra (see also Figures 4View Image and 7View Image).

Thus, the penumbral magnetic field consists of two major components: spines where the magnetic field is stronger and more vertical with respect to the direction perpendicular to the solar surface, and intraspines where the magnetic field is weaker and more horizontal. Whereas the magnetic field of the spines possibly connect with regions far from the sunspot to form coronal loops over the active region, the magnetic field in the intraspines turns back into the photosphere at the outer border of the sunspot or extend over the photosphere to form a canopy (Solanki et al., 1992; Rueedi et al., 1998). The filamentary structure of the penumbra persists even after averaging a time series of continuum images over 2 – 4.5 hours (Balthasar et al., 1996Jump To The Next Citation Point; Sobotka et al., 1999Jump To The Next Citation Point). This suggests that the two magnetic field components are more or less exclusive to each other (Thomas and Weiss, 2004; Weiss, 2006) except for a possible interaction through reconnection at the interface between them in the photosphere (Katsukawa et al., 2007). Such structure of the penumbral magnetic field, i.e., magnetic fields with two distinct inclinations interlaced with each other in the azimuthal direction, is referred to as uncombed penumbra (Solanki and Montavon, 1993Jump To The Next Citation Point) or interlocking comb structure (Thomas and Weiss, 1992). The fact that the magnetic field is weakened in the intraspines, as compared with the spines, can also be employed to deduce through total pressure balance considerations (as we already did in the case of umbral dots and light bridges; see Section 3.1.2) that the intraspines are elevated with respect to the spines.

To account for the filamentary structure of penumbra with the uncombed magnetic fields, some distinguished models, that are under a hot discussion nowadays, were proposed. One of these models, the embedded flux tube model is an empirical model proposed by Solanki and Montavon (1993Jump To The Next Citation Point), in which nearly horizontal magnetic flux tubes forming the intraspines are embedded in more vertical background magnetic fields (spines) in the penumbra (Figure 25View Image, left panel). The downward pumping mechanism (Thomas et al., 2002Jump To The Next Citation Point) was proposed to explain the origin of field lines that return back into the solar surface at the outer penumbra (Figures 4View Image, 7View Image, and 24View Image). In this scenario, submergence of the outer part of flux tubes occurs as a result of the downward pumping by the granular convection outside the sunspots, and such magnetic fields form the low-laying horizontal flux tubes. Another idea to account for the penumbral filaments is the field-free gap model initially proposed by Choudhuri (1986) and later refined by Spruit and Scharmer (2006Jump To The Next Citation Point) and Scharmer and Spruit (2006Jump To The Next Citation Point). Here, the penumbral bright filaments are regarded as manifestations of the protrusion of non-magnetized, convecting hot gas into the background oblique magnetic fields of the penumbra. Due to the continuity condition of the normal component of the magnetic field across the boundary between the background field and the protruding non-magnetized gas, the vertical component of the magnetic field vector, Bρ, in the background magnetic field must vanish right on top of the non-magnetic gas. This immediately yields a region, above the penumbral filaments, where the magnetic field is almost horizontal (i.e., intraspines).

All the aforementioned models attempt to explain, with different degrees of success, the configuration of the magnetic field in the penumbra. However, the appearance of a penumbra is always associated with a distinctive gas flow, i.e., the Evershed flow and, therefore, this must also be taken into account by these models. In the next section we will address this issue.

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Figure 25: Models for explaining the uncombed penumbral structure. Upper-left: embedded flux tube model (from Solanki and Montavon, 1993Jump To The Next Citation Point, reproduced by permission of the ESO); lower-left: rising flux tube model (from Schlichenmaier et al., 1998aJump To The Next Citation Point, reproduced by permission of the ESO); right: field-free gap model (from Spruit and Scharmer, 2006Jump To The Next Citation Point, reproduced by permission of the ESO).

3.2.2 Relation between the sunspot magnetic structure and the Evershed flow

The Evershed flow was discovered in 1909 by John Evershed at the Kodaikanal Observatory in India as red and blue wavelength shifts in the spectra of absorption lines in the limb-side and disk-center-side of the penumbra, respectively. This feature can be explained by a nearly horizontal outflow in the photosphere of the penumbra (Evershed, 1909). Under an insufficient spatial resolution, it appears as a stationary flow with typical speeds of 1 – 2 km s–1, where the magnitude of the flow velocity increases with optical depth τc (towards the deep photosphere; Bray and Loughhead, 1979).

An outstanding puzzle about the Evershed flow lies in the relation between the velocity vector and the magnetic field vector in the penumbra. Since the averaged magnetic field in the penumbra has a significant vertical component, with an angle with respect to the normal vector on the solar surface between ∘ ζ ≈ 40 – 80 (see Figure 11View Image in Section 2.1), and the Evershed flow is apparently horizontal, this would mean that the flow would move across the magnetic field. Under these circumstances, the sunspot’s magnetic field (which is frozen-in to the photospheric gas) would be removed away within a few hours.

It is highly plausible that there is a close relationship between the Evershed flow and the filamentary structure of the penumbra. Indeed, it was recognized in the 1960s that the flow is not spatially uniform but concentrated in narrow channels in penumbra; e.g., Beckers (1968Jump To The Next Citation Point) reported that the flow originates primarily in dark regions between bright penumbral filaments. Two models were proposed to account for the nature of penumbral filaments and the Evershed flow before 1990. One is the elevated dark filament model in which the penumbral dark regions are regarded as elevated fibrils with nearly horizontal magnetic field overlaying the normal photosphere and carry the Evershed flow in them (Moore, 1981; Cram and Thomas, 1981; Thomas, 1988Jump To The Next Citation Point; Ichimoto, 1988). The other is the rolling convection model in which penumbral filaments are regarded as convective elements radially elongated by a nearly horizontal magnetic field in penumbra and where the Evershed flow is confined in dark lanes that are analogous to the intergranular dark lanes (Danielson, 1961b; Galloway, 1975). Both models assume a nearly horizontal magnetic field in the penumbra and, therefore, contradict the observational fact that a significant fraction of sunspot’s vertical magnetic flux comes out through the penumbra (Solanki and Schmidt, 1993).

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Figure 26: Geometry of magnetic field and Evershed flow in penumbra. Magnetic field lines are shown by inclined and colored cylinders, while the Evershed flow is indicated by white arrows in dark penumbral channels. Note that the Evershed flow concentrates along the more horizontal magnetic field lines (white cylinders) (from Title et al., 1993Jump To The Next Citation Point, reproduced by permission of the AAS).

The long-lasting enigma on the Evershed flow was finally solved by the discovery of the interlocking comb structure of the penumbral magnetic field (Section 3.2.1). Under this scenario, the Evershed flow is confined in nearly horizontal magnetic field channels in penumbra (i.e., intraspines), while out of the flow channels (i.e., in the spines) the magnetic field is more vertical. Both components, when averaged together, make the spatially averaged magnetic field far from completely horizontal (Figures 11View Image and 26View Image; see also Title et al., 1993Jump To The Next Citation Point). The relationship between the Evershed flow and the horizontal magnetic field in the penumbra has been highlighted in many works in the past: Stanchfield II et al. (1997Jump To The Next Citation Point, Figure 7) or Mathew et al. (2003, Figure 12). The latter two works were obtained with spectropolarimetric data at 1” resolution. A more updated result, employing Hinode/SP data with 0.3” resolution, has been presented by Borrero and Solanki (2008Jump To The Next Citation Point, see Figure 27View Image). This figure demonstrates that the Evershed flow (seen as large positive or redshifted line-of-sight velocities; middle panel) is concentrated along the intraspines: regions where the magnetic field is horizontal (γ ≈ 90∘; bottom panel) and weaker (upper panel).

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Figure 27: Variation of the physical parameters at τc = 1 (continuum) along an azimuthal cut around the limb-side penumbra (i.e., along one of the blue ellipses in Figure 10View Image). From top to bottom: magnetic field strength B, line-of-sight velocity Vlos, and inclination of the magnetic field γ. Dotted curves in each panel show the scattered light fraction obtained from the inversion algorithm. Note that the velocity (Evershed flow) is strongest in the regions where the magnetic field is weak and horizontal (intraspines), while it avoids the regions with more less inclined and stronger magnetic field (spines) (from Borrero and Solanki, 2008Jump To The Next Citation Point, reproduced by permission of the AAS).

In the embedded flux-tube model (Solanki and Montavon, 1993Jump To The Next Citation Point), the Evershed flow is supposed to be confined in the horizontal magnetic flux tubes embedded in more vertical background magnetic field of the penumbra. In such picture, the siphon flow mechanism was proposed as the driver of the Evershed flow (Meyer and Schmidt, 1968; Thomas, 1988; Degenhardt, 1991; Montesinos and Thomas, 1993): a difference in the magnetic field strength between two footpoints of a flux tube causes a difference of gas pressure, and drives the flow in a direction towards the footpoint with a higher field strength (i.e., the footpoint outside the sunspot to account for the Evershed outflow). Schlichenmaier et al. (1998b) and Schlichenmaier et al. (1998a) investigated the dynamical evolution of a thin magnetic flux tube8 embedded in a penumbral stratification (Jahn and Schmidt, 1994a) and proposed the hot rising flux tube model, in which a radial and thin flux tube containing hot plasma raises towards the solar surface due to buoyancy. As the flux tube reaches the τc = 1-level it cools down due to radiation, thereby producing a gradient on the gas pressure along the flux tube and, thus, driving the Evershed flow along the tube’s axis (i.e., radial direction in the penumbra).

By performing an inversion of Stokes profiles of three infrared spectral lines at 1565 nm and using a two component penumbral model in which two different magnetic atmospheres are interlaced horizontally, Bellot Rubio et al. (2004Jump To The Next Citation Point) found a perfect alignment of the magnetic field vector and the velocity vector in the component that contains the Evershed flow. This picture was supported by Borrero et al. (2004) who also performed Stokes inversions of the same infrared lines. With a further elaborated analysis, Borrero et al. (2005) found that the penumbral flux tubes are hotter and not completely horizontal in the inner part of the penumbra, while they become gradually more horizontal and cooler with increasing radial distance. This is accompanied by an increase in the flow velocity and a decrease of the gas pressure difference between flux tube and the background component, with the flow speed eventually exceeding the critical value to form a shock front at large radial distances (V  > 6 – 7 km s–1). They argued that these results strongly support the siphon flow as the physical mechanism responsible for the Evershed flow.

Until recently the relationship between the Evershed flow and the brightness of the penumbral filaments has been somewhat controversial. Many authors (Beckers, 1968; Title et al., 1993; Shine et al., 1994; Rimmele, 1995aJump To The Next Citation Point; Balthasar et al., 1996; Stanchfield II et al., 1997Jump To The Next Citation Point; Rouppe van der Voort, 2002) have presented evidence that the Evershed flow is concentrated in dark filaments, while some studies claimed that there is no correlation (Wiehr and Stellmacher, 1989; Lites et al., 1990; Hirzberger and Kneer, 2001). Rimmele (1995aJump To The Next Citation Point) showed that the correlation becomes better when one compares the intensity and velocity originating from the same height, and also gave a hint that the correlation is different between inner and outer penumbra. Schlichenmaier et al. (2005), Bellot Rubio et al. (2006), and Ichimoto et al. (2007aJump To The Next Citation Point) presented evidence that the Evershed flow takes place preferentially in bright filaments in the inner penumbra, but in dark filaments in the outer penumbra.

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Figure 28: Spatial correlation between penumbral filaments and the Evershed flow. Correlation coefficients between the Evershed flow and brightness (lower-right), and between the Evershed flow and the elevation angle of magnetic field from the solar surface (upper-right) are shown as a function of radial distance from the sunspot center. Arc-segments along which the correlation is calculated are shown in the left panel. The sunspot was located at the heliocentric angle of Θ = 31°. The direction to the center of the solar disk is shown by an arrow in the left panel. The data employed here was recorded with the spectropolarimeter on-board Hinode (SOT/SP). Red lines show the results for the limb-side penumbra, whereas blue corresponds to the center-side penumbra.

Figure 28View Image presents the spatial correlation between penumbral filaments and the Evershed flow. The correlation coefficient between the Doppler shift (Vlos) and the elevation angle of magnetic field vector from the solar surface9 as a function of the radial position in the penumbra, is displayed in the upper-right panel, whereas the correlation between the Doppler shift (Vlos) and continuum intensity is shown in the lower-right panel. In these plots, the results for limb-side penumbra are shown in red color and for disk-center-side are shown in blue color. The abscissa in this figure spans from the umbra-penumbra boundary (left) to the outer border of the penumbra (right). The curves along which the correlation coefficients are obtained are shown for both limb-side and disk-center-side penumbra in the left panels. The data employed for this figure was obtained by SOT/SP when the sunspot was located at the heliocentric angle of Θ = 31°, thus, the Doppler shift is mainly produced by the horizontal Evershed flow. In this plot, line-of-sight velocities are taken in absolute value such that there is no difference between the redshifts in the limb-side and the blueshifts in the center side that are characteristic of the Evershed flow. In Figure 28View Image, we notice that the Evershed flow correlates with more horizontal magnetic fields throughout the entire penumbra, while it correlates with bright filaments in the inner penumbra but with dark filaments in the outer penumbra. These results are consistent with the idea that penumbral filaments, which harbor a nearly horizontal magnetic field, are brighter in inner penumbra but darker in outer penumbra. The correlation between Doppler shift and intensity shows an asymmetric distribution between the disk center-side and limb-side penumbra (lower right). This suggests that overposed to the Evershed flow, which is mainly horizontal, there exists a vertical component in the velocity vector in the penumbra. This vertical component will be discussed in detail in Sections 3.2.3 and 3.2.4.

So far we have discussed only investigations that were carried out with ME-inversion codes and, therefore, referred only to the physical parameters of the sunspot penumbra at a constant τ-level (see Sections 1.3.1 and 2.1). In order to investigate the depth dependence of the line-of-sight velocity and magnetic field vector in the penumbra, τ-dependent inversion codes (see Sections 1.3.1 and 2.2) must be applied to observations of the polarization signals in spectral lines. This has been addressed by a number of authors, such as Jurčák et al. (2007Jump To The Next Citation Point), who applied the SIR inversion code (Ruiz Cobo and del Toro Iniesta, 1992) to the spectropolarimetric data obtained by SOT/SP on Hinode and found that a weaker and more horizontal magnetic field is associated with an increased line-of-sight velocity in the deep layers of the bright filaments in the inner penumbra. In the outer penumbra, however, stronger flows and more horizontal magnetic fields tend to be located in dark filaments (Jurčák and Bellot Rubio, 2008). With a further application of the SIR inversion on SOT/SP data, Borrero et al. (2008Jump To The Next Citation Point) found that the magnetic field in the spines wraps around the horizontal filaments (i.e., intraspines). Some results from the latter two works are presented in Figures 29View Image and 30View Image.

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Figure 29: Depth structure of penumbra derived from Stokes inversions of spectro-polarimetric data. Showns are vertical cuts across the penumbral filaments. On the left, from top to bottom, are temperature T, field strength B, field inclination γ, and line-of-sight velocity Vlos (from Jurčák et al., 2007, reproduced by permission of the PASP).
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Figure 30: Vertical stratification (optical depth τc) of the physical parameters in the penumbra. The horizontal axis is the azimuthal direction around the penumbra and, therefore, it is perpendicular to the radial penumbral filaments. Upper-left panel: line-of-sight velocity Vlos. Upper-right: total magnetic field strength B. Lower-left: inclination of the magnetic field vector with respect to the normal vector to the solar surface ζ (see Equation 10View Equation). Lower-right: azimuth of the magnetic field vector Ψ (Equation 11View Equation). This plot demonstrates that the strong and vertical magnetic field of the spines extends above the intraspines (indicated by the index i), where the Evershed flow is located where the magnetic field is rather horizontal and weak. It also shows that the azimuth of the magnetic field changes sign above the intraspines, indicating that the magnetic field of the spines wraps around the intraspines. The arrows in this figure show the direction of the magnetic field in the plane perpendicular to the axis of the penumbral filaments (from Borrero et al., 2008, reproduced by permission of the ESO).

3.2.3 The problem of penumbral heating

One important issue that needs to be addressed to understand the origin of the penumbra is how the energy transport takes place. Whatever mechanism exists, it must supply enough energy to maintain the penumbral surface brightness to a level of 70 – 80% of the quiet Sun granulation (see Sections 2.5 and 3). Since the most efficient form of energy transport in the solar photosphere is convection, the key question is therefore to identify how convective motions in the presence of a rather strong, B ≈ 1500 G, and horizontal, ∘ ζ ≈ 40– 80, magnetic field (see Figure 11View Image) occur.

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Figure 31: Possible patterns of convection present in the sunspot penumbra. The upper panel corresponds to a pattern of radial convection, where upflows are presented at the inner footpoints of the penumbral filaments and downflows at the outer footpoints. This pattern is predicted by the embedded flux-tube model and the hot rising flux-tube model. The lower panel shows a pattern of azimuthal or overturning convection, where the upflows/downflows alternate in the direction perpendicular to the filaments’ axis. This is the flow pattern predicted by the field-free gap model.

In search for these convective motions we have to examine the predictions that the different models (see Section 3.2.2) make about vertical flows in the penumbra. The hot rising flux-tube model (see Section 3.2.2) predicts the presence of upflows at the inner footpoints of the flux tubes and downflows at their outer footpoints (Schlichenmaier, 2002Jump To The Next Citation Point). Provided that the flux tubes are evenly distributed, this would yield a preference for upflows in the inner penumbra, whereas downflows would dominate at large radial distances. Because of these features, we will refer to this form of convection as radial convection, with convective flows occurring along the penumbral filament (see upper panel in Figure 31View Image). Schlichenmaier and Solanki (2003Jump To The Next Citation Point) examined the possible heat transport in the context of this model and found that the heat supplied by this model is sufficient only if the upflowing hot plasma at the inner flux tube’s footpoint travels only a small radial distance ℒ before tuning into a downflow (see upper panel in Figure 31View Image), with a new flux tube appearing immediately after. This implies that there should be a significant magnetic flux and mass flux returning to and emerging from the photosphere in penumbra. This is a natural consequence of the rapid cooling that the hot rising plasma suffers once it reaches the τc = 1-level (Schlichenmaier et al., 1999).

Contrary to the aforementioned models, the field-free gap model provides a very efficient heat transport since here convective motions are present over the entire length along the bright penumbral filaments, with upflows at the center of the filaments and downflows at the filaments’ edges. This strongly resembles to the convective motions discussed in Sections 3.1.3 and 3.1.4 in the context of umbral dots and light bridges and, therefore, provides a connection between the different small-scale features in sunspots. The field-free gap model does not predict any particular radial preference for upflows and downflows in the penumbra. It however predicts that alternating upflows/downflows should be detected in the direction perpendicular to the filaments (i.e., azimuthally around the penumbra). Because of this feature we will refer to this type of convection as azimuthal convection or overturning convection (see lower panel in Figure 31View Image). Note that the field-free gap model does not readily offer an explanation for the Evershed flow. This is an important point that will be addressed in Section  3.2.7.

3.2.4 Vertical motions in penumbra and signature of convection

In order to distinguish between the different proposed models that attempt to explain the heating mechanism in the penumbra (Section 3.2.3) we need to address the origin of the Evershed flow, as well as identifying the sources and sinks associated with this flow and its mass balance. Since its discovery, the Evershed flow has been recognized as a horizontal motion of the photospheric gas. However, as already mentioned in Section 3.2.2, the Evershed flow is not purely horizontal as it possesses a vertical component. A clear evidence for the vertical motions appeared only after the 1990s when high spatial resolution became available in spectroscopic observations.

Since then, a number of observations have been reported regarding the vertical component of the flow in the penumbra. On the one hand, upflows in the penumbra have been reported by Johannesson (1993), Schlichenmaier and Schmidt (1999, 2000Jump To The Next Citation Point), and Bellot Rubio et al. (2005Jump To The Next Citation Point), and with much higher spatial resolution by Rimmele and Marino (2006Jump To The Next Citation Point). On the other hand, downflows have been observed in and around the outer edge of penumbra by, among others, Rimmele (1995b), Westendorp Plaza et al. (1997Jump To The Next Citation Point), del Toro Iniesta et al. (2001Jump To The Next Citation Point), Schlichenmaier et al. (2004), Bellot Rubio et al. (2004Jump To The Next Citation Point), and Sánchez Cuberes et al. (2005). Both down- and upflows have been simultaneously observed in the penumbra by Schmidt and Schlichenmaier (2000), Schlichenmaier and Schmidt (2000), Westendorp Plaza et al. (2001aJump To The Next Citation Point), Tritschler et al. (2004), Sánchez Almeida et al. (2007), Ichimoto et al. (2007aJump To The Next Citation Point), and Franz and Schlichenmaier (2009Jump To The Next Citation Point). Figure 32View Image highlights some observations that clearly show the downflow patches around a sunspot with an opposite polarity (Westendorp Plaza et al., 1997Jump To The Next Citation Point, left panel) and upflow patches at the leading edge of penumbral bright filaments (Rimmele and Marino, 2006Jump To The Next Citation Point, right panel). These last features correspond to the solar called bright penumbral grains and they are related to the peripheral umbral dots (Sobotka et al., 1999).

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Figure 32: Selected observations of vertical motions in sunspots. Left panels: Discovery of downflows around the outer border of a sunspot. The sunspot is located near the center of the solar disk. Top is the continuum image and bottom is the magnetic field inclination overlaid with velocity contours. Blue regions have a magnetic field polarity opposite to the sunspot, while white contours associated with these regions show downflows with + 3 km s–1 (from Westendorp Plaza et al., 1997, reproduced by permission of Macmillan Publishers Ltd: Nature). Right panels: Close-up of the inner part of a limb-side penumbra. Top and bottom are filtergram (intensity) and Dopplergram (V los) in the Fe i 5576 Å line. Each Evershed flow channel (white filaments in the Dopplergram) is associated with a bright grain and upflow (dark point in the Dopplergram) (from Rimmele and Marino, 2006, reproduced by permission of the AAS).

According to the picture drawn by the present observations, most material carried by the Evershed flow is, thus, supposed to flow back into the photosphere at the downflow patches (Westendorp Plaza et al., 2001a), while some fraction (∼ 10%) of the material may continue to flow across the penumbral outer edge along the elevated magnetic field to form a canopy (Solanki and Bruls, 1994; Solanki et al., 1999). The mass flux balance between up- and downflows in a sunspot observed near the disk center was also inferred under the MISMA hypothesis, though individual flow regions were not spatially resolved (Sánchez Almeida, 2005b). This model postulates that the magnetic field varies rapidly (in all three directions) at scales much smaller than the mean free path of the photon (Sánchez Almeida and Landi Degl’Innocenti, 1996; Sánchez Almeida et al., 1996).

The configuration of the magnetic field is affected by the aforementioned vertical flows. In fact, some of the magnetic field lines plunge back into the deep photosphere at the outer edge of the sunspot and its surroundings (see Figures 4View Image and 7View Image). The relationship between vertical motions and the magnetic field vector in the penumbra is clearly demonstrated by spectropolarimetric data of a sunspot near disk center. Figure 33View Image shows maps of Stokes V (circular polarization) at ˚ Δλ = ±(100, 300) m A away from the line center of the Fe i 6302.5 Å spectral line. The sunspot in this figure is the same one as in Figure 24View Image (Θ = 2.9°). The sign of Stokes V is reversed for Δλ = (100,300) m ˚A. If there are no mass motions in the sunspot, Stokes V maps in the blue and red wings are expected to be identical since the Zeeman effect produces anti-symmetric Stokes V profiles around the line center. This is the case of the main lobes of the Stokes V profiles at ± 100 mÅ. However, the maps in ± 300 mÅ are remarkably different from each other: a number of small and elongated structures with the same polarity of the sunspot are visible in the –300 mÅ V map (middle-left panel in Figure 33View Image) over the penumbra, but with a slight preference to appear in inner penumbra, whereas a number of patches with the opposite polarity of the sunspot are seen in the +300 mÅ V map (middle-right panel in Figure 33View Image), preferentially in the mid and outer parts of the penumbra. As is confirmed by the Dopplergram in the line-wing of Stokes I (bottom-left panel in Figure 33View Image), the former features are associated with upward motions while the later correspond to strong downflows. The typical line-of-sight velocities of the blueshifted regions and the redshifted regions are approximately 1 km s–1 and ∼ 4 – 7 km s–1, respectively. The presence of very fast downflows in the mid and outer regions of the penumbra has been reported previously by del Toro Iniesta et al. (2001) and Bellot Rubio et al. (2004), who suggest that many of those downflow patches (where the magnetic field also turns back into the solar photosphere) harbor supersonic velocities.

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Figure 33: Upper and middle panels: Stokes V maps of a sunspot near the solar disk center (Θ = 2.9°; same sunspot as in Figures 24View Image and 34View Image) at two different wavelengths (shown in each panel) from the center of the Fe i 6302.5 Å spectral line. The sign of Stokes V is reversed for +(100,300) mÅ. Bottom panels: line-of-sight velocity (Doppler velocity) measured in the wings (left) and on the core (right) of the spectral line.
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Figure 34: Continuum intensity Ic (panel a) and field inclination γ (panel b) in the penumbral region shown as a yellow box in Figure 24View Imagea. Overlaid are contours for upflow regions with 0.8 km s–1 (blue) and downflow regions with V ∕Ic = 0.01 in the far red wing of Fe i 6302.5 Å line (red). The sunspot shown here was located almost at disk center: Θ = 2.9°.

Figure 34View Image shows enlargement of a penumbral region indicated by a box in Figure 24View Imagea, where contours for upflow and downflow regions are overlaid on continuum intensity (panel a) and field inclination γ (panel b) maps. It is obvious in panel b that the upflow and downflow patches are aligned with nearly horizontal field channels (filaments with light appearance in the inclination map) that carry the Evershed flow, and that small-scale upflows are preferentially located near the inner penumbra, while downflows dominate at the outer ends of the horizontal field channels (see also Figure 1 in Ichimoto, 2010Jump To The Next Citation Point and Figure 5 in Franz and Schlichenmaier, 2009Jump To The Next Citation Point). Thus, the upflow and downflow patches seen here can be regarded as the sources and sinks of the elementary Evershed flow embedded in deep penumbral photosphere. When all the aforementioned results and observations for the velocity and the magnetic field vector are put together, the picture of the penumbra that emerges is that of Figure 35View Image.

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Figure 35: Cartoon of penumbral magnetic field and the Evershed flow structure.

These results seem to support the idea that convective motions occur in a radial pattern along the penumbral filaments (radial convection; see upper panel in Figure 31View Image). In principle, this lends a strong support to the hot rising flux-tube model (see Section 3.2.1). However, a closer look at Figure 34View Image reveals that the radial distance ℒ between upflows in the inner penumbra and downflows in the outer penumbra is typically several megameters. This would imply that the energy carried by the upflows is not sufficient to heat the penumbra according to the argument of Schlichenmaier and Solanki (2003). Interestingly, Ruiz Cobo and Bellot Rubio (2008Jump To The Next Citation Point) revisited this problem with the embedded flux-tube model, and argued that the significant portion of brightness of the penumbra can be explained with the hot Evershed flow taking place at the inner footpoints of rather thick flux tubes (> 200 km; see Footnote 8) even for large values of ℒ.

Notice, however, that the fact that one type of convection is detected, does not immediately rule out the existence of the other type, azimuthal/overturning convection which is proposed by the field-free gap model (lower panel in Figure 31View Image). Indeed, the search for an azimuthal convective pattern, i.e., upflows at the center of penumbral filaments and downflows at their edges, has intensified in the past few years. Some works, employing continuum images, have provided compelling evidence that the azimuthal/overturning convection does indeed also exist (Márquez et al., 2006; Ichimoto et al., 2007bJump To The Next Citation Point; Bharti et al., 2010Jump To The Next Citation Point, see also Section 3.2.5), at least in the inner penumbra. Unfortunately, results based on spectroscopic measurements have been contradictory so far. Whereas some works (Rimmele, 2008Jump To The Next Citation Point; Zakharov et al., 2008Jump To The Next Citation Point) report on positive detections of such downflows and upflows (of up to 1 km s–1), others claim that at the present resolution that convective pattern does not exist (Franz and Schlichenmaier, 2009Jump To The Next Citation Point; Bellot Rubio et al., 2010Jump To The Next Citation Point). It is, therefore, of the uttermost importance to provide a conclusive detection (or ruling out) of an azimuthal/overturning convective flow in the penumbra that might help us settle once and for all the problem of the penumbral heating. A number of reasons have been put forward in order to explain the lack of evidence supporting an azimuthal/overturning convection. One of the reasons is the lack of sufficient spatial resolution to resolve the velocity fields inside the penumbral filaments. However, this does not explain why azimuthal/overturning convective motions are already detected in umbral dots at the present resolution (see Section 3.1.3) but not in penumbral filaments. Another reason that has been advocated, within the context of the field-free gap model, has been that the τc = 1 level is formed above the convective flow rendering it invisible to spectropolarimetric observations. This argument, however, fails to explain why is then the azimuthal/overturning convective pattern seen in umbral dots since there the τc = 1 level should be formed even higher above the convective flow than in penumbral filaments (Borrero, 2009Jump To The Next Citation Point). The most adequate explanation, therefore, for the lack of evidence supporting an azimuthal/overturning convective flow pattern (if it exists) lies in the large magnitude of the Evershed flow, which overshadows the contribution from the convective up/downflows on the measured line-of-sight velocity as soon as the observed sunspot is slightly away from disk-center (Θ = 0°).

Regardless of which form of convection takes place, it is very clear that this is indeed the mechanism that is responsible for the energy transport in the penumbra. This is emphasized by the very close relationship existing between upflows and bright grains in the penumbra as seen in Figure 34View Image. In this figure we display the continuum intensity Ic (panel a) and the inclination of the magnetic field vector γ (panel b) for the southern part of the sunspot shown in Figure 24View Image overlaid with contours showing the vertical motions. The blue contours show blueshifts equal or larger than 0.8 km s–1 in the wing of Stokes I of Fe i 6301.5 Å, while the red contours show V ∕Ic > 0.01 at Fe i 6302.5 Å + 0.365 Å representing strong downflow regions with opposite magnetic polarity to the spot. The fact that upflows (blue contours) correlate so well with bright penumbral regions, strongly suggest that the vertical component of the Evershed flow supplies the heat to maintain the penumbral brightness even though a quantitative evaluation of the heat flux is not available (Ichimoto et al., 2007a). Puschmann et al. (2010b) supports this scenario in a more quantitative manner based on their 3D empirical penumbral model derived from the Stokes inversion of the Hinode/SP data, i.e., the penumbral brightness can be explained by the energy transfer of the ascending mass carried by the Evershed flow if the obtained physical quantities are extrapolated to slightly deeper layer below the observable depth (τc = 1).

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Figure 36: Stokes profiles (observed with Hinode/SP) of Fe i 6301.5 Å and 6302.5 Å spectral lines in an upflow (panel a) and downflow (panel b) regions in the penumbra. Solid curves show results of a Milne–Eddington fitting algorithm (see Section 1.3.). These profiles correspond to the sunspot observed very close to disk center (Θ = 1.1°) on February 28, 2007 (AR 10944).

Figure 36View Image shows typical Stokes profiles in upflow (panel a) and downflow (panel b) regions in the penumbra, respectively. These profiles correspond to the sunspot AR 10944 on February 28, 2007 very close to the center of the solar disk: Θ = 1.1°. It is noticeable that the upflow region shows a blue hump in Stokes V with the same polarity of the main lobe in the blue wing, while the downflow region shows a strong third lobe with opposite polarity in the far red wing of Stokes V . These asymmetric Stokes V profiles imply the presence of a strong velocity (and magnetic field) gradient along the line-of-sight: Vlos(τc) and B (τc). The solid curves show the best-fit profiles produced by a Milne–Eddington (ME) inversion algorithm. A ME-inversion assumes that the physical parameters are constant with optical depth (see Section 1.3) and, therefore, it always produces anti-symmetric Stokes V profiles, thereby failing to properly fit the highly asymmetric observed circular polarization signals. Sánchez Almeida and Ichimoto (2009) reproduced the red-lobe profiles using the MISMA model (see Section 3.2.4), and suggested that reverse polarity patches result from aligned magnetic field lines and mass flows that bend over and return to the solar interior at very small scales all throughout the penumbra. While this scenario does not help to distinguish between a radial or azimuthal/overturning form of convection (Figure 31View Image) it certainly emphasizes the presence of small-scale convection, which in turn is needed to sustain the penumbral brightness. Other works have also pointed out the relationship between the polarity of the vertical component of the magnetic field and the upflow/downflow regions in the penumbra. For instance, Sainz Dalda and Bellot Rubio (2008) found small-scale, radially elongated, bipolar magnetic structures in the mid-penumbra aligned with intraspines. They move radially outward and were interpreted by these authors as manifestations of the sea-serpent field lines that harbor the Evershed flow (Schlichenmaier, 2002) and, eventually, leave the spot to form moving magnetic features. Martínez Pillet et al. (2009) found a continuation of such magnetized Evershed flow outside sunspots at supersonic speeds.

3.2.5 Inner structure of penumbral filaments

Improvements in the spatial resolution in ground-based optical observations revealed further details about the rich variety of fine-scale structures in the penumbra. Scharmer et al. (2002Jump To The Next Citation Point) discovered a notable feature in penumbral filaments at 0.1” resolution with the Swedish 1-m Solar Telescope at La Palma, i.e., bright penumbral filaments in the inner penumbra often show internal substructure in the form of two bright edges separated by a central dark core (Figure 37View Image). The temporal evolution of these structures shows that the dark core and lateral bright edges move together as a single entity. Some of the dark features are not in parallel to penumbral filaments, but they form oblique dark streaks crossing penumbral filaments. These streaks make the filaments look as if they are twisting with several turns along their length.

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Figure 37: Bright penumbral filaments showing a dark central core. Image was taken in G-band at 430.5 nm with the Swedish 1-m Solar Telescope. Tickmarks have a scaling of 1000 km on the Sun (from Scharmer et al., 2002, reproduced by permission of Macmillan Publishers Ltd: Nature).

The visibility of the dark cores in the filaments is not uniform over the penumbra when the sunspot is located outside the disk center: dark cores are more clearly identified in disk center-side penumbra while they are hardly seen in limb side penumbra (Sütterlin et al., 2004Jump To The Next Citation Point; Langhans et al., 2007Jump To The Next Citation Point). In addition, dark cores are better defined in G-band images than in continuum, which suggests that dark cores are structures that are elevated above the continuum formation height τc = 1 (Rimmele, 2008).

The typical lifetime of dark-cored penumbral filaments was estimated as < 45 minutes (Sütterlin et al., 2004) while some dark cores last longer than 90 minutes (Langhans et al., 2007Jump To The Next Citation Point). The first spectroscopic observation of dark cores was reported by Bellot Rubio et al. (2005), who found a significant enhancement of the Doppler shift which they interpreted as an upflow in the dark cores. It is also found by spectroscopic (Bellot Rubio et al., 2007a) and filtergram (Langhans et al., 2007) observations, that dark-cored filaments are more prominent in polarized light than in continuum intensity, and that dark cores are associated with a weaker and a more horizontal magnetic field than their lateral brightenings and harbor an enhanced radial Evershed outflow. These features are to be considered on top of the already weak and horizontal magnetic field that characterizes the penumbral intraspines (see Section 3.2.1).

Based on a stratified atmosphere consisting of nearly horizontal magnetic flux tubes embedded in a stronger and more vertical field Borrero (2007), as well as Ruiz Cobo and Bellot Rubio (2008Jump To The Next Citation Point), performed radiative transfer calculations to show that these models reproduce the appearance of the dark-cored penumbral filaments. In these models, the origin of the dark cores is attributed to the presence of the higher density region inside the tubes, which shifts the surface of optical depth unity towards higher (cooler) layers.

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Figure 38: A sunspot located at Θ = 30°, east-ward from the center of the solar disk. Space-time plots along the slits across inner penumbral filaments are shown on both sides. The position of the slits are indicated at the top of each space-time plot with partial images whose locations are shown by dashed lines on the sunspot image. Twist (or turning motion) of penumbral bright filament is seen as helical structures of bright filaments in the space-time plot.

From time series of continuum images taken by Hinode/SOT, Ichimoto et al. (2007bJump To The Next Citation Point) found a number of penumbral bright filaments revealing twisting motions around their axes. As it also happens with the dark core at the center of the penumbral filaments, the twisting motions are well observed only in particular portions in the penumbra but, in this case, the locations in which penumbral filaments are oriented parallel to the limb. The direction of the twist (lateral motions of dark streaks that run across filaments diagonally) is always from the limb-side to disk-center side (see Figure 38View Image). Therefore, the twisting feature is not likely a real twist or turn of filaments but, rather, are a manifestation of their dynamical nature, so that their appearance depends on the viewing angle. Overturning/azimuthal convection (see Figure 31View Image) at the source region of the Evershed flow (Ichimoto et al., 2007b; Zakharov et al., 2008) has been proposed as the origin of such features. Such picture with overturning/azimuthal convection causing the observed twisting motions is supported by a positive correlation between the speed of twisting motion and the brightness of penumbral filament in space and time (Bharti et al., 2010). Spruit et al. (2010) interpret the oblique striations that propagate outward to produce the twisting appearance of the filaments as a corrugation of the boundary between the convective flow inside the bright filament and the magnetic field wrapping around it. On the other hand, there are some arguments that some of filaments have intrinsic twist originated from the screw pinch instability (Ryutova et al., 2008; Su et al., 2010).

3.2.6 The Net Circular Polarization in sunspots

The net circular polarization (NCP) is defined as 𝒩 = R V (λ )dλ with the Stokes V signal integrated over a spectral line. 𝒩 = 0 in a perfectly anti-symmetric Stokes V profile, as the area of the blue lobe compensates the area of the red lobe (see solid lines in Figure 36View Image). However, the Stokes V profiles deviate from purely anti-symmetric (see dots in the same figure) and, therefore, 𝒩 ⁄= 0 if there exists a gradient of plasma motion along the line-of-sight: Vlos(τc). In general, Vlos(τc) can produce only small amounts of the NCP. For the large values observed in sunspots, a coupling of a velocity gradient and a gradient in the magnetic field vector B (τc) within the line-forming region ¯τ (see Section 1.3.1) is required (Auer and Heasley, 1978; Landolfi and Landi Degl’Innocenti, 1996Jump To The Next Citation Point). Consequently, the NCP provides a valuable tool to diagnose the magnetic field and velocity structures along the optical depth τc in the sunspot’s atmosphere. Observation of the NCP in sunspots were first reported by Illing et al. (1974a) and Illing et al. (1974b), and were followed by Henson and Kemp (1984) and Makita and Ohki (1986). From these early observations a number of basic features and properties of the NCP at low spatial resolution were inferred (see, e.g., Martínez Pillet, 2000Jump To The Next Citation Point):

  1. The largest NCP occurs in the limb-side penumbra around the apparent magnetic neutral line with the same sign as the umbra’s blue lobe of the Stokes V profile.
  2. The disk center-side penumbra also shows NCP but in the opposite sign to that of the limb-side penumbra and with less magnitude.
  3. The penumbra of sunspots at disk center show a NCP with the same sign to that of the limb-side penumbra.

Besides a gradient in Vlos(τc), which is a necessary condition to produce a non-vanishing NCP, the works from Sánchez Almeida and Lites (1992Jump To The Next Citation Point) and Landolfi and Landi Degl’Innocenti (1996) show that a gradient in any of the three components of the magnetic field vector will also enhance the amount of NCP. These gradients are often referred to as the ΔB, Δ γ and Δ φ mechanisms, with Δ indicating a variation of the physical quantity with optical depth τc. The NCP in sunspots was first interpreted in terms of the ΔB-effect by Illing et al. (1975), who employed a magnetic field strength and line-of-sight velocity that increased with optical depth in the penumbra. Makita (1986) interpreted the NCP in sunspots by means of the Δ φ-effect, i.e., the sunspot’s magnetic field is twisted and unwound along its axis, and has azimuthal rotation along the line-of-sight in the penumbra.

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Figure 39: Spatial distribution of the net circular polarization in sunspots reproduced by the embedded flux tube model and observed in Fe i 6302.5 Å and Fe i 15648 Å spectral lines (from Müller et al., 2002Jump To The Next Citation Point, 2006Jump To The Next Citation Point, reproduced by permission of the ESO).

Nowadays, the most successful scenario to reproduce the NCP of sunspots is based on the Δ γ-effect. Sánchez Almeida and Lites (1992) explained the NCP employing a penumbral model in which the Evershed flow increases with depth and where magnetic field lines become progressively more horizontal with depth in the penumbra. They argued, however, that the large gradient in the inclination of the magnetic field needed to explain the observed NCP was not consistent with a sunspot’s magnetic field in magnetohydrostatic equilibrium. Solanki and Montavon (1993Jump To The Next Citation Point) addressed this problem, and proposed that the needed gradients to reproduce the NCP could be achieved without affecting the sunspot’s equilibrium if they assumed the presence of a horizontal flux tube carrying the Evershed flow embedded in a more vertical background that wraps around it: embedded flux-tube model. In this model, the gradients in Vlos and γ are naturally produced as the line-of-sight crosses the boundary between the background and the horizontal flux tube. Those works were followed by more elaborated models by Martínez Pillet (2000Jump To The Next Citation Point) and Borrero et al. (2006Jump To The Next Citation Point). In order to explain the NCP observed in penumbra at disk-center, Solanki and Montavon (1993) and Martínez Pillet (2000) assumed an upflow in the background magnetic field to make the Δ γ-effect to operate with the deeply embedded flux tubes that carries the horizontal Evershed flow. Schlichenmaier et al. (2002), Müller et al. (2002), and Müller et al. (2006) further developed this idea and provided three-dimensional penumbral models in which horizontal flux tubes are embedded in a more vertical penumbral background magnetic field, to successfully account for the observed azimuthal (i.e., variation around the sunspot) distribution of the NCP at low spatial resolution (≃ 1”) over the penumbra located outside the disk center (see Figure 39View Image). In these models, the Δ φ-effect also plays an important role to reproduce the asymmetric distribution of the NCP around the line connecting the disk center and the sunspot’s center, in particular for the signals observed in the Fe i 15648 Å spectral line (Schlichenmaier and Collados, 2002). Additional improvements were implemented by Borrero et al. (2007Jump To The Next Citation Point), who incorporated a more realistic configuration of the flux tubes and surrounding magnetic fields. The azimuthal distribution and center-to-limb variation of NCP in Fe i 6302.5 Å and Fe i 15648 Å lines were again reproduced successfully. Borrero and Solanki (2010Jump To The Next Citation Point) further studied the effect of azimuthal/overturning convective motions in penumbral filaments on the NCP, and found that these convective motions are less significant than Evershed flow for the generation of net circular polarization.

High resolution observations (< 0.5”) of the NCP in sunspots were first reported by Tritschler et al. (2007Jump To The Next Citation Point). They demonstrated the filamentary distribution of NCP in the penumbra, although the spatial correlation with the Evershed flow channels was not conclusive. Using Hinode/SP data Ichimoto et al. (2008b) found that, as expected, the NCP with the same sign as the umbral blue-lobe is associated with the Evershed flow channels in limb-side penumbra. Remarkably these authors also found that the Evershed flow channels in the disk-center-side penumbra show again the same sign of the NCP, whereas the opposite sign was observed in disk-center-side penumbra in the inter-Evershed flow channels (spines; see Section 3.2.1). This is indicated in Figure 40View Image.

When the sunspot is close to disk center, the NCP in both upflow and downflow regions is associated with the same sign of the umbral blue-lobe (see panels a and b in Figure 40View Image). These results appear to be inconsistent with the current explanation of the NCP by means of the Δγ-effect associated with the presence of the Evershed flow in the deep layers of the penumbra. It rather suggests a positive correlation between the magnetic field strength and the flow velocity as the cause of the NCP, and also serves as a strong evidence for the presence of gas flows in inter-Evershed flow channels (spines). The presence of plasma motions in spines inferred from the net circular polarization informs us that the current simple two component penumbral models consisting of Evershed flow channels and the spines (with no mass motion) are not compatible with the observations, and strongly suggests that there are dynamic features in penumbra that remain unresolved with the current observations.

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Figure 40: Upper panels: spatial distribution of NCP observed by SOT/SP in the Fe i 6302.5 Å spectral line for a sunspot close to disk center (Θ = 2.9°; same as sunspot in Figure 24View Image). Lower panels: same as above but for a sunspot at Θ = 27.4°. In panels b) and d), the velocity contours are plotted over the orginal NCP distributions shown in panels a) and c), respectively. Color contours indicate velocities of –1.8 km s–1 (blue), –0.6 km s–1 (green), 0.6 km s–1 (pink), and 1.8 km s–1 (red), with negative and positive values indicating up- and downflows, respectively.

It is also noticeable that the NCP associated with the upflow regions in disk-center-side penumbra is created by a hump in blue wing of Stokes V profiles (see panel a in Figure 36View Image). Such Stokes V profiles obviously infer that the upflows in the inner penumbra posses a strong magnetic field compared with that in the surrounding penumbral atmosphere, and is apparently inconsistent with numerical simulations that infer upflowing gas with a weaker magnetic field (see Section 3.2.7). Similar conclusions have been reached by Tritschler et al. (2007) and Borrero and Solanki (2008Jump To The Next Citation Point) for the outer penumbra.

3.2.7 Unified picture and numerical simulations of the penumbra

As described in previous sections, two opposing ideas have been proposed to account for the penumbral uncombed structures (see Section 3.2.1): the embedded flux-tube model and the field-free gap model. The embedded flux-tube model, or the rising hot flux tube with the dynamic evolution of the flux tube, explains a number of observational aspects about the fine scale features of the penumbra such as the origin of Evershed flow, inward migration of penumbral grains, and asymmetric Stokes profiles observed in penumbra (see Section 3.2.6), but it faces difficulties when attempting to explain the heat transport to the penumbral surface (see Section 3.2.3). Some observational results support the finite vertical extension of the flux tube; i.e., Doppler shifts in multiple spectral lines formed at different height infer elevated Evershed flow channels (Ichimoto, 1987; Rimmele, 1995a; Stanchfield II et al., 1997), and some SIR-like inversions10 of Stokes profiles in spectral lines claim to have detected the lower boundary of the flux tube (Borrero et al., 2006). However, these results do not necessary provide a concrete evidence of the presence of thin and elevated flux tubes in the inner or middle penumbra and, actually, most observations suggest a monotonic increase in the magnitude of the Evershed flow towards the deeper photospheric layers, while finding no evidence for a lower discontinuity of the magnetic field in the observable layers (see also Figures 29View Image and 30View Image). The flux-tube models by Borrero et al. (2007) and Ruiz Cobo and Bellot Rubio (2008) also suggest that penumbral flux tubes are not necessarily thin since the τ = 1 c level is located inside the tubes and, therefore, the lower boundary will not be visible. Thus, there is no definite observational evidence for the presence of a lower boundary in the flux tubes, at least in inner and middle penumbra, and the concept of narrow, elevated flux tubes embedded in the penumbra is not a scheme with strong observational bases.

In the field-free gap penumbral model, the gap is formed by a convecting hot and field-free gas protruding upward into the background oblique magnetic fields of the penumbra, and is supposed to be the region that harbors the Evershed flow. Contrary to the previous flux-tube model, the field-free gap penumbral model (Spruit and Scharmer, 2006; Scharmer and Spruit, 2006) has an advantage in explaining the heat transport to penumbral surface and, possibly, in explaining the twisting appearance of penumbral bright filaments. It does so thanks to the azimuthal/overturning convective pattern described in Section 3.2.3 (see also lower panel in Figure 31View Image). However, it does not address the origin of the Evershed flow nor physical nature of the inner and outer ends of penumbral filaments. Furthermore, it is obvious from the highly Doppler-shifted polarization signals in spectral lines in penumbra that the flowing gas is not field-free. From the SIR inversion of a spectro-polarimetric data, Borrero and Solanki (2008) argued that the magnetic field strength in the Evershed flow channels (intraspines) increases with the depth below τc = 0.1 level, and there exist strong magnetic fields near the continuum formation level that is not compatible with the field-free gap model.

Thus, both the embedded flux-tube model and the field-free gap model have their own advantages but also have considerable shortcomings. It would be natural, therefore, to modify these two penumbral models as follows; in the flux-tube model, we may consider vertically elongated flux tubes (or slabs) rather than the round cross section, and add an azimuthal/overturning convective flow pattern inside the penumbral filaments in addition to the horizontal Evershed flow. This would allow this model to transport sufficient energy through convection as to explain the penumbral brightness. In the field-free gap model we propose to add a rather strong, ≃ 1000 G (yet still weaker than in the spines), and nearly horizontal magnetic field inside the field-free gap. This has the consequence that the presence of rather strong horizontal magnetic field inside the gap allows this model to explain the observed net circular polarization in sunspots (see Section 3.2.6; see also Borrero and Solanki, 2010) as well as featuring a magnetized Evershed flow.

After the proposed modifications, we find that there are no fundamental differences between the two pictures as far as geometry of the inner penumbra is concerned. Note that in both the rising flux-tube model and the field-free gap model, the rising motions of hot gas in the Evershed flow channels are driven by the buoyancy force in the superadiabatic stratification of the penumbral atmosphere. Such unified picture has been already discussed by Scharmer et al. (2008), Borrero (2009), and Ichimoto (2010). In this concept, the Evershed flow could be understood as a consequence of the thermal convection with the gas flow deflected horizontally outward under a strong and inclined magnetic field.

Although the proposed modifications are observationally driven, recent 3D MHD simulations of sunspots (Heinemann et al., 2007; Rempel et al., 2009a,b; Kitiashvili et al., 2009; Rempel, 2011Jump To The Next Citation Point) present a magnetic field configuration that closely follows the inner structure for penumbral filaments that we have proposed above. The results from these simulations are able to reproduce the radial filamentary structure of the penumbra as seen in continuum images, the uncombed structure of the magnetic field, Evershed outflows along the filaments with a nearly horizontal magnetic field, and overturning convective motions in upwelling plumes. In addition, a detailed inspection of the numerical simulations provides great insights on the physical processes taking place in the penumbra. According to Rempel (2011Jump To The Next Citation Point), the Evershed flow is driven by vertical pressure forces in upflows that are deflected into the horizontal direction through the Lorentz-force generated by the horizontally stretched magnetic fields in flow channels, and the radial flow velocity reaches up to 8 km s–1 at the depth of τc = 1 with a rapid decline toward the higher atmospheric layers.

Figure 41View Image shows a vertical cross section of the filaments in the inner penumbra from the MHD simulations by Rempel (2011Jump To The Next Citation Point). Remarkable features are the sharp enhancement of the radial component of the magnetic field around τ = 1 c level, where the upflow of convective gas protrudes and creates a narrow boundary layer with a concentration of a strong horizontal Lorenz force that acts as the engine that drives the horizontal Evershed flow. The connectivity, or the presence of the outer footpoints, of the magnetic field in the flowing channel are rather a consequence of the fast outflow than its cause as is assumed in the siphon-flow picture.

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Figure 41: Vertical variation, according to the MHD simulations by Rempel (2011Jump To The Next Citation Point), of the physical parameters across a cut perpendicular to the penumbral filaments in the inner penumbra. Displayed are: a) radial and b) vertical components of the magnetic field vector, c) inclination of the magnetic field vector with respect to the vertical direction z. The bottom panels show: d) radial and e) vertical components of the velocity vector, and f) the energy conversion by the component, along the direction of the filaments, of the Lorentz force. The two solid lines indicate the τc = 1 and τc = 0.01 levels (from Rempel, 2011, reproduced by permission of the AAS).

Thus, the recent MHD simulations have begun to reproduce many details of fine scale dynamics and structure of the magnetic field observed in the penumbra. The most essential physical processes that form the penumbra take place near or beneath the τc = 1. Therefore, the detection of the vertical gradients of the magnetic field and velocity vectors in the deep layers of the penumbra is an important target for future observations. Another important target for observations is to find the downflows or returning (inward) flows that could be associated with the azimuthal/overturning convection but still not have been detected yet at the spatial resolutions of Hinode (Franz and Schlichenmaier, 2009) and of the Swedish 1-m Solar Telescope (Bellot Rubio et al., 2010). This had been already discussed in Section 3.2.4.

The dynamic interaction between magnetic fields and granular convection around outer edge of penumbra is discussed as the formation mechanism of the interlocking comb structure of penumbra in the simulations by Thomas et al. (2002), Weiss et al. (2004), and Brummell et al. (2008). In this picture, the magnetic fields in the intraspines, which plunge below the solar surface near the edge of the spot (see Section 3.2.1), are created as a consequence of the submergence of the magnetic field lines due to the downward pumping-mechanism by small-scale granular convection outside the sunspot. Stochastic flows could be driven along such magnetic fields by the convective collapse caused by the the localized submergence of the magnetic fields. This scenario may capture an essential point of the dynamical convective process, but it is questionable if the entire penumbral structure is controlled by such processes taking place outside the sunspots.


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