Significant progress in our ability to simulate sunspots using realistic MHD simulations (i.e., MHD simulations that include the solar equation of state and multidimensional radiative transfer) was only possible over the past couple years. This is primarily due to the fact that pursuing radiative MHD simulations on the scale of sunspots with sufficient resolution to still capture the essential scales of magneto-convective energy transport requires fairly large computational domains and accordingly computing power. Additional to the computational domain size also the physical parameters encountered in and above the umbral region of a sunspot pose significant numerical challenges. The combination of several kG magnetic field with the rather small density scale height leads to a steep increase of the Alfvén velocity above the sunspot umbra reaching values in excess of a few 1000 km s–1. Such high velocities lead to severe time step constraints for explicit codes that make such a simulation almost impractical. A very low value of also leads to problems in codes that use the conservative formulation of the energy equation, since the determination of the internal energy requires to compute the small difference between the nearly equal values of the total and the magnetic energies. Hence, progress in this field calls for either restricting simulations to small domain sizes with focus on certain aspects of the problem, or by relaxing some of the most severe numerical constraints, primarily with respect to very large Alfvén velocities (see, e.g, appendix of Rempel et al., 2009b).
The first realistic radiative MHD simulation of a sunspot umbra was presented by Schüssler and Vögler (2006). The simulation setup is a computational domain of 5.76 Mm horizontal and 1.4 Mm vertical extent located entirely within the umbra region. In order to circumvent the severe Alfvénic time step constraint, the upper boundary was placed about 400 km above the Rosseland optical depth of unity () in the umbra (and, therefore, still beneath the level of in the quiet Sun). The average vertical field strength in the box was chosen to be 2500 G and the initial stratification as well as entropy at the bottom boundary condition were adjusted to yield a thermally relaxed state with about 17 – 18% of the quiet Sun radiative energy flux.
The simulations show a magneto-convective mode that consists of instationary almost field-free upflow plumes that form naturally even in a monolithic strong field region. The plumes start like an oscillatory convection mode as it has been proposed before. But owing to the sharp drop in density near the solar photosphere, upflowing plasma has to expand and weakens the vertical magnetic field to a degree that overturning convection can set in. The result are narrow almost field-free gaps filled with overturning convection that extent about 500 km downward in depth. The typical lifetime of about 30 min is closely related to the size of the energy reservoir these plumes have access to. The energy transported by the overturning plume and radiated away near leads to a decrease of the superadiabatic gradient of the stratification, which was the original driver of the instability. Once the stratification is sufficiently stable the overturning convection stops and the field-free gap closes again. The typical lifespan and photospheric appearance of an upflow plume is shown in Figure 5.
The photospheric manifestation of the upflow plumes is a umbral dot, visible for about 25 min, showing a substructure with a central dark lane, in some cases also a more complicated y-shape dark lane. The dark lane originates from a local density and pressure enhancement above the stagnation point at which most of the mass flux turn around. As a consequence the levels are elevated, leading to a shift of the line formation height into a lower temperature region.
While the overall photospheric appearance of this convection mode might look very similar to the intrusion of unmagnetized plasma in a shallow cluster model as proposed by Parker (1979a), the physical origin is fundamentally different: Almost field-free upflow plumes can form within a strong monolithic magnetic field as a consequence of convective instability (energy reservoir), strong stratification (expansion of upflows leading to weakening of field), and flux expulsion by convection within the upflow plume (cf. Weiss, 1964). In that sense the presence of umbral dots can not be used as a support for the cluster model. On the contrary, we may conceive that the umbra is an overall monolithic fully magnetic structure, in which the fine structure is a local disturbance. The dots are produced locally by magneto-convection processes, the latter being a necessity for the large amount of energy that needs to be transported.
In order to simulate the transition from umbra to penumbra, it is required to laterally extend the computational domain and also include regions of granulation. The inclusion of granulation automatically requires that the top boundary of the domain is placed a few 100 km above the quiet Sun photosphere and therefore almost 1000 km above the level in the umbra. Under these conditions a severe Alfvénic time step constraint becomes unavoidable and all of the magneto-convection simulations discussed here relax this constraint by artificially limiting the Lorentz-force in low regions. As long as the reduced Lorentz-force remains dominant over pressure forces, the overall field topology is not affected, but the computational expense is reduced by about 2 orders of magnitude.
To keep the computational expense at a minimum it is also beneficial to consider sunspots in ‘slab’ geometry, i.e., modeling a slender rectangular section of a sunspot. The first realistic radiative MHD simulation based on this concept was performed by Heinemann et al. (2007). They considered a rectangular section of a (slab-like) small sunspot of about 4 Mm diameter. The main result of this simulation is the formation of filamentary structures in the outer part of the spot, various properties of which (such as dark cores, inward propagation during formation phase, outflows, and strongly inclined magnetic field) are consistent with observational results
However, the filaments found by Heinemann et al. (2007) are much shorter than the typical lengths of real penumbral filaments, and the overall extension of the simulated penumbra is very small. While the magneto-convection mode they identified shares many similarities with the mode found by Schüssler and Vögler (2006), umbral dots were not present in their simulation.
Rempel et al. (2009b) presented a slab simulation based on the same concept, but allowing for a substantially larger sunspot of about 20 Mm diameter. Due to larger overall extent, the filamentary structure of the inner penumbra is more pronounced and individual filaments reach a length of up to 3 Mm and exhibit a clearly visible central dark lane (see Figure 6).
The umbral region also shows the formation of umbral dots similar to those found by Schüssler and Vögler (2006), which allows to clearly identify the common magneto-convective origin of both structures. Figure 7 summarizes the properties of magnetic field, temperature and velocity on a plane perpendicular to the filament highlighted in Figure 6. The vertical extent of the displayed region is 1.3 Mm and almost the same as in Figure 5, and the field and flow properties can be compared accordingly. In both cases energy transport takes place in form of hot rising plumes that lead to a strong reduction of the vertical magnetic field component, however, in the case of penumbral filaments the vertical extent of the region with reduced field strength and vertical flow is larger. The inclined magnetic field near the periphery of the spot causes a symmetry breaking, which leads to elongated filaments with strong outflows along flow tubes of nearly horizontal field near optical depth unity. In addition to the flow along the filament, the upflow also turns over into a motion perpendicular to the filament axis. Dark lanes appear above the strongest upflows owing to the upward bulging of the surface of optical depth unity and the piling up of plasma in a cusp-shaped region at the top of the filament, above which the less inclined field outside the filament becomes laterally fairly homogeneous. The horizontal outflows are concentrated along the dark lanes. All these properties are consistent with recent observational results (e.g., Bellot Rubio et al., 2005; Rimmele and Marino, 2006; Langhans et al., 2007; Ichimoto et al., 2007a; Borrero et al., 2008; van Noort and Rouppe van der Voort, 2008; Zakharov et al., 2008).
Most of the energy radiated away in the photosphere is provided by the deep reaching central upflow pattern, which connects to layers with substantially larger heat capacity. Shallow roll-type convection as indicated in Figure 7 is only of secondary importance for the energy transport and is observationally indistinguishable from the deep reaching component of the flow pattern.
Based on the simulations of Heinemann et al. (2007) it has been suggested by Scharmer et al. (2008) that the Evershed flow can be identified with the horizontal flow seen in the filaments and that it is identical with the horizontal flow component of the magneto-convection in the filament channels. An average outflow of about 1 – 2 km s–1 in the penumbral region is also present in the simulations of Rempel et al. (2009b). However, the overall appearance of the penumbral region seen in Heinemann et al. (2007) and Rempel et al. (2009b) seems more representative of an inner sunspot penumbra with highly intermittent filaments penetrating partially into the umbra region. Rempel et al. (2009b) also found weak inflows at the edges of filaments where the vertical flow is downward directed (see Figure 7). The indications of an outer penumbra with extended regions of horizontal field and strong radial outflows with large filling factor are very weak in slab simulations, however, they are present in simulations of full sunspots described in the following section.
Both, the simulations of Heinemann et al. (2007) and Rempel et al. (2009b) show some evidence of a weak moat flow diverging from the sunspot and transporting magnetic flux away from the sunspot. Due to the limited horizontal extent of the domain this flux accumulates near the domain boundaries and forms small pores as visible in Figure 6.
Kitiashvili et al. (2009) presented recently a numerical setup that can be considered as a model for magneto-convection within a sunspot penumbra. They focused on a small section of magneto-convection in strongly inclined magnetic field, where the inclination and field strength is imposed through the initial state and maintained through the boundary condition. They found a strong dependence of the average horizontal outflow on field strength and inclination angle. Average outflow speeds in the 1 – 2 km s–1 range required about 1.5 kG field strength combined with an imposed mean inclination of 85°. In addition, they reported on temporal variations of the simulated Evershed flow in the range from 15 – 40 min, that can be associated with Evershed clouds (Shine et al., 1994; Rimmele, 1994; Cabrera Solana et al., 2007). A further analysis of these simulations with focus on magnetic flux returning beneath the photosphere and forming so called “sea-serpent magnetic structures” was presented by Kitiashvili et al. (2010a).
While the slab geometry can capture many aspects of the transition from umbra to inner penumbra with a minimal computational expense, simulations of circular sunspots are a better setup for realistic simulations of the outer penumbra as well as the surrounding moat region. Since the slab geometry only allows expansion of the horizontal field in one dimension there is a tendency of too strong horizontal magnetic field in the penumbral regions when applied to large spots with diameters around or in excess of 20 Mm.
A simulation of a complete sunspots is much more demanding since the combination of large domain size and the required high resolution requires large computational grid sizes. This task was recently accomplished by Rempel et al. (2009a), who performed a simulation of a opposite polarity sunspot pair in a domain of 98.304 × 49.152 × 6.144 Mm at a resolution of 32 km in the horizontal directions and 16 km in the vertical requiring a total of 1.8 billion grid points. A pair of opposite polarity sunspots was chosen as setup in order to cover a variety of combinations of field strength and inclination angles in a single simulation run.
Figure 8 shows an intensity image from that simulation after about 5.75 hours of temporal evolution. While both spots have an identical flux (1.6 × 1022 Mx), the spot on the left (right) has a central field strength of about 3 (4) kG; owing to the periodic boundary conditions the magnetic field is more inclined along the x-direction. This underlying field geometry is also manifested in the fine structure: The weaker spot shows more umbral dots, most extended penumbrae are found in the x-direction, preferentially in the center of the simulation domain. The latter is due to the fact that the spot separation in the box is 43 Mm, while the separation across the periodic boundaries is 55 Mm. Figure 9 shows a magnetogram as well as subsurface field strength for the same snapshot.
In addition to umbral dots and dark cored filaments, which have been studied before in slab geometry, this simulation presents for the first time an extended outer penumbra with a strong radial outflow that has a filling factor close to unity and average velocities of up to 5 km s–1 (peak flow speeds can reach 14 km s–1). The location of regions with radial outflows is strongly related to the average inclination angle of the magnetic field. Figure 10 presents the relation between average flow velocity in the x-direction and field inclination in the center of the domain averaged over 8 Mm in the y-direction and 1 hour in time. Coherent outflows from the spots start where the inclination angle exceeds 45°. Note that this relation holds for the 4 penumbral regions present in this plot, despite their different appearance in intensity (Figure 8) and is, thus, of more universal nature. The Evershed flows are also clearly visible in the animation that is provided with Figure 9.
Recently, Rempel (2011a) presented a detailed analysis of this simulation with regard to the photospheric appearance of sunspot fine structure and the physical origin of large scale outflows. Figure 11 summarizes the fine structure of the penumbra at . Radially aligned filaments in intensity correspond to regions with moderately enhanced radial field and strongly reduced vertical field strength resulting in a strong variation of the inclination angle. Note that the enhancement of horizontal field is restricted to a narrow boundary layer forming around . Fast horizontal outflows are present along horizontal stretches of the field. Overturning convection is found everywhere in the penumbra, upflows are preferentially located in the center of filaments. The correlations between Evershed flow, intensity, and field strength are presented in Figure 12 and are discussed in more detail in Rempel (2011b) and Rempel (2011a). While in the inner penumbra strong outflows are found preferentially in bright filaments, the correlation disappears in the outer penumbra. This is consistent with findings from recent high resolution observations (Schlichenmaier et al., 2005; Ichimoto et al., 2007a). The correlation with field strength is negative in the inner and positive in the outer penumbra, the latter was also proposed by Tritschler et al. (2007) and Ichimoto et al. (2008b) as explanation for changes in the net circular polarization (NCP) at different viewing angles.
It is found that maintaining the penumbra brightness requires overturning motions at the level with a RMS value of 1 km s–1, about half of what is found in granulation (simulation results point toward an approximate relation of the form between azimuthally averaged intensity and vertical RMS velocity). Fast horizontal outflows are primarily driven in a narrow boundary layer found beneath in the central upflow regions of filaments. In this boundary layer the Lorentz force facilitates the energy exchange between pressure driving in upflows and the acceleration of fluid that takes place primarily in the horizontal direction. This process is not limited to filament heads and takes place along the full length of filaments. The resulting outflow reaches a maximum velocity close to the local Alfvén velocity in the deep photosphere and declines rapidly with height (see Rempel, 2011a, for details).
Most of these features are found to be robust with respect to numerical resolution within the currently accessible range. A recent simulation with 16 km horizontal and 12 km in the vertical resolution is presented in Figure 13 and further described in Rempel (2011b).
Altogether, the results of Schüssler and Vögler (2006), Heinemann et al. (2007), Scharmer et al. (2008), Rempel et al. (2009a,b), and Rempel (2011a,b) indicate a new level of realism in the theoretical modeling of sunspot structure. The basic properties of the simulated umbral dots and penumbral filaments are consistent with a variety of observational results and provide a basis for a physical understanding of umbral and penumbral structure in terms of a common magneto-convective process that is modulated by the varying inclination angle of the magnetic background field.
In the almost vertical magnetic field of the umbra upflow plumes strongly expand as consequence of stratification and reduce the field strength to a degree that overturning convection sets in. The observable manifestations are bright umbral dots with short dark lanes above the central upflow. With increasing inclination angle the symmetry between the horizontal directions becomes broken resulting in elongated peripheral umbral dots and dark cored filaments of the inner penumbra. The vertical magnetic field component is diminished to almost zero field strength similar to umbral dots as a consequence of expansion and overturning motions. In contrast to that the horizontal field component is moderately enhanced in a shallow boundary layer leading to the formation of an uncombed structure of filaments with almost horizontal field and more vertical background field in between. The central upflows in the filaments are deflected outward by the inclined field in a narrow boundary layer leading to outflows with almost Alfvénic velocity in the deep photosphere.
The common element in all these regimes is overturning convection on scales much shorter than the radial extent of the penumbra, which is responsible for most of the energy and mass transport. Deep reaching upflow plumes connect the photosphere to layers with substantial heat capacity and maintain the substantial brightness of the penumbra. The presence of the inclined magnetic field imposes a large degree of anisotropy as well as preferred direction leading to the appearance of large scale organized horizontal outflows.
Living Rev. Solar Phys. 8, (2011), 3
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