List of Figures

Watch/download Movie Figure 1: (mpg-Movie; 908 KB)
Movie: The change of the synthetic emergent Stokes profiles (I,Q,U,V ) when the magnetic field present in the solar plasma varies. The magnetic field vector is expressed in spherical coordinates: B moduli of the magnetic field vector, γ inclination of the magnetic field vector with respect to the observer’s line-of-sight (z-axis in this case), and φ azimuth of the magnetic field vector in the plane perpendicular to the observer’s line-of-sight. Results have been obtained under the Milne–Eddington approximation.
View Image Figure 2:
These plots show the magnetic field vector in the sunspot AR 10923, observed on November 14, 2006 close to disk center (Θ = 8.7° at the umbral center). The upper-left panel displays the normalized (to the quiet Sun value) continuum intensity at 630 nm. The upper-right panel displays the total magnetic field strength, whereas the lower-left and lower-right panels show the inclination of the magnetic field vector γ with respect to the observer’s line-of-sight, and the azimuth of the magnetic field vector in the plane perpendicular to the line-of-sight φ, respectively. The white contours on the colored panels indicate the umbral boundary, defined as the region in the top-left panel where I ∕Iqs < 0.3. These maps should be interpreted as the average over the optical depth range in which the employed spectral lines are formed: ¯τ ≃ [1,10−3].
View Image Figure 3:
Same as Figure 2 but for the sunspot AR 10933, observed on January 9, 2007 close to the limb (Θ = 49.0° at the umbral center).
View Image Figure 4:
Vertical component of the magnetic field Bρ in the local reference frame in two different sunspots: AR 10923 (top; Θ = 8.7°) and AR 10933 (bottom: Θ = 49.0°). The black contours highlight the regions where the magnetic field points downwards towards the solar center: B ρ < 0. The white contours surround the umbral region, defined as the region where the continuum intensity (normalized to the quiet Sun intensity) I∕I < 0.3 qs. The horizontal and vertical directions in these plots correspond to the eβ and eα directions, respectively.
View Image Figure 5:
Same as Figure 4 but for the B β component of the magnetic field vector in the local reference frame. The arrow field indicates the direction of the magnetic field vector in the plane tangential to the solar surface.
View Image Figure 6:
Same as Figure 5 but for the B α component of the magnetic field vector.
View Image Figure 7:
Same as Figure 4 but for the inclination of the magnetic field with respect to the normal vector to the solar surface eρ: ζ (see Equation (10View Equation)). The black contours indicate the regions where ∘ ζ > 90 and coincide with the regions, in Figure 4, where B ρ < 0.
View Image Figure 8:
Same as Figure 7 but for the azimuthal angle of the magnetic field in the plane of the solar surface: Ψ (see Equation (11View Equation)).
View Image Figure 9:
Map of the Wilson depression Zw (x, y) in a small region of the inner penumbra in AR 10953 observed on May 1, 2007 with Hinode/SP. The white contours enclose regions where upflows are present: Vlos > 0.3 km s–1. Negative values of Zw correspond to elevated structures. In this figure x and y correspond to our coordinates xβ and xα, respectively (from Puschmann et al., 2010b, reproduced by permission of the AAS).
View Image Figure 10:
Map of the continuum intensity for two sunspots. The top panel shows AR 10923 observed at Θ = 8.7°, whereas the bottom panel shows AR 10933, observed at Θ = 49.0°. These are the same sunspots as discussed in Sections 1.3 and 1.3.2. The blue ellipses are employed to determine the azimuthal averages (Ψ-averages) of the magnetic field vector. Note that the outermost ellipse tries to match the boundary between the penumbra and the quiet Sun. The orange arrow points towards the center of the solar disk.
View Image Figure 11:
Left panels: Azimuthally averaged components of the magnetic field vector as a function of the normalized radial distance r∕Rs from the sunspot’s center. The magnetic field corresponds to a constant τ-level. In green the total magnetic field strength Btot(r,¯τ) is presented while red and blue refer to the vertical B ρ and horizontal B h components of the magnetic field. Top panel shows the radial variations for AR 10923 and the bottom panel refers to AR 10933 (see Figure 10 for details). Right panels: inclination at a constant τ-level of the magnetic field vector with respect to the vertical direction on the solar surface, as a function of the normalized radial distance from the sunspot’s center: ζ (r,τ¯) (see Equation (10View Equation)). The horizontal dashed line is placed at ζ = 90°, indicating when the magnetic field points downwards on the solar surface. The vertical dashed line at r∕Rs ≃ 0.4 is placed at the boundary between the umbra and the penumbra.
View Image Figure 12:
Azimuthally averaged components of the magnetic field vector as a function of the normalized radial distance in the sunspot r∕Rs: total magnetic field strength B (upper-left), vertical component of the magnetic field Bρ (upper-right), horizontal component of the magnetic field B h (lower-left), inclination of the magnetic field vector with respect to the vertical direction on the solar surface ζ (lower-right). Each panel contains three curves, representing different optical depths: red is for the deep photosphere or continuum level (log τc = 0), blue is the mid-photosphere (logτc = − 1.5), and green is the upper-photosphere (logτc = − 3). The vertical dashed line at r ∕Rs ≈ 0.4 indicates the separation between the umbra and the penumbra. These results correspond to the sunspot AR 10923 observed on November 14, 2006 at Θ = 8.7° (see also Figure 2; upper panels in Figures 4, 5, 6, and 11).
View Image Figure 13:
Top panel: vertical derivatives of the different components of the magnetic field vector as a function of the normalized radial distance in a sunspot: r∕Rs. Total field strength dBtot∕dx ρ (green), horizontal component of the magnetic field dB ∕dx (r) h ρ (blue), vertical component of the magnetic field dBρ∕dx ρ (red). Bottom panel: same as above but for the inclination of the magnetic field vector with respect to the vector perpendicular to the solar surface: dζ∕dx ρ. The vertical dashed line at r∕Rs ≃ 0.4 represents the umbra-penumbra boundary. The vertical solid lines gives an idea about the standard deviation (from all pixels across a given ellipse in Figure 10). These results correspond to AR 10923, observed on November 14, 2006 at Θ = 8.7°.
View Image Figure 14:
Same as Figures 48 but for the vertical component of the current density vector jz (or jρ) in sunspot AR 10923.
View Image Figure 15:
Same as Figure 9 but for the vertical component of the current density vector: jz (or j ρ). The arrows indicate the horizontal component of the current density vector: j β and j α. The white contours enclose the area where the vertical component of the magnetic field, B ρ, is equal to 650 G (solid) and 450 G (dashed) (from Puschmann et al., 2010c, reproduced by permission of the AAS).
View Image Figure 16:
Similar to Figure 11 but for the plasma-β as a function of the normalized radial distance in the sunspot: r∕R s. The different curves refer to different optical depths in the sunspot: τc = 1 (yellow), τc = 0.1 (red), −2 τc = 10 (green), and − 3 τc = 10 (blue). The vertical dashed line at r∕Rs ≃ 0.4 indicates the umbra-penumbra boundary.
View Image Figure 17:
Same as Figures 9 and 15 but for the plasma-β at z = 0 in the inner penumbra of a sunspot. The white contours are the same as in Figure 15: B ρ = 650 (solid white) and B ρ = 450 (dashed white). This sunspot is AR 10953 observed on May 1st, 2007 with Hinode/SP (from Puschmann et al., 2010c, reproduced by permission of the AAS).
View Image Figure 18:
Upper-left: scatter plot of the total field strength vs. temperature at τc = 1. Upper-right: inclination of the magnetic field with respect to the vertical direction on the solar surface (ζ; see Equation (10View Equation)) versus temperature at τc = 1. Bottom-left: vertical component of the magnetic field Bz (called B ρ in our Section 2.1) vs. temperature at τc = 1. Bottom-right: horizontal component of the magnetic field Br (called Bh in Section 2.1) versus the temperature at τc = 1. In all these panels circles represent umbral points, whereas crosses and triangles correspond to points in the umbra-penumbra boundary and penumbral points, respectively (from Mathew et al., 2004, reproduced by permission of the ESO).
View Image Figure 19:
Same as Figure 7 but for twist angle of the magnetic field in the plane of the solar surface: Δ (Equation (26View Equation)).
View Image Figure 20:
Map of a sunspot (AR 11072) umbra and inner penumbra obtained with the Swedish 1-m Solar Telescope (SST). This sunspot was observed on May 23, 2010 at Θ = 15°. The image was taken with a 10 Å filter located between the Ca H and Ca K spectral lines. It was subsequently restored using Multi-Object Multi-Frame Blind Deconvolution (MOMFBD) technique. The red circles surround a local intensity enhancement in the umbral core: umbral dot (UD), and a portion of a light bridge (LB) (adapted from Henriques et al., 2011; in preparation).
View Image Figure 21:
Vertical stratification (in optical depth τc-scale) of the temperature (left panel) and the total magnetic field strength (right panel). The blue curves shows the stratification for the diffuse umbral background, where the red curves correspond to the vertical stratification along an umbral dot. Close to the continuum, τc = 1, the umbral dot is much hotter and possesses a weaker magnetic field than the umbra. These differences disappear about 100 – 200 km higher in the photosphere: log τc ∼ − 2 (from Riethmüller et al., 2008a, reproduced by permission of the AAS).
View Image Figure 22:
Results from spectropolarimetric observations. Left panel: continuum intensity map inside the umbra of a sunspot. The circles denote the location of several umbral dots (see as intensity enhancements; see also 20). The largest circle encircles two large umbral dots that show prominent central dark lanes. Right panel: map of the line-of-sight velocity in deep layers. This map shows an upflow (blueshift) along the central dark lane and downflows (redshift) at the footpoints of the dark lane (from Ortiz et al., 2010, reproduced by permission of the AAS).
View Image Figure 23:
Results from 3D MHD simulations. Left panel: continuum intensity in the umbra of a sunspot. Right panel: map of the line-of-sight velocity. This panel shows upflows (blueshift) along the central dark lane of umbral dots. Downflows (redshift) are also visible all around the central dark lane, although they are stronger at the footpoints of the dark lane (from Schüssler and Vögler, 2006, reproduced by permission of the AAS).
View Image 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 34.
View Image Figure 25:
Models for explaining the uncombed penumbral structure. Upper-left: embedded flux tube model (from Solanki and Montavon, 1993, reproduced by permission of the ESO); lower-left: rising flux tube model (from Schlichenmaier et al., 1998a, reproduced by permission of the ESO); right: field-free gap model (from Spruit and Scharmer, 2006, reproduced by permission of the ESO).
View Image 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., 1993, reproduced by permission of the AAS).
View Image 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 10). 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, 2008, reproduced by permission of the AAS).
View Image 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.
View Image 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).
View Image 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).
View Image 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.
View Image 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).
View Image 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 24 and 34) 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.
View Image Figure 34:
Continuum intensity Ic (panel a) and field inclination γ (panel b) in the penumbral region shown as a yellow box in Figure 24a. 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°.
View Image Figure 35:
Cartoon of penumbral magnetic field and the Evershed flow structure.
View Image 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).
View Image 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).
View Image 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.
View Image 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., 2002, 2006, reproduced by permission of the ESO).
View Image 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 24). 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.
View Image Figure 41:
Vertical variation, according to the MHD simulations by Rempel (2011), 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).
View Image Figure 42:
Sketch showing the geometry of the problem and the different reference frames employed in Section 1.3.2. The reference frame {ex,ey, ez} is centered at the Sun’s center ‘O’. The observed point at the Sun’s surface is denoted by ’P’. The observer is located at the point ‘E’, denoting the Earth. The vector OP is parallel to eρ, while EP is parallel to e∗ l.