4.4 Pores and sunspots

Recently, “realistic” radiative-convective MHD models of pores and sunspots have become possible. Bercik et al. (2003Jump To The Next Citation Point), Stein et al. (2003Jump To The Next Citation Point), and Vögler et al. (2005Jump To The Next Citation Point) found micropores forming spontaneously in magneto-convection simulations. Cameron et al. (2007a) modeled pores in magneto-convection simulations by imposing them as initial and boundary conditions. Stein et al. (2010b) found that large pores formed spontaneously in deep magneto-convection simulations. Schüssler and Vögler (2006Jump To The Next Citation Point) simulated a sunspot umbra where convection produced umbral dots. Heinemann et al. (2007Jump To The Next Citation Point), Scharmer et al. (2008Jump To The Next Citation Point), and Rempel et al. (2009bJump To The Next Citation Point) started with flaring, rectangular, slab magnetic concentrations. Rempel et al. (2009aJump To The Next Citation Point) and Rempel (2011Jump To The Next Citation Point) modeled sunspots in a magneto-convection simulation starting from a pair of axisymmetric, self-similar magnetic funnels. Cheung et al. (2010Jump To The Next Citation Point) modeled sunspots formed from an emerging, twisted half torus magnetic “flux tube”. An excellent review, especially of helioseismic applications, is Moradi et al. (2010). For more details on sunspots see the review by Rempel and Schlichenmaier (2011Jump To The Next Citation Point).
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Figure 25: Magnetic field (filled contours at 250 G intervals from 0 G to 3500 G) and temperature (top) (1000 K intervals from 4000 K to 16,000 K), and ln ρ (bottom) (in 0.5 intervals from –2 to 4). The τ = 1 depth is shown as the thick line around z = 0 Mm. The flux concentrations are significantly cooler than their surroundings and so have a smaller scale height. The established, strong “flux tube” in the center has been evacuated and is in equilibrium. The smaller flux concentrations on either side are in the process of being evacuated, starting above the surface and piling up plasma below the surface. Image reproduced by permission from Bercik (2002Jump To The Next Citation Point); Bercik et al. (2003Jump To The Next Citation Point).
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Figure 26: Image of vertical velocity (red/yellow down, blue/green up in km s–1) with magnetic field contours at 0.5 kG intervals at the surface (left) and 1.5 Mm below the surface (right). Image reproduced by permission from Bercik (2002Jump To The Next Citation Point).
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Figure 27: Spontaneously formed simulated pore. Clockwise from upper left: emergent intensity, vertical magnetic field at ⟨τ⟩ = 1, horizontal magnetic field at the same level, inclination angle to the vertical. The pore is edge brightened in part of its circumference. The field is vertical in the pore interior and becomes inclined more than 45° at the pore edge.

In magneto-convection simulations with initial vertical fields, micropores form spontaneously in vertices of the intergranular lanes where several lanes meet (Bercik et al., 2003Jump To The Next Citation Point; Stein et al., 2003; Vögler et al., 2005). In the typical formation scenario a small bright granule is surrounded by strong magnetic fields in the intergranular lanes. The upward velocity in the small granule reverses and it disappears with the area it occupied becoming dark. The surrounding strong fields move into the dark micropore area (Figure 28View Image).

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Figure 28: Micropore formation sequence. Left panels: images of the magnetic field strength, center panels: emergent intensity, and right panels: mask showing low intensity, strong field locations. Image reproduced by permission from Bercik (2002) and Bercik et al. (2003).

As the upflow velocity in a flux concentration slows and reverses, the upward heat flux decreases and the plasma inside the concentration cools by radiation through the surface (Figure 25View Image). As a result, the density scale height decreases and the plasma settles lower. Initially the material piles up below the surface until a new hydrostatic structure is established (Figure 25View Image). The micropores are also heated by radiation from their hotter sidewalls (Figure 21View Image, Spruit, 1976, 1977). On the order of a granular timescale the magnetic field is dispersed and the micropore disappears.

Micropores have an amoeba-like structure with arms extending along the intergranular lanes. Fluid flows are suppressed inside them and they are surrounded by downflowing plasma which is concentrated into a few downdrafts on their periphery (Figure 26View Image).

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Figure 29: mpg-Movie (24372 KB) Time sequence of temperature and density fluctuations during pore formation viewed from below (surface is at the bottom). Note the cooler temperatures extending downward from surface, followed by lower densities. Movie produced by Sandstrom, CSC, NASA Ames Res. Ctr.

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Figure 30: mpg-Movie (33906 KB) Time sequence of log |B | during pore formation. The flux concentration forms first at the surface and then extends downward. Near the surface the pore’s field is filamentary, but at large depths it becomes mostly coherent. Movie produced by Sandstrom, CSC, NASA Ames Res. Ctr.

Pores have developed spontaneously in magneto-convection emerging flux simulations when rising Ω-loops emerge through the surface and the upper boundary, leaving behind vertical magnetic field concentrations (Stein and Nordlund, 2006). The pores grow by accumulating flux from their surroundings. The pore pictured in Figure 27View Image has reached a flux of 2.4 × 1020 Mx and occupies an area of 6 Mm2. The flux concentration develops first near the surface. It cools and quickly becomes partially evacuated and flux concentration extends downward, reaching all the way to the bottom of the domain (at 20 Mm depth) – Figure 29Watch/download Movie and Figure 30Watch/download Movie, see also Kitiashvili et al. (2010Jump To The Next Citation Point). Most magnetic field lines in the pore connect to the end of a large scale loop rising from the bottom of the domain, although some connect to various other structures. Additional flux is being transported into the pore by horizontal flows along the intergranular lanes. These flows feeding the pore extend to depths of several megameters. The simulated pores have sometimes lasted for a long time – greater than 8 hrs (Kitiashvili et al., 2010) and 12 hrs in our case.

Pores, like micropores, are surrounded by downflows concentrated into a few downdrafts. The ubiquitous occurrence of downflows in the close vicinity but outside magnetic flux concentrations (see, for example, also Steiner et al., 1998) has been explained in terms of baroclinic flows by Deinzer et al. (1984). The effect has been observationally verified by Langangen et al. (2007). Pores are edge brightened (Figure 27View Image). Cameron et al. (2007b) explain this as due to the fact that the surface of unit optical depth occurs at slightly higher temperature at the edges of pores, possibly due to decreased overlying density because of the spreading magnetic field.

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Figure 31: mpg-Movie (15141 KB) Emergent intensity and |B | (kG) in opposite polarity spot pair initiated from a pair of axial symmetric, self-similar flaring magnetic field funnels. Each spot has the same flux, but the one on the left has a slightly weaker field. The simulation domain is 98 × 49 Mm by 6 Mm depth. The vertical dimension has been stretched by a factor of 2 in the bottom panel (from Rempel et al., 2009aJump To The Next Citation Point).

No one has yet produced a sunspot ab initio. Several “realistic” magneto-convection simulations of sunspots have been made starting with idealized, imposed initial magnetic field configurations. See review by Rempel and Schlichenmaier (2011). Cheung et al. (2010Jump To The Next Citation Point) has come closest, starting from an emerging, twisted, half torus magnetic “flux tube”. Others have started with monolithic, self-similar magnetic configurations. Schüssler and Vögler (2006Jump To The Next Citation Point) investigated magneto-convection in a uniform, vertical field representing a sunspot umbra. Heinemann et al. (2007Jump To The Next Citation Point), Scharmer et al. (2008Jump To The Next Citation Point), and Rempel et al. (2009b) studied flaring rectangular slab field configurations. Rempel et al. (2009a) and Rempel (2011Jump To The Next Citation Point) started with a pair of axisymmetric, self-similar funnels (Schüssler and Rempel, 2005), with the same flux but slightly different field strengths (Figure 31Watch/download Movie). All these simulations develop thin upflow plumes with surrounding downflows that are the observed umbral dots. The most challenging property of spots to model has been their penumbra, which are found to depend crucially on the existence of very inclined magnetic fields in the outer parts of the spots.

In the Cheung et al. (2010Jump To The Next Citation Point) simulation, spots form from an emerging Ω-loop (Figure 14Watch/download Movie). The field first emerges with mixed polarities. The opposite polarities then counterstream to collect into the opposite polarity sunspots. This counterstreaming of opposite polarities is due to the underlying large-scale structure of the coherent subsurface roots of the emerged “flux tube”, which influence the surface dynamics via the Lorentz force, especially magnetic tension (Cheung et al., 2010).

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Figure 32: Vertical slice through an umbral dot. Image is density fluctuation with respect to the surroundings. The solid line is Rossland optical depth unity. The dotted lines are isotherms. The arrows are velocity (longest is 2.7 km/s). Image reproduced by permission from Schüssler and Vögler (2006Jump To The Next Citation Point), copyright by AAS.

Although the strong magnetic fields in sunspots inhibit convection, they do not shut it down entirely. Umbral convection is observed as umbral dots and has been simulated by Schüssler and Vögler (2006). In such strong fields, convection manifests itself as very narrow upflow plumes of hot plasma with neighboring, narrow cool return downflows. The tendency of the magnetic field to expand as the gas pressure declines toward the surface pinches the rising plumes and accelerates the upward flow. As in normal convection, the upflows are braked rapidly near the surface where the plasma loses buoyancy due to radiative cooling. The plasma piles up, the gas pressure increases and makes the plasma expand latterly, which reduces the magnetic field strength. As a result of the enhanced density, the optical depth increases and photons can only escape from higher, cooler layers producing a dark lane through the bright umbral dot (Figure 32View Image). Above the plume, which has been decelerated, the magnetic field again closes in, arching over the gap in a cusp shape.

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Figure 33: Penumbral field lines color coded by velocity at several locations in a penumbra. Set 1 is in the umbra just adjacent to the penumbral head. Sets 2 – 6 are at increasing radial distance in the penumbra. Velocities are 0 – 2 km/s (red), 2 – 4 km/s (yellow), 4 – 8 km/s (green), > 8 km/s (blue). The cutting plane shows the vertical field magnitude at ⟨τ⟩ = 1. Image reproduced by permission from Rempel (2011Jump To The Next Citation Point), copyright by AAS.
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Figure 34: Work as a function of radius and depth in the penumbra. The contours are where the average outflow is more than 2 and 4 km/s. The Lorentz force accelerates the fluid in the Evershed flow along the penumbral filaments, while in the inner penumbra below the surface there is an approximate balance between pressure and Lorentz forces. Image reproduced by permission from Rempel (2011Jump To The Next Citation Point), copyright by AAS.

Heinemann et al. (2007), Scharmer et al. (2008), and Rempel (2011) have modeled sunspot penumbra (Figures 31Watch/download Movie, 33View Image, 34View Image). They find that penumbra are produced by overturning convective motions that occur in an inclined magnetic field and that the observed Evershed outflows (Evershed, 1909) are the horizontal flow of overturning convection channeled along the penumbral magnetic field.

However, unlike normal convection, it is the pressure force that that is pushing the upflow as well as the overturning horizontal flow. The downflows are in nearly hydrostatic equilibrium. Near the lower pressure photosphere the nearly vertical field lines of the penumbra spread outward (tending toward a potential field structure) and become more horizontal. The mass loading by the overturning, horizontal convective motions bends the magnetic field lines downward even more, when the initial inclination is more than 45°, which produces the nearly horizontal penumbral field (Figure 33View Image). Cooling near τ = 1 increases the plasma density and field line bending. The Lorentz force turns the flow along the magnetic field to produce the Evershed outflow (Figure 34View Image).


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